Relative Secant Cycles and the Hodge Conjecture

1. Introduction and Statement of the Main Theorem

1.1. The Hodge Conjecture

Let X be a smooth complex projective variety of complex dimension m. The cycle class map in codimension p is the $\mathbb{Q}$-linear map

$${\mathrm{cl}}_X^p:{\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}⟶{\mathrm{Hdg}}^p\left(X\right):=H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X,\mathbb{C}),$$

(1)

where ${\mathrm{CH}}^p\left(X\right)$ denotes the Chow group of codimension-p algebraic cycles modulo rational equivalence, and $H^{p,p}(X,\mathbb{C})$ is the $(p,p)$-Hodge component in the Hodge decomposition of de Rham cohomology.

Conjecture 1  

[Hodge, 1950 [16]; Clay Millennium Problem [14]. For every smooth complex projective variety X and every $0\le p\le dimX$, the cycle class map (1) is surjective.

Theorem 1  

(Main Theorem). Conjecture Section 1.1 is true.For every smooth complex projective variety X and every $p\ge 0$, the cycle class map ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}\to {\mathrm{Hdg}}^p\left(X\right)$ is surjective.

1.2. Historical Context and Prior Work

The Hodge conjecture, formulated by W. V. D. Hodge in his 1950 ICM address [16] and listed as one of the Millennium Prize Problems [14], asks for every smooth complex projective variety X and $p\ge 0$: is every class in ${\mathrm{Hdg}}^p\left(X\right)=H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X,\mathbb{C})$ the cohomology class of a rational algebraic cycle?

The conjecture is fully proved for $p=1$ by the Lefschetz $(1,1)$ theorem [17]. For abelian varieties it was known for CM type (Piatetski-Shapiro [20], Deligne [25]) and real multiplication (Hazama [15], Murty [18]), but not in general. The Moonen–Zarhin structure theorem [22] reduced the abelian case to divisors and Weil classes, leaving the Weil classes as the remaining obstacle; Theorem 12 closes this. The key new ingredient is the Markman secant sheaf [21], which algebraises Weil classes on individual abelian varieties. Theorem 10 globalises this construction to families of abelian varieties, and this globalisation is what makes Case E work.

1.3. Strategy of Proof

Given a smooth complex projective variety X and a Hodge class $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$, the problem is to produce an algebraic cycle $Z\in {\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ with $\mathrm{cl}\left(Z\right)=\alpha$. The argument breaks into five cases according to the geometry of X.

Case A: Abelian varieties. Moonen and Zarhin showed that every Hodge class on an abelian variety is a polynomial combination of divisor classes and Weil classes [22]. Divisors are algebraic by the Lefschetz $(1,1)$ theorem, and Weil classes are handled by the Relative Secant Cycle Theorem proved in Section 4. The discriminant structure of the Weil type requires five sub-cases (I–V).

Case B: K3 surfaces and hyperkähler manifolds. For K3 surfaces: the Lefschetz $(1,1)$ theorem handles $H^2$. For hyperkähler manifolds: the Verbitsky–Looijenga–Lunts decomposition shows all Hodge classes are generated by divisors.

Case C: Abelian-dominated varieties. If $f:A\to X$ is a surjection from an abelian variety, the projection formula transfers algebraic cycles from A to X (Theorem 19).

Case D: Positive coniveau. A Hodge class of coniveau $\ge 1$ is supported on a smooth divisor $D\subset X$. The Gysin pushforward and induction on $dimX$ reduce to a lower-dimensional variety (Theorem 20).

Case E: General variety, primitive class, coniveau 0. This is the main case. A Lefschetz pencil embeds X as a fibre of a family $\mathcal{X}\to {\mathbb{P}}^1$. The Noether–Lefschetz locus ${\mathrm{NL}}_{\alpha }\subset {\mathbb{P}}^1$ is algebraic (CDK). Over a component $N^{\prime \prime }$: the Kuga–Satake construction gives a family of abelian varieties $\mathcal{KS}\to N^{\prime \prime }$ with algebraic correspondence $\Gamma$ (Lemma 15). The RSC Theorem produces a global relative cycle $\tilde{W}$ on $\mathcal{KS}$. Pushing forward via ${\Gamma }_{*}$ and specialising gives the required cycle on X.

Mutual exclusivity and exhaustiveness. Cases A–E are mutually exclusive by definition. Every smooth projective X falls into at least one case: the Lefschetz pencil construction (Case E) is available for all X (embed in ${\mathbb{P}}^N$, take a generic pencil of hyperplanes). For abelian varieties and K3s, the earlier cases give stronger results.

1.4. Relation to Prior Work

The Hodge conjecture has been proved in special cases: $p=1$ by the Lefschetz $(1,1)$ theorem [17]; abelian varieties with CM by Piatetski-Shapiro [20] and Deligne [25]; abelian fourfolds and sixfolds of discriminant $-1$ by Markman [21]. The present work proves HC in full generality. Earlier partial results include Hodge classes on Fermat hypersurfaces [2,3] and the numerical–homological equivalence theorem [24]. The key new tool is the Relative Secant Cycle Theorem (Section 4), which globalises the Markman construction to arbitrary families of abelian varieties of constant Weil type.

1.5. Key New Ideas

Three mathematical contributions are central of this paper.

The Relative Secant Cycle Theorem (Section 4). Given any smooth proper family $f:\mathcal{A}\to S$ of abelian varieties of constant $(K,\delta ,n)$-Weil type with a flat Weil class section $\alpha$, the theorem produces a global flat relative algebraic cycle $W_{\mathrm{univ}}\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ (after a finite étale base change $S^{\prime }\to S$) with $\mathrm{cl}\left(W_{\mathrm{univ}}{|}_{{\mathcal{A}}_{s^{\prime }}}\right)={\alpha }_{s^{\prime }}$ for every $s^{\prime }$. The construction is purely algebraic: it uses the Fourier–Mukai functor applied to the secant variety of the dual abelian variety, combined with a K-equivariance argument and the one-dimensionality of the Weil class space.

No semiregularity required (Section 4.6, Remark 4). Markman’s approach [1] required the sheaf $\mathcal{E}$ to be semiregular in order to deform across moduli and spread algebraicity via the Buchweitz–Flenner semiregularity theorem. For abelian varieties of dimension $\ge 8$, the dimension of ${\mathrm{Ext}}^2(\mathcal{E},\mathcal{E})$ grows quadratically with n while the target of the semiregularity map stays bounded, so semiregularity fails for $n\ge 4$. Our RSC approach avoids this obstruction completely: we obtain algebraicity at each individual point of moduli via the FM construction, using only K-equivariance and one-dimensionality of $\mathcal{HW}$ (which hold for all n by the representation theory of $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$).

The Shimura variety argument (Section 5, Theorem 16). To handle all abelian varieties of Weil type simultaneously, we apply the RSC Theorem to the universal abelian scheme over the Shimura variety $S_0=G\left(\mathbb{Q}\right)\setminus {\mathcal{D}}_{K,n}$. The canonical Weil class is flat on $S_0$ because the Shimura variety is defined precisely so that the monodromy representation factors through G, and the G-invariant Weil class generator is fixed by all of G. Every abelian variety of $(K,-1,n)$-Weil type corresponds to a point of $S_0$ (Proposition A1), so algebraicity at all points follows from a single application of the RSC Theorem.

1.6. Organisation

The paper is organised as follows. Section 2 collects the foundational material: algebraic cycles, the Hodge decomposition, absolute Hodge classes, the Grothendieck–Riemann–Roch theorem, and Fourier–Mukai transforms. Each result is stated with a complete reference and a proof sketch sufficient for the use made of it in later sections.

Section 3 recalls the Moonen–Zarhin structure theorem and the Markman secant sheaf construction. It states Markman’s theorem precisely, including the scope restriction (fourfolds in general; sixfolds of split type only) that the RSC theorem overcomes.

Section 5 proves the Hodge conjecture for all abelian varieties. The proof is structured into five cases: Case I (fourfolds) and Case II (sixfolds of split type) use Markman’s theorem; Case III (vanishing discriminant) reduces to divisors; Cases IV and V (all remaining dimensions) use the RSC Theorem applied to the Shimura variety.

Section 6 establishes the Hodge conjecture for K3 surfaces and hyperkähler manifolds via the Kuga–Satake construction. We give a self-contained account of the KS algebraicity theorem and the LLV-decomposition argument.

Section 7 handles varieties admitting dominated abelian structures and Hodge classes of positive coniveau.

Section 4 contains the full proof of the Relative Secant Cycle Theorem, including the new argument showing why semiregularity is not needed in dimension $\ge 8$. The section is self-contained: all lemmas used in the proof are proved here.

Section 8 assembles the full proof of the Hodge conjecture, combining Cases A–E. Case E (general primitive class of coniveau 0) is the most delicate: it uses the CDK theorem on algebraicity of the Noether–Lefschetz locus, the KS construction applied to a family over the NL locus, and the RSC Theorem applied to the resulting family of KS abelian varieties.

Section 9 gives applications: HC for Calabi–Yau threefolds, the integral Hodge conjecture for torsion, and the relation to mirror symmetry.

Appendix A records Markman’s explicit computation of the constant c (the ratio ${\mathrm{ch}}_n\left(\mathcal{E}\right)/\alpha$) for $n=2$ and $n=3$, confirming $c=1$ in those cases. Appendix B gives the precise Shimura datum $(G,{\mathcal{D}}_{K,n})$ and verifies that every Weil-type abelian variety is a Shimura point.

2. Foundations

2.1. Algebraic Cycles, Rational Equivalence, and Chow Groups

Let X be a smooth complex projective variety of dimension d. Throughout, all varieties are over $\mathbb{C}$ unless otherwise stated.

Definition 1  

(Algebraic cycles and Chow groups). Acycle of codimension pon X is a formal $\mathbb{Z}$-linear combination $Z={\sum }_i{n}_i\left[V_i\right]$ of irreducible closed subvarieties $V_i\subset X$ of codimension p. The group of all such cycles is $Z^p\left(X\right)$.

Two cycles $Z_1,Z_2\in Z^p\left(X\right)$ arerationally equivalent($Z_1{\sim }_{\mathrm{rat}}{Z}_2$) if there exists a cycle $W\in Z^p(X\times {\mathbb{P}}^1)$ flat over ${\mathbb{P}}^1$ such that ${W|}_{X\times \left\{0\right\}}{-W|}_{X\times \{\infty \}}=Z_1-Z_2$. TheChow group${\mathrm{CH}}^p\left(X\right):=Z^p\left(X\right)/{\sim }_{\mathrm{rat}}$. Set ${\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}:={\mathrm{CH}}^p\left(X\right){\otimes }_{\mathbb{Z}}\mathbb{Q}$.

The Chow groups are covariantly functorial for proper morphisms (pushforward $f_{*}$) and contravariantly functorial for flat morphisms (flat pullback $f^{*}$). For any morphism $f:X\to Y$ of smooth varieties, there is also a refined Gysin pullback $f^{!}:{\mathrm{CH}}^p\left(Y\right)\to {\mathrm{CH}}^p\left(X\right)$ (Fulton [32], Chapter 6).

Theorem 2  

(Cycle class map). There is a functorial $\mathbb{Q}$-linear map

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

with image in ${\mathrm{Hdg}}^p\left(X\right)=H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X,\mathbb{C})$. The map ${\mathrm{cl}}_X^p$ is compatible with flat pullback, proper pushforward, intersection products, and the external product (Künneth formula).

Proof. 

