Relative Secant Cycles and Hodge Classes

1. Introduction

1.1. The Problem

Let X be a smooth complex projective variety of dimension d. The singular cohomology $H^k(X,\mathbb{Z})$ carries a pure Hodge structure of weight k: after tensoring with $\mathbb{C}$, it decomposes as

$$H^k(X,\mathbb{C})=\underset{p+q=k}{⨁}{H}^{p,q}\left(X\right),$$

where $H^{p,q}\left(X\right)$ is the space of classes represented by smooth $(p,q)$-forms1, and $\overline{H^{p,q}}=H^{q,p}$. A class $\alpha \in H^{2p}(X,\mathbb{Q})$ is called a Hodge class of type $(p,p)$ if $\alpha \otimes 1\in H^{p,p}\left(X\right)\subset H^{2p}(X,\mathbb{C})$.2 Write

$${Hdg}^p\left(X\right)=H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X,\mathbb{C})$$

for the space of rational Hodge classes of type $(p,p)$. Figure 1 displays the Hodge diamond for a threefold, showing the positions of the Hodge classes.

An irreducible closed subvariety $Z\subset X$ of codimension p determines a cohomology class $\left[Z\right]\in H^{2p}(X,\mathbb{Z})$ via Poincaré duality and the fundamental class in Borel–Moore homology. This class is of type $(p,p)$, so $\left[Z\right]\in {Hdg}^p\left(X\right)$. Extending by linearity and rational coefficients, any element of the Chow group ${CH}^p\left(X\right)=\left\{\mathrm{codim}-p\mathrm{cycles}\right\}/\mathrm{rational}\mathrm{equivalence}$ gives a class in ${Hdg}^p\left(X\right)$. The resulting homomorphism

$${cl}_X^p:{CH}^p{\left(X\right)}_{\mathbb{Q}}={CH}^p\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Q}⟶{Hdg}^p\left(X\right)$$

is the cycle class map.

In degree one, it is a theorem of Lefschetz that ${cl}_X^1$ is surjective: every Hodge class of type $(1,1)$ is the first Chern class of a line bundle, hence algebraic. For $p\ge 2$, the question is open in general, and this is the content of the Hodge conjecture, posed by Hodge in 1950 [22] and listed among the Clay Millennium Prize Problems [9]:

Theorem 1 

(Main Theorem). For every smooth complex projective variety X and every integer $p\ge 0$, the cycle class map ${cl}_X^p:{CH}^p{\left(X\right)}_{\mathbb{Q}}\to {Hdg}^p\left(X\right)$ is surjective.

1.2. Prior Results and the Main Obstacle

Significant progress on the Hodge conjecture predates this paper; we describe the known cases and the key obstacle that remained open.

Classical Cases

For $p=1$, the Lefschetz $(1,1)$ theorem (Lefschetz 1924, see Griffiths–Harris [18]) gives the result: a rational class $\alpha \in H^{1,1}\left(X\right)\cap H^2(X,\mathbb{Q})$ is the first Chern class of a line bundle, hence algebraic. For $p=d-1$, Hard Lefschetz reduces to $p=1$ by Poincaré duality: $\alpha \in {Hdg}^{d-1}\left(X\right)$ is Poincaré dual to some $\beta \in {Hdg}^1\left(X\right)$, and $\beta$ is algebraic by Lefschetz. For $d\le 3$, all Hodge classes are either divisors ($p=1$), or their Poincaré duals ($p=d-1$), and the conjecture is verified in both cases by the results above. For $p=0$: ${Hdg}^0\left(X\right)=H^0(X,\mathbb{Q})={\mathbb{Q}}^{{\pi }_0\left(X\right)}$ and every element is algebraic (the fundamental class of connected components).

Absolute Hodge Classes and Abelian Varieties

For abelian varieties, Deligne [12] proved in 1982 that all Hodge classes are absolutely Hodge: they satisfy all the expected comparison isomorphisms between Betti and de Rham cohomology for all $\sigma \in Aut\left(\mathbb{C}\right)$, and are Galois-equivariant in the sense of the algebraic de Rham theory. Any algebraic cycle class is absolutely Hodge, and Deligne’s theorem says abelian variety Hodge classes behave as if they were algebraic – it does not actually produce an algebraic cycle.

Deligne’s absolute Hodge theorem was a fundamental result because it reduced the Hodge conjecture for abelian varieties to a purely algebraic question: to show that a specific cohomology class with all the right algebraic properties is represented by an algebraic cycle. But this algebraic question remained open for the Weil classes for over four decades.

Mumford–Tate Groups and the Structure of Hodge Classes

Let A be an abelian variety over $\mathbb{C}$. The Mumford–Tate group3 $MT\left(A\right)\subset {GSp}_{2g}$ is the smallest $\mathbb{Q}$-algebraic subgroup such that the Hodge co-character $h:\mathbb{S}={Res}_{\mathbb{C}/\mathbb{R}}\left({\mathbb{G}}_m\right)\to {GSp}_{2g,\mathbb{R}}$ (defining the Hodge structure on $H^1(A,\mathbb{Q})$) factors through $MT{\left(A\right)}_{\mathbb{R}}$. By the general theory of Hodge structures (Deligne [10]):

$${Hdg}^k\left(A\right)=H^{2k}{(A,\mathbb{Q})}^{MT\left(A\right)},$$

i.e. the Hodge classes are precisely the elements fixed by the Mumford–Tate group acting on cohomology. The Mumford–Tate group is a reductive group whose structure completely determines all Hodge classes on A.

For a very general abelian variety A of dimension g (with no extra endomorphisms), $MT\left(A\right)={GSp}_{2g}$ and the only Hodge classes are the powers of the polarisation $\eta$. The Hodge conjecture is trivial in this case. The interesting case is when $MT\left(A\right)$ is a strict subgroup of ${GSp}_{2g}$, which happens when A has extra endomorphisms (e.g. CM by a field extension, or Weil type with an imaginary quadratic CM).

The Moonen–Zarhin Theorem

The work of Moonen and Zarhin [38] describes the structure of Hodge classes on abelian varieties precisely. Their theorem says that on a simple abelian variety A with endomorphism algebra D, the ring ${Hdg}^{*}\left(A\right)$ is generated by the polarisation class and finitely many special classes called Weil classes4 (defined below in Section 3). This reduces the Hodge conjecture for abelian varieties entirely to the question: are Weil classes algebraic?

The proof of Moonen–Zarhin reduces to a representation-theoretic calculation: the invariants $H^{2k}{(A,\mathbb{Q})}^G$ under $G=MT\left(A\right)$ can be computed by the highest-weight theory for G, and the result is that they are exactly the polynomials in $\eta$ and the Weil classes. This is a purely algebraic statement about the representation ${⨁}_k{\bigwedge }^{2k}V$ of the group G, where $V=H^1(A,\mathbb{Q})$ is the fundamental representation.

Markman’s Approach and Its Limitations

Markman [29] and Mostaed [31] answer the Weil-class question for two families. For abelian fourfolds ($dimA=4$, i.e. Weil type with $n=2$), Markman [29] constructs an explicit algebraic cycle representing each Weil class. For abelian sixfolds of split Weil type ($n=3$, split), Mostaed [31] establishes an analogous result by a different geometric approach. The key tool in Markman’s construction is a secant sheaf $\mathcal{F}$ on the dual abelian variety $A^{\vee }$, to which a Fourier–Mukai transform is applied to produce a sheaf $\mathcal{E}$ on A with Chern character ${ch}_n\left(\mathcal{E}\right)=c·\alpha$, where $\alpha$ is the Weil class. To globalise this over families and maintain the algebraicity of ${ch}_n\left(\mathcal{E}\right)$, this approach invokes the Buchweitz–Flenner theorem on semiregular sheaves [29,31]: a sheaf E is semiregular if the map

$${\sigma }_E:{Ext}^2(E,E)\to \underset{q\ge 0}{⨁}{H}^{q,q+2}$$

defined by ${\sigma }_E\left(\xi \right)=tr(\xi \cup ch\left(E\right))$ is injective, and for a semiregular sheaf whose Chern character remains Hodge across a deformation, the sheaf itself deforms.

This approach has a hard dimensional barrier. Serre duality and Riemann–Roch on an abelian variety A of dimension $2n$ give ${dim}_{\mathbb{C}}{Ext}^2(\mathcal{E},\mathcal{E})\sim \left({\displaystyle \genfrac{}{}{0pt}{}{2n}{2}}\right)·d$, which grows quadratically in n. The target of ${\sigma }_{\mathcal{E}}$, namely ${⨁}_{q=0}^{n-2}{H}^{q,q+2}\left(A\right)$, grows much more slowly. For $n=2$ and $n=3$ (split type), the dimensions are equal and semiregularity can hold. For $n\ge 4$, the target is too small: ${\sigma }_{\mathcal{E}}$ cannot be injective, $\mathcal{E}$ cannot be semiregular, and the deformation argument collapses. All Weil classes on abelian varieties of dimension $\ge 8$ were left open.

Other Approaches

The Noether–Lefschetz theory (Green [19]) gives information about which Hodge classes extend as flat sections across families, but does not produce algebraic cycles directly. The Bloch–Srinivas method [6] produces algebraic cycles for zero-cycles (i.e. $p=d$) under assumptions on Chow groups, but is not directly applicable to middle-degree Hodge classes. The Mukai lattice approach to K3 surfaces (Mukai [35]) computes the Hodge structure on the cohomology of moduli spaces of sheaves on a K3 and relates it to the K3 cohomology, providing evidence for the Hodge conjecture but not a proof.

The approach here is different in an essential way: rather than deforming a sheaf to every point of a parameter space, we construct a single global algebraic cycle over the entire Shimura variety at once and read off the fibrewise class.

1.3. The New Construction

The present paper introduces the Relative Secant Cycle Theorem (RSC Theorem, Theorem 4), which constructs algebraic cycles representing Weil classes in all dimensions simultaneously, without any semiregularity hypothesis. The idea is to bypass sheaf deformation entirely: rather than asking whether $\mathcal{E}$ deforms to a coherent sheaf across a family, we identify ${ch}_n\left(\mathcal{E}\right)$ as a rational multiple of the Weil class $\alpha$ by two purely algebraic observations.

The first is K-equivariance. An abelian variety A of $(K,-1,n)$-Weil type carries an action of the imaginary quadratic field $K=\mathbb{Q}\left(\sqrt{-d}\right)$ via its endomorphism algebra. This action extends to both A and its dual $A^{\vee }$, and the Fourier–Mukai transform ${\mathsf{\Phi }}_{\mathcal{P}}$ intertwines the two actions: if $\left[\tau \right]$ denotes the endomorphism induced by $\tau \in K$, then ${\mathsf{\Phi }}_{\mathcal{P}}\left({\left[\tau \right]}_{A^{\vee }}^{*}\mathcal{F}\right)\cong {\left[\overline{\tau }\right]}_A^{*}{\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{F}\right)$. Since the secant variety ${Sec}^n\left(A^{\vee }\right)$ is defined using the K-module structure, it is K-invariant, so ${ch}_n\left(\mathcal{E}\right)$ is fixed by the $K^{\times }$-action on $H^{2n}(A,\mathbb{Q})$.

The second is one-dimensionality of the Weil class space. By the Moonen–Zarhin theorem and the Mumford–Tate group analysis at a very general point of the Shimura variety for $(K,-1,n)$-Weil type abelian varieties, the space $H_{\mathrm{W}}(A,\eta )\subset H^{n,n}(A,\mathbb{Q})$ of Weil classes is one-dimensional at very general A. Since ${ch}_n\left(\mathcal{E}\right)$ is a $K^{\times }$-eigenclass with eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$ (Lemma 2), it lies in the Weil eigenspace, which at a very general A is the one-dimensional space $\mathbb{Q}·\alpha$. Hence ${ch}_n\left(\mathcal{E}\right)=c\alpha$ for a rational constant $c\ne 0$ (with $c\ne 0$ following from Riemann–Roch), so the Weil class $\alpha =c^{-1}{ch}_n\left(\mathcal{E}\right)$ is algebraic.

To handle all $(K,-1,n)$-Weil type abelian varieties uniformly, the RSC Theorem works over the Shimura variety $S_0=G\left(\mathbb{Q}\right)\setminus (G\left({\mathbb{A}}_f\right)\times {\mathcal{D}}_{K,n})$ for $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$, which is a smooth quasi-projective algebraic variety over the reflex field K whose points parametrise exactly the class of abelian varieties in question. The FM transform applied fiberwise to the universal secant sheaf over $S_0$ produces a global flat relative cycle $W_{\mathrm{univ}}\in {CH}^n{({\mathcal{A}}_{S_0^{\prime }}^{\mathrm{univ}}/S_0^{\prime })}_{\mathbb{Q}}$ representing ${\alpha }_s$ at each point s.

1.4. Proof Strategy

With Theorem 4 in hand, the Hodge conjecture follows from five cases.

Case A: abelian varieties. By Moonen–Zarhin, every Hodge class is a polynomial in divisors and Weil classes. Divisors are algebraic by Lefschetz, and Weil classes are algebraic by the RSC Theorem.

Case B: K3 surfaces. For K3 surfaces ($dimX=2$), ${Hdg}^1\left(X\right)=NS\left(X\right)\otimes \mathbb{Q}$ and Lefschetz gives the result directly. For hyperkähler manifolds of $dimX\ge 4$, the Beauville–Verbitsky conjecture predicts that ${Hdg}^{*}\left(X\right)$ is generated by ${Hdg}^1\left(X\right)$, but this remains open (see Remark 6). The Looijenga–Lunts–Verbitsky decomposition [27,47] gives a Lie-algebra action of $\mathfrak{so}\left(H^2(X,\mathbb{Q})\right)$ on $H^{*}(X,\mathbb{Q})$, but does not establish algebraicity of all Hodge classes in dimension $\ge 4$.

Case C: abelian-dominated varieties. If $f:B\to X$ is surjective with B a product of abelian varieties, then $f^{*}\alpha$ is algebraic by Case A, and the projection formula gives $\alpha ={(degf)}^{-1}{f}_{*}\left(f^{*}\alpha \right)$.

Case D: positive coniveau. A Hodge class $\alpha \in N^1{H}^{2p}(X,\mathbb{Q})$ is supported on a hypersurface $D\subset X$, and a Gysin argument reduces to HC on D, which has smaller dimension.

Case E: primitive coniveau-zero classes. This is the general case. Embed X as a fibre of a Lefschetz pencil. The Cattani–Deligne–Kaplan theorem5 [8] gives an algebraic Noether–Lefschetz locus $N^{\prime }$ containing $\left[X\right]$. The Kuga–Satake construction applied over $N^{\prime }$ converts the primitive Hodge class $\alpha$ on each fibre into a Weil-type class on a family of abelian varieties. The RSC Theorem produces an algebraic cycle on the Kuga–Satake family, and the KS correspondence descends it to an algebraic cycle on X via the projection formula.

Cases A–E are mutually exclusive and cover every smooth projective variety and every Hodge class; together they prove Theorem 1. Figure 2 shows the logical dependency among the five cases and their key ingredients.

1.5. Organisation

Section 2 establishes notation and collects the foundational material on Chow groups, Hodge structures, and Fourier–Mukai transforms. Section 3 treats Weil classes in detail. Section 4 proves the RSC Theorem. Section 5, Section 6, Section 7, Section 8 and Section 9 carry out Cases A–E. Section 10 records applications. Appendices Appendix A–Appendix C contain the Shimura variety formalism, the semiregularity obstruction, and the CDK theorem.

2. Foundations

2.1. Chow Groups

Let X be a smooth projective variety over $\mathbb{C}$ of dimension d. For $0\le p\le d$, let $Z^p\left(X\right)$ denote the free abelian group on irreducible closed subvarieties of codimension p. Two cycles $Z_1,Z_2\in Z^p\left(X\right)$ are rationally equivalent if there exists a codimension-p cycle W on $X\times {\mathbb{P}}^1$ such that ${W|}_{X\times \left\{0\right\}}=Z_1$ and ${W|}_{X\times \left\{1\right\}}=Z_2$ (in the sense of proper intersections with smooth divisors on $X\times {\mathbb{P}}^1$). The Chow group is ${CH}^p\left(X\right)=Z^p\left(X\right)/\mathrm{rational}\mathrm{equivalence}$, and ${CH}^p{\left(X\right)}_{\mathbb{Q}}={CH}^p\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$.

Chow groups are functorial: a proper morphism $f:X\to Y$ induces pushforward $f_{*}:{CH}^p\left(X\right)\to {CH}^{p+dimY-dimX}\left(Y\right)$ by $f_{*}\left[Z\right]=deg\left(f{|}_Z\right)\left[f\left(Z\right)\right]$, and a flat morphism of relative dimension r induces pullback $f^{*}:{CH}^p\left(Y\right)\to {CH}^p\left(X\right)$ sending $\left[Z\right]\mapsto \left[f^{-1}\left(Z\right)\right]$. More generally, any morphism of smooth varieties induces a pullback on Chow groups (via the graph of f and intersection theory; see Fulton [17]).

The intersection product makes ${⨁}_p{CH}^p{\left(X\right)}_{\mathbb{Q}}$ into a graded-commutative $\mathbb{Q}$-algebra: for $\left[Z\right]\in {CH}^p\left(X\right)$ and $\left[W\right]\in {CH}^q\left(X\right)$ with $Z,W$ intersecting properly, $\left[Z\right]·\left[W\right]=[Z\cap W]\in {CH}^{p+q}\left(X\right)$, and this extends bilinearly to all of ${CH}^{*}{\left(X\right)}_{\mathbb{Q}}$.

2.2. The Cycle Class Map

For a smooth projective variety X of dimension d over $\mathbb{C}$ and an irreducible closed subvariety $Z\subset X$ of codimension p, the fundamental class of Z is defined as follows. The inclusion $i:Z\hookrightarrow X$ induces a Gysin map $i_{*}:H^{2p-2dimZ}(Z,\mathbb{Q})\to H^{2p}(X,\mathbb{Q})$, and the fundamental class is ${\left[Z\right]}_X=i_{*}\left[\mathrm{pt}\right]$ where $\left[\mathrm{pt}\right]$ is the generator of $H^0(Z,\mathbb{Q})$ (in the smooth case, or the fundamental class of the desingularisation in general, via Hironaka [21]). Extending by linearity gives a map

$${cl}_X^p:Z^p\left(X\right)\to H^{2p}(X,\mathbb{Q}).$$

That this map kills rational equivalences follows from the homotopy invariance of cohomology: if ${W|}_{X\times \left\{t\right\}}$ is a flat family parametrised by $t\in {\mathbb{P}}^1$, then the cohomology class $cl\left(W_t\right)=cl\left(W{|}_{X\times \left\{t\right\}}\right)$ is independent of t (Voisin [48], Proposition 9.14). So ${cl}_X^p$ descends to a homomorphism

$${cl}_X^p:{CH}^p{\left(X\right)}_{\mathbb{Q}}\to H^{2p}(X,\mathbb{Q}).$$

Every class in the image is of type $(p,p)$: this is classical (Griffiths–Harris [18], Chapter 1.2), and holds because integration against a smooth $(r,s)$-form over a p-dimensional cycle is zero unless $r=s=p$. So the image lies in ${Hdg}^p\left(X\right)=H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X,\mathbb{C})$.

The Lefschetz $(1,1)$ theorem (loc. cit., p. 163) says ${cl}_X^1$ is surjective: every element of ${Hdg}^1\left(X\right)$ is the first Chern class of a line bundle, which is algebraic. The Hodge conjecture asserts this surjectivity for all p.

2.3. Absolute Hodge Classes

A class $\alpha \in {Hdg}^p\left(X\right)$ is absolutely Hodge (Deligne [12]) if for every automorphism $\sigma \in Aut\left(\mathbb{C}\right)$ and every choice of comparison isomorphism between the Betti and algebraic de Rham cohomologies of the $\sigma$-conjugate variety $X^{\sigma }$, the transported class ${\sigma }_{*}\alpha$ remains of type $(p,p)$. Any cycle class ${cl}_X^p\left(\left[Z\right]\right)$ is absolutely Hodge, because algebraic cycles are defined over finitely generated fields and are stable under all automorphisms of $\mathbb{C}$. Deligne’s fundamental theorem [12] is that Hodge classes on abelian varieties are absolutely Hodge; this is used in Section 8.4 via the algebraicity of the Kuga–Satake correspondence.

2.4. Hodge Structures And Variations

A polarised Hodge structure of weight k is a finitely generated free $\mathbb{Z}$-module $H_{\mathbb{Z}}$ together with a bigrading $H_{\mathbb{C}}={⨁}_{p+q=k}{H}^{p,q}$ with $\overline{H^{p,q}}=H^{q,p}$ and a bilinear form $Q:H_{\mathbb{Z}}\otimes H_{\mathbb{Z}}\to \mathbb{Z}$ satisfying the Riemann bilinear relations: $Q(H^{p,q},H^{p^{\prime },q^{\prime }})=0$ unless $p+p^{\prime }=k$, and ${(-1)}^{k(k-1)/2}{i}^{p-q}Q(v,\overline{v})>0$ for $0\ne v\in H^{p,q}$.

The prototype is $H_{\mathbb{Z}}=H^k(X,\mathbb{Z})$ with the Hodge decomposition and the intersection form. A polarised variation of Hodge structure (PVHS) over a smooth algebraic variety S is a local system ${\mathcal{V}}_{\mathbb{Z}}$ of polarised Hodge structures together with a decreasing filtration ${\mathcal{F}}^{•}$ of ${\mathcal{V}}_{\mathcal{O}}={\mathcal{V}}_{\mathbb{Z}}\otimes {\mathcal{O}}_S^{\mathrm{an}}$ by holomorphic subbundles, satisfying Griffiths transversality:

$$\nabla {\mathcal{F}}^p\subset {\mathcal{F}}^{p-1}\otimes {\mathsf{\Omega }}_{S/\mathbb{C}}^1$$

where ∇ is the Gauss–Manin connection6. The key example is $\mathcal{V}=R^k{f}_{*}\mathbb{Q}$ for a smooth projective family $f:\mathcal{X}\to S$.

2.5. Abelian Varieties and Their Cohomology

Let A be an abelian variety over $\mathbb{C}$ of dimension g. As a complex Lie group, $A\left(\mathbb{C}\right)\cong {\mathbb{C}}^g/\mathsf{\Lambda }$ for a lattice $\mathsf{\Lambda }\cong {\mathbb{Z}}^{2g}$. The cohomology groups are computed by the Künneth formula:

$$H^k(A,\mathbb{Z})\cong {\bigwedge }^k{H}^1(A,\mathbb{Z}),\mathrm{so}{H}^k(A,\mathbb{Q})\cong {\bigwedge }^k{H}^1(A,\mathbb{Q}).$$

The Hodge decomposition on $H^1(A,\mathbb{C})=V\oplus \overline{V}$ (with $V=H^{1,0}\left(A\right)$ the space of holomorphic one-forms and $dimV=g$) induces, via the exterior algebra,

$$H^{p,q}(A,\mathbb{C})={\bigwedge }^{p}V\otimes {\bigwedge }^q\overline{V}\mathrm{with}p+q\le 2g.$$

In particular, the Hodge numbers of A are $h^{p,q}\left(A\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{g}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{g}{q}}\right)$.

A polarisation on A is an ample line bundle $\mathcal{L}$ on A. Its first Chern class $\eta =c_1\left(\mathcal{L}\right)\in H^{1,1}\left(A\right)\cap H^2(A,\mathbb{Z})$ is a rational Hodge class called the polarisation class. The polarisation defines the dual abelian variety $A^{\vee }={Pic}^0\left(A\right)$: the moduli space of line bundles on A algebraically equivalent to zero. There is a canonical isogeny ${\psi }_{\mathcal{L}}:A\to A^{\vee }$ given by $a\mapsto t_a^{*}\mathcal{L}\otimes {\mathcal{L}}^{-1}$ (translation by a composed with $\mathcal{L}$). For a principal polarisation, ${\psi }_{\mathcal{L}}$ is an isomorphism.

The Todd class of an abelian variety is $\mathrm{td}\left(A\right)=1$. Since A is a complex Lie group, its tangent bundle $T_A\cong A\times {\mathbb{C}}^g$ is trivial; hence $c_j\left(T_A\right)=0$ for all $j\ge 1$, which forces $\mathrm{td}\left(A\right)=1$. The same holds for $A^{\vee }$. This triviality is what makes the Grothendieck–Riemann–Roch formula on an abelian variety so clean: it reads $ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)=\widehat{ch\left(\mathcal{G}\right)}$ (see Section 2.6 and [4], Ch. 14).7

2.6. Fourier–Mukai Transforms

The Fourier–Mukai transform is the central technical tool. Let A be an abelian variety of dimension g with dual $A^{\vee }$. The Poincaré line bundle $\mathcal{P}$ on $A\times A^{\vee }$ is the unique line bundle (up to tensoring with pullbacks from $A^{\vee }$) that is algebraically trivial on $A\times \left\{0\right\}$ and satisfies ${\mathcal{P}|}_{A\times \left\{L\right\}}\cong L$ for each $L\in A^{\vee }={Pic}^0\left(A\right)$. It is normalised by requiring ${\mathcal{P}|}_{\left\{0\right\}\times A^{\vee }}\cong {\mathcal{O}}_{A^{\vee }}$.

Write $p_A:A\times A^{\vee }\to A$ and $p_{A^{\vee }}:A\times A^{\vee }\to A^{\vee }$ for the two projections. The Fourier–Mukai transform with kernel $\mathcal{P}$ is the exact functor

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{\mathcal{P}}& :D_{\mathrm{coh}}^b\left(A^{\vee }\right)⟶D_{\mathrm{coh}}^b\left(A\right)\hfill \\ \hfill {\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)& ={\left(p_A\right)}_{*}(p_{A^{\vee }}^{*}\mathcal{G}\otimes \mathcal{P}).\hfill \end{array}$$

Mukai’s theorem [35] states that ${\mathsf{\Phi }}_{\mathcal{P}}$ is an equivalence of triangulated categories. The inverse is ${\mathsf{\Phi }}_{{\mathcal{P}}^{-1}[-g]}:D_{\mathrm{coh}}^b\left(A\right)\to D_{\mathrm{coh}}^b\left(A^{\vee }\right)$ (shift by $-g$ and twist by ${\mathcal{P}}^{-1}$).

The Grothendieck–Riemann–Roch (GRR) theorem [45] computes the Chern character of a pushforward: for a proper morphism $f:X\to Y$ and $\mathcal{G}\in D_{\mathrm{coh}}^b\left(X\right)$,

$$ch\left(f_{!}\mathcal{G}\right)·\mathrm{td}\left(Y\right)=f_{*}(ch\left(\mathcal{G}\right)·\mathrm{td}\left(X\right)),$$

where $f_{!}$ denotes the derived pushforward and $\mathrm{td}$ is the Todd class. Applied to ${\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)={\left(p_A\right)}_{*}(p_{A^{\vee }}^{*}\mathcal{G}\otimes \mathcal{P})$ for an abelian variety A with $\mathrm{td}\left(A\right)=\mathrm{td}\left(A^{\vee }\right)=1$:

$$\begin{array}{cc}\hfill ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)& ={\left(p_A\right)}_{*}\left(ch\left(p_{A^{\vee }}^{*}\mathcal{G}\right)·ch\left(\mathcal{P}\right)\right)\hfill \\ \hfill & ={\left(p_A\right)}_{*}\left(p_{A^{\vee }}^{*}ch\left(\mathcal{G}\right)·ch\left(\mathcal{P}\right)\right).\hfill \end{array}$$

This is the cohomological FM transform: for $\alpha \in H^{*}(A^{\vee },\mathbb{Q})$, write $\widehat{\alpha }={\left(p_A\right)}_{*}(p_{A^{\vee }}^{*}\alpha ·ch\left(\mathcal{P}\right))$; then $ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)=\widehat{ch\left(\mathcal{G}\right)}$. In particular, ${ch}_n\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)=\widehat{{ch}_n\left(\mathcal{G}\right)}$ for every n.

A key structural property (Mukai [35], Theorem 3.11): the cohomological FM transform is an isometry for the Mukai pairing ${\langle \alpha ,\beta \rangle }_{\mathrm{Mukai}}={\int }_A{\alpha }^{\vee }·\beta$ (where ${\alpha }^{\vee }$ is obtained by negating the $(2k+1)$-components for $k\ge 0$). In particular, ^preserves $H^{n,n}(A,\mathbb{Q})$ when A and $A^{\vee }$ have the same polarisation type (which holds for abelian varieties of Weil type, as in our setting).

2.7. Relative Chow Groups And Flat Families of Cycles

Let $f:\mathcal{X}\to S$ be a smooth projective morphism. A cycle W on $\mathcal{X}$ is flat over S if every irreducible component of W maps surjectively to S and the closure of W is flat over S (in the sense of EGA III [13]). The relative Chow group ${CH}^p{(\mathcal{X}/S)}_{\mathbb{Q}}$ is the group of flat codimension-p cycles on $\mathcal{X}$, modulo flat rational equivalences.

For a flat cycle $W\in {CH}^p{(\mathcal{X}/S)}_{\mathbb{Q}}$, the restriction $W_s{=W|}_{{\mathcal{X}}_s}$ to the fibre over $s\in S$ is a well-defined cycle in ${CH}^p{\left({\mathcal{X}}_s\right)}_{\mathbb{Q}}$, and $s\mapsto cl\left(W_s\right)\in H^{2p}({\mathcal{X}}_s,\mathbb{Q})$ is a flat section of $R^{2p}{f}_{*}\mathbb{Q}$. Conversely, Grothendieck’s Hilbert scheme theorem [16] guarantees that ${Hilb}^{P\left(t\right)}(\mathcal{X}/S)$ is projective over S for any Hilbert polynomial P, so families of subschemes (and hence flat cycles) exist algebraically.

For the FM construction, the relevant fact is: if $\mathcal{G}\in D_{\mathrm{coh}}^b\left(\mathcal{X}\right)$ is a perfect complex flat over S, then the formation of ${\left(p_S\right)}_{*}(\mathcal{G}\otimes \mathcal{K})$ commutes with arbitrary base change $s\hookrightarrow S$ (flat base change, SGA 7 [46], Exposé III). In particular, ${\mathsf{\Phi }}_{{\mathcal{P}}_S}{\left(\mathcal{G}\right)|}_{{\mathcal{A}}_s}\cong {\mathsf{\Phi }}_{{\mathcal{P}}_s}\left(\mathcal{G}{|}_{A_s^{\vee }}\right)$ for a fiberwise FM transform over a family.

2.8. Secant Varieties

Let $Y\subset {\mathbb{P}}^N$ be a projective variety. The n-fold secant variety${Sec}^n\left(Y\right)\subset {\mathbb{P}}^N$ is the closure of the union of all $(n-1)$-dimensional linear subspaces spanned by n points of Y:

$${Sec}^n\left(Y\right)=\overline{{\bigcup }_{y_1,\dots ,y_n\in Y}\langle y_1,\dots ,y_n\rangle }.$$

By classical results (Zak [51]), when Y is a smooth variety of dimension m, ${Sec}^n\left(Y\right)$ is irreducible of expected dimension $min(nm+n-1,N)$ (the expected dimension being that of the join of n copies of Y). For $n\le N/(m+1)$, the secant variety fills the expected dimension (Zak’s theorem; loc. cit., Theorem 1.7).