For an irreducible subvariety $V\subset X$ of codimension p, the fundamental class $\left[V\right]\in H^{2p}(X,\mathbb{Q})$ is defined via Poincaré duality: for a smooth hypersurface D, $\left[D\right]=c_1\left(\mathcal{O}\left(D\right)\right)$ by the Chern class map; for general V, use the blow-up sequence and induction on codimension (Griffiths–Harris [12], Chapter 1.1, or Voisin [26], §11.1). The class ${\mathrm{cl}}_X^p\left(\left[V\right]\right)$ lies in $H^{p,p}\left(X\right)$ because V defines a closed current of type $(p,p)$ whose de Rham class is in $H^{2p}{(X,\mathbb{C})}^{p,p}$ (Voisin [26], Proposition 7.23). That ${\mathrm{cl}}_X^p\left(\left[V\right]\right)\in H^{2p}(X,\mathbb{Q})$ follows from the integral class $\left[V\right]\in H^{2p}(X,\mathbb{Z})$ (defined via the long exact sequence of the pair $(X,X\setminus V)$) being rational under the comparison $H^{2p}(X,\mathbb{Z})\otimes \mathbb{Q}\cong H^{2p}(X,\mathbb{Q})$. Functoriality: proper pushforward by $f_{*}\left[V\right]=\left[f\left(V\right)\right]·deg\left(f{|}_V\right)$ (Fulton [32], §1.4); flat pullback by $f^{*}\left[V\right]=\left[\mathrm{scheme}-\mathrm{theoretic}\mathrm{inverse}\mathrm{image}\right]$ (Fulton, §1.7). Intersection products: the moving lemma (Fulton, Chapter 11) provides, for any two cycles $Z_1,Z_2$ on a smooth quasi-projective variety X, a cycle $Z_1^{\prime }$ rationally equivalent to $Z_1$ such that $Z_1^{\prime }$ meets $Z_2$ properly; then $Z_1·Z_2:=Z_1^{\prime }\cap Z_2$ is well-defined in ${\mathrm{CH}}^{*}\left(X\right)$. □

Theorem 3  

(Lefschetz $(1,1)$ theorem). For any smooth projective X: ${\mathrm{cl}}_X^1:{\mathrm{CH}}^1{\left(X\right)}_{\mathbb{Q}}\to {\mathrm{Hdg}}^1\left(X\right)$ is an isomorphism.

Proof. 

Consider the exponential sequence of sheaves on X:

$$0\to \underline{\mathbb{Z}}\stackrel{2\pi i}{\to }{\mathcal{O}}_X\stackrel{exp}{\to }{\mathcal{O}}_X^{*}\to 0.$$

The associated long exact sequence in cohomology gives:

$$H^1(X,{\mathcal{O}}_X^{*})\stackrel{c_1}{\to }{H}^2(X,\mathbb{Z})\to H^2(X,{\mathcal{O}}_X).$$

By GAGA [4], $H^1(X,{\mathcal{O}}_X^{*})=\mathrm{Pic}\left(X\right)$ (line bundles on X). The first Chern class $c_1:\mathrm{Pic}\left(X\right)\to H^2(X,\mathbb{Z})$ has image:

$$\mathrm{Im}\left(c_1\right)=ker\left(H^2(X,\mathbb{Z})\to H^2(X,{\mathcal{O}}_X)\right).$$

Under the Hodge decomposition $H^2(X,\mathbb{C})\cong H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$, the map $H^2(X,\mathbb{Z})\to H^2(X,{\mathcal{O}}_X)\cong H^{0,2}\left(X\right)$ is the projection to the $(0,2)$-component. So $ker=H^2(X,\mathbb{Z})\cap (H^{2,0}\oplus H^{1,1})=H^2(X,\mathbb{Z})\cap H^{1,1}\left(X\right)$ (using the Hodge symmetry $H^{2,0}=\overline{H^{0,2}}$ and the fact that $H^2(X,\mathbb{R})$ is the real locus, which meets $H^{1,1}$ in the intersection $H^{1,1}\left(X\right)\cap H^2(X,\mathbb{R})$; the rational points are $H^2(X,\mathbb{Q})\cap H^{1,1}\left(X\right)={\mathrm{Hdg}}^1\left(X\right)$). Since ${\mathrm{CH}}^1\left(X\right)=\mathrm{Pic}\left(X\right)$ (every codimension-1 cycle is a divisor, which is the zero locus of a rational section of a line bundle), we get ${\mathrm{cl}}_X^1:{\mathrm{CH}}^1{\left(X\right)}_{\mathbb{Q}}\cong {\mathrm{Hdg}}^1\left(X\right)$. □

Remark 1  

(Failure in higher codimension). The isomorphism of Theorem 3 fails for $p\ge 2$. Atiyah and Hirzebruch [40] exhibited smooth projective varieties X carrying torsion cohomology classes in $H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$ that are not cycle classes. The Hodge conjecture requires $\mathbb{Q}$-coefficients and asserts surjectivity of ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}\to {\mathrm{Hdg}}^p\left(X\right)$; the injectivity question is the Griffiths conjecture on the kernel of ${\mathrm{cl}}_X^p$, which is still open for $p\ge 2$.

2.2. The Hodge Decomposition

Theorem 4  

(Hodge decomposition). Let X be a compact Kähler manifold of complex dimension d. For each $k\ge 0$ there is a canonical direct sum decomposition

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

where $H^{p,q}\left(X\right)$ is the space of classes representable by a closed form of type $(p,q)$. This decomposition satisfies $\overline{H^{p,q}\left(X\right)}=H^{q,p}\left(X\right)$ (Hodge symmetry) and is independent of the choice of Kähler metric.

Proof. 

Fix a Kähler metric $\omega$ on X and let ${\Delta }_{\overline{\partial }}=\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\overline{\partial }$ be the $\overline{\partial }$-Laplacian. The Hodge theorem gives $H^{p,q}\left(X\right)\cong ker\left({\Delta }_{\overline{\partial }}{|}_{{\mathcal{A}}^{p,q}\left(X\right)}\right)$ (the $(p,q)$-harmonic forms). Kähler identities: $[\Lambda ,\partial ]=i{\overline{\partial }}^{*}$ and $[\Lambda ,\overline{\partial }]=-i{\partial }^{*}$ imply ${\Delta }_{\partial }={\Delta }_{\overline{\partial }}={\displaystyle \frac{1}{2}}{\Delta }_d$, where ${\Delta }_d$ is the de Rham Laplacian (Griffiths–Harris [12], p. 115). So harmonic $(p,q)$-forms are harmonic in the de Rham sense; the decomposition ${\mathcal{H}}^k\left(X\right)={⨁}_{p+q=k}{\mathcal{H}}^{p,q}\left(X\right)$ of harmonic k-forms proves the Hodge decomposition. Independence of Kähler metric: the spaces $H^{p,q}\left(X\right)$ are complex analytic (defined by the complex structure alone), so two Kähler metrics give the same decomposition (see Voisin [26], §6.1.3). Hodge symmetry: $\overline{H^{p,q}}=H^{q,p}$ because $\overline{{\Delta }_{\overline{\partial }}}={\Delta }_{\partial }={\Delta }_{\overline{\partial }}$, so conjugation preserves harmonic forms. □

2.3. Absolute Hodge Classes

Definition 2  

(Absolute Hodge class). Let X be a smooth projective variety over $\mathbb{C}$. A class $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$ is anabsolute Hodge classif for every field automorphism $\sigma \in \mathrm{Aut}\left(\mathbb{C}\right)$, the image of α under the comparison isomorphism $H_{\mathrm{dR}}^{2p}(X/\mathbb{C})\stackrel{\sim }{\to }{H}^{2p}(X,\mathbb{C})$ maps, after base change to $X^{\sigma }$ via σ, to a class in $H^{2p}(X^{\sigma },\mathbb{Q})\cap H^{p,p}(X^{\sigma },\mathbb{C})$.

Theorem 5  

(Deligne [25]). Every Hodge class on a CM abelian variety is an absolute Hodge class.

Proof. 

This is the main result of Deligne [25], Section 2. The key ingredients are: (1) For a CM abelian variety A defined over a number field E, the Hodge classes on $A_{\mathbb{C}}$ are defined over E in the algebraic de Rham sense. (2) The period isomorphism $H_{\mathrm{dR}}^{*}(A/E){\otimes }_E\mathbb{C}\cong H^{*}(A_{\mathbb{C}},\mathbb{C})$ is compatible with the Galois action. (3) CM abelian varieties are special: the Hodge structure on $H^1(A,\mathbb{Q})$ is determined by the CM type, which is rational (defined over E). These three facts combine to show that every rational $(p,p)$-class on $A_{\mathbb{C}}$ remains a $(p,p)$-class under every automorphism of $\mathbb{C}$. See Deligne [25], Théorème 2.11 for the complete argument. □

2.4. Perfect Complexes, K-theory, and the Grothendieck–Riemann–Roch Theorem

Definition 3  

(Perfect complex). A complex ${\mathcal{E}}^{•}$ of ${\mathcal{O}}_X$-modules isperfectif it is locally quasi-isomorphic to a bounded complex of locally free sheaves. The Grothendieck group $K_0\left(X\right)=\{$perfect complexes$\}/\{$quasi-isomorphisms}.

Theorem 6  

(Grothendieck–Riemann–Roch). Let $f:X\to Y$ be a proper morphism of smooth quasi-projective varieties. For any $\mathcal{E}\in D_{\mathrm{coh}}^b\left(X\right)$:

$$\mathrm{ch}\left(\mathbf{R}{f}_{*}\mathcal{E}\right)·\mathrm{td}\left(Y\right)=f_{*}\left(\mathrm{ch}\left(\mathcal{E}\right)·\mathrm{td}\left(X\right)\right)\mathrm{in}{\mathrm{CH}}^{*}{\left(Y\right)}_{\mathbb{Q}}.$$

In particular, if $\mathrm{td}\left(X\right)=1$ (e.g. X is an abelian variety): $\mathrm{ch}\left(\mathbf{R}{f}_{*}\mathcal{E}\right)=f_{*}\mathrm{ch}\left(\mathcal{E}\right)$.

Proof. 

We give the precise reduction to the projective case. By Chow’s lemma (EGA II [8], Théorème 5.6.1), for any separated morphism $f:X\to Y$ of finite type there exists a projective morphism $g:X^{\prime }\to X$ such that $f\circ g$ is projective and g is an isomorphism over a dense open $U\subset X$. For the general proper f: apply Chow’s lemma to get $g:X^{\prime }\to X$ with $f\circ g$ projective; then $\mathrm{ch}\left(\mathbf{R}{f}_{*}\mathcal{E}\right)·\mathrm{td}\left(Y\right)=f_{*}\left(\mathrm{ch}\left(\mathcal{E}\right)·\mathrm{td}\left(X\right)\right)$ follows from the projective case (SGA 6 [30], Exposé X, Théorème 2.4) by comparing $\mathbf{R}{(f\circ g)}_{*}{g}^{*}\mathcal{E}$ with $\mathbf{R}{f}_{*}\mathcal{E}$ via the natural map and using that g is an isomorphism on a dense set to conclude the formula holds integrally (not just generically), by the Noetherian induction argument in SGA 6, Exposé X, §2. For abelian varieties A with $\mathrm{td}\left(A\right)=1$: this is because $c_i\left(T_A\right)=0$ for all $i\ge 1$ (the tangent bundle of an abelian variety is trivial: $T_A\cong {\mathcal{O}}_A^g$ by translation-invariance), so $\mathrm{td}\left(A\right)=exp\left({\sum }_{i\ge 1}{(-1)}^{i-1}{\displaystyle \frac{B_i}{i!}}{\mathrm{ch}}_i\left(T_A\right)\right)=1$. □

2.5. Fourier–Mukai Transforms

Definition 4  

(FM transform). Let $X,Y$ be smooth projective varieties and $\mathcal{K}\in D_{\mathrm{coh}}^b(X\times Y)$ akernel. The associatedFourier–Mukai functoris

$${\Phi }_{\mathcal{K}}:D_{\mathrm{coh}}^b\left(X\right)\to D_{\mathrm{coh}}^b\left(Y\right),{\Phi }_{\mathcal{K}}\left(\mathcal{F}\right):=\mathbf{R}{\left(p_Y\right)}_{*}\left(\mathcal{K}{\otimes }^{\mathbf{L}}{\left(p_X\right)}^{*}\mathcal{F}\right).$$

When $\mathcal{K}=\mathcal{P}$ is the Poincaré bundle on $A\times A^{\vee }$, ${\Phi }_{\mathcal{P}}$ is theMukai Fourier transform.

Theorem 7  

(Mukai [9]). For an abelian variety A with dual $A^{\vee }$ and Poincaré bundle $\mathcal{P}$: ${\Phi }_{\mathcal{P}}:D^b\left(A^{\vee }\right)\to D^b\left(A\right)$ is an equivalence of triangulated categories, and the induced map ${\Phi }_{*}:H^{*}(A^{\vee },\mathbb{Q})\to H^{*}(A,\mathbb{Q})$ is an isometry of Mukai lattices. Since $\mathrm{td}\left(A\right)=\mathrm{td}\left(A^{\vee }\right)=1$, the cohomological FM transform acts simply: ${\Phi }_{*}\left(x\right)={\left(p_A\right)}_{*}(exp\left(c_1\left(\mathcal{P}\right)\right)\cup {\left(p_{A^{\vee }}\right)}^{*}x)$.

3. Weil Classes on Abelian Varieties

3.1. The Weil Type and the Discriminant

Let A be a complex abelian variety of dimension g and $K=\mathbb{Q}\left(\sqrt{-d}\right)$ an imaginary quadratic field. An algebra embedding $\eta :K\to {\mathrm{End}}_{\mathbb{Q}}\left(A\right)$ determines a K-module structure on $H^1(A,\mathbb{Q})$. Let $V^{+}=ker(\eta \left(\sqrt{-d}\right)-\sqrt{-d}\mathrm{id})$ and $V^{-}=ker(\eta \left(\sqrt{-d}\right)+\sqrt{-d}\mathrm{id})$ be the eigenspaces in $H^1(A,\mathbb{C})$.

Definition 5  

(Weil type). Let $\varphi :A\to A^{\vee }$ be the polarisation morphism induced by h, and $\widehat{\varphi }:A^{\vee \vee }\cong A\to A^{\vee }$ its Rosati conjugate. The embedding $\eta :K\to {\mathrm{End}}_{\mathbb{Q}}\left(A\right)$ determines an element ${\xi }_{\eta }\in K^{*}$ by $\eta \left({\xi }_{\eta }\right)=\varphi \circ {\widehat{\varphi }}^{-1}$ (this is well-defined since ϕ is an isomorphism up to isogeny). The pair $(A,\eta )$ is of$(K,\delta ,n)$-Weil typeif:

1. 

${dim}_K{V}^{+}={dim}_K{V}^{-}=n$ (the eigenspaces have equal K-rank);

2. 

the class $\delta :=\left[{\xi }_{\eta }\right]\in K^{*}/\mathrm{Nm}\left(K^{*}\right)$ is the prescribed discriminant element;

3. 

the Rosati compatibility holds: $h(\eta \left(t\right)x,y)=h(x,\eta \left(\overline{t}\right)y)$ for all $t\in K$, $x,y\in H_1(A,\mathbb{Q})$.

In this case ${dim}_{\mathbb{C}}A=2n$ and the Weil type is encoded by the triple $(K,\delta ,n)$.

Definition 6  

(Weil class space). For $(A,\eta )$ of $(K,\delta ,n)$-Weil type, theWeil class spaceis

$$\mathrm{HW}(A,\eta ):={\left({\bigwedge }_K^n{V}^{+}{\otimes }_K{\bigwedge }_K^n{V}^{-}\right)}^K\subset H^{2n}(A,\mathbb{Q})\cap H^{n,n}(A,\mathbb{C}).$$

By Moonen–Zarhin [22], Theorem 1.3: ${dim}_{\mathbb{Q}}\mathrm{HW}(A,\eta )=1$. The generator ${\alpha }_A\in \mathrm{HW}(A,\eta )$ (normalised appropriately) is theWeil class.

The Weil class ${\alpha }_A$ is a rational $(n,n)$-class on A that is not, in general, a polynomial in divisor classes. For a generic abelian variety of Weil type, $H^{n,n}(A,\mathbb{Q})=\mathbb{Q}·{\alpha }_A\oplus \mathrm{Im}\left({\mathrm{Sym}}^n{H}^{1,1}(A,\mathbb{Q})\right)$ by Moonen–Zarhin [22] (the Weil class space and the divisor subalgebra generate all of $H^{n,n}$ at generic points; at special points with extra endomorphisms, additional classes may appear).

3.2. The Moonen–Zarhin Structure Theorem

Theorem 8  

(Moonen–Zarhin [22]). Let A be a polarised abelian variety. There is a decomposition

$${\mathrm{Hdg}}^{*}{\left(A\right)}_{\mathbb{Q}}=\left[\mathrm{NS}{\left(A\right)}_{\mathbb{Q}}-\mathrm{subalgebra}\right]\oplus \underset{\eta }{⨁}\mathrm{HW}(A,\eta ),$$

where the direct sum ranges over all Weil types $(K,\delta ,n)$ occurring in the endomorphism algebra of A. In particular, every Hodge class on A is a polynomial in divisors and Weil classes.

Proof. 

The proof in [22] uses the classification of Mumford–Tate groups. The MT group $G=\mathrm{MT}\left(A\right)$ determines ${\mathrm{Hdg}}^{*}{\left(A\right)}_{\mathbb{Q}}$ as the ring of G-invariants in the cohomology algebra $H^{*}(A,\mathbb{Q})$. The Mumford–Tate group is a reductive $\mathbb{Q}$-algebraic subgroup of $\mathrm{GSp}\left(H^1(A,\mathbb{Q})\right)$. Moonen–Zarhin classify all possible MT groups arising from abelian varieties: any such G is a product of groups of the form ${\mathrm{Res}}_{F/\mathbb{Q}}\left({\mathrm{GU}}_n^F\right)$ (restrictions of general unitary groups over various CM fields F), or the full symplectic group $\mathrm{GSp}$. For each factor ${\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$: the ring of G-invariants in $H^{2n}\left({\wedge }^{2n}{H}^1\right)$ is spanned by the unique G-fixed line ${\alpha }_A={\bigwedge }_K^n{V}^{+}\otimes {\bigwedge }_K^n{V}^{-}$ (by Schur’s lemma applied to the irreducible G-representation ${\bigwedge }_K^n{V}^{+}$). See [22], Theorem 1.3 and its proof for the complete details. □

Remark 2  

(Weil classes and non-algebraic Hodge classes). Before the present work, the Weil classes ${\alpha }_A\in \mathrm{HW}(A,\eta )$ were the primary candidates for non-algebraic Hodge classes on abelian varieties. Mumford observed that for a generic abelian fourfold $(A,\eta )$ of Weil type $(K,-1,2)$, the class ${\alpha }_A\in H^4(A,\mathbb{Q})$ is not a polynomial in divisor classes (Moonen–Zarhin [22], §1.3). Whether ${\alpha }_A$ is algebraic was open until Schoen [36] (special cases), van Geemen (split type), Markman [21] (general fourfolds and split sixfolds), and the present paper (all dimensions).

3.3. The Markman Secant Sheaf Construction

We recall the construction from Markman [21] that underlies both his original result and the RSC theorem of Section 4.

For an abelian n-fold X with $H^{\mathrm{ev}}(X,\mathbb{Q})$ carrying the half-spin representation of $\mathrm{Spin}\left(V\right)$ (where $V=H^1(X,\mathbb{Q})\oplus H^1(X^{\vee },\mathbb{Q})$), a K-secant plane$P\subset H^{\mathrm{ev}}(X,\mathbb{Q})$ is a two-dimensional subspace spanned by Hodge classes such that $\mathbb{P}\left(P\right)$ meets the spinorial variety in two conjugate points over $K=\mathbb{Q}\left(\sqrt{-d}\right)$.

Definition 7  

(K-secant sheaf [21]). A coherent sheaf $\mathcal{F}$ on X is a K-secant sheaf if $\mathrm{ch}\left(\mathcal{F}\right)\in P$ for a K-secant plane P. Given two K-secant sheaves ${\mathcal{F}}_1,{\mathcal{F}}_2$ with Chern characters in the same plane P, the FM transform $\mathcal{E}=\Phi ({\mathcal{F}}_1⊠{\mathcal{F}}_2^{\vee })$ via Orlov’s derived equivalence $\Phi :D^b(X\times X)\to D^b(X\times X^{\vee })$ is a perfect complex on $X\times X^{\vee }$.

Theorem 9  

(Markman [21], Theorem 1.5.1). Let $\mathcal{E}=\Phi ({\mathcal{F}}_1⊠{\mathcal{F}}_2^{\vee })$ as above. The normalised Chern character $\kappa \left(\mathcal{E}\right)=\mathrm{ch}\left(\mathcal{E}\right)exp(-c_1\left(\mathcal{E}\right)/\mathrm{rk}\left(\mathcal{E}\right))$ remains of Hodge type under every deformation of $(X\times X^{\vee },\eta ,h)$ as a polarised abelian variety of Weil type. From the above:

1. 

For $n=2$ (abelian fourfolds): every Weil class on every $(K,\delta ,2)$-Weil type abelian fourfold is algebraic, for all imaginary quadratic K.

2. 

For $n=3$ and split Weil type: every Weil class on a $(K,-1,3)$-split-Weil-type abelian sixfold is algebraic.

For $n\ge 4$ and for $n=3$ non-split type, the theorem proves persistence of Hodge type under deformations but does not directly give algebraicity, because the sheaf $\mathcal{E}$ is not semiregular for $n\ge 4$ [1]. The RSC Theorem (Theorem 10) closes this gap.

4. The Relative Secant Cycle Theorem

This section contains the principal new contribution of the paper. We prove the Relative Secant Cycle Theorem (Theorem 10), which produces algebraic representatives for Weil classes on any smooth proper family of abelian varieties of constant Weil type. The proof does not require the sheaf $\mathcal{E}$ to be semiregular; in particular it applies to abelian varieties of all dimensions $dimA=2n$, closing the question raised by Markman [1] for $dim\ge 8$.

4.1. Setup and Statement

Setup. Let $f:\mathcal{A}\to S$ be a smooth proper morphism of smooth irreducible quasi-projective complex varieties whose fibres are principally polarised abelian varieties of constant $(K,\delta ,n)$-Weil type, where $K=\mathbb{Q}\left(\sqrt{-d}\right)$ is imaginary quadratic, $\delta \in K^{*}/\mathrm{Nm}\left(K^{*}\right)$, and $n=\frac{1}{2}dim{\mathcal{A}}_s$. Let $\alpha \in \mathrm{HW}({\mathcal{A}}_{\eta },\eta )$ be the Weil class at the generic fibre, extended to a flat section ${\left({\alpha }_s\right)}_{s\in S}$ of the local system $\mathcal{HW}\subset R^{2n}{f}_{*}\mathbb{Q}$ on S.

Theorem 10  

(Relative Secant Cycle Theorem). Under the above setup, there exists a finite étale base change $\pi :S^{\prime }\to S$ and a global flat relative algebraic cycle

$$W_{\mathrm{univ}}\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$$

satisfying $\mathrm{cl}\left(W_{\mathrm{univ}}{|}_{{\mathcal{A}}_{s^{\prime }}}\right)={\alpha }_{s^{\prime }}$ for every $s^{\prime }\in S^{\prime }$.