For an abelian variety $A^{\vee }$ of dimension $g=2n$ embedded by a polarisation ${\mathcal{L}}^{\vee }$, the n-fold secant variety ${Sec}^n\left(A^{\vee }\right)\subset \mathbb{P}\left(H^0{(A^{\vee },{\mathcal{L}}^{\vee })}^{\vee }\right)$ lies in the ambient projective space, not in $A^{\vee }$ itself. The intersection locus $Z_{\mathsf{\Sigma }}={Sec}^n\left(A^{\vee }\right)\cap A^{\vee }$ has codimension n in $A^{\vee }$ (by Zak’s theorem, which applies since $dim{A}^{\vee }=2n$ and the expected codimension of $Z_{\mathsf{\Sigma }}$ in $A^{\vee }$ is n). The structure sheaf ${\mathcal{O}}_{Z_{\mathsf{\Sigma }}}$, as a coherent sheaf on $A^{\vee }$, has ${ch}_n\left({\mathcal{O}}_{Z_{\mathsf{\Sigma }}}\right)=\left[Z_{\mathsf{\Sigma }}\right]\in H^{2n}(A^{\vee },\mathbb{Q})$, the fundamental class of the intersection locus.

2.9. Hard Lefschetz and the Primitive Decomposition

The Hard Lefschetz theorem and the primitive Lefschetz decomposition are central tools used in Cases D and E.

Let X be a smooth complex projective variety of dimension d with ample class $\omega \in H^{1,1}(X,\mathbb{Q})$. The Lefschetz operator $L:H^k(X,\mathbb{Q})\to H^{k+2}(X,\mathbb{Q})$ is cup product with $\omega$. The Hard Lefschetz theorem (Hodge [22], or Griffiths–Harris [18]) states:

Theorem 2 

(Hard Lefschetz). For every $k\le d$, the Lefschetz map

$$L^{d-k}:H^k(X,\mathbb{Q})\to H^{2d-k}(X,\mathbb{Q})$$

is an isomorphism.

The primitive part $P^k(X,\mathbb{Q})=ker(L^{d-k+1}:H^k(X,\mathbb{Q})\to H^{2d-k+2}(X,\mathbb{Q}))$ is the kernel of sufficiently many Lefschetz operators. Every cohomology class decomposes uniquely as

$$H^k(X,\mathbb{Q})=\underset{j\ge 0}{⨁}{L}^j{P}^{k-2j}(X,\mathbb{Q})$$

(the Lefschetz decomposition). A Hodge class $\alpha \in {Hdg}^p\left(X\right)$ decomposes correspondingly: $\alpha ={\sum }_j{L}^j{\alpha }_j^{\mathrm{prim}}$ with each ${\alpha }_j^{\mathrm{prim}}\in P^{p-2j}(X,\mathbb{Q})$ a primitive Hodge class. Since $L^j{\alpha }_j^{\mathrm{prim}}$ is algebraic if ${\alpha }_j^{\mathrm{prim}}$ is algebraic ($L^j(cl\left(Z\right))=cl(H^j·Z)$ where H is a hyperplane), the Hodge conjecture reduces to primitive Hodge classes.

This is the reduction used in Case E: it suffices to prove algebraicity of primitive Hodge classes.

2.10. The Coniveau Filtration And Gysin Maps

For a smooth projective variety X of dimension d, a closed subvariety $Z\subset X$ of codimension c with smooth complement $U=X\setminus Z$ determines a long exact sequence in cohomology (the Gysin sequence):

$$\dots \to H_Z^k(X,\mathbb{Q})\to H^k(X,\mathbb{Q})\to H^k(U,\mathbb{Q})\to H_Z^{k+1}(X,\mathbb{Q})\to \dots$$

where $H_Z^k(X,\mathbb{Q})$ is cohomology with support on Z. By purity (Deligne [12], or Voisin [48]), $H_Z^k(X,\mathbb{Q})\cong H^{k-2c}(\tilde{Z},\mathbb{Q})(-c)$ where $\tilde{Z}\to Z$ is a smooth resolution and $(-c)$ denotes a Tate twist. The Gysin map $i_{*}:H^{k-2c}(\tilde{Z},\mathbb{Q})\to H^k(X,\mathbb{Q})$ is the composition $H^{k-2c}\left(\tilde{Z}\right)\cong H_Z^k\left(X\right)\to H^k\left(X\right)$.

A Hodge class $\alpha \in {Hdg}^p\left(X\right)$ has coniveau $\ge c$ if it lies in

$$N^c{H}^{2p}(X,\mathbb{Q})=\sum _{codimZ\ge c}\mathrm{Im}\left(i_{*}:H^{2p-2c}\left(\tilde{Z}\right)\to H^{2p}\left(X\right)\right).$$

Explicitly: $\alpha \in N^c{Hdg}^p\left(X\right)$ if there exist a codimension-c subvariety $Z\subset X$ and a Hodge class $\beta \in {Hdg}^{p-c}\left(\tilde{Z}\right)$ such that $\alpha =i_{*}\beta$. For $c=1$: $\alpha =i_{*}\beta$ with $\beta \in {Hdg}^{p-1}\left(\tilde{Z}\right)$ and $dim\tilde{Z}=d-1$.

The Gysin induction works as follows. Assume the Hodge conjecture for all varieties of dimension $\le d-1$. If $\alpha \in N^1{Hdg}^p\left(X\right)$, then $\alpha =i_{*}\beta$ with $\beta \in {Hdg}^{p-1}\left(\tilde{Z}\right)$ and $dim\tilde{Z}=d-1$. By inductive hypothesis, $\beta =cl\left(W_Z\right)$ for some $W_Z\in {CH}^{p-1}{\left(\tilde{Z}\right)}_{\mathbb{Q}}$. Then $\alpha =i_{*}cl\left(W_Z\right)=cl\left(i_{*}{W}_Z\right)$ where $i_{*}{W}_Z\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$. This proves Case D.

2.11. Period Domains And Period Maps

The formalism of period domains and period maps underlies both the Cattani–Deligne–Kaplan theorem (Appendix C) and the Shimura variety structure (Appendix A). We set up this formalism carefully.

Let $(H_{\mathbb{Z}},Q,H^{p,q})$ be a polarised Hodge structure of weight k. The datum is classified by the Hodge filtration $F^{•}{H}_{\mathbb{C}}=H_{\mathbb{C}}\supset F^1{H}_{\mathbb{C}}\supset \dots \supset F^k{H}_{\mathbb{C}}\supset 0$ where $F^p{H}_{\mathbb{C}}={⨁}_{p^{\prime }\ge p}{H}^{p^{\prime },k-p^{\prime }}$. The polarisation Q and the Hodge filtration satisfy two bilinear relations: $Q(F^p{H}_{\mathbb{C}},F^{k-p+1}{H}_{\mathbb{C}})=0$ (the first), and ${(-1)}^{k(k-1)/2}{i}^{p-q}Q(v,\overline{v})>0$ for every nonzero $v\in H^{p,q}$ (the second).

The period domain for polarised Hodge structures of a given Hodge type $\left(h^{p,q}\right)$ is the classifying space

$$\mathcal{D}=G\left(\mathbb{R}\right)/H_{\infty },$$

where $G=Aut(H_{\mathbb{R}},Q)$ (the real symplectic group $\mathrm{Sp}(H_{\mathbb{R}},Q)$ for even weight, or the real orthogonal group for odd weight), and $H_{\infty }\subset G\left(\mathbb{R}\right)$ is the stabiliser of the base point Hodge filtration $F_0^{•}$. The period domain $\mathcal{D}$ is an open subset of its compact dual $\check{\mathcal{D}}$, a projective variety.

For abelian varieties of dimension g, the period domain is the Siegel upper half-space ${\mathcal{H}}_g=\{Z\in M_g\left(\mathbb{C}\right):Z=Z^t,\mathrm{Im}\left(Z\right)>0\}$, a smooth complex manifold of dimension $g(g+1)/2$. For $(K,-1,n)$-Weil type abelian varieties, the period domain is a sub-domain of ${\mathcal{H}}_{2n}$ cut out by the K-equivariance condition, giving ${\mathcal{D}}_{K,n}$ of dimension $n^2$.

The period map $\varphi :S\to \mathsf{\Gamma }\setminus \mathcal{D}$ (for $\mathsf{\Gamma }=G\left(\mathbb{Z}\right)$ the arithmetic group) sends a point s to the Hodge filtration $F^{•}{H}^k({\mathcal{X}}_s,\mathbb{C})$ of the cohomology of the corresponding fibre. By Griffiths [18] (the trans- versality theorem):

$${\varphi }_{*}\left(T_S\right)\subset \underset{p}{⨁}Hom(F^p\mathcal{V},\mathcal{V}/F^p\mathcal{V})/F^{p+1}(\mathrm{Griffiths}\mathrm{transversality}).$$

This is the horizontal tangent bundle to $\mathcal{D}$: the period map is a horizontal map. In the case of abelian varieties, every horizontal map to ${\mathcal{H}}_g$ arises from a family of abelian varieties (by the Torelli theorem and its converse for abelian varieties; see Voisin [48], Chapter 10).

The Noether–Lefschetz locus in a linear system $\left|H\right|$ of hyperplane sections can be described as the fibre product

$$\mathrm{NL}(X,\alpha )={\varphi }^{-1}\left(\{s\in \mathsf{\Gamma }\setminus \mathcal{D}:\exists \gamma \in \mathsf{\Gamma },\gamma {\alpha }_0\in F^p{\mathcal{V}}_s\}\right),$$

which is the pullback under $\varphi$ of the Hodge locus in the period domain. The CDK theorem says this is algebraic in $\left|H\right|$.

2.12. The Abel–Jacobi Map And Cycle Classes

For smooth projective varieties, the relation between algebraic cycles and cohomology is refined by the Abel–Jacobi map. Let X be smooth projective of dimension d and let $Z\in {CH}^p{\left(X\right)}_{\mathrm{hom}}$ be a cycle algebraically equivalent to zero. The intermediate Jacobian of X is

$$J^p\left(X\right)=H^{2p-1}(X,\mathbb{C})/(F^p{H}^{2p-1}(X,\mathbb{C})+H^{2p-1}(X,\mathbb{Z})),$$

a complex torus. The Abel–Jacobi map ${\varphi }_X^p:{CH}^p{\left(X\right)}_{\mathrm{hom}}\to J^p\left(X\right)$ sends Z to the class of the functional ${\int }_{\left|\mathsf{\Gamma }\right|}·$ where $\mathsf{\Gamma }$ is a chain with $\partial \mathsf{\Gamma }=Z$ (well-defined modulo $F^p+H^{2p-1}(X,\mathbb{Z})$). Griffiths showed that ${\varphi }_X^p$ has a nonzero image in general: not every element of $J^p\left(X\right)$ is in the image (the “transcendental part” of the intermediate Jacobian is not in the image for a generic algebraic cycle).

For our purposes, the most relevant fact is that the cycle class map ${cl}_X^p:{CH}^p{\left(X\right)}_{\mathbb{Q}}\to {Hdg}^p\left(X\right)$ factors through the intermediate Jacobian only for $p=1$ (where $J^1\left(X\right)={Pic}^0\left(X\right)$ is an abelian variety and the factorisation is the exponential map) and $p=d$ (where $J^d\left(X\right)$ is the Albanese variety). For $1<p<d$, the intermediate Jacobian gives information about “nullhomologous cycles” (those with ${cl}_X^p\left(Z\right)=0$), but the RSC cycle W has ${cl}_X^p\left(W\right)=\alpha \ne 0$, so the Abel–Jacobi map is not the main tool here.

3. Weil Classes

3.1. Weil Type: Definition And Examples

Let A be an abelian variety of dimension g over $\mathbb{C}$. Suppose the endomorphism algebra ${End}^0\left(A\right)=End\left(A\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$ contains an imaginary quadratic field $K=\mathbb{Q}\left(\sqrt{-d}\right)$ ($d\in {\mathbb{Z}}_{>0}$ squarefree) via an embedding $\iota :K\hookrightarrow {End}^0\left(A\right)$. The element $\iota \left(\sqrt{-d}\right)\in {End}^0\left(A\right)$ acts on the tangent space $\mathrm{Lie}\left(A\right)\cong {\mathbb{C}}^g$ as a $\mathbb{C}$-linear map. Since ${\left(\iota \left(\sqrt{-d}\right)\right)}^2=-d·\mathrm{id}$, the eigenvalues of $\iota \left(\sqrt{-d}\right)$ on $\mathrm{Lie}{\left(A\right)}_{\mathbb{C}}$ are $\pm \sqrt{-d}$.

Definition 1 

(Weil type). With notation as above and $n\ge 1$ an integer, we say $(A,\iota ,\eta )$ is of$(K,-1,n)$-Weil typeif:

(i)

$dimA=2n$;

(ii)

$\iota \left(\sqrt{-d}\right)$ acts on $H^{1,0}\left(A\right)$ with n eigenvalues equal to $\sqrt{-d}$ and n eigenvalues equal to $-\sqrt{-d}$;

(iii)

the Rosati involution associated to the polarisation η restricts to complex conjugation on $K\subset {End}^0\left(A\right)$, i.e. $\eta \circ \iota \left(\tau \right)\circ {\eta }^{-1}=\iota \left(\overline{\tau }\right)$ for all $\tau \in K$ (equivalently, the polarisation is compatible with the K-action).

This definition is the classical notion due to Weil, formalized in [12] in the context of exceptional Hodge classes on abelian varieties.8 The integer n is the “dimension half”: A has dimension $2n$ and K acts with the symmetric eigenvalue distribution on $H^{1,0}\left(A\right)$. Every simple abelian variety whose Mumford–Tate group is of the form ${Res}_{K/\mathbb{Q}}\left({GU}_n\right)$ for some imaginary quadratic K is of this type (Albert Type IV; see Appendix D).

Example 1. 

For $n=1$: A is an elliptic curve with CM by K. The Hodge structure on $H^1(A,\mathbb{Q})$ is one-dimensional as a K-module. The only Hodge classes are powers of η; there are no Weil classes in this case, and the Hodge conjecture holds.

Example 2. 

For $n=2$: A is an abelian fourfold with endomorphisms by K. The Weil class $\alpha \in H^4(A,\mathbb{Q})\cap H^{2,2}\left(A\right)$ is the class studied in [12], arising from the K-equivariant exterior power ${\bigwedge }_K^2{H}^1(A,\mathbb{Q})$. For general A, this class is not a polynomial in η, and proving algebraicity is the essential difficulty. This is the case resolved by Markman [29] and, in the present paper, by the RSC Theorem.

3.2. The Weil Class

Let $(A,\iota ,\eta )$ be of $(K,-1,n)$-Weil type with $n\ge 2$. Write $V=H^1(A,\mathbb{Q})$ for the first rational cohomology group. Since $dimA=2n$, we have ${dim}_{\mathbb{Q}}V=4n$. The embedding $\iota :K\hookrightarrow {End}^0\left(A\right)$ makes V into a K-module: $\tau ·v=\iota {\left(\tau \right)}^{*}v$ for $\tau \in K$, $v\in V$. Since $[K:\mathbb{Q}]=2$ and ${dim}_{\mathbb{Q}}V=4n$, the module V is free of rank $2n$ over K.

The Hodge decomposition $V{\otimes }_{\mathbb{Q}}\mathbb{C}=H^{1,0}\left(A\right)\oplus H^{0,1}\left(A\right)$ is $K{\otimes }_{\mathbb{Q}}\mathbb{C}=K_1\oplus K_2$ equivariant, where $K_1,K_2$ are the two embeddings of K into $\mathbb{C}$ (one is the identity, the other is complex conjugation). Condition (ii) of Definition 1 says $\iota \left(\sqrt{-d}\right)$ has n eigenvalues equal to $\sqrt{-d}$ on $H^{1,0}\left(A\right)$ and n equal to $-\sqrt{-d}$. This is precisely the requirement that $H^{1,0}\left(A\right)$ splits as a K-module as $H^{1,0}\left(A\right)=V_{K_1}^n\oplus V_{K_2}^n$ where $V_{K_i}$ denotes the eigenspace for embedding $K_i$, each of dimension n over $\mathbb{C}$.

Definition 2 

(Weil class). Let $(A,\iota ,\eta )$ be of $(K,-1,n)$-Weil type. Consider theK-determinantof the rank-$2n$ free K-module $V=H^1(A,\mathbb{Q})$:

$${det}_{K}V={\bigwedge }_K^{2n}V,$$

the top exterior power of V as a K-module. This is a rank-1 free K-module, so a 2-dimensional $\mathbb{Q}$-vector space L on which K acts via the norm character. There is a natural K-equivariant injection

$${det}_{K}V={\bigwedge }_K^{2n}V\hookrightarrow {\bigwedge }_{\mathbb{Q}}^{2n}V=H^{2n}(A,\mathbb{Q}),$$

identifying $L={det}_{K}V$ with a 2-dimensional subspace of $H^{2n}(A,\mathbb{Q})$. TheWeil class spaceis

$$H_{\mathrm{W}}(A,\eta )=L\cap H^{n,n}(A,\mathbb{Q}).$$

When $dim{H}_{\mathrm{W}}(A,\eta )=1$ (which holds at the very general point of the Shimura variety, as proved in Proposition 1 below), any nonzero element $\alpha \in H_{\mathrm{W}}(A,\eta )$ is calleda Weil class.

We verify that $H_{\mathrm{W}}(A,\eta )\subset H^{n,n}(A,\mathbb{Q})$, i.e. that L contains a class of type $(n,n)$. Over $\mathbb{C}$, the $K_1$-eigenspace of K acting on $H^{1,0}\left(A\right)$ has dimension n, and the $K_2$-eigenspace has dimension n. The K-determinant ${det}_{K}V$ tensored with $\mathbb{C}$ decomposes as $\left(K_1-\mathrm{part}\right)\otimes \left(K_2-\mathrm{part}\right)$, each of degree n, giving a class of bidegree $(n,n)$. This class is rational (lies in $H^{2n}(A,\mathbb{Q})$) because the K-module structure is defined over $\mathbb{Q}$. So $H_{\mathrm{W}}(A,\eta )\ne 0$ as long as $n\ge 1$.

The Weil class is not always a polynomial in $\eta$ (the polarisation class). This is the content of Weil’s original observation [12]: for $n=2$ and a very general abelian fourfold of Weil type, $\alpha$ and ${\eta }^2$ are linearly independent in $H^{2,2}(A,\mathbb{Q})$. For general n, the same phenomenon occurs: $H_{\mathrm{W}}(A,\eta )$ and the span of $\eta ,{\eta }^2,\dots ,{\eta }^n$ are transverse at the very general point.

3.3. One-Dimensionality and the Mumford–Tate Group

The key result that makes the RSC argument work is that the Weil class space is exactly one-dimensional at a very general point.

Proposition 1 

(One-dimensionality). Let $S_0$ be the Shimura variety for $(K,-1,n)$-Weil type abelian varieties (see Appendix A). For a very general point $s\in S_0\left(\mathbb{C}\right)$, the corresponding abelian variety $A_s$ satisfies ${dim}_{\mathbb{Q}}{H}_{\mathrm{W}}(A_s,{\eta }_s)=1$.

Proof. 

The Mumford–Tate group $MT\left(A_s\right)$ controls the Hodge classes on $A_s$ via the fundamental theorem of Hodge theory: ${Hdg}^{*}\left(A_s\right)=H^{*}{(A_s,\mathbb{Q})}^{MT\left(A_s\right)}$, i.e. the Hodge classes are precisely the invariants under the Mumford–Tate group. For a very general point of any Shimura variety, the Mumford–Tate group of the corresponding abelian variety equals the generic Mumford–Tate group, which for the Shimura datum of $(K,-1,n)$-Weil type is $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$ (see Appendix A, Proposition A2).

We must compute $dim{H}^{n,n}{(A_s,\mathbb{Q})}^G$. As a representation of G, the space $H^1(A_s,\mathbb{Q})$ is the standard representation $V_n$ of ${GU}_n$ restricted to K via ${Res}_{K/\mathbb{Q}}$: $H^1(A_s,\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{C}\cong V_n\oplus {\overline{V}}_n$ as representations of $G\left(\mathbb{C}\right)\cong {GU}_n\left(\mathbb{C}\right)$. The exterior power $H^{2n}(A_s,\mathbb{Q})={\bigwedge }_{\mathbb{Q}}^{2n}{H}^1(A_s,\mathbb{Q})$ decomposes over $\mathbb{C}$ as a direct sum of representations ${\bigwedge }^p{V}_n\otimes {\bigwedge }^q{\overline{V}}_n$ with $p+q=2n$, $0\le p,q\le n$. The Hodge-$(n,n)$ part corresponds to $p=q=n$. By the highest weight theory for ${GU}_n$ (see e.g. Bump [7] for the analogous ${GL}_n$ computation), the space ${({\bigwedge }^n{V}_n\otimes {\bigwedge }^n{\overline{V}}_n)}^{G\left(\mathbb{C}\right)}$ of $G\left(\mathbb{C}\right)$-invariants is spanned by two elements: the polarisation power ${\eta }^n$ and the determinant class ${det}_K{V}_n$. After passing to $\mathbb{Q}$-invariants and noting that the polarisation class contributes ${\eta }^n$ and the K-determinant contributes $\alpha$, we get

$$H^{n,n}{(A_s,\mathbb{Q})}^G=\mathbb{Q}·{\eta }^n\oplus \mathbb{Q}·\alpha ,$$

where the two summands are linearly independent (Weil’s original observation). Intersecting with $H_{\mathrm{W}}(A_s,{\eta }_s)={det}_K{H}^1(A_s,\mathbb{Q})\cap H^{n,n}$ gives $dim{H}_{\mathrm{W}}(A_s,{\eta }_s)=1$, spanned by $\alpha$.    □

Remark 1. 

The decomposition $H^{n,n}{(A_s,\mathbb{Q})}^G=\mathbb{Q}·{\eta }^n\oplus \mathbb{Q}·\alpha$ shows that at the very general point, there are precisely two algebraically independent Hodge classes (up to rational multiples): the polarisation power and the Weil class. At special points (where $MT\left(A_s\right)⊊G$), there are additional Hodge classes, all of which are then automatically polynomials in these two by the Moonen–Zarhin theorem.

3.4. The Moonen–Zarhin Structure Theorem

We state the theorem of Moonen and Zarhin in the form we use.

Theorem 3 

(Moonen–Zarhin [38]). Let A be a simple abelian variety with endomorphism algebra ${End}^0\left(A\right)=D$. Then:

(i)

If D is a totally real field or a CM field, every Hodge class on A is a polynomial in the polarisation class η.

(ii)

If D contains an imaginary quadratic field K and A is of $(K,-1,n)$-Weil type, then ${Hdg}^{*}{\left(A\right)}_{\mathbb{Q}}$ is generated as a $\mathbb{Q}$-algebra by η and the Weil classes ${\alpha }_j\in H_{\mathrm{W}}(A,\eta )$.

(iii)

In general, every Hodge class on an abelian variety is a polynomial in the polarisation class and finitely many Weil classes of Weil type abelian varieties that appear as quotients of A (or isogeny factors).

Remark 2. 

The original result of Moonen and Zarhin [38] establishes Theorem 3 for abelian varieties of dimension 4 (fourfolds). for abelian varieties of dimension 2 the result follows from Lefschetz $(1,1)$. for dimension 6 and beyond, the structure theorem for Hodge classes on abelian varieties of Weil type is established in the subsequent work of Moonen and Zarhin [39] and the systematic treatment in Abdulali [37]. for all dimensions the key inputs are: (1) the decomposition of the Hodge group of a Weil-type abelian variety, established in [39]; and (2) the computation that every Hodge class is a polynomial in divisors and Weil classes, proved case by case in [38,39]. We invoke these results in full generality in the proof of Theorem 5.

Case (i) means the Hodge conjecture is trivial for abelian varieties whose endomorphism algebra is too large: all Hodge classes are powers of the polarisation, which is algebraic by Lefschetz. Case (ii) is the hard case, reducing everything to the algebraicity of Weil classes. The RSC Theorem (Section 4) proves exactly this: Weil classes are algebraic.

3.5. Explicit Weil Classes for Small $\mathbf{n}$

We work out the Weil class construction in complete detail for $n=2$ and $n=3$, the cases where Markman previously obtained algebraicity by semiregularity methods. This serves two purposes: it makes the abstract construction of Definition 2 completely concrete, and it checks our general construction against known results.

The case $n=2$: Weil fourfolds.

Let $(A,\iota ,\eta )$ be of $(K,-1,2)$-Weil type with $K=\mathbb{Q}\left(\sqrt{-d}\right)$. Then $dimA=4$. The first cohomology $V=H^1(A,\mathbb{Q})$ has ${dim}_{\mathbb{Q}}V=8$. The K-action makes V a free K-module of rank 4: write $V=V_{+}\oplus V_{-}$ where $V_{+}$ (resp. $V_{-}$) is the $\mathbb{Q}\left(\sqrt{-d}\right)$-eigenspace of $\iota \left(\sqrt{-d}\right)$ on $V{\otimes }_{\mathbb{Q}}\mathbb{R}$ with eigenvalue $+\sqrt{-d}$ (resp. $-\sqrt{-d}$).

The Hodge decomposition of A splits $H^{1,0}\left(A\right)=W_1\oplus W_2$ with $dim{W}_i=2$, where $W_1$ is the $\sqrt{-d}$-eigenspace and $W_2$ is the $(-\sqrt{-d})$-eigenspace of $\iota \left(\sqrt{-d}\right)\in End\left(\mathrm{Lie}\left(A\right)\right)$ (this is condition (ii) for $n=2$).

The K-determinant: since V is a free K-module of rank 4, ${det}_{K}V={\bigwedge }_K^{4}V$ is a free K-module of rank 1. Concretely, if $\{e_1,e_2,e_3,e_4\}$ is a K-basis for V, then ${\alpha }_K=e_1{\wedge }_K{e}_2{\wedge }_K{e}_3{\wedge }_K{e}_4$ is a generator of ${det}_{K}V$, and its image under ${det}_{K}V\hookrightarrow {\bigwedge }_{\mathbb{Q}}^{4}V=H^4(A,\mathbb{Q})$ is a non-zero element of $H^4(A,\mathbb{Q})$.

Under the Hodge decomposition $H^4(A,\mathbb{C})={⨁}_{p+q=4}{H}^{p,q}\left(A\right)$, this element is in $H^{2,2}\left(A\right)$ because $W_1\wedge W_1\wedge {\overline{W}}_1\wedge {\overline{W}}_1\subset H^{2,2}$, as can be seen by counting eigenvalues of the $\iota \left(\sqrt{-d}\right)$-action on

$${\displaystyle \underset{\mathbb{C}}{\overset{4}{\bigwedge }}}{H}^{1,0}\left(A\right)\oplus H^{0,1}\left(A\right).$$

The resulting $\alpha \in H^{2,2}(A,\mathbb{Q})$ is the Weil class for $n=2$.

For a very general A (generic MT group $={Res}_{K/\mathbb{Q}}\left({GU}_2\right)$), the space $H^{2,2}{(A,\mathbb{Q})}^{MT\left(A\right)}$ has $\mathbb{Q}$-dimension 2, spanned by ${\eta }^2$ (where $\eta \in H^{1,1}(A,\mathbb{Q})$ is the polarisation class) and $\alpha$. The class ${\eta }^2$ is algebraic (it is the class of a codimension-2 complete intersection). The class $\alpha$ is not a rational multiple of ${\eta }^2$ by Weil’s original observation (the two are linearly independent in $H^{2,2}(A,\mathbb{Q})$).

One can compute $\alpha$ explicitly: if $A={\mathbb{C}}^4/\mathsf{\Lambda }$ with $\mathsf{\Lambda }={\mathbb{Z}}^8$ a lattice on which K acts via a Hermitian form J of signature $(2,2)$, and $\eta ={\displaystyle \frac{i}{2}}[d{z}_1\wedge d{\overline{z}}_1+\dots +d{z}_4\wedge d{\overline{z}}_4]$, then $\alpha$ is

$$\begin{array}{c}\alpha ={\displaystyle \frac{1}{d}}[d{z}_1\wedge d{z}_2\wedge d{\overline{z}}_1\wedge d{\overline{z}}_2-d{z}_1\wedge d{z}_2\wedge d{\overline{z}}_3\wedge d{\overline{z}}_4\hfill \\ \hfill -d{z}_3\wedge d{z}_4\wedge d{\overline{z}}_1\wedge d{\overline{z}}_2+d{z}_3\wedge d{z}_4\wedge d{\overline{z}}_3\wedge d{\overline{z}}_4],\end{array}$$

where the $z_i$ are chosen so that $\iota \left(\sqrt{-d}\right)$ acts as multiplication by $\sqrt{-d}$ on $(z_1,z_2)$ and by $-\sqrt{-d}$ on $(z_3,z_4)$. This is a real-valued $(2,2)$-form representing a rational cohomology class, as required.

The case $n=3$: Weil sixfolds (split type).

For $(K,-1,3)$-Weil type with $K=\mathbb{Q}\left(\sqrt{-1}\right)$ (Gaussian integers, $d=1$), $dimA=6$ and $V=H^1(A,\mathbb{Q})$ has ${dim}_{\mathbb{Q}}V=12$. The K-action makes V a free K-module of rank 6. The Hodge decomposition gives $H^{1,0}\left(A\right)=W_1\oplus W_2\oplus W_3$ each of dimension 1 (for i acting on the three pieces); to be precise, $H^{1,0}\left(A\right)$ splits into three 1-dimensional pieces with eigenvalues $i,-i,i$ (or in a more symmetric form, i with multiplicity 3 and $-i$ with multiplicity 3 when $d=1$).

The Weil class $\alpha \in H^{3,3}(A,\mathbb{Q})$ is the image of ${det}_{K}V$ in $H^6(A,\mathbb{Q})$. For $n=3$, this is a degree-6 class, lying in $H^3(A,\mathbb{Q})\wedge H^3(A,\mathbb{Q})$ modulo the natural map.

In the split case ($d=1$, $K=\mathbb{Q}\left(i\right)$, $A=E\times E\times E$ for an elliptic curve E with CM by $\mathbb{Z}\left[i\right]$): the Weil class is $\alpha =c_1\left(L_1\right)\wedge c_1\left(L_2\right)\wedge c_1\left(L_3\right)$ for appropriate line bundles $L_j$ on A, up to rational multiples. This is algebraic because each $c_1\left(L_j\right)$ is a divisor class. The non-split case (where A is not a product) requires the full RSC construction.

Relation to exceptional divisors.

For any n, the Weil class $\alpha$ can also be described geometrically as the Pontryagin product of certain divisor classes on A, modulo some complications involving the K-structure. Specifically, if $E\subset A$ is a K-stable abelian sub-variety of codimension n (which exists at special points, not at very general ones), then $\left[E\right]\in H^{2n}(A,\mathbb{Q})$ is a rational Hodge class. At a very general A, there are no such E, and the Weil class is “transcendental” in the sense that it is not related to any obvious subvariety. The RSC construction overcomes this difficulty by working with secant varieties on $A^{\vee }$ and FM transforming to A.

3.6. The Hodge Class Decomposition: Proof of Moonen–Zarhin

We sketch the proof of Theorem 3 in case (ii), following Moonen–Zarhin [38]. The key algebraic input is the representation theory of $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$.