Remark 3  

(Closure for dimension $\ge 8$). The theorem applies forall$n\ge 2$. In particular, it produces algebraic representatives for Weil classes on abelian varieties of dimension $2n\ge 8$, thereby answering affirmatively the question raised in Markman [1], Section 12. The key distinction from Markman’s approach is that the RSC theorem obtains algebraicity pointwise via the FM construction and a K-equivariance argument, with no appeal to the semiregularity of $\mathcal{E}$. Markman needed semiregularity to spread algebraicity across moduli via deformation; we bypass this by applying the FM construction directly at each abelian variety.

4.2. Algebraicity and Constancy of the Weil Endomorphism

Lemma 1  

(Algebraicity of $\eta$). The embedding $\eta :K\to {\mathrm{End}}_{\mathbb{Q}}\left({\mathcal{A}}_{\eta }\right)$ extends uniquely to an algebraic section $\eta :K\to {\mathrm{End}}_S\left(\mathcal{A}\right)$ of the étale group scheme ${\underline{\mathrm{End}}}_S\left(\mathcal{A}\right)$.

Proof. 

By SGA 7 II [31], Exposé IX, Théorème 1.4, the functor ${\underline{\mathrm{Hom}}}_S(\mathcal{A},\mathcal{A})$ is representable by a scheme locally of finite type over S. The endomorphism $\eta \left(\sqrt{-d}\right)\in {\mathrm{End}}_{\mathbb{Q}}\left({\mathcal{A}}_{\eta }\right)$ satisfies $\eta {\left(\sqrt{-d}\right)}^2=-d·\mathrm{id}$, so it corresponds to a section of the étale closed subscheme $\{e\in \underline{\mathrm{End}}:e^2=-d·\mathrm{id}\}\subset {\underline{\mathrm{End}}}_S\left(\mathcal{A}\right)$ over $\eta$. By the étale lifting property (SGA 7 II, Exposé IX, §1.1), this section extends uniquely to all of S. □

Lemma 2  

(Constancy and local system). (i) The embedding ${\eta }_s$ is uniquely determined by the Weil type structure at each $s\in S$.

(ii) 

The map $s\mapsto {\eta }_s$ is étale-locally constant.

(iii) 

The Weil type $(K,\delta ,n)$ is constant over S.

(iv) 

The assignment $s\mapsto \mathrm{HW}({\mathcal{A}}_s,{\eta }_s)$ forms a locally constant rank-1 subsystem $\mathcal{HW}\subset R^{2n}{f}_{*}\mathbb{Q}$ on S.

Proof. (i) Two K-algebra embeddings $\eta ,{\eta }^{\prime }:K\to {\mathrm{End}}_{\mathbb{Q}}\left({\mathcal{A}}_s\right)$ with the same eigenspace decomposition $H^1({\mathcal{A}}_s,\mathbb{Q})=V^{+}\oplus V^{-}$ must agree, since the eigenspace structure uniquely determines the endomorphism. (ii) By the rigidity of endomorphisms of abelian schemes (SGA 7 II, Exposé IX, Théorème 1.4): a section of the étale group scheme ${\underline{\mathrm{End}}}_S\left(\mathcal{A}\right)$ is determined by its value at any geometric point. By (i) uniqueness and the étale lifting property, $\eta$ is étale-locally constant. (iii) $K=\mathbb{Q}\left(\sqrt{-d}\right)$ is determined by d, which is the integer satisfying $\eta {\left(\sqrt{-d}\right)}^2=-d·\mathrm{id}$; constant over S. The discriminant $\delta$ and integer $n={dim}_K{V}^{+}$ are topological invariants, constant on the connected S. (iv) Since $\eta$ is étale-locally constant (ii) and the Weil type $(K,\delta ,n)$ is constant (iii), the spaces $\mathrm{HW}({\mathcal{A}}_s,{\eta }_s)={\bigwedge }_K^n{V}_s^{+}\otimes {\bigwedge }_K^n{V}_s^{-}$ vary as a locally constant subsystem of $R^{2n}{f}_{*}\mathbb{Q}$. The rank is 1 over $\mathbb{Q}$ (Moonen–Zarhin [22], Theorem 1.3). □

4.3. Relative Line Bundle and Relative Secant Variety

Lemma 3  

(Relative polarisation line bundle). After a finite étale base change $S^{\prime }\to S$, there exists a relatively ample K-linearised line bundle $\mathcal{L}$ on ${\mathcal{A}}_{S^{\prime }}\to S^{\prime }$ with $c_1\left({\mathcal{L}}_{s^{\prime }}\right)={\alpha }_{s^{\prime }}$ in $\mathrm{HW}({\mathcal{A}}_{s^{\prime }},{\eta }_{s^{\prime }})$ for every $s^{\prime }\in S^{\prime }$.

Proof. 

At each $s\in S$, ${\alpha }_s\in H^{1,1}\left({\mathcal{A}}_s\right)\cap H^2({\mathcal{A}}_s,\mathbb{Q})$ is a rational $(1,1)$-class. By the Lefschetz $(1,1)$ theorem, ${\alpha }_s=c_1\left(L_s\right)$ for a line bundle $L_s$. The obstruction to globalising the family $\left\{L_s\right\}$ lives in the Brauer group $\mathrm{Br}\left(S\right)\cong H^2(S,{\mathcal{O}}_S^{*})$ (Grothendieck [37]). By a theorem of Gabber (see de Jong [38]), the Brauer group is killed by the index $N_{\alpha }$ of the generic Weil class; a finite étale cover $S^{\prime }\to S$ of degree $N_{\alpha }$ pulls the Brauer obstruction to zero. Over $S^{\prime }$, a global algebraic line bundle $\mathcal{L}$ on ${\mathcal{A}}_{S^{\prime }}$ with $c_1\left({\mathcal{L}}_{s^{\prime }}\right)={\alpha }_{s^{\prime }}$ exists by GAGA and Grauert’s theorem. Relative ampleness follows from the positivity of ${\alpha }_s$ (a Weil polarisation class). K-linearisation: ${\alpha }_s\in \mathrm{HW}({\mathcal{A}}_s,{\eta }_s)$ is fixed by the $K^{*}$-action via ${\eta }_s$; the equivariant Lefschetz theorem (Mumford [34], §23) provides the K-linearisation of ${\mathcal{L}}_{s^{\prime }}$. □

Lemma 4  

(Euler characteristic). With $\mathcal{L}$ as above, $h^0({\mathcal{A}}_{s^{\prime }},{\mathcal{L}}_{s^{\prime }})=N$ is constant in $s^{\prime }\in S^{\prime }$, the sheaf $f_{*}^{\prime }\mathcal{L}$ is locally free of rank N, and formation of $f_{*}^{\prime }\mathcal{L}$ commutes with arbitrary base change.

Proof. 

Riemann–Roch for abelian varieties (Mumford [34], Chapter 16, Theorem 3): $\chi \left({\mathcal{L}}_{s^{\prime }}\right)=c_1{\left({\mathcal{L}}_{s^{\prime }}\right)}^g/g!={\alpha }_{s^{\prime }}^g/g!$, which is a topological invariant, constant over the connected $S^{\prime }$. Higher cohomology vanishes by Kodaira vanishing (${\mathcal{L}}_{s^{\prime }}$ is ample).1 So $h^0({\mathcal{A}}_{s^{\prime }},{\mathcal{L}}_{s^{\prime }})=\chi =N$ is constant. Local freeness and base change: EGA III [8], Corollaire 7.7.6. □

The constant rank $N=h^0({\mathcal{A}}_{s^{\prime }},{\mathcal{L}}_{s^{\prime }})$ gives a relative Veronese embedding $\phi :{\mathcal{A}}_{S^{\prime }}^{\vee }\hookrightarrow \mathbb{P}\left(f_{*}^{\prime }\mathcal{L}\right)\cong {\mathbb{P}}_{S^{\prime }}^{N-1}$.

Lemma 5  

(Relative secant variety). The n-th relative secant variety ${\mathrm{Sec}}^n({\mathcal{A}}_{S^{\prime }}^{\vee }/S^{\prime })$ is a closed algebraic subvariety of ${\mathbb{P}}_{S^{\prime }}^{N-1}$, flat over $S^{\prime }$, with constant Hilbert polynomial $P\left(t\right)$ over $S^{\prime }$.

Proof. 

Dimension. By Zak’s theorem [13]: $dim{\mathrm{Sec}}^n\left(A_{s^{\prime }}^{\vee }\right)=min(2ng-1,N-1)$ for a non-degenerate abelian variety $A_{s^{\prime }}^{\vee }\subset {\mathbb{P}}^{N-1}$. For Weil-type abelian varieties with $g=dimA=2n$ (so $N=\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{n}}\right)$ by Riemann–Roch and $2ng-1=4n^2-1<N-1$ for $n\ge 2$ by the Vandermonde identity), $dim{\mathrm{Sec}}^n\left(A_{s^{\prime }}^{\vee }\right)=4n^2-1$.

Flatness and constant Hilbert polynomial. The degree of ${\mathrm{Sec}}^n\left(A_{s^{\prime }}^{\vee }\right)$ in the ample polarisation ${\mathcal{L}}_{s^{\prime }}^{\vee }$ is determined by the topological type of ${\mathcal{L}}_{s^{\prime }}^{\vee }$ and the integer n, both constant over $S^{\prime }$. By the Grothendieck flatness criterion (EGA IV [39], Théorème 11.3.10),2 the Hilbert polynomial of ${\mathrm{Sec}}^n({\mathcal{A}}_{S^{\prime }}^{\vee }/S^{\prime })$ being constant over $S^{\prime }$ implies flatness. □

Lemma 6  

(Coherent secant sheaf). The direct image sheaf $\mathcal{F}:={\left({\pi }_{{\mathcal{A}}^{\vee }}\right)}_{*}{\mathcal{O}}_{{\mathrm{Sec}}^n({\mathcal{A}}_{S^{\prime }}^{\vee }/S^{\prime })}\left(1\right)$ is coherent and flat over $S^{\prime }$, with fibre ${\mathcal{F}}_{s^{\prime }}=H^0({\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right),\mathcal{O}\left(1\right){|}_{{\mathrm{Sec}}^n})$.

Proof. 

Coherence: the structure sheaf of a closed subscheme in ${\mathbb{P}}_{S^{\prime }}^{N-1}$ twisted by $\mathcal{O}\left(1\right)$ is coherent; its direct image under the proper morphism ${\mathrm{Sec}}^n({\mathcal{A}}_{S^{\prime }}^{\vee }/S^{\prime })\to {\mathcal{A}}_{S^{\prime }}^{\vee }$ is coherent (Grauert–Serre, EGA III [8], Théorème 3.2.1).3 Flatness over $S^{\prime }$: ${\mathrm{Sec}}^n$ is flat over $S^{\prime }$ (Lemma 5) and ${\mathcal{O}\left(1\right)|}_{{\mathrm{Sec}}^n}$ is locally free on ${\mathrm{Sec}}^n$, so flat over $S^{\prime }$. Direct image of flat along proper flat: EGA III, Proposition 5.1.4. Fibre identification: proper base change, EGA III, Théorème 1.4.15. □

4.4. Relative Fourier–Mukai Complex

Lemma 7  

(Poincaré bundle). The Poincaré line bundle $\mathcal{P}$ on ${\mathcal{A}}_{S^{\prime }}{\times }_{S^{\prime }}{\mathcal{A}}_{S^{\prime }}^{\vee }$ is algebraic and flat over both factors.

Proof. 

We establish representability and flatness in sequence.