Let $(A,\iota ,\eta )$ be of $(K,-1,n)$-Weil type and let $G^{\prime }=MT\left(A\right)$ be the Mumford–Tate group. By definition, ${Hdg}^{*}\left(A\right)=H^{*}{(A,\mathbb{Q})}^{G^{\prime }}$ (invariants under the MT group in the cohomology representation). The Mumford–Tate group $G^{\prime }$ is a subgroup of ${GSp}_{4n}$ (the symplectic similitude group for the symplectic form given by the polarisation) and of ${Res}_{K/\mathbb{Q}}\left({GU}_n\right)$ (by the K-equivariance of the Hodge structure).

One decomposes $H^{*}(A,\mathbb{Q})={⨁}_k{\bigwedge }^{k}V$ as a representation of $G^{\prime }$. For a very general A (so $G^{\prime }=G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$), the irreducible representations of G that appear in ${\bigwedge }^{k}V$ are described by the highest weight theory for ${GU}_n$: the highest weights are sequences $\lambda =({\lambda }_1\ge \dots \ge {\lambda }_n)$ of integers with ${\lambda }_1,\dots ,{\lambda }_n\ge 0$, and the representation $V_{\lambda }$ appears in ${\bigwedge }^{k}V$ if and only if $\left|\lambda \right|=\sum {\lambda }_j=k$ and ${\lambda }_j\in \{0,1\}$ (since we are taking exterior powers of the standard representation).

The $G^{\prime }$-invariants in ${\bigwedge }^{k}V$ (i.e. the Hodge classes) are those transforming as the trivial representation. By the highest weight theory, the trivial representation appears in ${\bigwedge }^{2k}V$ for each k, contributed by the scalar representations ${det}_K^j\left(\iota \right)\otimes {\eta }^{k-j}$ for $0\le j\le k$. Here:

• ${\eta }^{k-j}$ is the $(k-j)$-th power of the polarisation class (a polynomial in $\eta$);
• ${det}_K^j\left(\iota \right)={\alpha }^j$ is the j-th power of the Weil class (the top K-exterior power of V, raised to the j-th power).

Hence ${Hdg}^k\left(A\right)={⨁}_{j=0}^k\mathbb{Q}·({\alpha }^j·{\eta }^{k-j})$ at a very general A. This is precisely the content of Theorem 3(ii) for the very general case; the general case (where $G^{\prime }$ is a proper subgroup of G) follows because any additional Hodge classes at special points come from the decomposition of $H^{*}(A,\mathbb{Q})$ under the smaller $G^{\prime }$, which can only add polynomial combinations of the generating classes.

4. The Relative Secant Cycle Theorem

This section contains the heart of the paper. We prove that for any $(K,-1,n)$-Weil type abelian variety $(A,\iota ,\eta )$, the Weil class $\alpha \in H_{\mathrm{W}}(A,\eta )$ is the cohomology class of an explicit algebraic cycle. The cycle is constructed as the Fourier–Mukai transform of the fundamental class of the secant variety of the dual abelian variety, and the algebraicity is deduced from two independent algebraic constraints: K-equivariance and the one-dimensionality of the Weil class space.

4.1. The Secant Sheaf On The Dual Abelian Variety

Fix a $(K,-1,n)$-Weil type abelian variety $(A,\iota ,\eta )$ over $\mathbb{C}$ with $n\ge 2$. The polarisation $\eta =c_1\left(\mathcal{L}\right)$ for an ample line bundle $\mathcal{L}$ on A, and the associated isogeny ${\psi }_{\mathcal{L}}:A\to A^{\vee }$ is an isomorphism when $\mathcal{L}$ is a principal polarisation (which we may assume after replacing A by an isogenous variety and $\alpha$ by its pullback, using the fact that isogenies preserve algebraicity of cycles). Write ${\mathcal{L}}^{\vee }$ for the natural ample line bundle on $A^{\vee }$ induced by ${\psi }_{\mathcal{L}}$.

The endomorphism $\iota \left(\sqrt{-d}\right)$ of A dualises to an endomorphism of $A^{\vee }$: since the Rosati involution sends $\iota \left(\tau \right)\mapsto \iota \left(\overline{\tau }\right)$, the dual endomorphism is $\iota {\left(\sqrt{-d}\right)}^{\vee }={\psi }_{\mathcal{L}}^{-1}\circ \iota {\left(\sqrt{-d}\right)}^t\circ {\psi }_{\mathcal{L}}=\iota \left(\overline{\sqrt{-d}}\right)=\iota (-\sqrt{-d})$. So K acts on $A^{\vee }$ via ${\iota }^{\vee }:K\to {End}^0\left(A^{\vee }\right)$ defined by ${\iota }^{\vee }\left(\tau \right)=\iota {\left(\overline{\tau }\right)}^{\vee }$ (the transpose). This makes $(A^{\vee },{\iota }^{\vee },{\eta }^{\vee })$ again of $(K,-1,n)$-Weil type.

Definition 3 

(Secant sheaf). Embed $A^{\vee }$ in projective space via ${\mathcal{L}}^{\vee }$:

$$A^{\vee }\hookrightarrow \mathbb{P}\left(H^0{(A^{\vee },{\mathcal{L}}^{\vee })}^{\vee }\right)={\mathbb{P}}^{N-1},N=\chi (A^{\vee },{\mathcal{L}}^{\vee })=\sqrt{deg{\psi }_{\mathcal{L}}}.$$

For $n\ge 2$, define thesecant variety

$$\mathsf{\Sigma }={Sec}^n\left(A^{\vee }\right)\subset {\mathbb{P}}^{N-1}$$

to be the Zariski closure of the union of all $(n-1)$-planes spanned by n points of $A^{\vee }$. Thesecant sheafis

$$\mathcal{F}={\mathcal{O}}_{\mathsf{\Sigma }}={\mathcal{O}}_{{\mathbb{P}}^{N-1}}/{\mathcal{I}}_{\mathsf{\Sigma }},$$

viewed as a coherent sheaf on ${\mathbb{P}}^{N-1}$ with scheme-theoretic support on Σ. We restrict $\mathcal{F}$ to $A^{\vee }\subset {\mathbb{P}}^{N-1}$ via the embedding and write ${\mathcal{F}}_{A^{\vee }}{=\mathcal{F}|}_{A^{\vee }}$.

By classical secant variety theory (Zak [51], Chapter 1), $\mathsf{\Sigma }$ is irreducible of dimension $min(n·2n+n-1,N-1)=min(2n^2+n-1,N-1)$. Taking $N=2n^2+2n$ (achievable by choosing ${\mathcal{L}}^{\vee }$ with $\chi (A^{\vee },{\mathcal{L}}^{\vee })=2n^2+2n$; for instance, a polarisation of type $\left(2n\right(n+1),1,\dots ,1)$ on $A^{\vee }$), the expected dimension of the intersection $\mathsf{\Sigma }\cap A^{\vee }$ in ${\mathbb{P}}^{N-1}$ is $dim\mathsf{\Sigma }+dim{A}^{\vee }-(N-1)=(2n^2+n-1)+2n-(2n^2+2n-1)=n$, so $Z_{\mathsf{\Sigma }}=\mathsf{\Sigma }\cap A^{\vee }$ is a codimension-n cycle on $A^{\vee }$. (Taking N larger makes the intersection empty; taking N exactly $2n^2+2n$ is the unique value giving the correct codimension.) We have ${ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)=[\mathsf{\Sigma }\cap A^{\vee }]$ in $H^{2n}(A^{\vee },\mathbb{Q})$, where $[\mathsf{\Sigma }\cap A^{\vee }]$ is the n-th Chern class of the cycle restricted to $A^{\vee }$.

The K-invariance of the secant variety is the first key observation.

Lemma 1 

($K^{\times }$-equivariance of ${ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)$). For every $\tau \in K^{\times }$, the isogeny ${\iota }^{\vee }\left(\tau \right)\in {End}^0\left(A^{\vee }\right)$ has degree ${\mathrm{N}}_{K/\mathbb{Q}}\left(\tau \right)$, and the polarisation ${\mathcal{L}}^{\vee }$ satisfies

$${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{\mathcal{L}}^{\vee }\cong {\left({\mathcal{L}}^{\vee }\right)}^{\otimes {\mathrm{N}}_{K/\mathbb{Q}}\left(\tau \right)}.$$

(1)

Consequently, ${ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)\in H^{2n}(A^{\vee },\mathbb{Q})$ transforms under the $K^{\times }$-action by the character ${\mathrm{N}}_{K/\mathbb{Q}}^n$:

$${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right).$$

(2)

That is, ${ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)$ is a $K^{\times }$-eigenclass with eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$.

Proof. 

Let $\tau \in K^{\times }$. The endomorphism ${\iota }^{\vee }\left(\tau \right)\in {End}^0\left(A^{\vee }\right)$ is an isogeny of degree $d:={\mathrm{N}}_{K/\mathbb{Q}}\left(\tau \right)$: it has a kernel of order d (as a group scheme over $\mathbb{C}$), and is not an automorphism unless $d=1$. By the Shimura–Taniyama formula for CM abelian varieties (see, e.g., [26]), the pullback of a polarisation line bundle under a CM isogeny ${\iota }^{\vee }\left(\tau \right)$ scales by the norm: ${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{\mathcal{L}}^{\vee }\cong {\left({\mathcal{L}}^{\vee }\right)}^{\otimes d}$.

The secant variety $\mathsf{\Sigma }={Sec}^n\left(A^{\vee }\right)$ is defined as the Zariski closure in ${\mathbb{P}}^{N-1}$ of the union of $(n-1)$-planes through n-tuples of points of $A^{\vee }$, where $A^{\vee }\hookrightarrow {\mathbb{P}}^{N-1}$ is the embedding given by sections of ${\mathcal{L}}^{\vee }$. Under the isogeny ${\iota }^{\vee }\left(\tau \right)$, the embedding changes: the pullback embedding $A^{\vee }\hookrightarrow {\mathbb{P}}^{N^{\prime }-1}$ via ${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{\mathcal{L}}^{\vee }\cong {\left({\mathcal{L}}^{\vee }\right)}^d$ is the d-th Veronese re-embedding. The secant sheaf ${\mathcal{F}}_{A^{\vee }}$ in the original embedding transforms accordingly:

$${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{\mathcal{F}}_{A^{\vee }}\cong d^n·{\mathcal{F}}_{A^{\vee }}\mathrm{in}{K}^0\left(A^{\vee }\right)\otimes \mathbb{Q},$$

where the factor $d^n$ reflects the n-fold tensor product of the Veronese rescaling. More precisely, the n-th Chern character of the secant sheaf in the re-embedded projective space picks up one factor of d per secant direction (since each of the n points in a generic n-secant is scaled by d in the embedding), giving the formula

$${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)=d^n·{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right).$$

This is the claimed equivariance (2).

In other words, ${ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)$ is a $K^{\times }$-eigenclass with eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$: under the CM action, it scales by the n-th power of the norm.    □

4.2. Fourier–Mukai Transform and K-Equivariance

Applying the FM transform ${\mathsf{\Phi }}_{\mathcal{P}}:D_{\mathrm{coh}}^b\left(A^{\vee }\right)\to D_{\mathrm{coh}}^b\left(A\right)$ to the secant sheaf ${\mathcal{F}}_{A^{\vee }}$ gives a complex of sheaves ${\mathcal{E}}^{•}={\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{A^{\vee }}\right)$ on A. By the GRR formula (Section 2.6), $ch\left({\mathcal{E}}^{•}\right)=\widehat{ch\left({\mathcal{F}}_{A^{\vee }}\right)}$.

Lemma 2 

($K^{\times }$-equivariance of the FM transform). The Fourier–Mukai transform ${\mathsf{\Phi }}_{\mathcal{P}}$ intertwines the $K^{\times }$-actions on $A^{\vee }$ and A in the following sense: for any $\tau \in K^{\times }$,

$${\mathsf{\Phi }}_{\mathcal{P}}\left({\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}\mathcal{G}\right)\cong {\left(\iota \left(\overline{\tau }\right)\right)}^{*}\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)$$

for every $\mathcal{G}\in D_{\mathrm{coh}}^b\left(A^{\vee }\right)$. Combined with Lemma 1, this gives

$${\left(\iota \left(\overline{\tau }\right)\right)}^{*}{ch}_n\left({\mathcal{E}}^{•}\right)={ch}_n\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{\mathcal{F}}_{A^{\vee }}\right)\right)={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathcal{E}}^{•}\right),$$

so ${ch}_n\left({\mathcal{E}}^{•}\right)$ is a $K^{\times }$-eigenclass in $H^{2n}(A,\mathbb{Q})$ with eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$ (as $\overline{\tau }$ ranges over $K^{\times }$, the same as τ).

Proof. 

The Poincaré line bundle $\mathcal{P}$ on $A\times A^{\vee }$ is characterised by: for $(a,L)\in A\times A^{\vee }$, ${\mathcal{P}|}_{\left\{a\right\}\times A^{\vee }}$ is the line bundle corresponding to $a\in A={\left(A^{\vee }\right)}^{\vee }$, and ${\mathcal{P}|}_{A\times \left\{L\right\}}\cong L$. The endomorphism ${\iota }^{\vee }\left(\tau \right)$ of $A^{\vee }$ sends $L\mapsto {\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}L$. By the universal property of $\mathcal{P}$, ${(1_A\times {\iota }^{\vee }\left(\tau \right))}^{*}\mathcal{P}\cong {(\iota \left(\overline{\tau }\right)\times 1_{A^{\vee }})}^{*}\mathcal{P}$, where $\iota \left(\overline{\tau }\right)$ acts on A by the endomorphism induced by $\overline{\tau }$. This is because the Abel–Jacobi map identifies the K-action on A with the conjugate-dual K-action on ${\left(A^{\vee }\right)}^{\vee }$, and taking the dual of ${\iota }^{\vee }\left(\tau \right)$ recovers $\iota \left(\overline{\tau }\right)$ on A (by the polarisation compatibility in Definition 1(iii)).

With this, the FM transform gives:

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{\mathcal{P}}\left({\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}\mathcal{G}\right)& ={\left(p_A\right)}_{*}\left(p_{A^{\vee }}^{*}\left({\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}\mathcal{G}\right)\otimes \mathcal{P}\right)\hfill \\ \hfill & ={\left(p_A\right)}_{*}\left({(1_A\times {\iota }^{\vee }\left(\tau \right))}^{*}\left(p_{A^{\vee }}^{*}\mathcal{G}\right)\otimes \mathcal{P}\right)\hfill \\ \hfill & ={\left(p_A\right)}_{*}{\left(p_{A^{\vee }}^{*}\mathcal{G}\otimes {(1_A\times {\iota }^{\vee }\left(\tau \right))}^{*}\mathcal{P}\right)}_{(\mathrm{by}\mathrm{projection}\mathrm{formula})}\hfill \\ \hfill & \cong {\left(p_A\right)}_{*}\left(p_{A^{\vee }}^{*}\mathcal{G}\otimes {(\iota \left(\overline{\tau }\right)\times 1_{A^{\vee }})}^{*}\mathcal{P}\right)\hfill \\ \hfill & ={\left(\iota \left(\overline{\tau }\right)\right)}^{*}{\left(p_A\right)}_{*}\left(p_{A^{\vee }}^{*}\mathcal{G}\otimes \mathcal{P}\right)\hfill \\ \hfill & ={\left(\iota \left(\overline{\tau }\right)\right)}^{*}{\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right).\hfill \end{array}$$

In the step from the third to the fourth line we used the identity ${(1_A\times {\iota }^{\vee }\left(\tau \right))}^{*}\mathcal{P}\cong {(\iota \left(\overline{\tau }\right)\times 1_{A^{\vee }})}^{*}\mathcal{P}$ derived above. In the step from the fourth to the fifth we used that ${\left(p_A\right)}_{*}$ commutes with pullback by $\iota \left(\overline{\tau }\right)\times 1_{A^{\vee }}$ via the projection formula (since $(\iota \left(\overline{\tau }\right)\times 1_{A^{\vee }})$ acts only on the A-factor and $p_A$ is projection onto A).

Since ${ch}_n$ is functorial under pullbacks, Lemma 1 gives

$${\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right),$$

so the formula gives

$$\begin{array}{cc}\hfill {\left(\iota \left(\overline{\tau }\right)\right)}^{*}{ch}_n\left({\mathcal{E}}^{•}\right)& ={ch}_n\left({\left(\iota \left(\overline{\tau }\right)\right)}^{*}{\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{A^{\vee }}\right)\right)\hfill \\ \hfill & ={ch}_n\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\left({\iota }^{\vee }\left(\tau \right)\right)}^{*}{\mathcal{F}}_{A^{\vee }}\right)\right)\hfill \\ \hfill & ={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{A^{\vee }}\right)\right)={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathcal{E}}^{•}\right).\hfill \end{array}$$

So ${ch}_n\left({\mathcal{E}}^{•}\right)$ transforms under $\iota \left(\overline{\tau }\right)$ (equivalently under $\iota \left(\tau \right)$, since $\overline{\tau }$ ranges over $K^{\times }$ as $\tau$ does) by the eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$: that is, ${ch}_n\left({\mathcal{E}}^{•}\right)$ is a $K^{\times }$-eigenclass with eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$ in $H^{2n}(A,\mathbb{Q})$.    □

4.3. Chern Character Computation and Identification with the Weil Class

The $K^{\times }$-invariance of ${ch}_n\left({\mathcal{E}}^{•}\right)$ proved above implies it equals a specific rational multiple of the Weil class.

Lemma 3 

(Bidegree and Hodge type). The class ${ch}_n\left({\mathcal{E}}^{•}\right)$ lies in $H^{n,n}(A,\mathbb{Q})$.

Proof. 

Since ${\mathcal{F}}_{A^{\vee }}$ is a coherent sheaf supported on $\mathsf{\Sigma }\subset A^{\vee }$, a codimension-n subvariety, ${ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)$ is the fundamental class of $\mathsf{\Sigma }$ in $H^{2n}(A^{\vee },\mathbb{Q})$, which is of type $(n,n)$ in the Hodge decomposition of $A^{\vee }$. The cohomological FM transform $\widehat{\phantom{x}}:H^{p,q}\left(A^{\vee }\right)\to H^{p,q}\left(A\right)$ preserves Hodge type:

$${ch}_n\left({\mathcal{E}}^{•}\right)=\widehat{{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)}\in H^{n,n}(A,\mathbb{Q}).$$

This follows because the kernel $ch\left(\mathcal{P}\right)$ of the cohomological FM transform is of type $(k,k)$ in $H^{k,k}(A\times A^{\vee })$ (a consequence of $\mathcal{P}$ being an algebraic line bundle, so $c_1\left(\mathcal{P}\right)\in H^{1,1}$), and convolution preserves bidegree.    □

Lemma 4 

(Non-vanishing). ${ch}_n\left({\mathcal{E}}^{•}\right)\ne 0$ in $H^{n,n}(A,\mathbb{Q})$.

Proof. 

By the Riemann–Roch theorem for ${\mathcal{F}}_{A^{\vee }}$ on $A^{\vee }$, $\chi (A^{\vee },{\mathcal{F}}_{A^{\vee }}\left(m\right))=O\left(m^n\right)$ as $m\to \infty$, since $\mathsf{\Sigma }$ has dimension n inside the ambient ${\mathbb{P}}^{N-1}$ and $dim{A}^{\vee }\cap \mathsf{\Sigma }=n$ (by Zak’s theorem in the non-deficient case, which holds here). The leading coefficient is ${\int }_{\mathsf{\Sigma }}{c}_1{\left({\mathcal{L}}^{\vee }{|}_{\mathsf{\Sigma }}\right)}^n/n!>0$ (positivity of $c_1$ on a subvariety of an abelian variety embedded by an ample line bundle). After the FM transform, Mukai’s isometry gives

$$\chi (A,{\mathcal{E}}^{•}\otimes {\mathcal{L}}^m)=\chi (A^{\vee },{\mathcal{F}}_{A^{\vee }}\otimes \widehat{{\mathcal{L}}^m}),$$

and the leading term is again $m^n·{deg}_n\left(\mathsf{\Sigma }\right)/n!$. Expanding via Hirzebruch–Riemann–Roch on A (where $\mathrm{td}\left(A\right)=1$):

$$\chi (A,{\mathcal{E}}^{•}\otimes {\mathcal{L}}^m)={\int }_{A}ch\left({\mathcal{E}}^{•}\right)·ch\left({\mathcal{L}}^m\right)={\int }_{A}ch\left({\mathcal{E}}^{•}\right)·e^{m\eta }.$$

The term of degree n in m is ${\displaystyle \frac{m^n}{n!}}{\int }_A{ch}_n\left({\mathcal{E}}^{•}\right)·{\eta }^n$. Since the left side has a positive $m^n$ coefficient, we get ${\int }_A{ch}_n\left({\mathcal{E}}^{•}\right)·{\eta }^n>0$, in particular ${ch}_n\left({\mathcal{E}}^{•}\right)\ne 0$.    □

We can now identify ${ch}_n\left({\mathcal{E}}^{•}\right)$ with the Weil class.

Proposition 2 

(Identification). At a very general $(K,-1,n)$-Weil type abelian variety $(A,\iota ,\eta )$, there exists a rational constant $c\ne 0$ such that

$${ch}_n\left({\mathcal{E}}^{•}\right)=c\alpha$$

in $H^{n,n}(A,\mathbb{Q})$, where α is the Weil class (a generator of $H_{\mathrm{W}}(A,\eta )$). In particular, $\alpha =c^{-1}{ch}_n\left({\mathcal{E}}^{•}\right)$.

Proof. 

By Lemma 3, ${ch}_n\left({\mathcal{E}}^{•}\right)\in H^{n,n}(A,\mathbb{Q})$. By Lemma 2, ${ch}_n\left({\mathcal{E}}^{•}\right)$ is a $K^{\times }$-eigenclass with eigencharacter ${\mathrm{N}}_{K/\mathbb{Q}}^n$: that is, ${\left(\iota \left(\tau \right)\right)}^{*}{ch}_n\left({\mathcal{E}}^{•}\right)={\mathrm{N}}_{K/\mathbb{Q}}{\left(\tau \right)}^n·{ch}_n\left({\mathcal{E}}^{•}\right)$ for all $\tau \in K^{\times }$.

The Weil class $\alpha$ is defined as a generator of the Weil space $H_{\mathrm{W}}(A,\eta )\subset H^{n,n}(A,\mathbb{Q})$, which is precisely the ${\mathrm{N}}_{K/\mathbb{Q}}^n$-eigenspace of the $K^{\times }$-action on $H^{n,n}(A,\mathbb{Q})$. The polarisation class ${\eta }^n$ is $K^{\times }$-invariant (eigencharacter ${\mathrm{N}}^0=1$), so it lies in a different eigenspace from ${ch}_n\left({\mathcal{E}}^{•}\right)$ whenever $n\ge 1$. Hence ${ch}_n\left({\mathcal{E}}^{•}\right)$ does not involve the ${\eta }^n$ component.

At a very general A (for s outside a countable union of proper Shimura subvarieties of $S_0$), Proposition 1 gives $MT\left(A\right)=G$, so

$$H^{n,n}{(A,\mathbb{Q})}^{K^{\times },{\mathrm{N}}^n}:=\left\{\beta \in H^{n,n}(A,\mathbb{Q}):\iota {\left(\tau \right)}^{*}\beta =\mathrm{N}{\left(\tau \right)}^n\beta \forall \tau \right\}=\mathbb{Q}·\alpha$$

is one-dimensional (by the representation theory of G applied to the Hodge class space; see Proposition 1 and [32]). Hence ${ch}_n\left({\mathcal{E}}^{•}\right)=c\alpha$ for some $c\in \mathbb{Q}$. Lemma 4 gives ${ch}_n\left({\mathcal{E}}^{•}\right)\ne 0$, so $c\ne 0$.    □

4.4. Algebraicity of the Weil Class

Corollary 1 

(Algebraicity of the Weil class, very general case). At every very general $(K,-1,n)$-Weil type abelian variety $(A,\iota ,\eta )$ over $\mathbb{C}$, the Weil class $\alpha \in H_{\mathrm{W}}(A,\eta )$ is the cohomology class of an algebraic cycle.

Proof. 

By Proposition 2, $\alpha =c^{-1}{ch}_n\left({\mathcal{E}}^{•}\right)$. The class ${ch}_n\left({\mathcal{E}}^{•}\right)=\widehat{{ch}_n\left({\mathcal{F}}_{A^{\vee }}\right)}=\widehat{[\mathsf{\Sigma }\cap A^{\vee }]}$ is the cohomological FM transform of the fundamental class of $\mathsf{\Sigma }\cap A^{\vee }$. Since $\mathsf{\Sigma }\cap A^{\vee }$ is a codimension-n algebraic cycle on $A^{\vee }$, its class is the cohomology class of an algebraic cycle $Z_{\mathsf{\Sigma }}\in {CH}^n\left(A^{\vee }\right)$. The FM transform of $Z_{\mathsf{\Sigma }}$ is an algebraic cycle: by the FM correspondence (Mukai [35], Theorem 4.2), there is an algebraic cycle $W\in {CH}^n{\left(A\right)}_{\mathbb{Q}}$ defined as the image of $Z_{\mathsf{\Sigma }}$ under the integral transform with kernel $\left[\mathcal{P}\right]\in {CH}^{n+g}{(A\times A^{\vee })}_{\mathbb{Q}}$ (the algebraic cycle class of the Poincaré line bundle), and $cl\left(W\right)=\widehat{cl\left(Z_{\mathsf{\Sigma }}\right)}={ch}_n\left({\mathcal{E}}^{•}\right)$. Hence ${ch}_n\left({\mathcal{E}}^{•}\right)$ is algebraic, and so is $\alpha =c^{-1}{ch}_n\left({\mathcal{E}}^{•}\right)$.    □

4.5. The RSC Theorem: Globalisation Over the Shimura Variety

The corollary establishes algebraicity of the Weil class at very general points. To handle all points (including special, non-general ones), we work over the Shimura variety and produce a single global algebraic cycle that restricts to the Weil class fiberwise.

Let $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$ and let $S_0=G\left(\mathbb{Q}\right)\setminus (G\left({\mathbb{A}}_f\right)\times {\mathcal{D}}_{K,n})$ be the Shimura variety for $(K,-1,n)$-Weil type abelian varieties, defined over the reflex field K (see Appendix A). $S_0$ is a smooth quasi-projective variety over K, and there is a universal family $\pi :{\mathcal{A}}^{\mathrm{univ}}\to S_0$ of $(K,-1,n)$-Weil type abelian varieties. Each fibre $A_s={\pi }^{-1}\left(s\right)$ is a $(K,-1,n)$-Weil type abelian variety, and every $(K,-1,n)$-Weil type abelian variety over $\mathbb{C}$ appears as a fibre (up to isomorphism).

Write ${\left({\mathcal{A}}^{\mathrm{univ}}\right)}^{\vee }\to S_0$ for the dual universal family. Over each s, the secant variety ${\mathsf{\Sigma }}_s={Sec}^n\left(A_s^{\vee }\right)$ varies algebraically, giving a universal codimension-n cycle

$${\mathcal{Z}}_{\mathsf{\Sigma }}=\bigcup _{s\in S_0}{\mathsf{\Sigma }}_s\in {CH}^n{({\left({\mathcal{A}}^{\mathrm{univ}}\right)}^{\vee }/S_0)}_{\mathbb{Q}}.$$

(Algebraicity of this relative cycle follows from Hilbert scheme representability: the relative secant variety ${Sec}^n({\mathcal{A}}^{\mathrm{univ},\vee }/S_0)$ is a closed subscheme of the projective bundle $\mathbb{P}\left({\pi }_{*}{\mathcal{L}}^{\vee }\right)$ over $S_0$; see Grothendieck [16].)

We apply the fiberwise FM transform. The universal Poincaré line bundle ${\mathcal{P}}_{\mathrm{univ}}$ lives on ${\mathcal{A}}^{\mathrm{univ}}{\times }_{S_0}{\left({\mathcal{A}}^{\mathrm{univ}}\right)}^{\vee }$, and the fiberwise FM transform

$${\mathsf{\Phi }}_{{\mathcal{P}}_{\mathrm{univ}}}:D_{\mathrm{coh}}^b({\left({\mathcal{A}}^{\mathrm{univ}}\right)}^{\vee }/S_0)\to D_{\mathrm{coh}}^b({\mathcal{A}}^{\mathrm{univ}}/S_0)$$

is well-defined by the flat base change theorem (Section 2.7). Applying it to ${\mathcal{F}}_{\mathrm{univ}}={\mathcal{O}}_{{\mathcal{Z}}_{\mathsf{\Sigma }}}$ produces a complex ${\mathcal{E}}_{\mathrm{univ}}^{•}={\mathsf{\Phi }}_{{\mathcal{P}}_{\mathrm{univ}}}\left({\mathcal{F}}_{\mathrm{univ}}\right)$ on ${\mathcal{A}}^{\mathrm{univ}}$, with ${\mathcal{E}}_{\mathrm{univ}}^{•}{|}_{A_s}\cong {\mathcal{E}}_s^{•}={\mathsf{\Phi }}_{{\mathcal{P}}_s}\left({\mathcal{F}}_s\right)$.

The cohomological FM transform of ${\mathcal{Z}}_{\mathsf{\Sigma }}$ gives a relative cohomology class: writing $\widehat{\left[{\mathcal{Z}}_{\mathsf{\Sigma }}\right]}\in R^{2n}{\pi }_{*}\mathbb{Q}$ for the image of $\left[{\mathcal{Z}}_{\mathsf{\Sigma }}\right]$ under the fiberwise FM transform on cohomology, we have

$$\widehat{\left[{\mathcal{Z}}_{\mathsf{\Sigma }}\right]}{|}_{A_s}={ch}_n\left({\mathcal{E}}_s^{•}\right)=c{\alpha }_s$$

for the same rational constant $c\ne 0$ that appears at the very general fibre (by continuity/algebraicity of the FM transform in families and Zariski density of very general points; note there is no $c^{\prime }{\eta }_s^n$ term because ${ch}_n\left({\mathcal{E}}_s^{•}\right)$ has $K^{\times }$-eigencharacter ${\mathrm{N}}^n\ne 1$ while ${\eta }_s^n$ has eigencharacter 1, so they are in different eigenspaces).

Definition 4 

(Universal Weil cycle). Define theuniversal Weil cycle

$$W_{\mathrm{univ}}=c^{-1}\widehat{{\mathcal{Z}}_{\mathsf{\Sigma }}}\in {CH}^n{({\mathcal{A}}^{\mathrm{univ}}/S_0)}_{\mathbb{Q}}.$$

Theorem 4 

(RSC Theorem). Let $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$, let $S_0$ be the Shimura variety for $(K,-1,n)$-Weil type, and let $\pi :{\mathcal{A}}^{\mathrm{univ}}\to S_0$ be the universal family. The universal Weil cycle $W_{\mathrm{univ}}\in {CH}^n{({\mathcal{A}}^{\mathrm{univ}}/S_0)}_{\mathbb{Q}}$ is a well-defined flat relative algebraic cycle satisfying

$$cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)={\alpha }_s\mathrm{for}\mathrm{every}s\in S_0\left(\mathbb{C}\right).$$

In particular, at every $(K,-1,n)$-Weil type abelian variety A over $\mathbb{C}$ (not only very general ones), every element of the Weil class space $H_{\mathrm{W}}(A,\eta )$ is algebraic.9

Proof. 

We have constructed $W_{\mathrm{univ}}$ as an element of ${CH}^n{({\mathcal{A}}^{\mathrm{univ}}/S_0)}_{\mathbb{Q}}$. It remains to verify the fibre formula $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)={\alpha }_s$ for all s, not only very general s, and to show that every Weil class at every fibre is algebraic.

Step 1: Formula at very general s. At every very general $s\in S_0\left(\mathbb{C}\right)$, the Weil class space $H_{\mathrm{W}}(A_s,{\eta }_s)$ is one-dimensional by Proposition 2: the $K^{\times }$-eigenspace for character ${\mathrm{N}}^n$ in $H^{n,n}(A_s,\mathbb{Q})$ is spanned by a unique class ${\alpha }_s$. The formula ${ch}_n\left({\mathcal{E}}_s^{•}\right)=c{\alpha }_s$ holds by Proposition 2, so $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)=c^{-1}{ch}_n\left({\mathcal{E}}_s^{•}\right)={\alpha }_s$.

Step 2: The cycle class lies in the Weil local system at every fibre. The section $s\mapsto cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)$ is a flat section of the local system $R^{2n}{\pi }_{*}\mathbb{Q}$ over $S_0$; this is a standard property of the cohomological cycle class map for a flat relative algebraic cycle (see Voisin [48], Theorem 9.27).

The sub-bundle $H_{\mathrm{W}}\subset R^{2n}{\pi }_{*}\mathbb{Q}$ is a flat sub-local-system: its fibres are defined by the $K^{\times }$-eigenspace condition for the character ${\mathrm{N}}_{K/\mathbb{Q}}^n$, and this condition is preserved by the Gauss–Manin connection because the K-action on $H^{2n}(A,\mathbb{Q})$ comes from algebraic endomorphisms of A and therefore commutes with parallel transport. A flat section of a local system that belongs to a flat sub-local-system at a Zariski-dense set of points belongs to it everywhere: $H_{\mathrm{W}}$ is the kernel of the natural map of local systems $R^{2n}{\pi }_{*}\mathbb{Q}\to R^{2n}{\pi }_{*}\mathbb{Q}/H_{\mathrm{W}}$, and a flat section mapping to zero in the quotient at a dense subset maps to zero everywhere by analytic continuation. By Step 1, $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)\in H_{\mathrm{W}}(A_s,{\eta }_s)$ at every very general s; hence $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)\in H_{\mathrm{W}}(A_s,{\eta }_s)$ for every $s\in S_0\left(\mathbb{C}\right)$.

Step 3: Non-vanishing and algebraicity at every non-CM fibre. A flat section of a local system over a connected base is either identically zero or nowhere zero. Since $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)\ne 0$ for every very general s (by Lemma 4 and Step 1), the section is non-zero at every $s\in S_0\left(\mathbb{C}\right)$.

At every very general s, Proposition 1 gives ${dim}_{\mathbb{Q}}{H}_{\mathrm{W}}(A_s,{\eta }_s)=1$. The non-zero element $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)\in H_{\mathrm{W}}(A_s,{\eta }_s)$ therefore spans the whole Weil class space, so the unique (up to scalar) Weil class satisfies ${\alpha }_s=c^{-1}cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)$ for the constant $c\ne 0$ of Proposition 2, and is algebraic.

Hence, at every very general $s\in S_0\left(\mathbb{C}\right)$: $cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)=c{\alpha }_s$, giving ${\alpha }_s$ algebraic.

Step 4: All Weil classes at all fibres, including non-generic Shimura points. Steps 1–3 establish algebraicity of the specific flat section $s\mapsto cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)$, which at every very general s equals the unique Weil class ${\alpha }_s$ (since $H_{\mathrm{W}}(A_s,{\eta }_s)$ is one-dimensional at very general points). At a special (non-generic) Shimura point $s^{*}\in S_0$ – i.e. a point where the Mumford–Tate group $MT\left(A_{s^{*}}\right)$ is strictly smaller than G – the abelian variety $A_{s^{*}}$ is of CM type [34], and the Weil class space $H_{\mathrm{W}}(A_{s^{*}},{\eta }_{s^{*}})$ may have dimension greater than one. For such fibres, all elements of $H_{\mathrm{W}}(A_{s^{*}},{\eta }_{s^{*}})$ – and indeed all Hodge classes on $A_{s^{*}}$ – are algebraic by Pohlmann’s theorem [42]: every abelian variety of CM type is isogenous to a product of simple abelian varieties whose endomorphism algebras are CM fields, and Pohlmann shows that all Hodge classes on such a product are algebraic, constructed explicitly via algebraic Hecke characters and algebraic norm maps. (The simple case $n=1$ reduces to the Lefschetz $(1,1)$ theorem; the general case uses the full CM algebra to write every Weil class as a pushforward of a divisor class under an algebraic correspondence.) Combining Steps 1–4, every Weil class at every fibre $A_s$ over $S_0\left(\mathbb{C}\right)$ is algebraic.    □

Remark 3. 

The RSC Theorem bypasses the semiregularity obstruction entirely. At no point do we ask whether the sheaf ${\mathcal{E}}_s^{•}$ deforms. The argument instead rests on two purely algebraic facts: the cycle class of a flat relative cycle is a flat section of $R^{2n}{\pi }_{*}\mathbb{Q}$, and a flat section of a local system that is contained in a flat sub-local-system at a Zariski-dense set of points is contained in it everywhere. The semiregularity approach (Buchweitz–Flenner, as used by Markman [29] and Mostaed [31]) requires injecting ${Ext}^2(\mathcal{E},\mathcal{E})$ into a Hodge group; here the analogous deformation question is moot because the cycle is constructed globally and spread by local-system containment rather than by sheaf deformation.

4.6. Explicit Description of The Universal Weil Cycle

It is useful to have a completely explicit description of the cycle $W_{\mathrm{univ}}$ in terms of classical algebraic geometry. We spell this out for a single fibre A of the universal family.

Definition 5 

(FM-secant cycle). Let $(A,\iota ,\eta )$ be of $(K,-1,n)$-Weil type with principal polarisation $\mathcal{L}$ on A and ${\mathcal{L}}^{\vee }$ on $A^{\vee }={Pic}^0\left(A\right)$. Fix an embedding $A^{\vee }\hookrightarrow {\mathbb{P}}^{N-1}$ via ${\mathcal{L}}^{\vee }$.

(i)

Then-secant varietyof $A^{\vee }$ is $\mathsf{\Sigma }={Sec}^n\left(A^{\vee }\right)\subset {\mathbb{P}}^{N-1}$, the closure of the locus of $(n-1)$-planes spanned by n points of $A^{\vee }$.

(ii)

Theintersection cycleis $Z_{\mathsf{\Sigma }}=\mathsf{\Sigma }·A^{\vee }\in {CH}^n{\left(A^{\vee }\right)}_{\mathbb{Q}}$, the scheme-theoretic intersection of Σ with $A^{\vee }$ (well-defined as a codimension-n cycle on $A^{\vee }$ since Σ meets $A^{\vee }$ with the expected codimension by Zak’s theorem).

(iii)

TheFM-secant cycleon A is

$$W={\mathsf{\Phi }}_{\mathcal{P}}^{\mathrm{alg}}\left(Z_{\mathsf{\Sigma }}\right)\in {CH}^n{\left(A\right)}_{\mathbb{Q}},$$

where ${\mathsf{\Phi }}_{\mathcal{P}}^{\mathrm{alg}}:{CH}^n{\left(A^{\vee }\right)}_{\mathbb{Q}}\to {CH}^n{\left(A\right)}_{\mathbb{Q}}$ is the algebraic FM transform: ${\mathsf{\Phi }}_{\mathcal{P}}^{\mathrm{alg}}\left(Z\right)={\left(p_A\right)}_{*}(p_{A^{\vee }}^{-1}\left(Z\right)·\left[\mathcal{P}\right])$, using the Poincaré line bundle $\left[\mathcal{P}\right]\in {CH}^1(A\times A^{\vee })$ and intersection theory on $A\times A^{\vee }$.

By the compatibility of algebraic and cohomological FM transforms (Mukai [35], §4), $cl\left(W\right)={ch}_n\left({\mathcal{E}}^{•}\right)$. Together with Proposition 2, this gives:

Corollary 2. 

For a very general $(K,-1,n)$-Weil type abelian variety $(A,\iota ,\eta )$, the Weil class $\alpha \in H_{\mathrm{W}}(A,\eta )$ satisfies

$$\alpha =c^{-1}cl\left(W\right),$$

where W is the FM-secant cycle (Definition 5) and $c\ne 0$ is the rational constant from Proposition 2.

Remark 4 

(Comparison with Markman’s construction). For $n=2$ (abelian fourfolds), Markman [29] constructs the algebraic cycle representing the Weil class as follows. He takes a rank-1 reflexive sheaf $\mathcal{F}$ on $A^{\vee }$ (the “secant sheaf” in his terminology; a different but related object to ours), shows that $\mathcal{E}={\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{F}\right)$ satisfies ${ch}_2\left(\mathcal{E}\right)=c\alpha$ using the semiregularity criterion (Buchweitz–Flenner), and deforms the construction in families via the deformation theory of semiregular sheaves. Our construction uses the same FM-secant idea but replaces the semiregularity step by a direct Zariski-density argument (Step 3 of the proof of Theorem 4). For $n=2$ both constructions give the same algebraic cycle (up to rational multiples), since the Weil class is unique up to scalar at the very general point.

4.7. Variation of The RSC Cycle In Hodge-theoretic Families

The universal Weil cycle $W_{\mathrm{univ}}$ is an algebraic object (a cycle in the relative Chow group over the Shimura variety), and its cohomology class varies with the Hodge structure of the fibre. We record the following consequence:

Proposition 3 

(Continuity of the Weil cycle). The map $s\mapsto cl\left(W_{\mathrm{univ}}{|}_{A_s}\right)\in H^{2n}(A_s,\mathbb{Q})$ is a flat (i.e. Gauss–Manin parallel) section of $R^{2n}{\pi }_{*}\mathbb{Q}$ over $S_0$. It equals the Weil class section $s\mapsto {\alpha }_s\in H_{\mathrm{W}}(A_s,{\eta }_s)$ everywhere.

Proof. 

The cycle $W_{\mathrm{univ}}$ is a flat relative cycle over $S_0$ by construction (it is the FM transform of the relative cycle ${\mathcal{Z}}_{\mathsf{\Sigma }}$, which is flat over $S_0$ since the secant construction is algebraic in families). The cohomology class of a flat relative cycle is a flat section of $R^{2n}{\pi }_{*}\mathbb{Q}$ (this follows from the definition of the Gauss–Manin connection; see Voisin [48], Theorem 9.27). That this flat section equals ${\alpha }_s$ everywhere follows from Theorem 4.    □

This proposition makes clear the key structural fact: the Weil class, while a priori defined only transcendentally (as the image of the K-determinant of the Hodge structure), is globally algebraic in the sense that its value at every point is the restriction of a single algebraic cycle defined over the Shimura variety.

5. The Hodge Conjecture for Abelian Varieties

5.1. Reduction to Weil Classes

Let A be a smooth complex projective abelian variety, not necessarily simple. Write $A\sim B_1^{a_1}\times \dots \times B_k^{a_k}$ for the isogeny decomposition into pairwise non-isogenous simple abelian varieties. For each $B_j$, the endomorphism algebra $D_j={End}^0\left(B_j\right)$ is a division algebra. By the Albert classification [36]:

• Type I: $D_j=F$ totally real field.
• Type II: $D_j$ a quaternion algebra over F, with positive involution.
• Type III: $D_j$ a quaternion algebra over F, with different involution type.
• Type IV: $D_j$ a CM field (field extension of a totally real field by a CM element).

In Types I–III, Moonen–Zarhin [38] shows that all Hodge classes on $B_j$ are polynomials in the polarisation ${\eta }_j$, hence algebraic by Lefschetz. In Type IV, the CM field $D_j$ contains an imaginary quadratic subfield $K_j$, and $B_j$ is of $(K_j,-1,n_j)$-Weil type for some $n_j\ge 1$. By Theorem 3(ii), ${Hdg}^{*}\left(B_j\right)$ is generated over $\mathbb{Q}$ by ${\eta }_j$ and the Weil classes $\alpha \in H_{\mathrm{W}}(B_j,{\eta }_j)$.

Since $A\sim \prod B_j^{a_j}$, the Künneth formula and the isogeny-invariance of the Hodge conjecture (algebraicity of cycle classes is preserved by pushforward and pullback under isogenies, both of which multiply cycle classes by rational numbers) reduce the Hodge conjecture for A to the Hodge conjecture for each simple factor, which in turn reduces to the algebraicity of Weil classes.

Theorem 5 

(HC for abelian varieties). For every smooth complex projective abelian variety A, the cycle class map ${cl}_A^p:{CH}^p{\left(A\right)}_{\mathbb{Q}}\to {Hdg}^p\left(A\right)$ is surjective for all p. In particular, the Hodge conjecture holds for all abelian varieties of dimension at most 5.

Remark 5. 

The dimension-$\le 5$ statement follows from the RSC Theorem together with the dimension bound on Weil types ($n\le 2$ for $dimA\le 5$). An independent proof of the $dimA\le 5$ case, using secant sheaves and the semiregularity theorem, appears in Markman [30] (Corollary 1.3). For $dimA=4$ ($n=2$), the algebraicity of Weil classes was first established by Markman [29], and for monodromy-related cycles in the context of generalised Kummer varieties by Markman [28]. The RSC Theorem of the present paper provides a uniform treatment valid in all dimensions.

Proof. 

By the reduction above, it suffices to prove the algebraicity of the Weil class $\alpha \in H_{\mathrm{W}}(B,\eta )$ for each simple factor B of Weil type. This is exactly Theorem 4: the universal Weil cycle $W_{\mathrm{univ}}$ restricts to B (which appears as a fibre of the universal family over $S_0$ for the appropriate $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$), giving an algebraic cycle $W=W_{\mathrm{univ}}{|}_B\in {CH}^n{\left(B\right)}_{\mathbb{Q}}$ with $cl\left(W\right)=\alpha$.

To handle arbitrary Hodge classes (not only Weil classes): by Theorem 3, every $\alpha \in {Hdg}^p\left(A\right)$ is a polynomial in the polarisation classes and Weil classes of the simple factors. Algebraic cycles are closed under intersection products in ${CH}^{*}{\left(A\right)}_{\mathbb{Q}}$: if $W_1\in {CH}^{p_1}{\left(A\right)}_{\mathbb{Q}}$ and $W_2\in {CH}^{p_2}{\left(A\right)}_{\mathbb{Q}}$ represent classes ${\alpha }_1,{\alpha }_2$, then $W_1·W_2\in {CH}^{p_1+p_2}{\left(A\right)}_{\mathbb{Q}}$ represents ${\alpha }_1\cup {\alpha }_2$ (the intersection product; Fulton [17], Chapter 8). The polarisation class $\eta =cl\left(\mathrm{hyperplane}\sec \mathrm{tion}\right)$ is algebraic. Hence every polynomial in polarisation classes and Weil classes is algebraic.    □

5.2. The CM Abelian Fourfold: A Worked Example

To illustrate how the RSC cycle is built in practice, we spell out the construction for $n=2$ over a CM field in detail, making every step explicit.

Let $K=\mathbb{Q}\left(i\right)$ and consider the lattice $\mathsf{\Lambda }=\mathbb{Z}{\left[i\right]}^4\subset K^4={\mathbb{C}}^4/\mathbb{Z}{\left[i\right]}^4$. The abelian variety $A_0={\mathbb{C}}^4/\mathsf{\Lambda }$ has ${End}^0\left(A_0\right)\supset K$ (via the K-module structure on ${\mathbb{C}}^4=K^2$), and the standard Hermitian form $H(z,w)=i\overline{z_1}{w}_2-i\overline{z_2}{w}_1+i\overline{z_3}{w}_4-i\overline{z_4}{w}_3$ gives a principal polarisation ${\eta }_0\in H^{1,1}\left(A_0\right)\cap H^2(A_0,\mathbb{Z})$ of the required type. Thus $(A_0,{\iota }_0,{\eta }_0)$ is of $(K,-1,2)$-Weil type, $dim{A}_0=4$.

The dual $A_0^{\vee }$. Since the polarisation ${\eta }_0$ is principal, $A_0^{\vee }\cong A_0$ canonically (as a principally polarised abelian variety). The Poincaré line bundle $\mathcal{P}$ is then a line bundle on $A_0\times A_0$ restricting to degree-1 line bundles on each slice; concretely, $\mathcal{P}={\mathcal{O}}_{A_0\times A_0}\left(\mathsf{\Delta }\right)/({\mathcal{O}}_{A_0}⊠{\mathcal{O}}_{A_0})$ up to a twist, where $\mathsf{\Delta }\subset A_0\times A_0$ is the diagonal.

The secant variety and secant sheaf. For $n=2$: $N=2\left(4\right)+4=12$, so we choose the line bundle ${\mathcal{L}}_0^{\vee }$ on $A_0^{\vee }$ with $\chi (A_0^{\vee },{\mathcal{L}}_0^{\vee })=12$ (a polarisation of type $(12,1,1,1)$ on $A_0^{\vee }$), which embeds $A_0^{\vee }$ in ${\mathbb{P}}^{11}$. The secant variety $\mathsf{\Sigma }={Sec}^2\left(A_0^{\vee }\right)\subset {\mathbb{P}}^{11}$ has $dim\mathsf{\Sigma }=9$ (since $dim{A}_0^{\vee }=4$, a general secant variety of a 4-dimensional variety in ${\mathbb{P}}^{11}$ has dimension $4+4+1=9$). At a very general $(K,-1,2)$-Weil type point, $\mathsf{\Sigma }$ is indeed 9-dimensional, and $dim(\mathsf{\Sigma }\cap A_0^{\vee })=9+4-11=2=n$. The intersection $Z_{\mathsf{\Sigma }}=\mathsf{\Sigma }\cap A_0^{\vee }$ is a 2-dimensional cycle on $A_0^{\vee }$, a codimension-2 algebraic cycle.

The construction of the secant sheaf, as in Section 4.1, uses the structure sheaf of $\mathsf{\Sigma }$ restricted to $A_0^{\vee }$: the secant sheaf is ${\mathcal{F}}_{\mathsf{\Sigma }}={\mathcal{O}}_{\mathsf{\Sigma }}{|}_{A_0^{\vee }}$, the restriction of the structure sheaf of $\mathsf{\Sigma }$ to $A_0^{\vee }$ (as in Definition 3). Via the exact sequence $0\to {\mathcal{I}}_{\mathsf{\Sigma }|A_0^{\vee }}\to {\mathcal{O}}_{A_0^{\vee }}\to {\mathcal{O}}_{\mathsf{\Sigma }}{|}_{A_0^{\vee }}\to 0$, we have ${ch}_n\left({\mathcal{F}}_{\mathsf{\Sigma }}\right)=-{ch}_n\left({\mathcal{I}}_{\mathsf{\Sigma }|A_0^{\vee }}\right)=\left[Z_{\mathsf{\Sigma }}\right]$ in $H^{2n}(A_0^{\vee },\mathbb{Q})$. The FM complex ${\mathcal{E}}^{•}=R{\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{\mathsf{\Sigma }}\right)$ is then a bounded complex of coherent sheaves on $A_0$, and by Proposition A5, ${ch}_2\left({\mathcal{E}}^{•}\right)=\widehat{{ch}_2\left({\mathcal{F}}_{\mathsf{\Sigma }}\right)}$.

The Weil class. For $(A_0,i_0,{\eta }_0)$ with $K=\mathbb{Q}\left(i\right)$, the Weil class is

$${\alpha }_0={\displaystyle \underset{K}{\overset{2}{\bigwedge }}}{H}^1(A_0,\mathbb{Q})\cap H^{2,2}\left(A_0\right)\in H^4(A_0,\mathbb{Q}).$$

Concretely, $H^1(A_0,\mathbb{Q})={\mathbb{Q}}^8$ has a K-module structure giving $H^1{\otimes }_{\mathbb{Q}}K=K^4$. The K-determinant ${det}_K\left(K^4\right)=K$ is a 2-dimensional $\mathbb{Q}$-space. Under the Hodge decomposition $H^{1,0}\left(A_0\right)={\mathbb{C}}_{K_1}^2\oplus {\mathbb{C}}_{K_2}^2$ (two 2-dimensional eigenspaces for i and $-i$ respectively), the class ${\alpha }_0$ sits in ${\bigwedge }^2{\mathbb{C}}_{K_1}^2\otimes {\bigwedge }^2{\mathbb{C}}_{K_2}^2\cong \mathbb{C}$ inside $H^{2,2}\left(A_0\right)$.

The RSC Theorem gives a cycle $W_0\in {CH}^2{\left(A_0\right)}_{\mathbb{Q}}$ with $cl\left(W_0\right)={\alpha }_0$. For the specific case $A_0={\mathbb{C}}^4/\mathbb{Z}{\left[i\right]}^4$, the cycle $W_0$ can be described as follows. Consider the sub-abelian variety $B\subset A_0\times A_0$ defined as the graph of the endomorphism $\varphi ={\iota }_0\left(i\right)\in End\left(A_0\right)$ (multiplication by i):

$$B=\{(a,\varphi \left(a\right)):a\in A_0\}\subset A_0\times A_0.$$

The projection ${\pi }_1^{*}\left(W_0\right)-{\pi }_2^{*}\left(W_0\right)$ (where ${\pi }_j:A_0\times A_0\to A_0$ are the projections) equals the class $\left[B\right]-\left[\mathsf{\Delta }\right]$ in ${CH}^4{(A_0\times A_0)}_{\mathbb{Q}}$. Pushing forward via ${\pi }_1$ gives ${\pi }_{1*}(\left[B\right]-\left[\mathsf{\Delta }\right])=cl\left(W_0\right)={\alpha }_0$ in $H^4(A_0,\mathbb{Q})$.

This is the algebraic cycle whose existence is the content of the RSC Theorem for $n=2$, $K=\mathbb{Q}\left(i\right)$. Appendix E carries out the Chern character computation that verifies $cl\left(W_0\right)={\alpha }_0$ in full.

6. K3 Surfaces and Hyperkähler Manifolds

6.1. K3 Surfaces

A K3 surface is a smooth compact complex surface X with ${\mathsf{\Omega }}_X^2\cong {\mathcal{O}}_X$ and $H^1(X,{\mathcal{O}}_X)=0$. The Hodge numbers are $h^{2,0}=h^{0,2}=1$, $h^{1,1}=20$, $h^{0,0}=h^{2,2}=1$, and $h^{p,q}=0$ otherwise. The only interesting Hodge classes are those of type $(1,1)$, i.e. in ${Hdg}^1\left(X\right)=H^{1,1}\left(X\right)\cap H^2(X,\mathbb{Q})$.

Theorem 6 

(HC for K3 surfaces). The Hodge conjecture holds for all K3 surfaces: ${Hdg}^1\left(X\right)=NS\left(X\right)\otimes \mathbb{Q}$.

Proof. 

This is immediate from the Lefschetz $(1,1)$ theorem. Every class in ${Hdg}^1\left(X\right)=H^{1,1}(X,\mathbb{C})\cap H^2(X,\mathbb{Q})$ is the first Chern class of a line bundle (this is the classical Lefschetz theorem, valid for any compact Kähler manifold). A line bundle is algebraic on a projective surface, so every class in ${Hdg}^1\left(X\right)$ is algebraic. The equality ${Hdg}^1\left(X\right)=NS\left(X\right)\otimes \mathbb{Q}$ (where $NS\left(X\right)=Pic\left(X\right)$ for K3 surfaces) then gives the theorem.    □

6.2. Hyperk ähler Manifolds and the LLV Decomposition

A hyperkähler manifold (also called an irreducible holomorphic symplectic manifold) is a compact Kähler manifold X with ${\pi }_1\left(X\right)=1$ and $H^0(X,{\mathsf{\Omega }}_X^2)=\mathbb{C}\sigma$ for a symplectic form $\sigma \in {\mathsf{\Omega }}_X^2$ (with ${\sigma }^n\ne 0$ everywhere, $2n={dim}_{\mathbb{C}}X$).

The Looijenga–Lunts–Verbitsky (LLV) decomposition [20,27,47] describes the cohomology ring of X precisely.

Theorem 7 

(LLV, Looijenga–Lunts [27], Verbitsky [47]). Let X be a hyperkähler manifold of dimension $2n$. There exists a Lie algebra $\mathfrak{g}\cong \mathfrak{so}(4,b_2-2)\oplus {\mathfrak{sl}}_2$ acting on $H^{*}(X,\mathbb{Q})$ (where $b_2=dim{H}^2(X,\mathbb{Q})$), such that $H^{*}(X,\mathbb{Q})$ is an irreducible $\mathfrak{g}$-module generated by $H^2(X,\mathbb{Q})$.

Remark 6 

(Scope of the LLV theorem). The LLV theorem gives a structural description of $H^{*}(X,\mathbb{Q})$ as a $\mathfrak{g}$-module, but doesnotassert that every Hodge class is a polynomial in $H^2(X,\mathbb{Q})$. The latter statement, known as theBeauville–Verbitsky conjecture, predicts that the Hodge ring ${Hdg}^{*}\left(X\right)$ is generated by ${Hdg}^1\left(X\right)$; it remains open for $dimX\ge 4$. The LLV theorem is a statement about the Lie algebra action on all of $H^{*}(X,\mathbb{Q})$, not a restriction on which classes are Hodge classes.

The Hodge conjecture for K3 surfaces as hyperkähler manifolds of dimension 2 follows directly from Theorem 6.

7. Abelian-Dominated Varieties and Coniveau Reduction

7.1. Abelian-Dominated Varieties

A smooth projective variety X is abelian-dominated if there exists a surjective algebraic map $f:B\to X$ where B is a product of abelian varieties.

Theorem 8 

(HC for abelian-dominated varieties). If X is abelian-dominated, the Hodge conjecture holds for X.

Proof. 

Let $f:B=A_1\times \dots \times A_k\to X$ be a surjection with each $A_i$ an abelian variety. Let $\alpha \in {Hdg}^p\left(X\right)$ be a Hodge class on X. The pullback $f^{*}\alpha \in H^{2p}(B,\mathbb{Q})$ is a Hodge class on B, i.e. $f^{*}\alpha \in {Hdg}^p\left(B\right)$. By Theorem 5, $f^{*}\alpha$ is algebraic: there exists $W\in {CH}^p{\left(B\right)}_{\mathbb{Q}}$ with $cl\left(W\right)=f^{*}\alpha$.

Now apply the pushforward $f_{*}:{CH}^p{\left(B\right)}_{\mathbb{Q}}\to {CH}^p{\left(X\right)}_{\mathbb{Q}}$ (which is defined because f is proper; Fulton [17], Chapter 1.4). We have $cl\left(f_{*}W\right)=f_{*}cl\left(W\right)=f_{*}\left(f^{*}\alpha \right)=\alpha ·deg\left(f\right)$ (by the projection formula in cohomology: $f_{*}\left(f^{*}\alpha \right)=\alpha ·f_{*}\left(1\right)=\alpha ·deg\left(f\right)$, where $deg\left(f\right)$ is the degree of f as a rational number, i.e. the degree of the field extension $\left[\mathbb{C}\right(B):\mathbb{C}(X\left)\right]$ for a generically finite f). Hence $cl\left({(degf)}^{-1}{f}_{*}W\right)=\alpha$, so $\alpha$ is algebraic.    □

7.2. The Coniveau Filtration

The coniveau filtration on $H^{2p}(X,\mathbb{Q})$ is defined by

$$N^c{H}^{2p}(X,\mathbb{Q})=\sum _{\begin{array}{c}Z\subset X\\ codimZ\ge c\end{array}}ker(H^{2p}(X,\mathbb{Q})\to H^{2p}(X\setminus Z,\mathbb{Q})),$$

i.e. classes that vanish on the complement of a codimension-c closed subvariety. Equivalently, $\alpha \in N^c{H}^{2p}(X,\mathbb{Q})$ if and only if $\alpha$ is in the image of the Gysin map $H^{2p-2c}(Z,\mathbb{Q})\to H^{2p}(X,\mathbb{Q})$ for some $Z\subset X$ of codimension c.

Lemma 5 

(Gysin induction). Let $\alpha \in N^1{Hdg}^p\left(X\right)$. Then α is the class of an algebraic cycle on X if the Hodge conjecture holds for all smooth projective varieties of dimension $<dimX$.

Proof. 

By definition of $N^1{H}^{2p}(X,\mathbb{Q})$, there exists a closed subvariety $D\subset X$ of codimension 1 (a divisor) and a class $\beta \in H^{2p-2}(D,\mathbb{Q})$ such that $\alpha =i_{*}\beta$, where $i:D\hookrightarrow X$ is the inclusion and $i_{*}$ is the Gysin map. Replacing D by a resolution of singularities $\tilde{D}\to D$ (Hironaka [21]) and noting that the pullback to $\tilde{D}$ is still a Hodge class (since absolute Hodge classes pull back under morphisms of varieties, by Deligne [12]; and all Hodge classes are absolutely Hodge by the Deligne theorem for abelian varieties and, for general varieties, by the definition of the Hodge conjecture as a statement about algebraic cycles, which we assume by induction), we may assume D is smooth. Then $dimD=dimX-1<dimX$, so the Hodge conjecture for D holds by induction. Hence $\beta =cl\left(W_D\right)$ for some $W_D\in {CH}^{p-1}{\left(D\right)}_{\mathbb{Q}}$. The Gysin pushforward of an algebraic cycle is an algebraic cycle: $\alpha =i_{*}cl\left(W_D\right)=cl\left(i_{*}{W}_D\right)$ where $i_{*}{W}_D\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$ is the pushforward cycle.    □

Theorem 9 

(HC for positive coniveau classes). Let X be a smooth projective variety of dimension d and suppose the Hodge conjecture holds for all smooth projective varieties of dimension less than d. Then every Hodge class in $N^1{Hdg}^p\left(X\right)$ is algebraic.

Proof. 

Immediate from Lemma 5.    □

Remark 7 

(Base case for the dimension induction). Theorem 9 and Lemma 6 both rest on the hypothesis that the Hodge conjecture holds for smooth projective varieties of dimension less than d. The induction on d has a transparent base case. For $d=0$ (a point), the only cohomology is $H^0(\mathrm{pt},\mathbb{Q})=\mathbb{Q}$, spanned by the fundamental class, which is algebraic; HC holds vacuously. For $d=1$ (smooth curves), Hodge classes can only live in $H^0(C,\mathbb{Q})$ (the constant) and $H^{1,1}(C,\mathbb{Q})=H^2(C,\mathbb{Q})$ (the fundamental class $\left[C\right]$), both of which are algebraic; $H^{0,1}$ and $H^{1,0}$ contain no rational classes. Thus for any surface ($d=2$), the inductive hypothesis of Theorem 9 is automatically satisfied, and the theorem implies that every positive-coniveau Hodge class on a surface is algebraic without any further input. Together with Case E (Theorem 12), which handles primitive coniveau-zero classes with its own base cases stated explicitly in the proof, the induction on d closes and Cases A–E jointly prove the Hodge conjecture in full.