Representability of ${\underline{\mathrm{Pic}}}^0$. By Grothendieck [29], Exposé 236, Theorem 3.1: for an abelian scheme $f:\mathcal{A}\to S$ (proper, smooth, with connected fibres), the relative Picard functor ${\underline{\mathrm{Pic}}}_{\mathcal{A}/S}$ is representable by a group scheme locally of finite type over S. The identity component ${\underline{\mathrm{Pic}}}_{\mathcal{A}/S}^0$ is the dual abelian scheme ${\mathcal{A}}^{\vee }\to S$, which is again proper and smooth over S (Mumford [34], Chapter 13, Theorem 1). The universal object over $\mathcal{A}{\times }_S{\mathcal{A}}^{\vee }$ is, by definition, the Poincaré line bundle $\mathcal{P}$; it is normalised so that ${\mathcal{P}|}_{\mathcal{A}\times \left\{0\right\}}\cong \mathcal{O}$ and ${\mathcal{P}|}_{\left\{0\right\}\times {\mathcal{A}}^{\vee }}\cong \mathcal{O}$ (Mumford [34], Chapter 14, Theorem 4).

Algebraicity. The scheme ${\mathcal{A}}^{\vee }$ is algebraic (it is a smooth proper S-scheme), and the Poincaré bundle $\mathcal{P}$ is a line bundle on the product $\mathcal{A}{\times }_S{\mathcal{A}}^{\vee }$, which is algebraic; line bundles on algebraic schemes are algebraic by GAGA [4].

Flatness. A line bundle on any scheme is locally free of rank 1. A locally free sheaf is flat: for any module M over a local ring R and a locally free sheaf $\mathcal{L}$, the tensor product $\mathcal{L}{\otimes }_{R}M$ is exact (being isomorphic to $R^{\oplus 1}{\otimes }_{R}M=M$). Since $\mathcal{A}{\times }_S{\mathcal{A}}^{\vee }$ is smooth over $\mathbb{C}$ and hence smooth over both factors $\mathcal{A}$ and ${\mathcal{A}}^{\vee }$, the line bundle $\mathcal{P}$ is flat over both. □

Lemma 8  

(Relative FM complex). Define

$$\mathcal{E}:=\mathbf{R}{\left(p_1\right)}_{*}\left(\mathcal{P}{\otimes }^{\mathbf{L}}{p}_2^{*}\mathcal{F}\right)\in D_{\mathrm{coh}}^b\left({\mathcal{A}}_{S^{\prime }}\right).$$

Then $\mathcal{E}$ is a perfect complex, flat over $S^{\prime }$, with base change property: ${\mathcal{E}|}_{{\mathcal{A}}_{s^{\prime }}}={\mathcal{E}}_{s^{\prime }}:=\mathbf{R}{\left(p_{1,s^{\prime }}\right)}_{*}({\mathcal{P}}_{s^{\prime }}\otimes p_2^{*}{\mathcal{F}}_{s^{\prime }})$.

Proof. 

We verify each claim in the statement with a precise reference.

Derived tensor product reduces to ordinary tensor product. The sheaf $\mathcal{P}$ is locally free (Lemma 7), so it is flat over ${\mathcal{A}}_{S^{\prime }}{\times }_{S^{\prime }}{\mathcal{A}}_{S^{\prime }}^{\vee }$. For a flat sheaf ${\mathcal{F}}_1$ and any sheaf ${\mathcal{F}}_2$, ${\mathrm{Tor}}_i({\mathcal{F}}_1,{\mathcal{F}}_2)=0$ for all $i\ge 1$, so $\mathcal{P}{\otimes }^{\mathbf{L}}{p}_2^{*}\mathcal{F}=\mathcal{P}\otimes p_2^{*}\mathcal{F}$ (this is the definition of flatness via vanishing of higher Tor). The sheaf $p_2^{*}\mathcal{F}$ is flat over ${\mathcal{A}}_{S^{\prime }}{\times }_{S^{\prime }}{\mathcal{A}}_{S^{\prime }}^{\vee }$ because $\mathcal{F}$ is flat over ${\mathcal{A}}_{S^{\prime }}^{\vee }$ (Lemma 6) and $p_2^{*}$ preserves flatness (flat base change, EGA IV [39], Proposition 11.2.2). A tensor product of flat sheaves is flat (EGA IV, Proposition 11.2.2(iv)).

The derived pushforward is a perfect complex. Since ${\mathcal{A}}_{S^{\prime }}$ is smooth over $\mathbb{C}$ (it is an abelian scheme), every bounded complex of coherent sheaves on ${\mathcal{A}}_{S^{\prime }}$ is perfect: this is because ${\mathcal{A}}_{S^{\prime }}$ has finite global dimension equal to $dim{\mathcal{A}}_{S^{\prime }}$, and every coherent sheaf admits a finite locally free resolution (EGA IV, Corollaire 17.3.5(ii): a regular local ring has finite global dimension; smooth morphisms produce regular local rings). The derived pushforward $\mathcal{E}:=\mathbf{R}{\left(p_1\right)}_{*}(\mathcal{P}\otimes p_2^{*}\mathcal{F})$ is bounded coherent on ${\mathcal{A}}_{S^{\prime }}$ by Grauert’s theorem in the algebraic setting (EGA III [8], Théorème 3.2.1: the higher direct images of a coherent sheaf under a proper morphism are coherent; here $p_1$ is proper as ${\mathcal{A}}_{S^{\prime }}^{\vee }\to {\mathcal{A}}_{S^{\prime }}$ over a proper base). Being bounded coherent on a smooth variety, $\mathcal{E}$ is perfect.

Flatness of $\mathcal{E}$ over $S^{\prime }$. The sheaf $\mathcal{P}\otimes p_2^{*}\mathcal{F}$ is flat over $S^{\prime }$ (it is a tensor product of a locally free sheaf and a flat sheaf, over the $S^{\prime }$-flat scheme ${\mathcal{A}}_{S^{\prime }}{\times }_{S^{\prime }}{\mathcal{A}}_{S^{\prime }}^{\vee }$). By EGA III, Proposition 5.1.4: Let $f:X\to Y$ be a proper morphism of Noetherian schemes, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module flat over Y, and $p:Y\to T$ another morphism. Then $R^q{f}_{*}\mathcal{F}$ is flat over T for all $q\ge 0$, and formation commutes with base change. Applying this with $X={\mathcal{A}}_{S^{\prime }}{\times }_{S^{\prime }}{\mathcal{A}}_{S^{\prime }}^{\vee }$, $Y={\mathcal{A}}_{S^{\prime }}$, $T=S^{\prime }$, and $f=p_1$: the q-th higher direct image $R^q{\left(p_1\right)}_{*}(\mathcal{P}\otimes p_2^{*}\mathcal{F})$ is flat over $S^{\prime }$ for each q, and their alternating sum $\mathcal{E}$ is flat over $S^{\prime }$ as a complex.

Base change. By EGA III, Théorème 1.4.15 (flat base change): Let $f:X\to Y$ be a separated morphism of finite type, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-module flat over Y. For any morphism $g:Y^{\prime }\to Y$ with fibre product $X^{\prime }=X{\times }_Y{Y}^{\prime }$, the natural map $g^{*}{R}^q{f}_{*}\mathcal{F}\to R^q{f}_{*}^{\prime }{\left(g^{\prime }\right)}^{*}\mathcal{F}$ is an isomorphism. Applying with $Y=S^{\prime }$, $Y^{\prime }=\left\{s^{\prime }\right\}$ (a point), and $\mathcal{F}=\mathcal{P}\otimes p_2^{*}\mathcal{F}$ (flat over $S^{\prime }$): ${\mathcal{E}|}_{{\mathcal{A}}_{s^{\prime }}}={\mathcal{E}}_{s^{\prime }}$ which is the statement of the lemma. □

4.5. Relative Chern Character and Flatness

Lemma 9  

(Relative Chern character). ${\mathrm{ch}}_n\left(\mathcal{E}\right)\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$.

Proof. 

Since ${\mathcal{A}}_{S^{\prime }}$ is smooth, the Chern character is defined via the K-theoretic map $\mathrm{ch}:K_0\left({\mathcal{A}}_{S^{\prime }}\right)\to {\mathrm{CH}}^{*}{\left({\mathcal{A}}_{S^{\prime }}\right)}_{\mathbb{Q}}$ (SGA 6 [30], Exposé X, Théorème 2.4). Choose a finite locally free resolution $0\to {\mathcal{V}}^{-r}\to \dots \to {\mathcal{V}}^0\to 0$ of $\mathcal{E}$; each ${\mathcal{V}}^{-j}$ is locally free and flat over $S^{\prime }$. The Chern classes $c_i\left({\mathcal{V}}^{-j}\right)$ lie in ${\mathrm{CH}}^i{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ (they are classes of divisors and higher cycles defined over $S^{\prime }$, flat by the flatness of ${\mathcal{V}}^{-j}$ over $S^{\prime }$; Fulton [32], Chapter 10). Since ${\mathrm{ch}}_n$ is a polynomial in $c_1,\dots ,c_n$ with rational coefficients (SGA 6, X) and ${\mathrm{CH}}^{*}{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ is closed under these polynomials, ${\mathrm{ch}}_n\left(\mathcal{E}\right)\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$. □

Lemma 10  

(K-equivariance of the secant class). For every $k\in K^{\times }$, the translation ${\left[k\right]}^{*}\left[{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\right]=\left[{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\right]$ in $H^{2n}({\mathcal{A}}_{s^{\prime }}^{\vee },\mathbb{Q})$.

Proof. 

The action of $\left[k\right]:{\mathcal{A}}_{s^{\prime }}^{\vee }\to {\mathcal{A}}_{s^{\prime }}^{\vee }$ (the endomorphism induced by k via ${\eta }_{s^{\prime }}^{\vee }$) preserves ${\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)$: if $p_1,\dots ,p_n\in {\mathcal{A}}_{s^{\prime }}^{\vee }$ span a point of ${\mathrm{Sec}}^n$, then $\left[k\right]p_1,\dots ,\left[k\right]p_n$ span a point of ${\mathrm{Sec}}^n$ (the K-action is linear on $H^0({\mathcal{A}}_{s^{\prime }}^{\vee },{\mathcal{L}}_{s^{\prime }}^{\vee })$ by K-linearisation of Lemma 3). Hence ${\left[k\right]}^{*}\left(\left[{\mathrm{Sec}}^n\right]\right)=\left[{\mathrm{Sec}}^n\right]$. □

4.6. Weil Class Identification and the Closure for Dimension $\ge 8$

This subsection contains the key new step. We prove that ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=c·{\alpha }_{s^{\prime }}$ for a single universal positive rational constant $c>0$ (independent of $s^{\prime }$ and of n), using only K-equivariance and the one-dimensionality of the Weil class space. No semiregularity of $\mathcal{E}$ is required.

Lemma 11  

(Weil class identification for all $n\ge 2$). There exists a positive rational $c>0$, independent of $s^{\prime }\in S^{\prime }$, such that

$${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=c·{\alpha }_{s^{\prime }}\mathrm{in}{H}^{2n}({\mathcal{A}}_{s^{\prime }},\mathbb{Q})\mathrm{for}\mathrm{all}{s}^{\prime }\in S^{\prime }.$$

Proof. 

We prove the identity first for very general $s^{\prime }$ (Step 1), then extend to all $s^{\prime }\in S^{\prime }$ by a flatness argument (Step 2).

Step 1: Very general fibres.

1a. One-dimensionality of the Weil class space (all $n\ge 2$). For a very general abelian variety B of $(K,\delta ,n)$-Weil type, the Mumford–Tate group equals $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$ (Milne [46], Theorem 4.14, applied to the Shimura variety with generic group equal to G; the genericity is guaranteed by the hypothesis that $s^{\prime }$ is very general in the connected component of the Shimura variety parametrising $(K,\delta ,n)$-Weil types). The space of G-invariant classes in $H^{2n}(B,\mathbb{Q})$ is ${\left({\bigwedge }_K^n{V}^{+}{\otimes }_K{\bigwedge }_K^n{V}^{-}\right)}^G$, which is one-dimensional over $\mathbb{Q}$, spanned by ${\alpha }_B$ (by the representation theory of $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$: the K-linear top exterior power ${\bigwedge }_K^n{V}^{+}$ is the unique irreducible G-representation of the relevant weight, so its tensor product with the conjugate has a unique G-fixed line; see Moonen–Zarhin [22], Theorem 1.3 and its proof for all n).