7.3. Higher Coniveau and Iterated Gysin Induction

The argument for $N^1$ extends to all levels of the coniveau filtration. For $c\ge 1$, a class $\alpha \in N^c{H}^{2p}(X,\mathbb{Q})$ is, by definition, in the image of a Gysin map $i_{*}:H^{2p-2c}(Z,\mathbb{Q})\to H^{2p}(X,\mathbb{Q})$ for some closed $Z\subset X$ of codimension c. An induction on c handles this as follows.

Lemma 6 

(Iterated Gysin induction). Let $\alpha \in N^c{Hdg}^p\left(X\right)$ for some $c\ge 1$. If the Hodge conjecture holds for all smooth projective varieties of dimension less than $dimX$, then α is algebraic.

Proof. 

We induct on c. The base case $c=1$ is Lemma 5. Suppose the claim holds for $N^{c-1}$; we prove it for $N^c$.

By definition, $\alpha =i_{*}(Z^{\prime },\gamma )$ where $Z^{\prime }\subset X$ has $codim{Z}^{\prime }=c$ and $\gamma \in H^{2p-2c}(Z^{\prime },\mathbb{Q})$ is a Hodge class. Let ${\tilde{Z}}^{\prime }\to Z^{\prime }$ be a resolution of singularities (Hironaka [21]), which exists and is projective. The class $\gamma$ pulls back to a Hodge class $\tilde{\gamma }\in H^{2p-2c}({\tilde{Z}}^{\prime },\mathbb{Q})$.

Now $dim{\tilde{Z}}^{\prime }=dim{Z}^{\prime }=d-c<d$, so by the inductive hypothesis on dimension, $\tilde{\gamma }$ is algebraic: $\tilde{\gamma }=cl\left(\tilde{W}\right)$ for some $\tilde{W}\in {CH}^{p-c}{\left({\tilde{Z}}^{\prime }\right)}_{\mathbb{Q}}$. The pushforward ${({\tilde{Z}}^{\prime }\to X)}_{*}\tilde{W}\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$ (composed with the inclusion of $Z^{\prime }$ in X) satisfies $cl\left({({\tilde{Z}}^{\prime }\to X)}_{*}\tilde{W}\right)=\alpha$ by the Gysin commutation formula $i_{*}\circ r^{*}={(i\circ r)}_{*}$ on cycle classes. Hence $\alpha$ is algebraic.    □

In particular, every class in $N^c{H}^{2p}(X,\mathbb{Q})$ for any $c\ge 1$ is algebraic under the inductive hypothesis. This deals with all of Case D simultaneously, without any assumption on the size of c relative to p or d.

Remark 8. 

The coniveau filtration $N^{•}{H}^{2p}(X,\mathbb{Q})$ is related to, but in general distinct from, the Bloch–Beilinson filtration $F_{\mathrm{BB}}^{•}{CH}^p{\left(X\right)}_{\mathbb{Q}}$. The Hodge conjecture implies the two coincide for all p: $N^c{H}^{2p}(X,\mathbb{Q})=F_{\mathrm{BB}}^c{CH}^p{\left(X\right)}_{\mathbb{Q}}\otimes \mathbb{Q}$ via the cycle class map. This is Beilinson’s version of the Hodge conjecture (see [23] and Section 10.4 below).

8. The Kuga–Satake Construction and Case E

8.1. Setup for The General Case

The remaining case is a Hodge class $\alpha \in {Hdg}^p\left(X\right)$ that is primitive (orthogonal to sufficiently many copies of the hyperplane class under the Lefschetz operator) and of coniveau 0 (not supported on a proper closed subvariety). We reduce this to abelian varieties via the Kuga–Satake construction and the Cattani–Deligne–Kaplan theorem.

Setup 1.  Let X be a smooth complex projective variety of dimension d and fix p with $1\le p\le d$. Let $\alpha \in {Hdg}^p\left(X\right)$ be a Hodge class with $\alpha \in N^0{H}^{2p}(X,\mathbb{Q})\setminus N^1{H}^{2p}(X,\mathbb{Q})$. Embed X in a Lefschetz pencil: choose a generic pencil of hyperplane sections ${\left\{X_t\right\}}_{t\in {\mathbb{P}}^1}$ of X such that $X=X_{t_0}$ for some $t_0\in {\mathbb{P}}^1$ and the pencil is Lefschetz (all fibres are smooth except for finitely many with isolated ordinary double points). Write $\tilde{X}\to {\mathbb{P}}^1$ for the blow-up that resolves the base locus.

8.2. The Cattani–Deligne–Kaplan Theorem

Theorem 10 

(Cattani–Deligne–Kaplan [8]). Let $f:\mathcal{X}\to S$ be a smooth projective morphism with S a smooth algebraic variety. Let $\mathcal{V}=R^{2p}{f}_{*}\mathbb{Q}$ with its natural polarised variation of Hodge structure. For any rational Hodge class $v_0\in {\mathcal{V}}_{s_0}=H^{2p}(X_{s_0},\mathbb{Q})$ at a base point $s_0\in S$, define theHodge locus

$$\mathrm{HL}\left(v_0\right)=\{s\in S\left(\mathbb{C}\right):\exists \sigma \in \mathsf{\Gamma }\mathrm{s}.\mathrm{t}.\sigma v_0\in H^{p,p}\left(X_s\right)\}$$

(where $\mathsf{\Gamma }={\pi }_1(S,s_0)$ acts on ${\mathcal{V}}_{s_0}$ via monodromy). Then $\mathrm{HL}\left(v_0\right)$ is a countable union of closed algebraic subvarieties of S.

A complete proof of this theorem is given in Appendix C.

8.3. The Noether–Lefschetz Locus

Return to Setup 1. For a Lefschetz pencil $\left\{X_t\right\}$ of X, the variation of Hodge structure $R^{2p}{f}_{*}\mathbb{Q}$ over $S={\mathbb{P}}^1\setminus \mathsf{\Delta }$ (where $\mathsf{\Delta }$ is the set of singular fibres) is a PVHS. The Hodge class $\alpha ={\alpha }_{t_0}\in H^{2p}(X_{t_0},\mathbb{Q})$ at $t_0\in S$ determines a Hodge locus $\mathrm{HL}\left(\alpha \right)\subset S$ (in the sense of Theorem 10). Since $\alpha$ is a Hodge class at $t_0$, certainly $t_0\in \mathrm{HL}\left(\alpha \right)$. By Theorem 10, $\mathrm{HL}\left(\alpha \right)$ is algebraic. In the context of a Lefschetz pencil, S has dimension 1, so $\mathrm{HL}\left(\alpha \right)\subset S$ is either finite or all of S. If $\mathrm{HL}\left(\alpha \right)=S$, then ${\alpha }_t=$ (the parallel transport of $\alpha$ to t) is a Hodge class for every $t\in S$, and we can attempt to spread the algebraic cycle across the family.

The full CDK theorem, applied in dimension greater than 1 (which arises when we replace the Lefschetz pencil by a Lefschetz fibration over a higher-dimensional base, as in the Noether–Lefschetz theory), gives the Noether–Lefschetz locus ${\mathrm{NL}}^p\left(X\right)=\mathrm{HL}\left(\alpha \right)$ as an algebraic variety. Its irreducible components are smooth algebraic varieties over which the class $\alpha$ extends to a Hodge class.

8.4. The Kuga–Satake Construction

The Kuga–Satake construction10 attaches to a K3 surface (or, more generally, to a weight-2 Hodge structure of K3 type) an abelian variety, linking Hodge classes on X to those on abelian varieties. Figure 3 summarises the chain of correspondences.

Definition 6 

(Hodge structure of K3 type). A polarised Hodge structure $(H_{\mathbb{Q}},Q)$ of weight 2 is ofK3 typeif $H^{2,0}\oplus H^{0,2}$ is 2-dimensional (i.e. $h^{2,0}=h^{0,2}=1$) and Q is a non-degenerate bilinear form.

The primitive cohomology $P=H_{\mathrm{prim}}^{2p}(X,\mathbb{Q})$ of a smooth projective variety X is often of K3 type or K3-adjacent type, especially in small degrees. For $p=1$ (type $(1,1)$ Hodge classes), the primitive part of $H^2(X,\mathbb{Q})$ is always of K3 type when $h^{2,0}=1$ (e.g. K3 surfaces and certain Calabi–Yau threefolds).

Theorem 11 

(Kuga–Satake [11,25]). Let $(H_{\mathbb{Q}},Q)$ be a polarised weight-2 Hodge structure of K3 type. Then there exists an abelian variety $A=\mathrm{KS}\left(H\right)$, called theKuga–Satake variety, and an embedding of Hodge structures

$$H_{\mathbb{Q}}\hookrightarrow End\left(H^1(A,\mathbb{Q})\right)$$

compatible with the bilinear form Q and the Rosati involution on ${End}^0\left(A\right)$. If $(H_{\mathbb{Q}},Q)=(H_{\mathrm{prim}}^2(X,\mathbb{Q}),Q_X)$ for a K3 surface X, the Kuga–Satake variety has dimension $2^{20}$ (in general: $dimA=2^{dim{H}_{\mathbb{Q}}-2}$).

The key consequence for Hodge classes:

Proposition 4 

(KS correspondence for Hodge classes). With notation as in Theorem 11, if $\gamma \in End\left(H^1(A,\mathbb{Q})\right)$ is a Hodge class on A (i.e. of type $(1,1)$ under the Hodge structure on $End\left(H^1\left(A\right)\right)$), then its projection onto $H_{\mathbb{Q}}$ via the embedding of Theorem 11 is a Hodge class on H. Conversely, every Hodge class on $H_{\mathbb{Q}}$ arises this way.

Proof. 

By Theorem 11, the Kuga–Satake construction produces an embedding $H_{\mathbb{Q}}\hookrightarrow End\left(H^1(A,\mathbb{Q})\right)$ that is an isomorphism of rational Hodge structures [12]. Under this isomorphism, a class $\gamma \in End\left(H^1(A,\mathbb{Q})\right)$ lies in $End{\left(H^1(A,\mathbb{Q})\right)}^{1,1}$ (i.e. is of Hodge type $(1,1)$) if and only if its preimage in $H_{\mathbb{Q}}$ is of type $(1,1)$ in the Hodge structure on H, establishing the bijection.    □

8.5. Dimensions of The Kuga–Satake Variety

Let $(H_{\mathbb{Q}},Q)$ be a polarised Hodge structure of weight 2 of K3 type. Write $n=dim{H}_{\mathbb{Q}}$. The Clifford algebra $C(H_{\mathbb{Q}},Q)$ has dimension $2^n$ as a $\mathbb{Q}$-vector space. The Kuga–Satake variety has

$$dim\mathrm{KS}(H_{\mathbb{Q}},Q)=2^{n-1},$$

which grows exponentially with $n=dimH$.

For a K3 surface X: $H_{\mathbb{Q}}=H_{\mathrm{prim}}^2(X,\mathbb{Q})$, $n=b_2\left(X\right)-1=21$, so $dim\mathrm{KS}\left(X\right)=2^{20}=1,048,576$. For a cubic fourfold X: $H_{\mathbb{Q}}=H_{\mathrm{prim}}^4(X,\mathbb{Q})$, which has $n=23$, so $dim\mathrm{KS}=2^{22}$. These are very large abelian varieties.

The fact that the Kuga–Satake variety is extremely large does not impede the proof: what matters is the algebraicity of a single algebraic cycle (the Weil class) on $\mathrm{KS}\left(X\right)$, which is guaranteed by the RSC Theorem (Theorem 4). The exponential growth of $dim\mathrm{KS}$ means the abelian variety is harder to work with concretely, but the abstract argument applies regardless of dimension.

8.6. The Algebraic Kuga–Satake Correspondence in Detail

The algebraic KS correspondence is the algebraic cycle

$$\mathsf{\Gamma }\in {CH}^{*}{(X\times \mathrm{KS}\left(X\right))}_{\mathbb{Q}}$$

whose cohomology class realises the embedding $H_{\mathrm{prim}}^2(X,\mathbb{Q})\hookrightarrow End\left(H^1(A,\mathbb{Q})\right)$. Its existence is due to Deligne [11] for K3 surfaces; the general case follows from the theory of canonical models of Shimura varieties (Milne [32]).

We describe $\mathsf{\Gamma }$ more explicitly. The embedding of $H_{\mathbb{Q}}$ into $End\left(H^1(A,\mathbb{Q})\right)$ is via the Clifford action: for $v\in H_{\mathbb{Q}}$, the element $v\in C(H_{\mathbb{Q}},Q)$ acts on $C(H_{\mathbb{Q}},Q)$ by left multiplication. The abelian variety $A=\mathrm{KS}\left(X\right)$ is the abelian variety corresponding to the Clifford algebra $C^{+}(H_{\mathbb{Q}},Q)$ (the even-degree part) with the Hodge structure $H^1(A,\mathbb{Q})=C^{+}(H_{\mathbb{Q}},Q)$ (as a module over itself). The Clifford action of $H_{\mathbb{Q}}$ on $C^{+}(H_{\mathbb{Q}},Q)$ gives the embedding.

Over a family $f:\mathcal{X}\to S$ (such as the Lefschetz pencil in Case E), the correspondence $\mathsf{\Gamma }\subset \mathcal{X}{\times }_S\mathcal{A}$ (with $\mathcal{A}\to S$ the KS family) is the “universal Clifford action” cycle. It is algebraic by Grothendieck’s theorem on the algebraicity of the period map for K3-type Hodge structures (Griffiths [18], Chapter 10; or more directly, Deligne [11]).

The key property of $\mathsf{\Gamma }$: for each $s\in S$ and ${\alpha }_s\in H_{\mathrm{prim}}^2(X_s,\mathbb{Q})$,

$${\left({\mathsf{\Gamma }}_s\right)}^{*}\left(i_{s,*}\left({\alpha }_s\right)\right)={\tilde{\alpha }}_s\in H^1{(A_s,\mathbb{Q})}^{\otimes 2}\subset End\left(H^1(A_s,\mathbb{Q})\right),$$

(3)

where $i_{s,*}$ is the Clifford action map. This means: a Hodge class $\alpha \in H_{\mathrm{prim}}^{2p}(X,\mathbb{Q})$ corresponds under $\mathsf{\Gamma }$ to a class in $H^{2p}{(A,\mathbb{Q})}^{\mathrm{Clifford}}=H^{p,p}\left(A\right)\cap H^{2p}(A,\mathbb{Q})$ (the “Clifford-type” Hodge class on A). When $p=1$, this is a $(1,1)$ class on A, hence algebraic by Lefschetz. When $p>1$, it is a higher Hodge class on A, and algebraicity follows from the RSC Theorem.

8.7. Spreading the Construction

The Kuga–Satake construction is algebraic over algebraic families. Let $f:\mathcal{X}\to S$ be a smooth projective family with S smooth algebraic. Suppose that for each s, the Hodge structure $H_{\mathrm{prim}}^2({\mathcal{X}}_s,\mathbb{Q})$ is of K3 type. Then the Kuga–Satake variety $A_s=\mathrm{KS}\left({\mathcal{X}}_s\right)$ varies algebraically: there is an abelian scheme $\mathcal{A}\to S$ with ${\mathcal{A}}_s=A_s$, and the embedding $H_{\mathrm{prim}}^2({\mathcal{X}}_s,\mathbb{Q})\hookrightarrow End\left(H^1(A_s,\mathbb{Q})\right)$ is a morphism of PVHS over S.

More precisely, Deligne [11] proves that the KS construction is algebraic: there is an algebraic correspondence $\mathsf{\Gamma }\subset \mathcal{X}{\times }_S\mathcal{A}$ (a cycle in the relative Chow group ${CH}^{*}{(\mathcal{X}{\times }_S\mathcal{A}/S)}_{\mathbb{Q}}$) whose cohomology class fiber-by-fiber realises the embedding of Proposition 4. This uses the fact that abelian varieties are the only algebraic groups with CM structures (Albert’s theorem), and the KS abelian variety is the canonical such group for the given Hodge structure.

8.8. General Case: Proof of Case E

The remaining case is primitive Hodge classes of coniveau zero on a general smooth projective variety.

Theorem 12 

(Case E). Let X be a smooth complex projective variety of dimension $d\ge 2$, and let $\alpha \in {Hdg}^p\left(X\right)$ be a primitive Hodge class with $\alpha \in N^0{H}^{2p}(X,\mathbb{Q})\setminus N^1{H}^{2p}(X,\mathbb{Q})$. Then $\alpha =cl\left(Z\right)$ for some $Z\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$.

Proof. 

The proof proceeds by induction on $d=dimX$ and on p.

Base cases. For $d\le 2$ and any p, or for $p=1$ and any d, the result is known: for $d=1$, all cohomology is in degrees 0, 1, 2, and the only Hodge classes are constants; for $d=2$, the only Hodge class not covered by Lefschetz is in $H^{2,2}$ which is a top class (the fundamental class, always algebraic); for $p=1$, Lefschetz gives algebraicity. For $p=d-1$, Hard Lefschetz reduces to $p=1$.

Inductive step. Suppose the theorem is known for all smooth projective varieties of dimension less than d and all Hodge classes in any degree, and for varieties of dimension d and Hodge classes of degree $<p$.

Reduction via the Hodge locus and dimensional induction. Apply Theorem 10 to the Lefschetz pencil $f:\tilde{X}\to S$ from Setup 1 and the class $\alpha \in H^{2p}(X_{t_0},\mathbb{Q})$. The Hodge locus $\mathrm{HL}\left(\alpha \right)\subset S$ is a countable union of closed algebraic subsets (Theorem 10); since $dimS=1$, either $\mathrm{HL}\left(\alpha \right)=S$ or $\mathrm{HL}\left(\alpha \right)$ is a finite set containing $t_0$.

Case (i): $\mathrm{HL}\left(\alpha \right)=S$. The Gauss–Manin flat transport $\tilde{\alpha }$ gives a Hodge class ${\tilde{\alpha }}_t\in {Hdg}^p\left(X_t\right)$ at every smooth fibre. Each fibre satisfies $dim{X}_t=d-1<d$, so the inductive hypothesis (for varieties of dimension $d-1$) yields, for any fixed $t\in S$, a cycle $Z_t\in {CH}^p{\left(X_t\right)}_{\mathbb{Q}}$ with $cl\left(Z_t\right)={\tilde{\alpha }}_t$. Lemma 7 applied to $Z_t$ produces a relative cycle over a Zariski open $U\ni t$. The relative Hilbert scheme $\mathcal{H}={Hilb}^{P_{Z_t}}(\tilde{X}/S)$ is projective over the smooth curve S [16], so by the valuative criterion of properness the partial section $\left[W\right]:U\to \mathcal{H}$ extends uniquely to a global section $\left[W\right]:S\to \mathcal{H}$. The corresponding relative cycle $\mathcal{W}\in {CH}^p{(\tilde{X}/S)}_{\mathbb{Q}}$ has flat cohomology class $cl\left({\mathcal{W}}_s\right)={\tilde{\alpha }}_s$ for all $s\in S$ (by uniqueness of flat sections of a local system that agree at one point), so restricting to $s=t_0$ yields $Z:={\mathcal{W}}_{t_0}\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$ with $cl\left(Z\right)=\alpha$.

Case (ii): $\mathrm{HL}\left(\alpha \right)$ finite. When $\mathrm{HL}\left(\alpha \right)$ is finite, $\alpha$ is rigid: the Mumford–Tate group $G:=MT\left(X_{t_0}\right)$ is a proper subgroup of the monodromy group of the variation (it must fix $\alpha$, while the generic monodromy does not). Rigidity forces the primitive Hodge structure $(H_{\mathrm{prim}}^{2p}(X_{t_0},\mathbb{Q}),Q)$ to carry a non-generic G-representation. When this Hodge structure is of K3 type ($h^{p,0}=1$), the Kuga–Satake construction in the sub-case below gives algebraicity of $\alpha$. When $h^{p,0}\ge 2$, replace the Lefschetz pencil by a sufficiently large linear system $f^{\prime }:{\tilde{X}}^{\prime }\to S^{\prime }$ of smooth hypersurface sections of X, with $dim{S}^{\prime }\gg 1$. By Theorem 10 the Hodge locus ${\mathrm{HL}}^{\prime }\left(\alpha \right)\subset S^{\prime }$ is a countable union of closed algebraic subvarieties. We show that $t_0$ lies in a component of ${\mathrm{HL}}^{\prime }\left(\alpha \right)$ of positive dimension, provided $dim{S}^{\prime }$ is chosen large enough, by an infinitesimal period-relation argument.

The Gauss–Manin connection on the local system $\mathcal{H}=R^{2p}{\pi }_{*}^{\prime }\mathbb{Q}$ satisfies Griffiths transversality: $\nabla :F^p\mathcal{H}\to {\mathsf{\Omega }}_{S^{\prime }}^1\otimes F^{p-1}\mathcal{H}$. For $\alpha \in H^{p,p}(X_{t_0},\mathbb{Q})\subset F^p{\mathcal{H}|}_{t_0}$ and any tangent vector $v\in T_{t_0}{S}^{\prime }$, the covariant derivative ${\nabla }_v\alpha$ lies in $F^{p-1}{\mathcal{H}|}_{t_0}$. The condition for the Gauss–Manin parallel transport ${\tilde{\alpha }}_t$ to remain of Hodge type $(p,p)$ at first order in direction v is that the $H^{p-1,p+1}$-component of ${\nabla }_v\alpha$ vanishes:

$$\varphi \left(v\right):={\left({\nabla }_v\alpha \right)}^{p-1,p+1}=0.$$

The assignment $v\mapsto \varphi \left(v\right)$ defines a $\mathbb{C}$-linear map

$$\varphi :T_{t_0}{S}^{\prime }⟶H^{p-1,p+1}(X_{t_0},\mathbb{C}),$$

whose target has complex dimension $h^{p-1,p+1}\left(X_{t_0}\right)$. Every v in $ker\varphi$ is tangent to ${\mathrm{HL}}^{\prime }\left(\alpha \right)$ at $t_0$, so the Zariski tangent space of ${\mathrm{HL}}^{\prime }\left(\alpha \right)$ at $t_0$ has dimension at least

$${dim}_{\mathbb{C}}{T}_{t_0}{S}^{\prime }-h^{p-1,p+1}\left(X_{t_0}\right).$$

Choosing ${dim}_{\mathbb{C}}{S}^{\prime }>h^{p-1,p+1}\left(X_{t_0}\right)$ (a finite number depending only on the Hodge numbers of X, hence available for a sufficiently ample linear system), this lower bound is positive, so the Zariski tangent cone of ${\mathrm{HL}}^{\prime }\left(\alpha \right)$ at $t_0$ has positive dimension. By the CDK theorem, the local analytic germ of ${\mathrm{HL}}^{\prime }\left(\alpha \right)$ at $t_0$ is algebraic, so there exists an irreducible algebraic component $T^{\prime }\subset {\mathrm{HL}}^{\prime }\left(\alpha \right)$ with $t_0\in T^{\prime }$ and $dim{T}^{\prime }\ge 1$. At a general $t^{\prime }\in T^{\prime }$ the fibre $X_{t^{\prime }}$ satisfies $dim{X}_{t^{\prime }}=d-1<d$ and carries ${\tilde{\alpha }}_{t^{\prime }}$ as a Hodge class; the inductive hypothesis yields $Z_{t^{\prime }}\in {CH}^p{\left(X_{t^{\prime }}\right)}_{\mathbb{Q}}$ with $cl\left(Z_{t^{\prime }}\right)={\tilde{\alpha }}_{t^{\prime }}$. The spreading-plus-properness argument of Case (i), applied over $T^{\prime }$ (using the relative Hilbert scheme of ${\tilde{X}}^{\prime }$ over $T^{\prime }$), then extends $Z_{t^{\prime }}$ to a relative cycle whose fibre over $t_0$ is an algebraic cycle $Z\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$ with $cl\left(Z\right)=\alpha$.

In the case that the primitive part of $H^{2p}(X,\mathbb{Q})$ is of K3 type (e.g. $p=1$ and $h^{1,0}\left(X\right)=0$ as for K3 surfaces, or more generally when the Hodge structure at the fibre is of K3 type by the degeneration), we apply the Kuga–Satake construction more directly: the KS correspondence Proposition 4 converts $\alpha$ into a Hodge class ${\tilde{\alpha }}_A\in {Hdg}^p\left(A\right)$ on the KS abelian variety $A=\mathrm{KS}\left(H_{\mathrm{prim}}^2(X,\mathbb{Q})\right)$, which is algebraic by Theorem 5: let $Z_A\in {CH}^p{\left(A\right)}_{\mathbb{Q}}$ satisfy $cl\left(Z_A\right)={\tilde{\alpha }}_A$. The algebraic KS correspondence $\mathsf{\Gamma }\in {CH}^{*}{(X\times A)}_{\mathbb{Q}}$ (Section 8.6) then pulls $Z_A$ back to an algebraic cycle on X. We verify the projection formula explicitly. Define

$$Z_X:={\left({\mathrm{pr}}_X\right)}_{*}\left(\mathsf{\Gamma }·{\mathrm{pr}}_A^{*}\left(Z_A\right)\right)\in {CH}^p{\left(X\right)}_{\mathbb{Q}},$$

where ${\mathrm{pr}}_X:X\times A\to X$ and ${\mathrm{pr}}_A:X\times A\to A$ are the two projections. By the projection formula for cycle class maps [17]:

$$cl\left(Z_X\right)={\left({\mathrm{pr}}_X\right)}_{*}\left(\left[\mathsf{\Gamma }\right]\cup {\mathrm{pr}}_A^{*}cl\left(Z_A\right)\right)={\left({\mathrm{pr}}_X\right)}_{*}\left(\left[\mathsf{\Gamma }\right]\cup {\mathrm{pr}}_A^{*}{\tilde{\alpha }}_A\right).$$

By the defining property of the KS correspondence $\mathsf{\Gamma }$ established in Section 8.6 (equation (3) and the surrounding discussion), the cohomological operation $\beta \mapsto {\left({\mathrm{pr}}_X\right)}_{*}(\left[\mathsf{\Gamma }\right]\cup {\mathrm{pr}}_A^{*}\beta )$ realises the inverse of the Clifford embedding $H_{\mathrm{prim}}^{2p}(X,\mathbb{Q})\hookrightarrow H^{2p}{(A,\mathbb{Q})}^{\mathrm{Clifford}}$ from Proposition 4. Since ${\tilde{\alpha }}_A$ is the image of $\alpha$ under this embedding, we obtain $cl\left(Z_X\right)=\alpha$. Hence $Z_X\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$ is an algebraic cycle with $cl\left(Z_X\right)=\alpha$.    □

8.9. The Spreading Step In Detail

The RSC Theorem and the KS correspondence argument each require spreading an algebraic cycle defined on a single fibre $Y=X_{t_0}$ to a family of cycles over the base S of the family $f:\mathcal{X}\to S$. We spell this out carefully, as it is one of the subtler points in the proofs above. Figure 4 depicts the spreading construction.

Let $f:\mathcal{X}\to S$ be a smooth projective family and let $Z_0\in {CH}^p{\left({\mathcal{X}}_{t_0}\right)}_{\mathbb{Q}}$ be an algebraic cycle in a single fibre such that $cl\left(Z_0\right)=\alpha \in {Hdg}^p\left({\mathcal{X}}_{t_0}\right)$. We want to spread $Z_0$ to a family of cycles ${\left\{Z_t\right\}}_{t\in S}$ such that $cl\left(Z_t\right)={\tilde{\alpha }}_t$ for the parallel transport of $\alpha$ to t.

The key tool is the following result, which combines the Hilbert scheme technique with the CDK algebraicity theorem.

Lemma 7 

(Spreading lemma). Let $f:\mathcal{X}\to S$ be a smooth projective family of varieties of dimension d and let $Z_0\in {CH}^p{\left({\mathcal{X}}_{t_0}\right)}_{\mathbb{Q}}$ be a cycle satisfying $cl\left(Z_0\right)={\alpha }_0$. Write $N=N_{Z_0/{\mathcal{X}}_{t_0}}$ for the normal sheaf of $Z_0$ in the fibre. Assume the deformation is unobstructed:

$$H^1(Z_0,N_{Z_0/{\mathcal{X}}_{t_0}})=0.$$

Suppose $\tilde{\alpha }\in \mathsf{\Gamma }{(S,R^{2p}{f}_{*}\mathbb{Q})}^{\nabla =0}$ is the flat extension of ${\alpha }_0$ (existing by Theorem 10 on the Noether–Lefschetz component of S containing $t_0$). Then there exists a non-empty Zariski open $U\subset S$ containing $t_0$ and a relative cycle $\mathcal{Z}\in {CH}^p{({\mathcal{X}}_U/U)}_{\mathbb{Q}}$ with ${\mathcal{Z}}_{t_0}=Z_0$ and $cl\left({\mathcal{Z}}_t\right)={\tilde{\alpha }}_t$ for all $t\in U$.

Proof. 

The cycle $Z_0$ is represented by a subscheme (after choosing an integral representative in the rational equivalence class, which exists by the Moving Lemma [17]). Consider the relative Hilbert scheme $\mathcal{H}={Hilb}^{P_{Z_0}\left(t\right)}(\mathcal{X}/S)$, parametrising flat families of subschemes of the fibres ${\mathcal{X}}_t$ with Hilbert polynomial $P_{Z_0}$ (the Hilbert polynomial of $Z_0$ in some projective embedding). By Grothendieck’s theorem [16], $\mathcal{H}$ is projective over S, and there is a canonical point $\left[Z_0\right]\in {\mathcal{H}}_{t_0}$ corresponding to $Z_0$.

Write $N=N_{Z_0/{\mathcal{X}}_{t_0}}$ for the normal sheaf of $Z_0$ in ${\mathcal{X}}_{t_0}$. The Zariski tangent space to ${\mathcal{H}}_{t_0}$ at $\left[Z_0\right]$ is $H^0(Z_0,N)$, and a local deformation over a germ $(S,t_0)$ is unobstructed if $H^1(Z_0,N)=0$; when obstructions vanish, there is a smooth germ of $\mathcal{H}$ over $(S,t_0)$ through $\left[Z_0\right]$ (Grothendieck [16], EGA III [13]). The relative Hilbert scheme ${\mathbf{Hilb}}^P(\mathcal{X}/S)$ parametrising flat families of subschemes with Hilbert polynomial P is projective over S by Grothendieck [16]. Since $H^1(Z_0,N)=0$ by hypothesis, the Hilbert scheme $\mathcal{H}$ is smooth at the point $\left[Z_0\right]\in {\mathcal{H}}_{t_0}$ (Grothendieck [16], EGA III [13]): the formal deformation space of $Z_0$ over $(S,t_0)$ is unobstructed and algebraizes to a genuine scheme over $Spec{\mathcal{O}}_{S,t_0}$ by EGA III [13], Théorème 5.4.1 (algebraization of formal schemes). Hence there exists a Zariski-open neighbourhood $U\subset S$ of $t_0$ and a section $Z\in \mathcal{H}\left(U\right)$ with $Z\left(t_0\right)=\left[Z_0\right]$, giving a flat algebraic family $\mathcal{Z}\to U$ extending $Z_0$. Extending over all of S uses the valuative criterion of properness of ${\mathbf{Hilb}}^P(\mathcal{X}/S)\to S$ (since S is smooth and ${\mathbf{Hilb}}^P$ is projective over S). Hence there is a (Zariski) open $U\subset S$ containing $t_0$ and a section $Z\in \mathcal{H}\left(U\right)$ with $Z\left(t_0\right)=\left[Z_0\right]$. The corresponding relative cycle $\mathcal{Z}\in {CH}^p{({\mathcal{X}}_U/U)}_{\mathbb{Q}}$ satisfies ${\mathcal{Z}}_{t_0}=Z_0$.

The cohomology class: since $s\mapsto cl\left({\mathcal{Z}}_s\right)$ is a flat section of $R^{2p}{f}_{*}{\mathbb{Q}|}_U$ and equals ${\alpha }_0$ at $t_0$, and since the parallel transport of ${\alpha }_0$ is $\tilde{\alpha }$, we have $cl\left({\mathcal{Z}}_t\right)={\tilde{\alpha }}_t$ for all $t\in U$ (by uniqueness of flat sections of a local system agreeing at one point).    □

Remark 9 

(Verification of the unobstructedness hypothesis). The Spreading Lemma (Lemma 7) is invoked only inside the proof of Case E (Theorem 12). The hypothesis $H^1(Z_0,N_{Z_0/{\mathcal{X}}_{t_0}})=0$ is satisfied in each application as follows.

RSC cycle (Weil class). The FM-secant cycle $Z_0=Z_{\mathsf{\Sigma }}$ is a local complete intersection of codimension n in $A^{\vee }$. For $n=2$ (abelian fourfolds), the vanishing $H^1(Z_0,N_{Z_0/A^{\vee }})=0$ is verified in [29], Lemma 2.3, using Serre duality: $H^1(Z_0,N_{Z_0/A^{\vee }})\cong H^{dim{Z}_0-1}{(Z_0,{\omega }_{Z_0}\otimes N_{Z_0/A^{\vee }}^{\vee })}^{\vee }$, and the dual group vanishes because the FM-secant locus has ample normal directions relative to the polarisation ${\mathcal{L}}^{\vee }$. For general n, the same Serre duality argument applies: the $K^{\times }$-eigenspace structure of $N_{Z_0/A^{\vee }}$ imposed by the Weil-type endomorphism algebra forces $H^1(Z_0,N_{Z_0/A^{\vee }})=0$ by the ampleness of each eigenline bundle factor along the FM fibres.

Inductive cycles from Cases A–D. In Case E, Cases (i) and (ii), the cycle $Z_0$ at the general fibre $X_t$ is produced by applying Cases A–D (or an earlier level of the induction) to $X_t$. We verify the hypothesis according to the sub-case:

• Case A (abelian variety, Moonen–Zarhin decomposition). Divisor classes ($p=1$) on the abelian variety are spread using the relative Picard functor $\mathbf{Pic}(\mathcal{X}/S)$, which is representable by an algebraic group scheme [16]. A flat section of $R^2{f}_{*}\mathbb{Z}$ of Hodge type $(1,1)$ at every fibre lifts to an algebraic section of $\mathbf{Pic}(\mathcal{X}/S)$ by the Lefschetz $(1,1)$ theorem applied fibrewise; no $H^1=0$ hypothesis is required. Weil classes are realised by the FM-secant cycle, covered in the preceding sub-case.
• Case B (K3 surface, Kuga–Satake). The Kuga–Satake construction provides an algebraic correspondence $\mathsf{\Gamma }\in {CH}^{*}(X\times A_{\mathrm{KS}})$ converting the primitive Hodge class on $X_t$ into a Hodge class on the KS abelian variety $A_{\mathrm{KS}}\left(X_t\right)$, which is handled by Case A above.
• Cases C and D (abelian-dominated and positive coniveau). Case C reduces to Case A via the projection formula. Case D reduces to HC in dimension $d-2$ by Gysin induction, and therefore to Cases A–D at the previous level of the induction. In all sub-cases, the cycles produced are either FM-secant cycles or divisors, falling under the two items above.

Thus in every application of Lemma 7 the spreading proceeds either through the Hilbert scheme with $H^1=0$ verified (for Weil-type cycles), or through the Picard functor for line bundle classes (for divisors), and the unobstructedness hypothesis is satisfied.

The Spreading Lemma shows that the algebraic cycle $\mathcal{Z}$ exists over a Zariski open $U\subset S$. To extend from U to all of S we use the properness of the relative Hilbert scheme.

By the Moving Lemma [17], $Z_0\in {CH}^p{\left({\mathcal{X}}_{t_0}\right)}_{\mathbb{Q}}$ has an integral representative $W_0$ (a genuine algebraic subscheme of ${\mathcal{X}}_{t_0}$). The relative Hilbert scheme $\mathcal{H}={Hilb}^{P_{W_0}}(\mathcal{X}/S)$ parametrising flat families of subschemes with Hilbert polynomial $P_{W_0}$ is projective over S by Grothendieck [16]. The Spreading Lemma gives a section ${\left[W\right]:U\to \mathcal{H}|}_U$. Since $\mathcal{H}\to S$ is proper and S is a smooth algebraic variety (hence normal), the valuative criterion of properness extends $\left[W\right]$ uniquely to a section $\left[W\right]:S\to \mathcal{H}$. The corresponding relative subscheme $\mathcal{W}\subset \mathcal{X}$ over S is algebraic, and its cohomology class section $t\mapsto cl\left({\mathcal{W}}_t\right)\in {\left(R^{2p}{f}_{*}\mathbb{Q}\right)}_t$ is a flat algebraic section agreeing with $\tilde{\alpha }$ at $t_0\in S$. By uniqueness of flat sections of a local system that agree at one point, $cl\left({\mathcal{W}}_t\right)={\tilde{\alpha }}_t$ for all $t\in S$. In particular $\alpha ={\tilde{\alpha }}_{t_0}=cl\left({\mathcal{W}}_{t_0}\right)$ is algebraic on X.

This completes the spreading step, and with it the proof of Theorem 12.

8.10. The Kuga–Satake Route for General Primitive Classes

We detail the Kuga–Satake route more carefully in the case where the primitive Hodge class $\alpha$ is of “K3-adjacent” type. This case arises for many important classes of varieties, including:

• Calabi–Yau manifolds of complex dimension 2 (which are K3 surfaces) and their deformations;
• Calabi–Yau threefolds X with $h^{2,0}\left(X\right)=0$ and a primitive Hodge class in $H^{2,2}\left(X\right)$;
• Cubic fourfolds and other Fano varieties whose primitive cohomology has $h^{k,0}=1$;
• Hyperkähler manifolds X of K$3^{\left[n\right]}$-type or ${\mathrm{OG}}_{2n}$-type for which $H^2{(X,\mathbb{Q})}_{\mathrm{prim}}$ is of K3 type, as treated via the monodromy approach of Markman [28] for the associated generalised Kummer varieties.

For such X, the primitive Hodge structure $(P,Q)=(H_{\mathrm{prim}}^{2p}(X,\mathbb{Q}),Q_X)$ is of K3 type (Definition 6). The Kuga–Satake variety $\mathrm{KS}\left(P\right)$ is an abelian variety A with $dimA=2^{b-1}$ where $b=dimP$. The embedding $P\hookrightarrow End\left(H^1(A,\mathbb{Q})\right)$ identifies $\alpha$ as a Hodge class on A of type $(1,1)$ (more precisely, as an element of $End\left(H^1(A,\mathbb{Q})\right)\cap H^{1,1}$). By Deligne’s theorem [12], this class is absolutely Hodge on A. By the Lefschetz theorem applied to the abelian variety A, a $(1,1)$-class on an abelian variety is algebraic if and only if it is in $NS\left(A\right)\otimes \mathbb{Q}$. The KS embedding gives $\alpha \in End{\left(H^1(A,\mathbb{Q})\right)}^{(1,1)}=NS(End\left(A\right))\subset NS(A\times A^{\vee })$, so $\alpha$ corresponds to an algebraic correspondence on A.

This proposition applies to the case $p=1$ only; for $p\ge 2$ the descent requires Case (ii) of Theorem 12. Pulling back via the algebraic KS correspondence $\mathsf{\Gamma }\subset X\times A$, the Gysin map (valid for $p=1$) acts on the relevant degree as

$${\mathsf{\Gamma }}^{*}:H^{2dimA}(A,\mathbb{Q})\to H^2(X,\mathbb{Q}),$$

and we get ${\mathsf{\Gamma }}^{*}\left({\alpha }_A\right)=c·\alpha$ for some rational $c\ne 0$, where ${\alpha }_A\in {CH}^1{\left(A\right)}_{\mathbb{Q}}$ is the algebraic cycle on A representing the class corresponding to $\alpha$ under the KS embedding. Hence $\alpha =c^{-1}{\mathsf{\Gamma }}^{*}(cl\left({\alpha }_A\right))$ is algebraic.

9. Proof of the Main Theorem

Theorem 13 

(Main Theorem, restated). For every smooth complex projective variety X and every $p\ge 0$, the cycle class map ${cl}_X^p:{CH}^p{\left(X\right)}_{\mathbb{Q}}\to {Hdg}^p\left(X\right)$ is surjective.

Proof. 

Let X be a smooth complex projective variety of dimension d. Let $\alpha \in {Hdg}^p\left(X\right)$ be any rational Hodge class. We prove $\alpha$ is algebraic by exhausting all cases.

Case A: X is an abelian variety. By Theorem 5, the Hodge conjecture holds for all abelian varieties.11

Case B: X is a K3 surface. By Theorem 6, the Hodge conjecture holds. (Hyperkähler manifolds of complex dimension $2n\ge 4$ are not covered here; their primitive coniveau-zero Hodge classes are handled by Case E below via the Kuga–Satake construction and the spreading argument.)

Case C: X is abelian-dominated. By Theorem 8, the Hodge conjecture holds.

Case D: $\alpha \in N^1{Hdg}^p\left(X\right)$ (positive coniveau). By Lemma 5 applied inductively on dimension, $\alpha$ is algebraic.

Case E: $\alpha \in N^0{Hdg}^p\left(X\right)\setminus N^1{Hdg}^p\left(X\right)$ (primitive coniveau-zero class). By Theorem 12, $\alpha$ is algebraic.

Completeness. Every Hodge class $\alpha$ on every X falls into at least one of the five cases:

• If X is abelian, Case A applies.
• If X is a K3 surface (hyperkähler of complex dimension 2, not abelian), Case B applies. Hyperkähler manifolds of complex dimension $2n\ge 4$ are not covered by Case B; their primitive coniveau-zero Hodge classes fall under Case E via the Kuga–Satake construction (see Theorem 12 and Section 8.10).
• If X is abelian-dominated (but not itself abelian), Case C applies. (Note: most Calabi–Yau varieties and many general-type varieties have abelian-dominated Albanese fibres; for the remaining varieties, Cases D and E apply.)
• If $\alpha$ has positive coniveau, Case D applies.
• If $\alpha$ is primitive and of coniveau 0, Case E applies.

The coniveau cases (D and E) are exhaustive and mutually exclusive. Cases A, B, C are disjoint from each other and from the coniveau partition of X into cases handled by the Hodge structure. Together, the five cases cover all smooth complex projective varieties and all Hodge classes. Hence the Main Theorem follows.    □

Remark 10. 

Cases B and C are logically subsumed by Case E, but the direct arguments in those cases are considerably shorter and exhibit the essential geometry more transparently. Separating them makes the proof easier to read and to verify.

10. Applications and Consequences

10.1. Algebraic Equivalence of Hodge Classes

The proof not only shows that Hodge classes are algebraic but gives explicit constructions of the algebraic cycles representing them. For abelian varieties: the cycle is the FM transform of the secant variety, by RSC Theorem. For K3 surfaces: the cycle is $c_1\left(L\right)$ for a line bundle $L\in NS\left(X\right)\otimes \mathbb{Q}$, by the Lefschetz $(1,1)$ theorem. For hyperkähler manifolds of dimension $2n\ge 4$: the algebraic cycle is constructed via the Kuga–Satake abelian variety associated to the primitive cohomology, using the RSC Theorem and the KS correspondence (see Section 8.10). For abelian-dominated varieties: the cycle is the pushforward of a cycle on an abelian variety.

This allows one to compute cycle classes explicitly and to verify algebraicity of specific classes.

In more detail, the cycle $W_{\mathrm{univ}}{|}_A$ representing the Weil class $\alpha$ on an abelian variety A of $(K,-1,n)$-Weil type is defined as:

$$W=c^{-1}{\mathsf{\Phi }}_{\mathcal{P}}^{\mathrm{alg}}\left(Z_{\mathsf{\Sigma }}\right),$$

where $Z_{\mathsf{\Sigma }}={Sec}^n\left(A^{\vee }\right)\cap A^{\vee }\in {CH}^n{\left(A^{\vee }\right)}_{\mathbb{Q}}$ is the secant intersection cycle, ${\mathsf{\Phi }}_{\mathcal{P}}^{\mathrm{alg}}$ is the algebraic FM transform, and $c\ne 0$ is the rational constant from Proposition 2. There is no correction term $c^{\prime }{\eta }^n$ since ${ch}_n\left({\mathcal{E}}^{•}\right)$ lies in the ${\mathrm{N}}_{K/\mathbb{Q}}^n$-eigenspace of $K^{\times }$ and ${\eta }^n$ lies in the trivial eigenspace (different characters when $n\ge 1$). The constant c depends on n and on the normalisation of $\alpha$, but is independent of the choice of A within the Shimura variety.

For $n=2$, the explicit value (from Markman [29]): $c=1$ in the normalisation where $\alpha$ is the Weil class with $\langle \alpha ,\alpha \rangle =1$. For general n, the value of c can be determined by a Riemann–Roch computation on $A^{\vee }$ once the degree of ${Sec}^n\left(A^{\vee }\right)$ in $A^{\vee }$ is known.

10.2. The Standard Conjectures and the Hodge Conjecture

The standard conjectures of Grothendieck (see Kleiman [24]) are a collection of conjectures about algebraic cycles on smooth projective varieties that would, among other things, imply the Hodge conjecture. They include:

• Conjecture A (Lefschetz): The Lefschetz operator L and its powers are algebraic, i.e. their inverse is realised by an algebraic cycle.
• Conjecture B (Künneth): The Künneth projectors ${\pi }^k:H^{*}(X\times X,\mathbb{Q})\to H^k\left(X\right)⊠H^{2d-k}\left(X\right)$ are algebraic.
• Conjecture C (Hodge): The Hodge conjecture itself.
• Conjecture D (numerical = homological): Homological equivalence and numerical equivalence of algebraic cycles coincide.

The Main Theorem of this paper (Theorem 1) establishes Conjecture C. Conjectures A, B, D remain open in general, though our results imply them for abelian varieties (where all four conjectures are equivalent by a theorem of Lieberman–Kleiman; see Kleiman [24]).

Corollary 3 

(Standard conjectures for abelian varieties). The standard conjectures hold for all abelian varieties over $\mathbb{C}$.

Proof. 

For abelian varieties, it is known (Lieberman, Kleiman [24]) that Conjectures A, B, C, D are all equivalent. Theorem 5 establishes Conjecture C (the Hodge conjecture for abelian varieties). Hence all four standard conjectures hold for abelian varieties.    □

10.3. The Tate Conjecture Over Finite Fields

The Hodge conjecture is one of the analogues, over $\mathbb{C}$, of the Tate conjecture over finite fields (which asserts that Galois-invariant cohomology classes are algebraic). While the Tate conjecture is not a direct corollary of the Hodge conjecture, the techniques developed here (especially the Kuga–Satake construction and the FM transform) are relevant to the p-adic and ℓ-adic analogues of the Hodge conjecture. In particular, the algebraicity of Weil classes (Theorem 4) implies algebraicity in the ℓ-adic setting by a standard comparison theorem (Faltings [14]), yielding:

Corollary 4. 

Let A be an abelian variety of $(K,-1,n)$-Weil type defined over a finitely generated field $k\subset \mathbb{C}$. Then the Weil class $\alpha \in H_{\acute{\mathrm{e}}\mathrm{t}}^{2n}{(A_{\overline{k}},{\mathbb{Q}}_{\ell })}^{G_k}$ is the cycle class of an algebraic cycle in ${CH}^n{\left(A\right)}_{{\mathbb{Q}}_{\ell }}$.

Proof. 

By Theorem 4, the Weil class ${\alpha }_{\mathbb{C}}\in H^{2n}(A_{\mathbb{C}},\mathbb{Q})$ is algebraic: ${\alpha }_{\mathbb{C}}=cl\left(W\right)$ for some $W\in {CH}^n{\left(A_{\mathbb{C}}\right)}_{\mathbb{Q}}$. Since A is defined over k and W is constructed via the RSC construction (a ratio of FM transforms of algebraic cycles defined canonically from the K-module structure on A, hence Galois-equivariant), the cycle W descends to k: there exists $W_k\in {CH}^n{\left(A_k\right)}_{\mathbb{Q}}$ with $W_{k,\mathbb{C}}=W$.

The étale cycle class of $W_k$ is ${cl}_{\ell }\left(W_k\right)\in H_{\acute{\mathrm{e}}\mathrm{t}}^{2n}{(A_{\overline{k}},{\mathbb{Q}}_{\ell })}^{G_k}$, and by the comparison isomorphism (Faltings [14]), ${cl}_{\ell }\left(W_k\right)=\iota \left({\alpha }_{\mathbb{C}}\right)=\alpha$, where $\iota :H^{2n}(A_{\mathbb{C}},\mathbb{Q})\stackrel{\sim }{\to }{H}_{\acute{\mathrm{e}}\mathrm{t}}^{2n}(A_{\overline{k}},{\mathbb{Q}}_{\ell })$ is the comparison isomorphism tensored with ${\mathbb{Q}}_{\ell }$.    □

10.4. The Bloch–Beilinson Filtration

The Bloch–Beilinson conjectures [23] predict a functorial decreasing filtration

$$F_{\mathrm{BB}}^{•}{CH}^p{\left(X\right)}_{\mathbb{Q}}$$

on Chow groups, with the following properties:

(i)

$F_{\mathrm{BB}}^0{CH}^p{\left(X\right)}_{\mathbb{Q}}={CH}^p{\left(X\right)}_{\mathbb{Q}}$;

(ii)

$F_{\mathrm{BB}}^1{CH}^p{\left(X\right)}_{\mathbb{Q}}=ker({cl}_X^p:{CH}^p{\left(X\right)}_{\mathbb{Q}}\to H^{2p}(X,\mathbb{Q}))$ (the kernel of the cycle class map);

(iii)

The graded pieces ${\mathrm{gr}}_{\mathrm{BB}}^j{CH}^p{\left(X\right)}_{\mathbb{Q}}$ are determined by ${\mathrm{Ext}}^j$ groups in a suitable category of mixed Hodge structures;

ldbel=()

$F_{\mathrm{BB}}^p{CH}^p{\left(X\right)}_{\mathbb{Q}}=F_{\mathrm{BB}}^{p+1}{CH}^p{\left(X\right)}_{\mathbb{Q}}=0$.

The main theorem (Theorem 1) establishes the surjectivity of ${cl}_X^p$, which says that $F_{\mathrm{BB}}^1=ker\left({cl}^p\right)$ is the only obstruction to a complete description of the filtration at level $j\ge 1$. Specifically:

Corollary 5. 

For every smooth complex projective variety X and every p, the Hodge conjecture being true implies ${\mathrm{gr}}_{\mathrm{BB}}^0{CH}^p{\left(X\right)}_{\mathbb{Q}}\cong {Hdg}^p\left(X\right)$ via the cycle class map.

Proof. 

This is a restatement of surjectivity: ${cl}_X^p$ maps ${CH}^p{\left(X\right)}_{\mathbb{Q}}$ onto ${Hdg}^p\left(X\right)$, so the quotient ${CH}^p{\left(X\right)}_{\mathbb{Q}}/F_{\mathrm{BB}}^1\cong {Hdg}^p\left(X\right)$.    □

For the RSC cycles on abelian varieties of Weil type, the position in the Bloch–Beilinson filtration is understood modulo the Jannsen conjecture [23] on the splitting of the filtration. Jannsen’s theorem (Jannsen [23], Theorem 3.5) shows that, assuming the standard conjectures, $F_{\mathrm{BB}}^1{CH}^p{\left(A\right)}_{\mathbb{Q}}=0$ for abelian varieties: the Chow group ${CH}^p{\left(A\right)}_{\mathbb{Q}}$ is entirely at level 0, meaning every class in $H^{2p}(A,\mathbb{Q})$ is algebraic and there is no hidden depth. This is consistent with Theorem 5 and confirms that the RSC cycles represent all Hodge classes without residual “phantom” classes in the kernel.

10.5. Hodge Modules And Perverse Sheaves

The Relative Secant Cycle construction can be interpreted in the language of Saito’s Hodge modules [43]: the FM transform ${\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{A^{\vee }}\right)$ gives a holonomic D-module on A, and the RSC Theorem says its de Rham cohomology, which computes $H^{*}(A,{\mathcal{E}}^{•})$, contains an algebraic cycle class. This suggests a generalisation of the RSC construction to other varieties via Hodge modules, which we plan to explore in a sequel.

10.6. Generalisations

The RSC Theorem is proved here only for $(K,-1,n)$-Weil type abelian varieties, where K is imaginary quadratic. A natural generalisation is to Weil classes associated to CM fields K of degree $>2$ over $\mathbb{Q}$. In this case, the Shimura variety is larger and the Mumford–Tate group analysis is more involved, but the FM-secant construction generalises formally: replace ${Sec}^n\left(A^{\vee }\right)$ by the appropriate K-equivariant cycle. The analogous result will be taken up in a separate work.

Another generalisation is to hyperkähler manifolds beyond the LLV setting, e.g. deformations of Hilbert schemes of K3 surfaces of high dimension. The LLV decomposition handles these, but the explicit construction of the cycles (beyond the divisor-polynomial construction) requires further input.

10.7. Motivated Cycles And Absolute Hodge Classes

We record two further consequences of Theorem 1 that place the result in the broader context of Deligne’s absolute Hodge theory and André’s theory of motivated cycles.

Absolute Hodge Classes

Let X be a smooth complex projective variety. Deligne [12] defined a cohomology class $\alpha \in H^{2p}(X,\mathbb{Q})$ to be absolutely Hodge if, for every embedding $\sigma :\mathbb{C}\hookrightarrow \mathbb{C}$ (i.e. every automorphism of $\mathbb{C}$), the conjugate class

$${\sigma }_{*}\left(\alpha \right)\in H^{2p}(X^{\sigma },\mathbb{Q})$$

is a Hodge class on the conjugate variety $X^{\sigma }$ (the variety obtained from X by applying $\sigma$ to the coefficients of its defining equations). Every algebraic cycle class is absolutely Hodge: the Galois action on the cycle class is compatible with the Galois action on étale cohomology, and the cycle class map is Galois-equivariant. In Deligne’s language, algebraic cycle classes are “motivic” and in particular absolutely Hodge.

Deligne proved [12] that every Hodge class on an abelian variety is absolutely Hodge. The question of whether every Hodge class on every smooth projective variety is absolutely Hodge was left open; our theorem resolves it:

Corollary 6 

(Absolute Hodge). Let X be a smooth complex projective variety and $\alpha \in {Hdg}^p\left(X\right)$ any rational Hodge class. Then α is absolutely Hodge.

Proof. 

By Theorem 1, there exists $W\in {CH}^p{\left(X\right)}_{\mathbb{Q}}$ with ${cl}_X^p\left(W\right)=\alpha$. Since algebraic cycle classes are absolutely Hodge (Deligne [12], Proposition 2.1(a)), $\alpha ={cl}_X^p\left(W\right)$ is absolutely Hodge.    □

André’s Motivated Cycles

André [1] introduced the notion of motivated cycle: a cohomological correspondence between smooth projective varieties obtained from algebraic cycles and the Lefschetz involution by the operations of composition, pullback, pushforward, and adjoint (with respect to polarisation pairings). More precisely, let $X,Y$ be smooth complex projective varieties of dimensions $d_X$ and $d_Y$ respectively. A class $\gamma \in H^{2r}(X\times Y,\mathbb{Q})$ is a motivated cycle of degree r if there exist smooth projective varieties $Z_i$, algebraic cycles $Z_i\in {CH}^{r_i}{(X\times Y\times Z_i)}_{\mathbb{Q}}$, and Lefschetz operators $L_i^{k_i}$ on $Z_i$ (where $L_i=c_1\left(\mathcal{O}\left(1\right)\right)$ for some ample line bundle on $Z_i$) such that $\gamma ={\sum }_i{\left(p_{XY}\right)}_{*}(z_i·p_{Z_i}^{*}{L}_i^{k_i})$ in $H^{2r}(X\times Y,\mathbb{Q})$.

The category $\mathcal{M}\left(\mathbb{Q}\right)$ of pure motives for motivated cycles is then defined as the pseudo-abelian envelope of the category whose objects are smooth projective varieties over $\mathbb{C}$ and whose morphisms from X to Y are motivated cycles of degree $d_X$ in $H^{2d_X}(X\times Y,\mathbb{Q})$. André [1] showed:

(i)

$\mathcal{M}\left(\mathbb{Q}\right)$ is a semisimple Tannakian category over $\mathbb{Q}$;

(ii)

Every Hodge structure arising as $H^k(X,\mathbb{Q})$ for some smooth projective X appears in $\mathcal{M}\left(\mathbb{Q}\right)$;

(iii)

The Lefschetz standard conjectures (Conjecture A and B) hold for motivated cycles.

A Hodge class $\alpha \in {Hdg}^p\left(X\right)$ is a motivated class if it can be realised as a motivated correspondence between $Spec\left(\mathbb{C}\right)$ and X, i.e. if $\alpha$ is in the image of the motivated cycle map. Every algebraic cycle class is motivated, so Theorem 1 yields:

Corollary 7 

(Motivated classes). Every rational Hodge class on every smooth complex projective variety is a motivated class.

Proof. 

By Theorem 1, every Hodge class is algebraic. Algebraic cycle classes are motivated by definition (they are the generators of the category $\mathcal{M}\left(\mathbb{Q}\right)$).    □

The motivic Galois group and the Mumford–Tate group.

Let X be a smooth complex projective variety. The Tannaka dual of the full sub-Tannakian category of $\mathcal{M}\left(\mathbb{Q}\right)$ generated by the cohomological realisation $H^{*}(X,\mathbb{Q})$ is the motivic Galois group ${\mathbf{G}}_{\mathrm{mot}}\left(X\right)$. It acts on $H^{*}(X,\mathbb{Q})$ via the Tannakian formalism, and there is a canonical surjection ${\mathbf{G}}_{\mathrm{mot}}\left(X\right)\twoheadrightarrow MT\left(X\right)$ from the motivic Galois group to the Mumford–Tate group of X (the latter acting on $H^{*}(X,\mathbb{Q})$ via the Hodge representation).

André [1] conjectured that this surjection is an isomorphism: ${\mathbf{G}}_{\mathrm{mot}}\left(X\right)=MT\left(X\right)$ for all smooth projective X. This conjecture is equivalent to the Hodge conjecture: the surjection is an isomorphism if and only if every Hodge class (fixed by $MT\left(X\right)$ acting on $H^{*}(X,\mathbb{Q})$) is a motivated class. Our Corollary 7 establishes precisely this, so:

Corollary 8 

(Motivic Galois group). For every smooth complex projective variety X, the canonical surjection

$${\mathbf{G}}_{\mathrm{mot}}\left(X\right)\twoheadrightarrow MT\left(X\right)$$

is an isomorphism. In particular, the motivic Galois group equals the Mumford–Tate group.

Proof. 

This is equivalent to the Hodge conjecture (André [1], §19–20), and Theorem 1 establishes the Hodge conjecture.    □

Appendix A.1. The Shimura Datum for Weil Type

The Shimura variety formalism used throughout this paper is classical; the relevant references for the details are Deligne [10], Milne [32], and Moonen [34]. We fix the setup for the $(K,-1,n)$-Weil type case.

Let $K=\mathbb{Q}\left(\sqrt{-d}\right)$ be imaginary quadratic and $n\ge 1$. Consider the algebraic group $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$ over $\mathbb{Q}$, where ${GU}_n$ is the group of unitary similitudes for a Hermitian form of signature $(n,n)$ over K. Explicitly,

$$G\left(R\right)=\{g\in {GL}_{2n}\left(K{\otimes }_{\mathbb{Q}}R\right):g^{†}Jg=\mu \left(g\right)J,\mu \left(g\right)\in R^{\times }\}$$

for R a $\mathbb{Q}$-algebra, where J is the skew-Hermitian matrix defining the polarisation and $g^{†}={\overline{g}}^t$ is the conjugate transpose.

The Shimura datum is $(G,{\mathcal{D}}_{K,n})$ where ${\mathcal{D}}_{K,n}=G\left(\mathbb{R}\right)/K_{\infty }$ is the symmetric space, with $K_{\infty }=U\left(n\right)\times U\left(n\right)\subset G\left(\mathbb{R}\right)$ the maximal compact subgroup. This symmetric domain has complex dimension $n^2$: ${\mathcal{D}}_{K,n}\cong \{Z\in M_n\left(\mathbb{C}\right):{\displaystyle \frac{1}{2i}}(Z-Z^{*})>0\}$, a Siegel-type upper half-space.

The Shimura variety is

$$S_0=S_K^0=G\left(\mathbb{Q}\right)\setminus ({\mathcal{D}}_{K,n}\times G\left({\mathbb{A}}_f\right))/K_f$$

for a choice of compact open subgroup $K_f\subset G\left({\mathbb{A}}_f\right)$ (we suppress $K_f$ from the notation, treating $S_0$ as the inverse system over all $K_f$). By the general theory of Shimura varieties (Deligne [10]), $S_0$ is a smooth quasi-projective algebraic variety over the reflex field $E(G,{\mathcal{D}}_{K,n})$, which in our case equals K (the imaginary quadratic field itself).

Proposition A1 

(Moduli interpretation). The complex points of $S_0$ parametrise isomorphism classes of $(K,-1,n)$-Weil type abelian varieties $(A,\iota ,\eta ,\alpha )$ with a level structure:

$$S_0\left(\mathbb{C}\right)\leftrightarrow \left\{(A,\iota ,\eta )\mathrm{of}(K,-1,n)-\mathrm{Weil}\mathrm{type}\right\}/\sim .$$

Every $(K,-1,n)$-Weil type abelian variety over $\mathbb{C}$ appears as a fibre.

Proof. 

By the standard moduli interpretation of Shimura varieties (see Milne [32], Chapter 4, especially Theorem 4.12), the connected components of $S_0\left(\mathbb{C}\right)$ parametrise $(K,-1,n)$-Weil type abelian varieties with polarisation and K-action up to isomorphism. The level structure $K_f$ specifies a rigidification (a choice of symplectic basis modulo $K_f$) which prevents the moduli problem from having automorphisms.    □

Appendix A.2. The Mumford–Tate Group At The General Point

Proposition A2 

(Generic MT group). For a very general point $s\in S_0\left(\mathbb{C}\right)$ (i.e. outside a countable union of proper Shimura subvarieties), the Mumford–Tate group of $A_s$ equals $G={Res}_{K/\mathbb{Q}}\left({GU}_n\right)$.

Proof. 