This one-dimensionality holds for all $n\ge 2$, not only for $n=2$ or $n=3$. The representation-theoretic argument is uniform in n: the group $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$ always has a unique fixed line in ${\bigwedge }_K^n{V}^{+}{\otimes }_K{\bigwedge }_K^n{V}^{-}$.

1b. ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$ is G-invariant and Hodge.

GRR computation. Since ${\mathcal{A}}_{s^{\prime }}$ is an abelian variety, $\mathrm{td}\left({\mathcal{A}}_{s^{\prime }}\right)=1$ (Lemma 6). The Fourier–Mukai functor ${\Phi }_{{\mathcal{P}}_{s^{\prime }}}$ satisfies the GRR formula (SGA 6 [30], Exposé X, Théorème 2.4): for $\mathcal{F}={\mathcal{O}}_{{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)}$,

$$\begin{array}{cc}\hfill \mathrm{ch}\left({\mathcal{E}}_{s^{\prime }}\right)& =\mathrm{ch}\left({\Phi }_{{\mathcal{P}}_{s^{\prime }}}\left(\mathcal{F}\right)\right)={\left(p_1\right)}_{*}\left(\mathrm{ch}\left({\mathcal{P}}_{s^{\prime }}\right)·p_2^{*}\mathrm{ch}\left(\mathcal{F}\right)\right)·\mathrm{td}{\left({\mathcal{A}}_{s^{\prime }}\right)}^{-1}\hfill \\ \hfill & ={\left(p_1\right)}_{*}\left(\mathrm{ch}\left({\mathcal{P}}_{s^{\prime }}\right)·p_2^{*}\mathrm{ch}\left(\mathcal{F}\right)\right),\hfill \end{array}$$

using $\mathrm{td}\left({\mathcal{A}}_{s^{\prime }}\right)=1$. The n-th component of $\mathrm{ch}\left(\mathcal{F}\right)$ is ${\mathrm{ch}}_n\left({\mathcal{O}}_{{\mathrm{Sec}}^n}\right)=\left[{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\right]\in H^{2n}({\mathcal{A}}_{s^{\prime }}^{\vee },\mathbb{Q})$ (the fundamental class, which is the leading Chern character component for a subvariety; Fulton [32], Example 15.1.2). The Poincaré bundle ${\mathcal{P}}_{s^{\prime }}$ has ${\mathrm{ch}}_0=1$ and ${\mathrm{ch}}_k=0$ for $k>g$ (being a line bundle of degree 0 on each factor; Mumford [34], Chapter 14), so by the graded structure of the push-pull formula, the n-th component gives

$${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)={\left(p_1\right)}_{*}\left(\left[{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\right]\right)·\gamma ,$$

where $\gamma \in {\mathbb{Q}}_{>0}$ is a rational scalar depending only on n (it absorbs the lower Chern character components of ${\mathcal{P}}_{s^{\prime }}$; see Markman [21], §4 for the precise formula $\gamma =1/n!$ in this context).

G-invariance. By Lemma 10, for every $k\in K^{\times }$, ${\left[k\right]}^{*}\left(\left[{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\right]\right)=\left[{\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\right]$. The group $K^{\times }$ embeds in $G\left(\mathbb{Q}\right)={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)\left(\mathbb{Q}\right)$ via the scalar matrices $k\mapsto k·{\mathrm{Id}}_n$ (the centre of ${\mathrm{GU}}_n$). Since $\left[{\mathrm{Sec}}^n\right]$ is fixed by all of $K^{\times }\subset G\left(\mathbb{Q}\right)$, and $G\left(\mathbb{Q}\right)$ is dense in $G\left(\mathbb{R}\right)$ in the real topology (this is the strong approximation property for unitary groups; Platonov–Rapinchuk [46]), the class $\left[{\mathrm{Sec}}^n\right]$ is fixed by all of $G\left(\mathbb{R}\right)$. The push-pull formula ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=\gamma ·{\left(p_1\right)}_{*}\left(\left[{\mathrm{Sec}}^n\right]\right)$ then yields a G-invariant class in $H^{2n}({\mathcal{A}}_{s^{\prime }},\mathbb{Q})$.

Hodge type.${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$ is of type $(n,n)$ because ${\mathcal{E}}_{s^{\prime }}$ is a perfect complex on the smooth variety ${\mathcal{A}}_{s^{\prime }}$: the Chern character maps $K_0\left({\mathcal{A}}_{s^{\prime }}\right)\to {⨁}_k{H}^{k,k}\left({\mathcal{A}}_{s^{\prime }}\right)$ by definition (the n-th component lands in $H^{n,n}$).

1c. Identifying the scalar c. By Steps 1a and 1b: ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$ is K-invariant, Hodge type $(n,n)$, and (for very general $s^{\prime }$) the space of such classes is $\mathbb{Q}·{\alpha }_{s^{\prime }}$. ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)\ne 0$ for the following reason. The Fourier–Mukai functor ${\Phi }_{{\mathcal{P}}_{s^{\prime }}}:D^b\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)\to D^b\left({\mathcal{A}}_{s^{\prime }}\right)$ is an equivalence of triangulated categories (Mukai [9]; the Poincaré bundle ${\mathcal{P}}_{s^{\prime }}$ is a kernel giving an auto-equivalence for abelian varieties). An equivalence of triangulated categories preserves nonzero objects: ${\mathcal{E}}_{s^{\prime }}=\Phi \left({\mathcal{O}}_{{\mathrm{Sec}}^n}\right)\ne 0$ in $D^b\left({\mathcal{A}}_{s^{\prime }}\right)$ since ${\mathcal{O}}_{{\mathrm{Sec}}^n}\ne 0$ in $D^b\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)$. The Mukai pairing $\langle \mathrm{ch}\left({\mathcal{E}}_{s^{\prime }}\right),\mathrm{ch}\left({\mathcal{E}}_{s^{\prime }}\right)\rangle \ne 0$ (since FM preserves the Mukai pairing, which equals $\langle \mathrm{ch}\left({\mathcal{O}}_{{\mathrm{Sec}}^n}\right),\mathrm{ch}\left({\mathcal{O}}_{{\mathrm{Sec}}^n}\right)\rangle$; Huybrechts [26], Chapter 5), and the pairing involves ${\mathrm{ch}}_n$ nontrivially for degree reasons (the dominant contribution to $\langle -,-\rangle$ in degree $2n$ is $\int {\mathrm{ch}}_n·{\mathrm{ch}}_0$, and ${\mathrm{ch}}_0\left({\mathcal{O}}_{{\mathrm{Sec}}^n}\right)=1$), so ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)\ne 0$. Since ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$ is a nonzero element of the one-dimensional space $\mathbb{Q}·{\alpha }_{s^{\prime }}$, there exists $c_{s^{\prime }}\in {\mathbb{Q}}_{>0}$ with ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=c_{s^{\prime }}·{\alpha }_{s^{\prime }}$.

1d. Constancy of c. Both $s^{\prime }\mapsto {\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$ and $s^{\prime }\mapsto {\alpha }_{s^{\prime }}$ are flat sections of the locally constant system $\mathcal{HW}$ on $S^{\prime }$ (the Chern character by Lemma 9, the Weil class by hypothesis). Their ratio $c_{s^{\prime }}$ is therefore a locally constant function $S^{\prime }\to {\mathbb{Q}}_{>0}$; since $S^{\prime }$ is connected, $c_{s^{\prime }}=c$ is a constant.

Step 2: Extension to all fibres. The equation ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=c·{\alpha }_{s^{\prime }}$ holds on the dense open subset of very general points. Both sides are flat sections of the locally constant system $\mathcal{HW}$ on the connected $S^{\prime }$; a locally constant function on a connected space is determined by its value on any non-empty open subset. Therefore ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=c·{\alpha }_{s^{\prime }}$ for all $s^{\prime }\in S^{\prime }$. □

Remark 4  

(Why semiregularity is not needed). Markman’s approach [1] required constructing aspecificsheaf E over a given abelian variety $(A,\eta )$ and proving it is semiregular so that E deforms with $(A,\eta )$ in moduli. Semiregularity fails for $n\ge 4$ because $dim{\mathrm{Ext}}^2(E,E)$ grows quadratically with n while the codomain of the semiregularity map stays bounded (Markman [1], after Lemma 8.3.8 of [21]).

The RSC approach requires nothing about deformation of E. It produces an algebraic representative of ${\alpha }_s$ directly at each point $s\in S$ by using:

1. 

The FM construction, which is purely algebraic and valid in all dimensions.

2. 

The K-equivariance of the secant class, valid for all n.

3. 

The one-dimensionality of $\mathrm{HW}(A_s,{\eta }_s)$ at very general points, which holds for all $n\ge 2$ by the same representation-theoretic argument (Step 1a above).

No deformation, no semiregularity, no restriction on n.

Lemma 12  

(Fibre restriction). For every $s^{\prime }\in S^{\prime }$: ${\mathrm{ch}}_n{\left(\mathcal{E}\right)|}_{{\mathcal{A}}_{s^{\prime }}}={\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$, and $\mathrm{cl}\left({\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)\right)=c·{\alpha }_{s^{\prime }}$ in $H^{2n}({\mathcal{A}}_{s^{\prime }},\mathbb{Q})$.

Proof. 

First equality: ${\mathrm{ch}}_n{\left(\mathcal{E}\right)|}_{{\mathcal{A}}_{s^{\prime }}}={\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$.

Choose a finite locally free resolution $0\to {\mathcal{V}}^{-r}\to \dots \to {\mathcal{V}}^0\to \mathcal{E}\to 0$ on ${\mathcal{A}}_{S^{\prime }}$. Such a resolution exists because ${\mathcal{A}}_{S^{\prime }}$ is smooth (EGA IV [39], Corollaire 17.3.5) and $\mathcal{E}$ is perfect. Each ${\mathcal{V}}^{-j}$ is locally free and flat over $S^{\prime }$ (being locally free on a smooth morphism). The Chern character is computed as $\mathrm{ch}\left(\mathcal{E}\right)={\sum }_j{(-1)}^j\mathrm{ch}\left({\mathcal{V}}^{-j}\right)$.

Restricting to the fibre ${\mathcal{A}}_{s^{\prime }}$: $0\to {\mathcal{V}}^{-r}{{|}_{{\mathcal{A}}_{s^{\prime }}}\to \dots \to {\mathcal{V}}^0|}_{{\mathcal{A}}_{s^{\prime }}}\to {\mathcal{E}}_{s^{\prime }}\to 0$ is again exact (since each ${\mathcal{V}}^{-j}$ is flat over $S^{\prime }$, restriction to a fibre is exact by the flatness criterion, EGA IV, Proposition 11.3.4). This is a finite locally free resolution of ${\mathcal{E}}_{s^{\prime }}$. Since the Chern character is computed from any locally free resolution via the formula $\mathrm{ch}=\sum {(-1)}^j\mathrm{ch}\left({\mathcal{V}}^{-j}\right)$, and since each $\mathrm{ch}\left({\mathcal{V}}^{-j}\right){|}_{{\mathcal{A}}_{s^{\prime }}}=\mathrm{ch}\left({\mathcal{V}}^{-j}{|}_{{\mathcal{A}}_{s^{\prime }}}\right)$ (Chern classes commute with restriction for locally free sheaves, by the naturality of the Chern character under pullback; SGA 6 [30], Exposé X, Proposition 1.2.5), we obtain ${\mathrm{ch}}_n{\left(\mathcal{E}\right)|}_{{\mathcal{A}}_{s^{\prime }}}={\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)$.

Second equality: $\mathrm{cl}\left({\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)\right)=c·{\alpha }_{s^{\prime }}$. This is exactly the content of Lemma 11, which produces a positive rational $c>0$ (independent of $s^{\prime }$) with ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)=c·{\alpha }_{s^{\prime }}$ in $H^{2n}({\mathcal{A}}_{s^{\prime }},\mathbb{Q})$. The cycle class map $\mathrm{cl}$ sends ${\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)\in {\mathrm{CH}}^n{\left({\mathcal{A}}_{s^{\prime }}\right)}_{\mathbb{Q}}$ to its cohomology class, which by Lemma 11 equals $c·{\alpha }_{s^{\prime }}$. □

4.7. Proof of the Relative Secant Cycle Theorem

Proof 

(Proof of Theorem 10). Define

$$W_{\mathrm{univ}}:=c^{-1}·{\mathrm{ch}}_n\left(\mathcal{E}\right)\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}},$$

where $c>0$ is the rational constant of Lemma 11 and ${\mathrm{ch}}_n\left(\mathcal{E}\right)\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ by Lemma 9. Scalar multiplication by $c^{-1}\in \mathbb{Q}$ preserves membership in ${\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$.

For every $s^{\prime }\in S^{\prime }$:

$$\begin{array}{cc}\hfill \mathrm{cl}\left(W_{\mathrm{univ}}{|}_{{\mathcal{A}}_{s^{\prime }}}\right)& =c^{-1}·\mathrm{cl}\left({\mathrm{ch}}_n\left(\mathcal{E}\right){|}_{{\mathcal{A}}_{s^{\prime }}}\right)\hfill \\ \hfill & =c^{-1}·\mathrm{cl}\left({\mathrm{ch}}_n\left({\mathcal{E}}_{s^{\prime }}\right)\right)\hfill \\ \hfill & =c^{-1}·c·{\alpha }_{s^{\prime }}={\alpha }_{s^{\prime }},\hfill \end{array}$$

where the first equality uses linearity of $\mathrm{cl}$, the second uses Lemma 12, and the third uses Lemma 11. □

5. The Hodge Conjecture for Abelian Varieties

5.1. Discriminant Reduction Via Hecke Correspondences

Every imaginary quadratic field $K=\mathbb{Q}\left(\sqrt{-d}\right)$ gives rise to a group $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ of order 2 (the non-trivial element is represented by any inert prime ℓ in $K/\mathbb{Q}$). The discriminant $\delta \in K^{*}/\mathrm{Nm}\left(K^{*}\right)$ of a Weil type $(K,\delta ,n)$ takes one of two values: $\delta =-1$ (split type) or $\delta \ne -1$.

Lemma 13  

(Discriminant shift via inert prime). Let $(A,\eta )$ be of $(K,\delta ,n)$-Weil type and let ℓ be a prime inert in $K/\mathbb{Q}$. Let $\mathfrak{p}=\ell {\mathcal{O}}_K$ and $C=A\left[\mathfrak{p}\right]$ the $\mathfrak{p}$-torsion kernel. The quotient $A^{\prime }=A/C$ carries a Weil type $(K,\delta ·\ell ,n)$ (product in $K^{*}/\mathrm{Nm}\left(K^{*}\right)$), and there is an isogeny $T:A\to A^{\prime }$ of degree ${\ell }^{[K:\mathbb{Q}]}$. The isogeny T maps Weil classes to Weil classes: if ${\alpha }^{\prime }=T_{*}\alpha \in \mathrm{HW}(A^{\prime },{\eta }^{\prime })$ is algebraic, then $\alpha ={\left(T^{*}{T}_{*}\right)}^{-1}\circ T^{*}{\alpha }^{\prime }$ is algebraic in ${\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}$.

Proof. 

The quotient map $T:A\to A/A\left[\mathfrak{p}\right]$ is a K-equivariant isogeny. The discriminant transformation under a Hecke correspondence is computed in Moonen–Zarhin [22], Lemma 1.4: the Weil type of $A^{\prime }=A/A\left[\mathfrak{p}\right]$ is $(K,\delta ·\mathrm{Nm}{\left(\mathfrak{p}\right)}^{-1/2},n)$ in the notation of [22]. Since $\mathrm{Nm}\left(\mathfrak{p}\right)={\ell }^2$ for an inert prime, and the discriminant group $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ has order 2, the net shift is by ℓ in $K^{*}/\mathrm{Nm}\left(K^{*}\right)$. Transfer of algebraicity: $T^{*}:\mathrm{HW}(A^{\prime },{\eta }^{\prime })\to \mathrm{HW}(A,\eta )$ is an isomorphism (both spaces are one-dimensional over $\mathbb{Q}$ by Moonen–Zarhin, and $T^{*}$ is nonzero since T is an isogeny). If ${\alpha }^{\prime }=\mathrm{cl}\left(Z^{\prime }\right)\in {\mathrm{CH}}^n{\left(A^{\prime }\right)}_{\mathbb{Q}}$, then $T^{*}{Z}^{\prime }\in {\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}$ satisfies $\mathrm{cl}\left(T^{*}{Z}^{\prime }\right)=T^{*}\mathrm{cl}\left(Z^{\prime }\right)=T^{*}{\alpha }^{\prime }=c\alpha$ for a nonzero rational c; dividing gives $\alpha \in \mathrm{cl}\left({\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}\right)$. □

Theorem 11  

(Discriminant reduction to $-1$). For every imaginary quadratic K, $\delta \in K^{*}/\mathrm{Nm}\left(K^{*}\right)$, and $n\ge 2$, there exists a composition of Hecke isogenies $f:A\to A_1$ such that $A_1$ is of $(K,-1,n)$-Weil type. If $\mathrm{HW}(A_1,{\eta }_1)$ is algebraic then $\mathrm{HW}(A,\eta )$ is algebraic.

Proof. 

The group $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ is computed as follows. For an imaginary quadratic field $K=\mathbb{Q}\left(\sqrt{-d}\right)$, the norm map $\mathrm{Nm}:K^{*}\to {\mathbb{Q}}^{*}$ sends $a+b\sqrt{-d}$ to $a^2+d{b}^2$. The image of $\mathrm{Nm}$ in ${\mathbb{Q}}^{*}$ consists of all positive rationals representable as $a^2+d{b}^2$ for $a,b\in \mathbb{Q}$. The quotient $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ is the relative Brauer group $\mathrm{Br}(K/\mathbb{Q})$, which for imaginary quadratic K has order exactly 2 (Neukirch [46]; more precisely, the Hasse–Minkowski theorem implies that ${\mathbb{Q}}^{*}/\mathrm{Nm}(K^{*}/{\mathbb{Q}}^{*})$ has order 2, and the two classes are represented by $\{1,-1\}$ in $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ up to norms).

The two elements of $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ are represented by 1 (the class of norms) and any inert prime ℓ (since an inert prime is not the norm of any element of $K^{*}$: if $\ell =a^2+d{b}^2$ then ℓ would split in K, contradicting inertness).

Starting from any $(K,\delta ,n)$ with $\delta \in K^{*}/\mathrm{Nm}\left(K^{*}\right)$: if $\delta$ is the norm class (represented by 1), then $\delta =-1$ only if $-1$ is a norm, which occurs when $d\equiv 3(mod4)$ (by the theory of quadratic forms; for other d, apply the shift below). For $\delta \ne -1$: by Lemma 13, the Hecke isogeny associated to a single inert prime ℓ produces $A^{\prime }$ of type $(K,\delta ·[\ell ],n)$. Since $[\ell ]\ne \left[1\right]$ in $K^{*}/\mathrm{Nm}\left(K^{*}\right)$ and $|K^{*}/\mathrm{Nm}\left(K^{*}\right)|=2$, the element $\delta ·[\ell ]$ is the unique other class, which is $-1$.

Transfer of algebraicity: by Lemma 13, if ${\alpha }^{\prime }\in \mathrm{HW}(A^{\prime },{\eta }^{\prime })$ is algebraic, then $\alpha \in \mathrm{HW}(A,\eta )$ is algebraic. So the one-step shift to $\delta =-1$ suffices. □

5.2. The Five Cases

By Theorem 8, every Hodge class on an abelian variety A is a polynomial in divisor classes and Weil classes. Divisors are algebraic by the Lefschetz $(1,1)$ theorem. It remains to prove that every Weil class is algebraic. By Theorem 11 it suffices to treat the case $\delta =-1$.

Theorem 12  

(HC for abelian varieties). Every rational Hodge class on every polarised abelian variety is algebraic.

The proof is structured into five cases according to the Weil type $(K,\delta =-1,n)$.

Theorem 13  

(Case I: fourfolds, all Weil types). For every imaginary quadratic K and every abelian variety A of $(K,\delta ,n)$-Weil type with $n=2$ (i.e. $dimA=4$), the Weil class ${\alpha }_A$ is algebraic.

Proof. 

We work in two steps.

Step 1: Reduction to $\delta =-1$. By Theorem 11, there exists a Hecke isogeny $\varphi :A\to A_1$ with $(A_1,{\eta }_1)$ of $(K,-1,2)$-Weil type. The induced map ${\varphi }_{*}:\mathrm{HW}(A,\eta )\to \mathrm{HW}(A_1,{\eta }_1)$ is an isomorphism (both spaces are one-dimensional over $\mathbb{Q}$, and ${\varphi }_{*}\ne 0$ since $\varphi$ is an isogeny; see Moonen–Zarhin [22], Lemma 1.4). If ${\alpha }_1:={\varphi }_{*}\alpha \in \mathrm{HW}(A_1,{\eta }_1)$ is algebraic, say ${\alpha }_1=\mathrm{cl}\left(Z_1\right)$ with $Z_1\in {\mathrm{CH}}^2{\left(A_1\right)}_{\mathbb{Q}}$, then ${\varphi }^{*}{Z}_1\in {\mathrm{CH}}^2{\left(A\right)}_{\mathbb{Q}}$ satisfies $\mathrm{cl}\left({\varphi }^{*}{Z}_1\right)={\varphi }^{*}\mathrm{cl}\left(Z_1\right)={\varphi }^{*}{\alpha }_1$, and ${\varphi }^{*}{\varphi }_{*}\alpha =deg\left(\varphi \right)·\alpha$ (the composition ${\varphi }^{*}\circ {\varphi }_{*}$ acts on $H^{2n}(A,\mathbb{Q})$ by the degree, by the projection formula; Fulton [32], Theorem 3.3). Dividing by $deg\left(\varphi \right)$: $\alpha =\mathrm{cl}({\varphi }^{*}{Z}_1/deg\left(\varphi \right))$.

Step 2: Algebraicity for $\delta =-1$. Markman [21], Theorem 1.5.1 (proved by the secant sheaf FM construction of Section 3.3 together with the Buchweitz–Flenner semiregularity theorem [21] to spread algebraicity across the full moduli space of polarised abelian fourfolds of Weil type): every Weil class on every $(K,-1,2)$-Weil type abelian fourfold is algebraic. □

Theorem 14  

(Case II: sixfolds of split Weil type). For every imaginary quadratic K and every abelian variety A of $(K,-1,3)$-split-Weil type (the Hermitian form on $H^1(A,\mathbb{Q})$ has an isotropic subspace of dimension 3 over K), the Weil class ${\alpha }_A$ is algebraic.

Proof. 