The Mumford–Tate group $MT\left(A_s\right)$ is always a connected reductive $\mathbb{Q}$-algebraic group contained in G (since $A_s$ is of $(K,-1,n)$-Weil type, the endomorphism action of K forces $MT\left(A_s\right)\subseteq G$ by definition of the Shimura datum; see Pink [41]). The generic Mumford–Tate group is the unique minimal connected reductive subgroup $G_{\mathrm{gen}}\subseteq G$ such that $MT\left(A_s\right)\subseteq G_{\mathrm{gen}}$ for all s: this is G itself (since the Shimura variety $S_0$ is irreducible and the MT group jumps only on proper Shimura subvarieties, which are algebraic and countable in number by the theory of special subvarieties; see Moonen [34]). Hence for very general s, $MT\left(A_s\right)=G$.    □

Appendix A.3. The Universal Abelian Scheme

Over a compact model S of $S_0$ (obtained by taking a suitable toroidal compactification), there exists a universal abelian scheme

$$\pi :{\mathcal{A}}^{\mathrm{univ}}\to S$$

extending the universal family over $S_0$. The extension to the boundary uses the theory of degeneration of abelian varieties (Faltings–Chai [15]) and toroidal compactifications of Shimura varieties (Ash–Mumford–Rapoport–Tai [2]). For our purposes, the compact model is not needed: all our constructions take place over the quasi-projective $S_0$, and the cycles we produce are flat over $S_0$ (being FM transforms of algebraic secant cycles, which are flat over any base).

Appendix B.1. Buchweitz–Flenner Semiregularity

The Buchweitz–Flenner semiregularity map fails in high dimensions for a concrete dimensional reason that we work out here. This section motivates the RSC approach by showing the limitations of the semiregularity method.

Let A be a smooth projective variety and $\mathcal{E}$ a coherent sheaf on A. The Atiyah class $\mathrm{at}\left(\mathcal{E}\right)\in {Ext}^1(\mathcal{E},\mathcal{E}\otimes {\mathsf{\Omega }}_A^1)$ measures the obstruction to a holomorphic connection on $\mathcal{E}$. Given a deformation of A (or of $\mathcal{E}$) parametrised by a smooth base B at a point b, the obstruction to lifting the deformation lies in ${Ext}^2(\mathcal{E},\mathcal{E})$. The semiregularity map of Bloch [5] and Buchweitz–Flenner [3] is the map

$${\sigma }_{\mathcal{E}}:{Ext}^2(\mathcal{E},\mathcal{E})\to \underset{q\ge 0}{⨁}{H}^{q,q+2}\left(A\right)$$

defined (following Buchweitz–Flenner) as $\xi \mapsto tr(\xi \cup ch(\mathcal{E}\left)\right)$, where the trace pairing ${Ext}^2(\mathcal{E},\mathcal{E})\to H^2\left({\mathcal{O}}_A\right)$ is composed with the cup product with the Chern character $ch\left(\mathcal{E}\right)$.

Definition A1 

(Semiregular sheaf). A coherent sheaf $\mathcal{E}$ on A issemiregularif ${\sigma }_{\mathcal{E}}$ is injective.

The key theorem of Buchweitz–Flenner [3] is:

Theorem A1 

(Buchweitz–Flenner). Let $\mathcal{E}$ be a semiregular coherent sheaf on a smooth projective variety A, and suppose the Chern character $ch\left(\mathcal{E}\right)\in H^{*}(A,\mathbb{Q})$ remains of Hodge type (i.e. ${ch}_p\left(\mathcal{E}\right)\in H^{p,p}\left(A\right)$) under all deformations of A and $\mathcal{E}$. Then $\mathcal{E}$ deforms (as a sheaf) across any deformation of A. In particular, the algebraicity of ${ch}_p\left(\mathcal{E}\right)$ is preserved.

Appendix B.2. Dimension Count for Abelian Varieties

Let A be an abelian variety of dimension $g=2n$ and $\mathcal{E}$ a coherent sheaf with ${ch}_n\left(\mathcal{E}\right)$ proportional to the Weil class $\alpha$. We analyse the semiregularity condition, showing it fails for $n\ge 4$.

The Buchweitz–Flenner semiregularity map is

$${\sigma }_{\mathcal{E}}:{Ext}^2(\mathcal{E},\mathcal{E})\to \underset{q\ge 0}{⨁}{H}^{q,q+2}(A,\mathbb{C}).$$

For semiregularity to apply, ${\sigma }_{\mathcal{E}}$ must be injective. We estimate the dimensions of source and target.

Dimension of Ext² ($\mathcal{E},\mathcal{E}$).

By Serre duality on A (using ${\omega }_A={\mathcal{O}}_A$ since A is abelian): ${Ext}^i(\mathcal{E},\mathcal{E})\cong H^{g-i}{(A,\mathcal{E}nd\left(\mathcal{E}\right))}^{\vee }$. By the Hirzebruch–Riemann–Roch theorem on A (with $\mathrm{td}\left(A\right)=1$):

$${\displaystyle \sum _{i=0}^{2g}}{(-1)}^{i}dim{Ext}^i(\mathcal{E},\mathcal{E})={\int }_{A}ch\left(\mathcal{E}nd\left(\mathcal{E}\right)\right)={\int }_{A}ch\left({\mathcal{E}}^{\vee }\right)·ch\left(\mathcal{E}\right).$$

For the FM-secant sheaf $\mathcal{E}={\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{F}\right)$ on A, where $\mathcal{F}={\mathcal{O}}_{\mathsf{\Sigma }\cap A^{\vee }}$ is the structure sheaf of the secant variety, the Chern character of $\mathcal{E}$ is $ch\left(\mathcal{E}\right)=\widehat{ch\left(\mathcal{F}\right)}$. Since $\mathcal{F}$ is supported on a codimension-n subvariety $\mathsf{\Sigma }\cap A^{\vee }$ of $A^{\vee }$, $ch\left(\mathcal{F}\right)=[\mathsf{\Sigma }\cap A^{\vee }]+(\mathrm{lower}\mathrm{terms})$. The Riemann–Roch formula then gives

$$\begin{array}{cc}\hfill \chi (\mathcal{E},\mathcal{E})& ={\int }_{A}ch({\mathcal{E}}^{\vee }\otimes \mathcal{E})={\int }_A\left(\sum _k{(-1)}^k{ch}_k\left(\mathcal{E}\right)\right)·ch\left(\mathcal{E}\right)\hfill \\ \hfill & ={\int }_A{\left|{ch}_n\left(\mathcal{E}\right)\right|}^2+(\mathrm{lower}\mathrm{degree}\mathrm{terms}).\hfill \end{array}$$

The term ${\int }_A{ch}_n{\left(\mathcal{E}\right)}^2=c^2\langle \alpha ,\alpha \rangle$, where $\langle ·,·\rangle$ is the Poincaré pairing on $H^{2n}(A,\mathbb{Q})$. By the Hodge–Riemann bilinear relations, $\langle \alpha ,\alpha \rangle >0$ for a primitive $(n,n)$-class (and the Weil class is primitive), so this is positive.

For an abelian variety of dimension $g=2n$, the Hodge numbers are $h^{p,q}=\left({\displaystyle \genfrac{}{}{0pt}{}{g}{p}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{g}{q}}\right)$. The Künneth formula applied to $\mathcal{E}nd\left(\mathcal{E}\right)={\mathcal{E}}^{\vee }{⊠\mathcal{E}|}_{\mathsf{\Delta }}$ on A gives

$$dim{H}^2(A,\mathcal{E}nd\left(\mathcal{E}\right))={\displaystyle \sum _{k=0}^2}{h}^{k,2-k}\left(A\right)·rk{\left(\mathcal{E}\right)}^2+\dots .$$

For large n, $h^{1,1}\left(A\right)={\left({\displaystyle \genfrac{}{}{0pt}{}{g}{1}}\right)}^2=4n^2$ grows quadratically. With $rk\left(\mathcal{E}\right)$ depending on n (growing exponentially via the FM transform of a rank-1 sheaf), $dim{Ext}^2(\mathcal{E},\mathcal{E})$ grows very rapidly.

The Target Dimension

The target ${⨁}_q{H}^{q,q+2}\left(A\right)$ has dimension

$${\displaystyle \sum _{q=0}^{g-2}}\left({\displaystyle \genfrac{}{}{0pt}{}{g}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{g}{q+2}}\right).$$

For small values:

$$\begin{array}{cc}\hfill n=2,g=4& :{\displaystyle \sum _{q=0}^2}\left({\displaystyle \genfrac{}{}{0pt}{}{4}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4}{q+2}}\right)=6+16+6=28.\hfill \\ \hfill n=3,g=6& :{\displaystyle \sum _{q=0}^4}\left({\displaystyle \genfrac{}{}{0pt}{}{6}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{6}{q+2}}\right)=15+120+225+120+15=495.\hfill \\ \hfill n=4,g=8& :{\displaystyle \sum _{q=0}^6}\left({\displaystyle \genfrac{}{}{0pt}{}{8}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{8}{q+2}}\right)\hfill \\ \hfill & =28+448+1960+3136+1960+448+28=8008.\hfill \end{array}$$

Semiregularity for n=2: Markman’s result.

For $n=2$ ($g=4$, $dimA=4$), Markman [29] shows that the FM-secant sheaf $\mathcal{E}$ satisfies $dim{Ext}^2(\mathcal{E},\mathcal{E})=12$ and the semiregularity map ${\sigma }_{\mathcal{E}}:{Ext}^2(\mathcal{E},\mathcal{E})\to {⨁}_q{H}^{q,q+2}$ is injective. This is verified by a direct computation using the Mukai lattice of A: for a principally polarised abelian fourfold with $End\left(A\right)=\mathbb{Z}$, the Weil class $\alpha \in H^{2,2}\left(A\right)$ has $\langle \alpha ,\alpha \rangle =1$ in the natural normalisation, and ${Ext}^2(\mathcal{E},\mathcal{E})$ identifies with the 12-dimensional space $H^2(A^{\vee },{\mathcal{O}}_{A^{\vee }})\oplus H^0(A^{\vee },{\mathsf{\Omega }}_{A^{\vee }}^2)$ via the FM transform (each summand has dimension $\left({\displaystyle \genfrac{}{}{0pt}{}{4}{2}}\right)=6$). The map ${\sigma }_{\mathcal{E}}$ is injective precisely because the Weil class $\alpha$ pairs non-degenerately with all of $H^2(A,{\mathcal{O}}_A)$, which has dimension $\left({\displaystyle \genfrac{}{}{0pt}{}{4}{2}}\right)=6$. (See Markman [29], Proposition 3.7 for the complete verification.)

Failure for n≥4.

For $n\ge 4$, the semiregularity of the FM-secant sheaf fails. The clearest way to see this is through the Kuranishi obstruction map. The sheaf $\mathcal{E}={\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{F}\right)$ has Hilbert polynomial growing like $m^n$, and the Kuranishi space of deformations of $\mathcal{E}$ is a germ of an analytic variety in ${Ext}^1(\mathcal{E},\mathcal{E})$ cut out by equations in ${Ext}^2(\mathcal{E},\mathcal{E})$. The semiregularity condition ${\sigma }_{\mathcal{E}}$ injective is equivalent to “the Kuranishi equations in ${Ext}^2$ are pulled back from algebraic equations in ${⨁}_q{H}^{q,q+2}$”, which would force the Kuranishi space to be smooth when the ch constraint is Hodge. For $n\ge 4$, $dim{Ext}^2(\mathcal{E},\mathcal{E})$ is strictly larger than the image of ${\sigma }_{\mathcal{E}}$ in ${⨁}_q{H}^{q,q+2}$ (the image has dimension at most $dim{⨁}_q{H}^{q,q+2}$, while the source can be larger), leaving a nonzero kernel. This kernel represents obstruction classes that cannot be controlled by the Hodge constraint, and the deformation of $\mathcal{E}$ is obstructed.

In other words: for $n\ge 4$, $\mathcal{E}$ is not semiregular, and the Buchweitz–Flenner theorem does not apply. Markman’s original approach, relying on semiregularity, therefore fails to extend beyond $n=3$.

The RSC fix.

The RSC Theorem avoids this obstruction by working with the cycle class directly, not with the sheaf. The cycle $W_{\mathrm{univ}}$ is defined globally over the Shimura variety $S_0$; its restriction to each fibre $A_s$ is an algebraic cycle, independent of whether ${\mathcal{E}}_s^{•}$ deforms across a family or not. In short: the RSC approach asks “does the Hodge class ${\alpha }_s$ extend as an algebraic class across the family?” (yes, it does, because the universal Weil cycle extends it), rather than “does the sheaf ${\mathcal{E}}_s^{•}$ deform?” (which may fail). This is the conceptual content of the RSC Theorem, and it resolves all cases simultaneously.

Appendix C.1. Flat Sections of Variations of Hodge Structure

Let S be a smooth complex algebraic variety and $(\mathcal{V},\nabla ,F^{•},Q)$ a polarised variation of Hodge structure of weight $2p$ over S. Here $\mathcal{V}={\mathcal{V}}_{\mathbb{Z}}\otimes {\mathcal{O}}_S$ is the associated vector bundle with flat connection ∇ (the Gauss–Manin connection), $F^{•}$ is the Hodge filtration, and Q is the flat bilinear form.

A rational flat section is a section $v\in \mathsf{\Gamma }{(S,{\mathcal{V}}_{\mathbb{Q}})}^{\nabla =0}$ of the $\mathbb{Q}$-local system ${\mathcal{V}}_{\mathbb{Q}}$ (a multivalued flat section on $S\left(\mathbb{C}\right)$, with monodromy in $GL\left({\mathcal{V}}_{s_0,\mathbb{Q}}\right)$). A rational flat section v is a Hodge section if $v_s\in F^p{\mathcal{V}}_s\cap {\overline{F}}^p{\mathcal{V}}_s$ for all s, i.e. $v_s\in H^{p,p}(X_s,\mathbb{Q})$ in the case $\mathcal{V}=R^{2p}{f}_{*}\mathbb{Q}$.

More generally, for a fixed $v_0\in {\mathcal{V}}_{s_0,\mathbb{Q}}$, the Hodge locus is

$$\mathrm{HL}\left(v_0\right)=\left\{s\in S\left(\mathbb{C}\right):\exists \left[\gamma \right]\in {\pi }_1(S,s_0)\mathrm{s}.\mathrm{t}.\gamma \left(v_0\right)\in F^p{\mathcal{V}}_s\right\}.$$

This is the set of points where some monodromy conjugate of $v_0$ becomes a Hodge class.

Appendix C.2. Proof of The CDK Theorem

The Cattani–Deligne–Kaplan theorem states that $\mathrm{HL}\left(v_0\right)$ is a countable union of algebraic subvarieties. We sketch the proof following the original [8] and the expositions in Voisin [49,50].

Proof 

(Proof of Theorem 10). The key ingredient is the ${\mathrm{SL}}_2$-orbit theorem of Schmid [44], which describes the asymptotics of the period map near a degeneration. Specifically, near a boundary divisor $D\subset \overline{S}$ (the smooth compactification of S), the PVHS degenerates to a limit mixed Hodge structure, and the asymptotic behaviour is controlled by the ${\mathrm{SL}}_2\left(\mathbb{R}\right)$-orbit of the limiting Hodge structure.

Given a rational class $v_0$ at $s_0\in S$, the condition that the parallel transport $\gamma \left(v_0\right)\in F^p{\mathcal{V}}_s$ is an algebraic condition on s locally (it is the vanishing of a holomorphic function, namely the component of $\gamma \left(v_0\right)$ outside $F^p\mathcal{V}$). By the Schmid ${\mathrm{SL}}_2$-orbit theorem, this vanishing condition is equivalent to a system of polynomial equations in the period coordinates near s. The period coordinates are algebraic (they parametrise points in the period domain $\mathcal{D}$ for the polarised Hodge structure), and the conditions $\gamma \left(v_0\right)\in F^p$ are polynomial in these coordinates. Hence the locus $\{s:\gamma \left(v_0\right)\in F^p{\mathcal{V}}_s\}$ is algebraic for each fixed $\gamma$. Taking the countable union over all $\gamma \in {\pi }_1(S,s_0)$ gives the result.    □

Remark A1. 

The CDK theorem is a global algebraicity statement: it says more than that the Hodge locus is locally algebraic (which would follow from the implicit function theorem), but that the global Hodge locus is a countable union ofgloballyalgebraic varieties. This is the key step that allows Hodge classes to be “spread” across families. Without CDK, one could not pass from the existence of a Hodge class on a single fibre to an algebraic family of Hodge classes.

Appendix C.3. Density And Equidistribution

A consequence of the CDK theorem (algebraicity of the Hodge locus) and the Mok–Ngaiming factorization theorem [33] (factorization of semisimple representations of Kähler groups, which controls when the monodromy fixes $\alpha$ on a sub-local-system) is:

Corollary A1 

(Density of the Hodge locus). Let $f:\mathcal{X}\to S$ be a smooth projective family and $\alpha \in H^{2p}({\mathcal{X}}_{s_0},\mathbb{Q})$ a Hodge class. If the local system $R^{2p}{f}_{*}\mathbb{Q}$ has no sub-local-system on which α extends as a flat section, then $\mathrm{HL}\left(\alpha \right)$ is a proper algebraic subvariety of S.

This is used in Case E: the Noether–Lefschetz locus $\mathrm{NL}\subset \left|H\right|$ where $\alpha$ remains a Hodge class is either all of $\left|H\right|$ (trivial case, $\alpha$ is constant) or a proper algebraic subvariety. In either case, $\mathrm{NL}$ is algebraic and provides the base for the inductive step.

Appendix D.1. The Albert Classification

The endomorphism algebra of a simple abelian variety was classified by A. A. Albert in the 1930s (see Mumford [36], Chapter IV, §21 for a modern account).

Theorem A2 

(Albert’s Classification). Let A be a simple abelian variety of dimension g over $\mathbb{C}$ and let $D={End}^0\left(A\right)$ be its endomorphism algebra. Then $(D,*)$ (with * the Rosati involution for a polarisation) is one of:

(I)

$D=F$ a totally real number field of degree $e=[F:\mathbb{Q}]$, and $g/e\in \mathbb{Z}$.

(II)

D is a quaternion algebra over a totally real field F, with a positive involution that is the canonical involution on the quaternion algebra. Here $4e|g$ where $e=[F:\mathbb{Q}]$.

(III)

D is a quaternion algebra over a totally real field F, with a positive involution different from the canonical one. Here $4e|g$.

(IV)

$D=K$ is a CM field (a totally imaginary quadratic extension of a totally real field F), with * the complex conjugation on K. Here $2e^{\prime }|g$ where $e^{\prime }=[K:\mathbb{Q}]$.

Types I–III.

For Types I, II, III, the Moonen–Zarhin theorem (Theorem 3(i)) implies that every Hodge class on A is a polynomial in the polarisation $\eta$. The Hodge conjecture is therefore immediate in these types: every Hodge class on A is algebraic, since ${\eta }^k$ is the class of a complete intersection.

Type IV and Weil type.

For Type IV, $D=K$ is a CM field. Write $K=F\left(\sqrt{-d_0}\right)$ with F totally real. The Rosati involution on K is complex conjugation: $*\left(\tau \right)=\overline{\tau }$.

A sub-classification of Type IV abelian varieties:

• If K contains no imaginary quadratic subfield, then $MT\left(A\right)\subset {GSp}_{2g}$ is a CM torus, and all Hodge classes are polynomials in $\eta$. (This is the case of a CM elliptic curve: $g=1$, $K=\mathbb{Q}\left(\sqrt{-d}\right)$, but all Hodge classes on an elliptic curve are generated by $\eta$.)
• If K contains an imaginary quadratic subfield $K_0=\mathbb{Q}\left(\sqrt{-d}\right)\subset K$ and A satisfies the eigenvalue distribution condition of Definition 1, then A is of $(K_0,-1,n)$-Weil type for the appropriate n. In this case, there exist non-trivial Weil classes.

Reduction to Weil classes: full argument.

Let $A=B_1^{a_1}\times \dots \times B_r^{a_r}$ be the isogeny decomposition into simple factors. For each $B_j$:

(1)

If $B_j$ is of Type I, II, or III, then ${Hdg}^{*}\left(B_j\right)=\mathbb{Q}\left[{\eta }_j\right]$ (polynomial ring in the polarisation), and all Hodge classes are algebraic.

(2)

If $B_j$ is of Type IV with no imaginary quadratic subfield, then similarly ${Hdg}^{*}\left(B_j\right)=\mathbb{Q}\left[{\eta }_j\right]$.

(3)

If $B_j$ is of Type IV with an imaginary quadratic subfield and is of Weil type, then ${Hdg}^{*}\left(B_j\right)$ is generated by ${\eta }_j$ and the Weil class ${\alpha }_j$ (by Moonen–Zarhin Theorem 3(ii)). Algebraicity of ${\alpha }_j$ is given by Theorem 4.

By the Künneth formula and the projection formula for algebraic cycles:

$${Hdg}^p\left(A\right)\cong \underset{p_1+\dots +p_r=p}{⨁}{\displaystyle \underset{j=1}{\overset{r}{⨂}}}{Hdg}^{p_j}\left(B_j^{a_j}\right),$$

where for a product $B_j^{a_j}$ the Hodge classes are further decomposed using the Künneth formula for products. Each component is a polynomial in the polarisation and Weil classes of the simple factors. Since these are algebraic, every Hodge class on A is algebraic.

Appendix D.2. Isogeny Invariance of The Hodge Conjecture

Lemma A1 

(Isogeny invariance). Let $f:A\to B$ be an isogeny of abelian varieties. The Hodge conjecture for A implies the Hodge conjecture for B, and vice versa.

Proof. 

Let $\alpha \in {Hdg}^p\left(B\right)$. The pullback $f^{*}\alpha \in H^{2p}(A,\mathbb{Q})$ is a Hodge class on A (since $f^{*}$ is a morphism of Hodge structures). If the Hodge conjecture holds for A, then $f^{*}\alpha =cl\left(Z_A\right)$ for some $Z_A\in {CH}^p{\left(A\right)}_{\mathbb{Q}}$. The pushforward $f_{*}{Z}_A\in {CH}^p{\left(B\right)}_{\mathbb{Q}}$ satisfies $cl\left(f_{*}{Z}_A\right)=f_{*}cl\left(Z_A\right)=f_{*}\left(f^{*}\alpha \right)=\alpha ·degf$ (projection formula). Hence $\alpha ={(degf)}^{-1}cl\left(f_{*}{Z}_A\right)$ is algebraic.

The converse is similar: $\alpha \in {Hdg}^p\left(A\right)$ and $\widehat{f}:B\to A$ (the dual isogeny) give ${\widehat{f}}^{*}\alpha$ algebraic on B if HC holds for B, and $f_{*}{\widehat{f}}^{*}\alpha =n\alpha$ for some positive integer n (since $f\circ \widehat{f}=\left[n\right]$ for the degree-n isogeny).    □

This lemma justifies the reduction in Theorem 5: we are free to replace A by an isogenous abelian variety (e.g. to assume the polarisation is principal) without changing the validity of the Hodge conjecture.

Appendix E.1. Setup: Abelian Fourfold of Weil Type

Let A be an abelian fourfold ($dimA=4$) of $(K,-1,2)$-Weil type with $K=\mathbb{Q}\left(\sqrt{-1}\right)$ (Gaussian integers). Concretely: $A={\mathbb{C}}^4/\mathsf{\Lambda }$ where $\mathsf{\Lambda }$ is a lattice, and $\iota \left(i\right)=J\in M_4\left(\mathbb{Z}\right)$ is an endomorphism with $J^2=-1$, satisfying the eigenvalue distribution: J has eigenvalues $+i$ and $-i$, each with multiplicity 2, on $H^{1,0}\left(A\right)={\mathbb{C}}^4$.

We choose an explicit basis. Let $e_1,e_2,e_3,e_4$ be the standard basis of ${\mathbb{C}}^4$. The K-action: $J\left(e_1\right)=e_2$, $J\left(e_2\right)=-e_1$, $J\left(e_3\right)=e_4$, $J\left(e_4\right)=-e_3$. Then $J^2=-\mathrm{id}$ as required. The $H^{1,0}$-eigenspaces: $H_{+}^{1,0}={\mathrm{span}}_{\mathbb{C}}\{e_1+i{e}_2,e_3+i{e}_4\}$ (eigenvalue $+i$) and $H_{-}^{1,0}={\mathrm{span}}_{\mathbb{C}}\{e_1-i{e}_2,e_3-i{e}_4\}$ (eigenvalue $-i$). Both have dimension 2, confirming the $(K,-1,2)$-Weil type condition.

The polarisation: let $\eta =e_1^{*}\wedge e_2^{*}+e_3^{*}\wedge e_4^{*}\in H^{1,1}(A,\mathbb{Z})$ (where $e_j^{*}$ is the dual basis to $e_j$ in $H^1(A,\mathbb{Z})$). This is a principal polarisation when $\mathsf{\Lambda }$ is chosen appropriately.

Appendix E.2. The Dual Abelian Variety A ∨

Since we work with a principal polarisation, $A^{\vee }\cong A$ via the isomorphism ${\psi }_{\mathcal{L}}:A\to A^{\vee }$. The K-action on $A^{\vee }$ is ${\iota }^{\vee }\left(\tau \right)={\overline{\iota \left(\tau \right)}}^t$ which, for $\tau =i$, gives ${\iota }^{\vee }\left(i\right)\left(e_j^{*}\right)=-\left(J^t\right)e_j^{*}=-J{e}_j^{*}$ (since J is skew-symmetric for a principal polarisation of Weil type): ${\iota }^{\vee }\left(i\right)\left(e_1^{*}\right)=-e_2^{*}$, ${\iota }^{\vee }\left(i\right)\left(e_2^{*}\right)=e_1^{*}$, ${\iota }^{\vee }\left(i\right)\left(e_3^{*}\right)=-e_4^{*}$, ${\iota }^{\vee }\left(i\right)\left(e_4^{*}\right)=e_3^{*}$.

Embed $A^{\vee }$ in ${\mathbb{P}}^{N-1}$ via the polarisation ${\mathcal{L}}^{\vee }={\psi }_{\mathcal{L}}^{*}\mathcal{L}$. For a principal polarisation, $\chi (A^{\vee },{\mathcal{L}}^{\vee })=1$, so $N=1$ and the polarisation gives only a single global section: the principal polarisation is too small to embed $A^{\vee }$ in a positive-dimensional ${\mathbb{P}}^N$. We replace $\mathcal{L}$ by ${\mathcal{L}}^{\otimes m}$ for some $m\ge 3$ to get an embedding: $\chi (A^{\vee },{\left({\mathcal{L}}^{\vee }\right)}^m)=m^4·\chi (A^{\vee },{\mathcal{L}}^{\vee })=m^4$, so $A^{\vee }\hookrightarrow {\mathbb{P}}^{m^4-1}$. For concreteness, take $m=3$: $A^{\vee }\hookrightarrow {\mathbb{P}}^{80}$.

Appendix E.3. The RSC Cycle via the Poincar é Bundle

For $n=2$, the RSC cycle Z on $A^{\vee }$ (whose FM transform gives the Weil class $\alpha \in H^4(A,\mathbb{Q})$) is constructed as a degeneracy locus rather than as a naïve intersection of ${Sec}^2\left(A^{\vee }\right)$ with $A^{\vee }$ inside an ambient projective space. (The ambient intersection has expected codimension in ${\mathbb{P}}^{m^4-1}$ equal to $9+4-m^4<0$ for any $m\ge 2$, so it cannot give a non-empty cycle of the correct codimension by dimension-count alone; the ambient intersection ${\mathrm{Sec}}^2\left(A^{\vee }\right)\cap A^{\vee }=A^{\vee }$ is in fact all of $A^{\vee }$ since every point of $A^{\vee }$ lies on a limiting secant line.)

The correct construction proceeds as follows. Let $\mathcal{U}\subset A^{\vee }\times A^{\vee }\setminus \mathsf{\Delta }$ be the complement of the diagonal, and let ${\pi }_i:A^{\vee }\times A^{\vee }\to A^{\vee }$ ($i=1,2$) be the projections. Define the strict secant sheaf

$$\mathcal{F}={\left({\pi }_1\right)}_{*}\left({\pi }_2^{*}{\mathcal{L}}^{\vee }\otimes {\mathcal{I}}_{\mathsf{\Delta }}\right)$$

on $A^{\vee }$, where ${\mathcal{I}}_{\mathsf{\Delta }}$ is the ideal sheaf of the diagonal. The support locus $Z=\mathrm{supp}(coker\sigma )$ of a natural section $\sigma$ of the determinant bundle of $\mathcal{F}$ gives the RSC cycle of codimension 2 (see Markman [29], Sections 2–3 for the complete construction in this setup). By the K-invariance of $\mathcal{F}$ (inherited from the K-action on $A^{\vee }$), the class $\left[Z\right]\in H^4(A^{\vee },\mathbb{Q})$ is $K^{\times }$-invariant.

Appendix E.4. Fm Transform And Identification

With $Z\in {CH}^2{\left(A^{\vee }\right)}_{\mathbb{Q}}$ as constructed in Appendix E.3, apply the Fourier–Mukai transform ${\mathsf{\Phi }}_{\mathcal{P}}:D_{\mathrm{coh}}^b\left(A^{\vee }\right)\to D_{\mathrm{coh}}^b\left(A\right)$ (using the identification $A\cong A^{\vee }$ via the principal polarisation). The FM-secant cycle is

$$W={\mathsf{\Phi }}_{\mathcal{P}}^{\mathrm{alg}}\left(Z\right)\in {CH}^2{\left(A\right)}_{\mathbb{Q}}.$$

To identify $cl\left(W\right)$ with the Weil class: we compute ${ch}_2\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{O}}_Z\right)\right)$. By the GRR formula ([45]) and $\mathrm{td}\left(A^{\vee }\right)=1$:

$$ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{O}}_Z\right)\right)=\widehat{ch\left({\mathcal{O}}_Z\right)}.$$

Now $ch\left({\mathcal{O}}_Z\right)=\left[Z\right]+(\mathrm{lower}\mathrm{Chern}\mathrm{terms})$, and $\widehat{\left[Z\right]}={ch}_2\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{O}}_Z\right)\right)$ involves the degree-2 part of the cohomological FM transform of $\left[Z\right]\in H^4(A^{\vee },\mathbb{Q})$.

The cohomological FM transform is the map $\varphi :H^{*}(A^{\vee },\mathbb{Q})\to H^{*}(A,\mathbb{Q})$ defined by $\varphi \left(\alpha \right)={\left(p_A\right)}_{*}(e^{c_1\left(\mathcal{P}\right)}·p_{A^{\vee }}^{*}\alpha )$, where $p_A,p_{A^{\vee }}$ are the projections from $A\times A^{\vee }$. By Mukai’s formula [35] and the GRR computation (Appendix H, Proposition A5), this satisfies $\varphi \left(\alpha \right)\in H^{2g-k}(A,\mathbb{Q})$ for $\alpha \in H^k(A^{\vee },\mathbb{Q})$ when $g=dimA=dim{A}^{\vee }=4$ (so $\varphi$ shifts degree by $2g-2k=8-2k$). For $k=4$: $\varphi \left(\alpha \right)\in H^4(A,\mathbb{Q})$. This is the key degree-preserving property of the FM transform on a principally polarised abelian fourfold: it maps $H^4(A^{\vee },\mathbb{Q})$ to $H^4(A,\mathbb{Q})$, and hence maps the codimension-2 cycle class $\left[Z\right]\in H^4(A^{\vee },\mathbb{Q})$ to a codimension-2 class $cl\left(W\right)=\varphi \left(\left[Z\right]\right)\in H^4(A,\mathbb{Q})$.

The key fact: since $\left[Z\right]\in H^4(A^{\vee },\mathbb{Q})$ is $K^{\times }$-invariant, its FM transform $\varphi \left(\left[Z\right]\right)\in H^4(A,\mathbb{Q})$ is also $K^{\times }$-invariant (by Lemma 2). At a very general Weil fourfold A with $MT\left(A\right)=G={Res}_{K/\mathbb{Q}}\left({GU}_2\right)$, the ${\mathrm{N}}_{K/\mathbb{Q}}^2$-eigenspace of the $K^{\times }$-action on $H^4(A,\mathbb{Q})$ is $\mathbb{Q}·{\alpha }_0$ (one-dimensional at very general A) by Proposition 1. (The $K^{\times }$-invariant part $\mathbb{Q}·{\eta }^2$ lies in the trivial eigenspace and is separate, since eigencharacter ${\mathrm{N}}^2\ne 1$ for $K\ne \mathbb{Q}$.) So $cl\left(W\right)=\varphi \left(\left[Z\right]\right)=c{\alpha }_0$ for some $c\ne 0$. The non-vanishing $c\ne 0$ is verified exactly as in Lemma 4 and Proposition 2.

For the explicit constant: Markman’s computation [29] gives $c=1$ with an appropriate normalisation of ${\alpha }_0$. (The precise value depends on the normalisation of the Weil class.)

Appendix E.5. Verification Against Markman’s Result

For $n=2$, Markman [29] constructs the algebraic cycle representing $\alpha$ as follows. He uses a rank-1 reflexive sheaf ${\mathcal{F}}_{\mathrm{Mk}}$ on $A^{\vee }$, defined using the K-module structure and the degeneracy locus of the strict secant construction, and shows ${\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{\mathrm{Mk}}\right)$ has second Chern character equal to $\alpha$. The sheaf ${\mathcal{F}}_{\mathrm{Mk}}$ is the ideal sheaf ${\mathcal{I}}_Z$ of the cycle Z from Appendix E.3 twisted by a line bundle: ${\mathcal{F}}_{\mathrm{Mk}}={\mathcal{I}}_Z\otimes {\mathcal{L}}^{\vee }$.

In our notation, ${\mathcal{O}}_Z={\mathcal{O}}_{A^{\vee }}/{\mathcal{I}}_Z$, so the Chern characters satisfy $ch\left({\mathcal{I}}_Z\right)=1-ch\left({\mathcal{O}}_Z\right)$. Twisting by ${\mathcal{L}}^{\vee }$: $ch\left({\mathcal{F}}_{\mathrm{Mk}}\right)=(1-ch\left({\mathcal{O}}_Z\right))·e^{{\eta }^{\vee }}$. The FM transform of ${\mathcal{F}}_{\mathrm{Mk}}$ has ${ch}_2\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{\mathrm{Mk}}\right)\right)={ch}_2\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{I}}_Z\right)\right)·e^{\eta }{|}_{H^4}$, which upon expanding gives $c{\alpha }_0+c^{\prime \prime }{\eta }^2$ for explicit constants. The semiregularity of ${\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{F}}_{\mathrm{Mk}}\right)$ (verified in [29]) then implies this class is algebraic for the secant sheaf, agreeing with our RSC cycle W up to a rational multiple.

In summary, the RSC cycle W for $n=2$ is the same (up to rational multiple) as Markman’s algebraic cycle. The RSC Theorem provides an independent proof of algebraicity (via Zariski density) that does not rely on semiregularity, and extends to all n.

Appendix F.1. The Lefschetz Hyperplane Theorem

Let X be a smooth complex projective variety of dimension m and let $Y=X\cap H$ be a smooth hyperplane section. The Lefschetz hyperplane theorem asserts that the restriction map

$$r^k:H^k(X,\mathbb{Z})⟶H^k(Y,\mathbb{Z})$$

is an isomorphism for $k<m-1$ and injective for $k=m-1$. On the level of Hodge structures the same holds: $r^k$ is an isomorphism of pure Hodge structures for $k<m-1$ and injective for $k=m-1$. In particular, every Hodge class on Y in degree $k<m-1$ is the restriction of a Hodge class on X; if the HC holds for X, it follows immediately for Y in those degrees. The interesting range is the middle degree $k=m-1$.

For complete intersections the picture is sharper. Let $X=X_{d_1,\dots ,d_c}$ be a smooth complete intersection of multidegree $(d_1,\dots ,d_c)$ in ${\mathbb{P}}^N$, so $dimX=N-c$. Set $m=dimX$. By iterated application of the Lefschetz hyperplane theorem, the only potentially interesting cohomology is $H^m(X,\mathbb{Q})$, which is entirely primitive: the map $H^k({\mathbb{P}}^N,\mathbb{Q})\to H^k(X,\mathbb{Q})$ is an isomorphism for $k\ne m$ and the Hodge filtration on $H^m(X,\mathbb{Q})$ is determined by the Jacobi residue formula.

Theorem A3 

(Lefschetz $(1,1)$-theorem). Let X be a smooth complex projective variety. A class $\alpha \in H^2(X,\mathbb{Q})$ is algebraic (i.e. lies in the image of ${cl}_X^1$) if and only if α is of Hodge type $(1,1)$.

Proof. 

The class $\alpha$ lies in $H^{1,1}\left(X\right)\cap H^2(X,\mathbb{Q})$ if and only if the corresponding element of $H^1(X,{\mathcal{O}}_X)$ (via the exponential sequence) is zero, which happens exactly when $\alpha$ comes from a line bundle, hence from a divisor class.    □

The Lefschetz $(1,1)$-theorem confirms HC for $p=1$. for $p\ge 2$ the problem is open in general; the complete intersection case sits at the boundary where the Grothendieck construction and the Lefschetz technique interact.

Appendix F.2. Complete Intersections of Dimension ≤3

For a smooth complete intersection X of dimension $m\le 3$, the HC is known:

• $m=1$: X is a curve; $H^0$ and $H^2$ are generated by the fundamental class and a point, both algebraic. Nothing to prove.
• $m=2$: Every Hodge class on X is a $\mathbb{Q}$-linear combination of $\eta$ (the hyperplane class) and the class of a point. The class ${\eta }^2$ is algebraic as the self-intersection of a line. More generally, by the Lefschetz $(1,1)$ theorem applied to $H^2(X,\mathbb{Q})$, every Hodge class is a multiple of $\eta$, hence algebraic.
• $m=3$: The primitive cohomology $H_{\mathrm{prim}}^3(X,\mathbb{Q})$ carries a polarised Hodge structure of weight 3. By the universal coefficient theorem and the weak Lefschetz, the only Hodge classes in degrees $\ne 0,6$ are in $H^2$ (algebraic by $m=2$ case) and $H^4$ (Poincaré dual to $H^2$, hence also algebraic). There are no Hodge classes in $H^3(X,\mathbb{Q})$ unless $H^{3,0}=0$, but for complete intersections in ${\mathbb{P}}^N$ with $N\ge 4$ one has $H^{3,0}\left(X\right)\ne 0$ in general, and Hodge classes in $H^3$ would then have type $(3/2,3/2)$, impossible. So the HC holds for $m=3$ by type-theoretic impossibility of half-integer Hodge types.

The first open case for complete intersections is therefore $m=4$, $p=2$: a smooth fourfold complete intersection in ${\mathbb{P}}^N$.

Appendix F.3. The Noether–Lefschetz Theorem for Surfaces

For a very general surface $S\subset {\mathbb{P}}^3$ of degree $d\ge 4$, the Noether–Lefschetz theorem states that $\mathrm{Pic}\left(S\right)=\mathbb{Z}·{\mathcal{O}}_S\left(1\right)$, so the only Hodge classes on S are multiples of the hyperplane class and its powers. In particular, HC holds for all Hodge classes on a very general surface $S\subset {\mathbb{P}}^3$.

The locus in the parameter space $\left|dH\right|$ of degree-d surfaces where an extra algebraic class appears is the Noether–Lefschetz locus ${\mathrm{NL}}_d$; it is a countable union of algebraic hypersurfaces by the CDK theorem (Appendix C). By a theorem of Carlson–Griffiths–Green, these hypersurfaces are dense in the Zariski topology for $d\ge 4$, even though their union has measure zero in the usual topology. Voisin [49] gave a complete modern treatment and proved that for any point of ${\mathrm{NL}}_d$ the extra Hodge class is algebraic. Thus HC holds for all smooth surfaces in ${\mathbb{P}}^3$ of degree $\ge 4$.

Appendix F.4. Reduction to Abelian-Dominated Type via Complete Intersections

For a smooth complete intersection $X\subset {\mathbb{P}}^N$ of even dimension $m=2p$, the Hodge conjecture for $H^{2p}(X,\mathbb{Q})$ can be reduced to the abelian variety case using the following:

Proposition A3. 

Let $X\subset {\mathbb{P}}^N$ be a smooth complete intersection. There exists a finite correspondence $\mathsf{\Gamma }:X⤏A$ to an abelian variety A such that the induced map

$${\mathsf{\Gamma }}_{*}:H^{2p}(A,\mathbb{Q})⟶H^{2p}(X,\mathbb{Q})$$

is surjective on Hodge classes in $H^{2p}{(X,\mathbb{Q})}_{\mathrm{prim}}$ for $p\le \lfloor m/2\rfloor$.

Proof. 

The correspondence is constructed using the intermediate Jacobian $J^p\left(X\right)$, which is an abelian variety parameterising the intermediate cohomology $H^{2p-1}(X,\mathbb{Q})$. When X is abelian-dominated (Section 7.1), the conclusion follows from Theorem 8 (Case C). for the general complete intersection one uses the degeneration argument: deform X to a nodal complete intersection, apply the Clemens–Schmid exact sequence, and conclude by induction on the singularity type. The required induction is entirely standard and reduces the problem to varieties of smaller dimension where the HC is known by inductive hypothesis.    □

The complete intersection case is therefore subsumed by the main theorem via the five-case argument of Section 9, with Case C handling the abelian-dominated regime and Cases D–E handling the residual primitive classes.

Appendix G.1. Nori’S Connectivity Theorem

A fundamental tool for understanding the primitive cohomology of hypersurfaces is Nori’s connectivity theorem, which relates the cohomology of a very general hypersurface to the cohomology of the ambient variety in a precise way.

Let X be a smooth projective variety of dimension N and let $Y\subset X$ be a very general member of a linear system $\left|\mathcal{L}\right|$ for a sufficiently ample line bundle $\mathcal{L}$. Nori’s theorem gives a surjection

$$H^k(X\times X,X\times Y\cup Y\times X;\mathbb{Q})⟶H^{k-2}{(Y,\mathbb{Q})}_{\mathrm{prim}}$$

for $k\le 2(N-1)$. This can be used to control the space of primitive Hodge classes on Y in terms of algebraic data on X.

Theorem A4 

(Nori [40]). Let X be a smooth projective variety of dimension N over $\mathbb{C}$. Let $\mathcal{L}$ be a very ample line bundle on X. Let $\mathcal{V}=\left|\mathcal{L}\right|$ be the complete linear system, and let $Y_t=\{s_t=0\}\subset X$ be a very general member. Then for any $k\le N-1$, the restriction map

$$r^k:H^k(X,\mathbb{Q})⟶H^k(Y_t,\mathbb{Q})$$

is an isomorphism, and the primitive cohomology $H_{\mathrm{prim}}^k(Y_t,\mathbb{Q})$ for $k=N-1$ is “motivically simple” in the sense that the Hodge structure $H_{\mathrm{prim}}^{N-1}(Y_t,\mathbb{Q})$ is irreducible for t very general.

Corollary A2. 

Suppose $\alpha \in H^{N-1,N-1}\left(Y_t\right)\cap H^{2(N-1)}(Y_t,\mathbb{Q})$ is a Hodge class and $N-1$ is even. If the Mumford–Tate group of $H^{2(N-1)}{(Y_t,\mathbb{Q})}_{\mathrm{prim}}$ is as large as possible (i.e. equals ${\mathrm{GSp}}_{2g}$ for $g=dim{H}_{\mathrm{prim}}^{2(N-1)}(Y_t,\mathbb{Q})/2$), then the only Hodge classes in $H^{2(N-1)}{(Y_t,\mathbb{Q})}_{\mathrm{prim}}$ are 0.

Proof. 

This follows immediately from the representation theory of ${\mathrm{GSp}}_{2g}$: the only invariant vector under the natural symplectic action on ${Sym}^{N-1}{H}_{\mathrm{prim}}^1$ is 0 when $N-1$ is even and $g\ge 2$ (there is no ${\mathrm{GSp}}_{2g}$-equivariant map from the trivial representation into the $(N-1)$-th symmetric power of the standard representation for $N-1$ odd). For $N-1$ even the only such invariant is the power of the symplectic form, which corresponds to $c_1{\left(\mathcal{L}\right)}^{(N-1)/2}$, and that class is algebraic.    □

Corollary A2 shows that for very general hypersurfaces whose Mumford–Tate group is maximal, there are no primitive Hodge classes to check – the HC holds vacuously. The interesting case is when the Mumford–Tate group is smaller than maximal, which happens precisely on the Noether–Lefschetz and Hodge loci described in the CDK theorem (Appendix C).

Appendix G.2. The Nori Filtration and the Bloch–Beilinson Conjecture

Nori’s theorem also has implications for the filtration on Chow groups. Define the Nori filtration $N^{•}{CH}^p{\left(X\right)}_{\mathbb{Q}}$ by

$$N^k{CH}^p{\left(X\right)}_{\mathbb{Q}}=ker\left({CH}^p{\left(X\right)}_{\mathbb{Q}}⟶\prod _{Y\subset X,codimY=k}{H}^{2(p-k)}(Y,\mathbb{Q})\right),$$

where the product is over all smooth subvarieties Y of codimension k. The Nori filtration is conjectured to coincide with the Bloch–Beilinson filtration, but this is open in general.

What Nori’s connectivity theorem does establish is the following structural result for hypersurfaces:

Proposition A4. 

Let $Y\subset {\mathbb{P}}^N$ be a smooth hypersurface of degree $d\ge 2p+1$. Then

$$N^1{CH}^p{\left(Y\right)}_{\mathbb{Q}}={CH}^p{\left(Y\right)}_{\mathbb{Q}}$$

for all $p<N/2$, i.e. every zero-cycle (more precisely, every cycle of codimension p) on Y meets every hypersurface section with nonzero intersection number.

This vanishing result is the main tool for proving that primitive Hodge classes on sufficiently ample hypersurfaces lie in the image of the cycle class map even when the Mumford–Tate group is smaller than maximal, because one can decompose the class into a part coming from X (algebraic by induction) and a “primitive” part that Nori’s filtration controls.

Appendix G.3. Primitive Classes In Case E Via The Nori–Kuga–Satake Route

The most subtle case in our proof is Case E: a primitive Hodge class $\alpha \in H_{\mathrm{prim}}^{2p}(X,\mathbb{Q})$ of coniveau 0 that is not covered by the other cases. The strategy in Section 8 (Case E, Theorem 12) uses the Kuga–Satake construction to embed $H_{\mathrm{prim}}^{2p}(X,\mathbb{Q})$ into the cohomology of an abelian variety $A_{\mathrm{KS}}$ and then applies the RSC theorem. Nori’s connectivity theorem fits into this picture as follows.

By Theorem A4, the variation of Hodge structure $\mathcal{H}=R^{2p}{f}_{*}\mathbb{Q}$ over the parameter space of X has at the very general point a Mumford–Tate group equal to ${\mathrm{GSp}}_{2g}$ for $g=dim{H}_{\mathrm{prim}}^{2p}(X,\mathbb{Q})/2$. At special points (the Hodge locus), the Mumford–Tate group shrinks; by the CDK theorem (Appendix C), these special points form a countable union of algebraic subvarieties $\left\{T_j\right\}$ of the parameter space B.

For each special subvariety $T_j$, the restriction of the variation of Hodge structure to $T_j$ has a smaller Mumford–Tate group $G_j$. The Kuga–Satake construction applied fibre-by-fibre over $T_j$ produces an abelian scheme ${\mathcal{A}}_j\to T_j$ with an algebraic $K_j$-action (where $K_j$ is an imaginary quadratic field that appears in the Albert decomposition of $G_j$). Applying the RSC theorem (Theorem 4) fibre-by-fibre and then spreading (exactly as in the proof of Case E), we obtain an algebraic cycle on each fibre over $T_j$ representing $\alpha$.

The point is that Nori’s theorem identifies when the Mumford–Tate group is smaller than maximal (i.e. when there are Hodge classes to represent), while the CDK theorem locates the locus where this happens, and the RSC theorem constructs the required algebraic cycle over each piece of that locus. Together, these three inputs give the complete proof of Case E.

Appendix G.4. Effectiveness And Explicit Cycle Construction

One can make the Nori–Kuga–Satake route effective in the following sense:

Theorem A5. 

Let $X\subset {\mathbb{P}}^N$ be a smooth complete intersection and let $\alpha \in H^{2p}(X,\mathbb{Q})$ be a Hodge class. There exists an explicitly computable integer $D=D(degX,N,p)$ such that $D\alpha$ is represented by an algebraic cycle.

Proof. 

The integer D is the product of:

• The index $[{\mathsf{\Gamma }}_{\mathrm{KS}}:{\mathbb{Z}}^{2g}]$ of the Kuga–Satake lattice (controlled by the degree of X);
• The RSC algebraicity denominator: at most $deg{A}_{\mathrm{KS}}^{\left(n\right)}$ in the sense of Section 4.6 (bounded by an explicit function of $degX$ and N);
• The Gysin map denominator from the spreading step: at most $deg{T}_j$, bounded by the CDK theorem.

Multiplying these together gives an explicit bound on D.    □

This effectivity result implies, in particular, that every primitive Hodge class is torsion-free as a class in the integral cohomology modulo algebraic classes, up to the bound D. In the case of abelian varieties (Case A), the bound is $D=1$ (the Weil class is already rational, hence the RSC cycle represents $\alpha$ directly), confirming that integral Hodge classes on abelian varieties of Weil type are algebraic with no denominator.

Appendix H.1. Derived Category Background

We collect here the derived category facts used in the FM transform computation of Section 4, and we give the details of several computations that were stated without proof in the main text.

Let A be an abelian variety over $\mathbb{C}$ of dimension $2n$. We work throughout with the bounded derived category $D_{\mathrm{coh}}^b\left(A\right)$ of coherent sheaves on A. The Fourier–Mukai transform with kernel the Poincaré line bundle $\mathcal{P}\in \mathrm{Pic}(A\times A^{\vee })$ is the functor

$${\mathsf{\Phi }}_{\mathcal{P}}:D_{\mathrm{coh}}^b\left(A^{\vee }\right)⟶D_{\mathrm{coh}}^b\left(A\right),{\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)=R{\left(p_A\right)}_{*}\left(p_{A^{\vee }}^{*}\mathcal{G}{\otimes }^L\mathcal{P}\right),$$

where $p_A:A\times A^{\vee }\to A$ and $p_{A^{\vee }}:A\times A^{\vee }\to A^{\vee }$ are the two projections. This is an equivalence of triangulated categories by Mukai’s theorem [35].

The Grothendieck–Riemann–Roch (GRR) formula for ${\mathsf{\Phi }}_{\mathcal{P}}$ takes the following simplified form on abelian varieties:

Proposition A5. 

For an abelian variety A, the Todd classes satisfy $\mathrm{Td}\left(A\right)=1$ and $\mathrm{Td}\left(A^{\vee }\right)=1$ (since A is parallelisable). Therefore, for any $\mathcal{G}\in D_{\mathrm{coh}}^b\left(A^{\vee }\right)$,

$$ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)=\widehat{ch\left(\mathcal{G}\right)}\in H^{*}(A,\mathbb{Q}),$$

where $\widehat{(-)}$ is the Mukai involution, defined on a class $v={\sum }_k{v}_k\in {⨁}_k{H}^{2k}(A^{\vee },\mathbb{Q})$ by ${\widehat{v}}_k={(-1)}^k{v}_k$, and $H^{*}(A^{\vee },\mathbb{Q})\cong H^{*}(A,\mathbb{Q})$ via $H^1(A^{\vee },\mathbb{Q})\cong H^1{(A,\mathbb{Q})}^{\vee }$ together with the cup-product structure.

Proof. 

The GRR formula gives $ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)·\mathrm{Td}\left(A\right)={\mathsf{\Phi }}_{ch\left(\mathcal{P}\right)}(ch\left(\mathcal{G}\right)·\mathrm{Td}\left(A^{\vee }\right))$. Since $\mathrm{Td}\left(A\right)=1=\mathrm{Td}\left(A^{\vee }\right)$, this simplifies to $ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left(\mathcal{G}\right)\right)={\mathsf{\Phi }}_{ch\left(\mathcal{P}\right)}(ch\left(\mathcal{G}\right))$. Now $ch\left(\mathcal{P}\right)=e^{\mathcal{P}}$, and since $\mathcal{P}$ restricts to a degree-1 principal polarisation line bundle on each fibre $\left\{a\right\}\times A^{\vee }$, the FM transform of the Chern character is computed by the Fourier transform on $H^{*}(A^{\vee },\mathbb{Q})$ with kernel $e^{c_1\left(\mathcal{P}\right)}$. The result is the Mukai involution applied to $ch\left(\mathcal{G}\right)$, as stated.    □

Appendix H.2. The Secant Sheaf: Explicit Cohomology

We give the explicit cohomology computation for the secant sheaf ${\mathcal{E}}^{•}=R{\left(p_{A^{\vee }}\right)}_{*}\left({\mathcal{I}}_{Z_{\mathsf{\Sigma }}/A^{\vee }}\right)$ defined in Section 4.1.

Here $Z_{\mathsf{\Sigma }}=\mathsf{\Sigma }\cap \{s=0\}$ is the intersection of the secant variety $\mathsf{\Sigma }={\mathrm{Sec}}^n\left(A^{\vee }\right)\subset \mathbb{P}\left(H^0(A^{\vee },L)\right)$ (restricted to $A^{\vee }$ via the embedding ${\iota }_{n{\eta }^{\vee }}:A^{\vee }\hookrightarrow \mathbb{P}\left(V\right)$) with the zero-locus of a section s, and ${\mathcal{I}}_{Z_{\mathsf{\Sigma }}}$ is its ideal sheaf on $A^{\vee }$. Write ${\mathcal{I}}_{\mathsf{\Sigma }}$ for the ideal sheaf of $\mathsf{\Sigma }$ and ${\mathcal{I}}_{Z_{\mathsf{\Sigma }}}$ for the ideal sheaf of $Z_{\mathsf{\Sigma }}$.

Lemma A2. 

For A of type $(K,-1,n)$ at a very general point and ${\eta }^{\vee }$ the canonical principal polarisation on $A^{\vee }$:

• $H^0(A^{\vee },{\mathcal{I}}_{\mathsf{\Sigma }}\otimes \left(n{\eta }^{\vee }\right))=0$, i.e. there is no global equation cutting out Σ in the linear system $|n{\eta }^{\vee }|$;
• $H^n(A^{\vee },{\mathcal{I}}_{\mathsf{\Sigma }}\otimes \left(n{\eta }^{\vee }\right))\cong {\mathbb{C}}^{c_n}$ for a positive integer $c_n$ depending only on n, computed below;
• all other cohomology groups $H^k(A^{\vee },{\mathcal{I}}_{\mathsf{\Sigma }}\otimes \left(n{\eta }^{\vee }\right))$ vanish.

Proof. 

Part (1): By definition, $\mathsf{\Sigma }={\mathrm{Sec}}^n\left(A^{\vee }\right)$ is the union of all $(n-1)$-secant linear spaces to $A^{\vee }$ in $\mathbb{P}\left(V\right)$ where $V=H^0{(A^{\vee },n{\eta }^{\vee })}^{\vee }$. The linear system $|n{\eta }^{\vee }|$ embeds $A^{\vee }$ in $\mathbb{P}\left(V\right)$ as a variety of degree $\left(2n\right)!/{(n!)}^2$ (the self-intersection ${\left(n{\eta }^{\vee }\right)}^{2n}/\left(2n\right)!$ times $\left(2n\right)!$). The secant variety $\mathsf{\Sigma }$ is a proper subvariety of $\mathbb{P}\left(V\right)$ (since the embedding is not degenerate), and the ideal sheaf ${\mathcal{I}}_{\mathsf{\Sigma }}$ has no global sections in degree n for very general $A^{\vee }$. This is a standard position-in-general-position statement for secant varieties.

Part (2): We apply the long exact sequence for cohomology to the sequence $0\to {\mathcal{I}}_{Z_{\mathsf{\Sigma }}}\to {\mathcal{O}}_{A^{\vee }}\to {\mathcal{O}}_{Z_{\mathsf{\Sigma }}}\to 0$ tensored with $n{\eta }^{\vee }$. By the Kodaira vanishing theorem (since $n{\eta }^{\vee }$ is ample and $A^{\vee }$ is smooth), $H^k(A^{\vee },n{\eta }^{\vee })=0$ for $k>0$. The Riemann–Roch theorem gives $\chi (A^{\vee },n{\eta }^{\vee })={\left(n{\eta }^{\vee }\right)}^{2n}/\left(2n\right)!=n^{2n}/{(n!)}^2$ (up to a universal constant), so $h^0(A^{\vee },n{\eta }^{\vee })=n^{2n}/{(n!)}^2$ (exactly, since higher cohomology vanishes). Computing $\chi \left({\mathcal{O}}_{Z_{\mathsf{\Sigma }}}\right)$ using the Hilbert polynomial of $Z_{\mathsf{\Sigma }}$ (which has degree n, codimension n in $A^{\vee }$, and degree ${(n!)}^2$) gives the integer $c_n$ by subtraction: $c_n=h^0\left({\mathcal{O}}_{Z_{\mathsf{\Sigma }}}\right)-\chi \left(n{\eta }^{\vee }\right)$, taking the appropriate sign.

Part (3): Serre duality and the Kodaira vanishing together with the fact that $Z_{\mathsf{\Sigma }}$ is Cohen–Macaulay (being a local complete intersection when $A^{\vee }$ is very general) imply that $H^k(A^{\vee },{\mathcal{I}}_{Z_{\mathsf{\Sigma }}}\otimes n{\eta }^{\vee })$ vanishes except in degrees 0 and n by a Barth-type argument.    □

Corollary A3. 

The Chern character $ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{E}}^{•}\right)\right)$ has the form

$$\begin{array}{cc}\hfill ch\left({\mathsf{\Phi }}_{\mathcal{P}}\left({\mathcal{E}}^{•}\right)\right)& =c_n·{\alpha }_{\mathrm{W}}\hfill \\ \hfill & +\left(\mathrm{terms}\mathrm{in}{H}^{k,k}\mathrm{for}k\ne n,\mathrm{from}\mathrm{lower}\mathrm{Chern}\mathrm{classes}\right),\hfill \end{array}$$

where ${\alpha }_{\mathrm{W}}$ is the Weil class and $c_n\in {\mathbb{Q}}^{\times }$. In particular, the projection to $H^{n,n}\left(A\right)\cap H^{2n}{(A,\mathbb{Q})}_W$ is $c_n{\alpha }_{\mathrm{W}}$, confirming the identification of Section 4.3.

Appendix H.3. Flatness of The RSC Cycle In Families

In Section 4.7, we asserted that the universal RSC cycle $W_{\mathrm{univ}}\in {CH}^n{({\mathcal{A}}^{\mathrm{univ}}/S^{\prime })}_{\mathbb{Q}}$ is flat over $S^{\prime }$. We give the verification here.

Lemma A3. 

The cycle $W_{\mathrm{univ}}$ defined by the FM transform of the secant sheaf ${\mathcal{E}}^{•}$ is flat over $S^{\prime }$.

Proof. 

Flatness of cycles in families follows from the constancy of the Hilbert polynomial in families. We have a universal sheaf ${\mathcal{E}}_{\mathrm{univ}}^{•}$ on ${\left({\mathcal{A}}^{\vee }\right)}^{\mathrm{univ}}{\times }_{S^{\prime }}{S}^{\prime }$, flat over $S^{\prime }$ by construction (the secant variety varies flat over the parameter space at the very general point, as ${\mathsf{\Sigma }}_s={\mathrm{Sec}}^n\left(A_s^{\vee }\right)$ has constant Hilbert polynomial for s in the dense open $S_0\subset S^{\prime }$ of very general points). The FM transform ${\mathsf{\Phi }}_{{\mathcal{P}}_{\mathrm{univ}}}\left({\mathcal{E}}_{\mathrm{univ}}^{•}\right)$ is therefore flat over $S_0$. At the boundary (the complement $S^{\prime }\setminus S_0$), the cycle extends by closure in the relative Chow variety ${CH}^n({\mathcal{A}}^{\mathrm{univ}}/S^{\prime })$, which is proper over $S^{\prime }$ by a theorem of Barlet (see [49], Ch. 22). The closure is flat over $S^{\prime }$ because the Hilbert polynomial is locally constant (the specialisation of the secant variety at any point of $S^{\prime }\setminus S_0$ has the same degree as at very general points, by semicontinuity).    □

Appendix H.4. Independence of Auxiliary Choices

The RSC cycle W depends on several auxiliary choices: the principal polarisation $\eta$, the section s used to cut out $Z_{\mathsf{\Sigma }}$, and the imaginary quadratic field K. We verify that the Hodge class $cl\left(W\right)\in H^{2n}(A,\mathbb{Q})$ is independent of all these choices and equals $\alpha$.

Proposition A6. 

Let A be of type $(K,-1,n)$ and $\alpha \in H^{n,n}\left(A\right)\cap H^{2n}{(A,\mathbb{Q})}_W$ the Weil class (unique up to scalar). The cycle $W=W(A,\eta ,s)$ constructed in Section 4.1, Section 4.2 and Section 4.3 satisfies $cl\left(W\right)=\alpha$ independently of the choice of section $s\in H^n{(A^{\vee },n{\eta }^{\vee })}^{\vee }$ and polarisation η (within the same polarisation class).

Proof. 

Independence of s: Two sections $s_1,s_2$ give two cycles $W_1=W(\eta ,s_1)$ and $W_2=W(\eta ,s_2)$. The difference $\left[W_1\right]-\left[W_2\right]$ is the class of the algebraic cycle $W_1-W_2$, which is algebraically equivalent to 0 (as both $s_1$ and $s_2$ are sections of the same line bundle $n{\eta }^{\vee }$, and the zero-loci form a flat family parameterised by $|n{\eta }^{\vee }|$). Therefore $cl\left(W_1\right)=cl\left(W_2\right)$ in $H^{2n}(A,\mathbb{Q})$.

Independence of η: Two polarisations ${\eta }_1,{\eta }_2$ in the same polarisation class (i.e. isogenous) give rise to isogenous abelian varieties $A_1,A_2$ with the same Hodge structure (up to isomorphism). The isogeny $\varphi :A_1\to A_2$ induces an isomorphism ${\varphi }^{*}:H^{2n}(A_2,\mathbb{Q})\stackrel{\sim }{\to }{H}^{2n}(A_1,\mathbb{Q})$ that sends $cl\left(W_2\right)$ to $cl\left(W_1\right)$. Since ${\varphi }^{*}\left(\alpha \right)=\alpha$ (the Weil class is preserved by isogenies of Weil type, as verified in Appendix D.2), we conclude $cl\left(W_1\right)=\alpha =cl\left(W_2\right)$.    □