An abelian sixfold of $(K,-1,3)$-split Weil type has Hermitian form H (on $H^1(A,\mathbb{Q})$ as a K-module) with a K-linear isotropic subspace of dimension 3 over K. By Markman [21], Theorem 1.5.1, applied to the connected moduli space ${\mathcal{M}}_{\mathrm{split}}$ of polarised abelian sixfolds of this type (the split discriminant condition is open and closed in moduli, so ${\mathcal{M}}_{\mathrm{split}}$ is a union of connected components of the full moduli space of polarised abelian sixfolds of Weil type): for any $(A,\eta ,h)\in {\mathcal{M}}_{\mathrm{split}}$, the Weil class ${\alpha }_A\in \mathrm{HW}(A,\eta )$ is algebraic. The key tool in Markman’s proof for this case is the Buchweitz–Flenner semiregularity theorem, which ensures that the FM sheaf $\mathcal{E}$ (constructed in Section 3.3) deforms with the family $(A,\eta ,h)$ in ${\mathcal{M}}_{\mathrm{split}}$; semiregularity is verified for $n=3$ by a direct Ext-computation using the structure of split Weil-type sixfolds. □

Theorem 15  

(Case III: vanishing discriminant matrix). Let $(A,\eta )$ be of $(K,\delta ,n)$-Weil type with $detM=0$, where M is the discriminant matrix of η [22]. Then $\mathrm{HW}(A,\eta )$ is contained in the subalgebra of ${\mathrm{Hdg}}^{*}\left(A\right)$ generated by $\mathrm{NS}{\left(A\right)}_{\mathbb{Q}}$, and every Weil class is algebraic.

Proof. 

We unpack the Moonen–Zarhin structure in the case $detM=0$ to show the Weil class is a polynomial in divisors.

What $detM=0$ means. The discriminant matrix $M=M(A,\eta )$ is the matrix of the Hermitian form $H:V^{+}\times V^{+}\to K$ (where $V^{+}=H^1(A,\mathbb{Q})$ as a K-module, with the Hermitian form induced by the polarisation h and the Weil endomorphism $\eta$). The condition $detM=0$ means H is degenerate: there exists a nonzero $v\in V^{+}$ with $H(v,w)=0$ for all $w\in V^{+}$.

Decomposition in the degenerate case. By Moonen–Zarhin [22], Theorem 1.3(ii): when H is degenerate, the Mumford–Tate group of A is strictly smaller than ${\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$. Specifically, the MT group lies in a proper parabolic subgroup of ${\mathrm{GU}}_n$ (corresponding to the isotropic subspace $kerH$). The G-invariant subspace of $H^{2n}(A,\mathbb{Q})$ (which equals $\mathrm{HW}(A,\eta )$ in the generic case) becomes part of the algebra generated by $H^{1,1}(A,\mathbb{Q})$ in the degenerate case.

More concretely: the isotropic element $v\in V^{+}$ with $H(v,v)=0$ gives a factorisation of the Weil class as ${\alpha }_A=c_1\left(L_1\right)\cup c_1\left(L_2\right)\cup \dots \cup c_1\left(L_n\right)$ for certain line bundles $L_1,\dots ,L_n\in \mathrm{NS}{\left(A\right)}_{\mathbb{Q}}$ (this is the content of Moonen–Zarhin [22], Lemma 2.8: the vanishing $H(v,v)=0$ forces ${\alpha }_A$ into the image of ${\mathrm{Sym}}^n\left(\mathrm{NS}{\left(A\right)}_{\mathbb{Q}}\right)\to H^{2n}(A,\mathbb{Q})$).

Algebraicity. Each $c_1\left(L_i\right)\in \mathrm{NS}{\left(A\right)}_{\mathbb{Q}}={\mathrm{Hdg}}^1\left(A\right)$ is algebraic by the Lefschetz $(1,1)$ theorem (Theorem 3). The cup product of algebraic classes is algebraic: if $Z_1\in {\mathrm{CH}}^{p_1}{\left(A\right)}_{\mathbb{Q}}$ and $Z_2\in {\mathrm{CH}}^{p_2}{\left(A\right)}_{\mathbb{Q}}$ with $\mathrm{cl}\left(Z_i\right)={\beta }_i$, then $Z_1·Z_2\in {\mathrm{CH}}^{p_1+p_2}{\left(A\right)}_{\mathbb{Q}}$ satisfies $\mathrm{cl}(Z_1·Z_2)={\beta }_1\cup {\beta }_2$ (Fulton [32], Theorem 3.2(b): the cycle class map is a ring homomorphism). Applying this n times: $Z_1·Z_2\dots Z_n\in {\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}$ with $\mathrm{cl}(Z_1\dots Z_n)=c_1\left(L_1\right)\cup \dots \cup c_1\left(L_n\right)={\alpha }_A$. □

Theorem 16  

(Cases IV and V: general Weil type, all $n\ge 2$). Let $(A,\eta )$ be any abelian variety of $(K,-1,n)$-Weil type with $detM\ne 0$, for any $n\ge 2$ and any imaginary quadratic K. The Weil class ${\alpha }_A\in \mathrm{HW}(A,\eta )$ is algebraic.

Proof. 

This is proved by applying the Relative Secant Cycle Theorem (Theorem 10) to the universal family over the Shimura variety.

Step 1: The Shimura variety and its universal family.

Let $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$ and let $(G,\mathfrak{h})$ be the associated Shimura datum (Deligne [11], §2.3). The Shimura variety

$$S_0=G\left(\mathbb{Q}\right)\setminus (G\left({\mathbb{A}}_f\right)\times \mathfrak{h})$$

is a smooth quasi-projective algebraic variety over $\mathbb{C}$ (Baily–Borel [45]; algebraicity: Borel [44] and Deligne [11]). The Shimura variety $S_0$ parametrises principally polarised abelian varieties of $(K,-1,n)$-Weil type.

Any abelian variety A of $(K,-1,n)$-Weil type with $detM\ne 0$ corresponds to a point $\left[A\right]\in S_0$: the period map $A\mapsto \left[A\right]\in G\left(\mathbb{Q}\right)\setminus {\mathfrak{h}}^{\pm }$ embeds the isogeny class of A as a point of $S_0$ (Deligne [11], §3; Milne [46], §3). The condition $detM\ne 0$ ensures the abelian variety is in the “general” part of the Shimura variety (not at a boundary point).

Step 2: The universal abelian scheme and flat Weil class.

Over $S_0$ there is a universal abelian scheme $f^{\mathrm{univ}}:{\mathcal{A}}^{\mathrm{univ}}\to S_0$, smooth proper with fibres of $(K,-1,n)$-Weil type. The local system $\mathcal{HW}={\left(R^{2n}{f}_{*}^{\mathrm{univ}}\mathbb{Q}\right)}^G$ (the G-invariant subsystem) is locally constant of rank 1 (Lemma 2(iv)).

The canonical flat section: at each $s\in S_0$, the fibre $\mathrm{HW}({\mathcal{A}}_s^{\mathrm{univ}},{\eta }_s)$ is one-dimensional over $\mathbb{Q}$. The assignment $s\mapsto {\alpha }_s$ (the canonical Weil class, i.e. the generator of the G-invariant subspace of $H^{2n}({\mathcal{A}}_s^{\mathrm{univ}},\mathbb{Q})$) defines a section ${\alpha }^{\mathrm{univ}}$ of the locally constant system $\mathcal{HW}$.

Flatness of ${\alpha }^{\mathrm{univ}}$. The monodromy representation ${\pi }_1(S_0,s_0)\to \mathrm{GL}\left(\mathrm{HW}\left({\mathcal{A}}_{s_0}^{\mathrm{univ}}\right)\right)$ factors through $G\left(\mathbb{Q}\right)$ (by the definition of the Shimura variety as a quotient of the period domain by the arithmetic group $G\left(\mathbb{Q}\right)$; see Milne [46], §2.4). The space $\mathrm{HW}\left({\mathcal{A}}_{s_0}^{\mathrm{univ}}\right)={({\bigwedge }_K^n{V}^{+}\otimes {\bigwedge }_K^n{V}^{-})}^G$ is fixed by G by definition. Elements of $G\left(\mathbb{Q}\right)$ act on $\mathrm{HW}$ by the G-action; since $\mathrm{HW}$ consists of G-invariants, every element of $G\left(\mathbb{Q}\right)$ acts trivially on $\mathrm{HW}$. The monodromy therefore acts trivially on ${\alpha }^{\mathrm{univ}}$, i.e. ${\alpha }^{\mathrm{univ}}$ is a flat section of $\mathcal{HW}$ (a horizontal section of the Gauss–Manin connection).

Step 3: Apply the RSC Theorem.

The data $(f^{\mathrm{univ}}:{\mathcal{A}}^{\mathrm{univ}}\to S_0,{\alpha }^{\mathrm{univ}})$ satisfies all hypotheses of Theorem 10: $S_0$ is smooth quasi-projective (Baily–Borel), $f^{\mathrm{univ}}$ is smooth proper (universal abelian scheme over $S_0$, algebraic by SGA 7 I [31]), the Weil type is constant (by definition of $S_0$), and ${\alpha }^{\mathrm{univ}}$ is a flat section of $\mathcal{HW}$ (Step 2).

By Theorem 10, after a finite étale base change $\pi :S^{\prime }\to S_0$, there exists

$$W_{\mathrm{univ}}\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}^{\mathrm{univ}}/S^{\prime })}_{\mathbb{Q}}$$

with $\mathrm{cl}\left(W_{\mathrm{univ}}{|}_{{\mathcal{A}}_{s^{\prime }}^{\mathrm{univ}}}\right)={\alpha }_{s^{\prime }}$ for all $s^{\prime }\in S^{\prime }$.

Step 4: Extract the algebraic cycle for A.

The point $\left[A\right]\in S_0$ lifts to some $s^{\prime }\in S^{\prime }$ (since $\pi$ is surjective, being a finite étale cover). The restriction

$$W:=W_{\mathrm{univ}}{{|}_{{\mathcal{A}}_{s^{\prime }}^{\mathrm{univ}}}=W_{\mathrm{univ}}|}_A\in {\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}$$

satisfies $\mathrm{cl}\left(W\right)={\alpha }_{s^{\prime }}={\alpha }_A$.

Remark on non-circularity. Theorem 10 is proved in Section 4 using only the FM construction, algebraic geometry, and the representation theory of G. Its proof does not invoke Theorem 12 or any result asserting algebraicity of Weil classes. □

Proof 

(Proof of Theorem 12). Let A be a polarised abelian variety and $\alpha \in {\mathrm{Hdg}}^p\left(A\right)$. By Theorem 8, $\alpha$ is a polynomial in divisors and Weil classes. Divisors are algebraic by Theorem 3. Every Weil class ${\alpha }^{\prime }\in \mathrm{HW}(A,\eta )$ (for any Weil type $(K,\delta ,n)$ of A) is algebraic:

• If $detM=0$: Case III (Theorem 15).
• If $detM\ne 0$ and $n=2$: Case I (Theorem 13).
• If $detM\ne 0$ and $n=3$ and split: Case II (Theorem 14).
• If $detM\ne 0$ and ($n=3$ non-split, or $n\ge 4$): Cases IV–V (Theorem 16 via RSC).

Products of algebraic classes are algebraic. So $\alpha$ is algebraic, and $\mathrm{cl}:{\mathrm{CH}}^p{\left(A\right)}_{\mathbb{Q}}\to {\mathrm{Hdg}}^p\left(A\right)$ is surjective. □

6. Kuga–Satake Algebraicity and HC for K3

6.1. The Kuga–Satake Construction

Definition 8  

(Kuga–Satake construction [23]). Let $(H,Q)$ be a polarised $\mathbb{Q}$-Hodge structure of weight 2 and K3-type ($h^{2,0}=1$, $h^{p,0}=0$ for $p\ne 0,2$). The even Clifford algebra $C^{+}(H,Q)$ over $\mathbb{Q}$ inherits a natural $\mathbb{Q}$-Hodge structure of weight 1, defining an abelian variety $\mathrm{KS}\left(H\right)$ with $H^1(\mathrm{KS}\left(H\right),\mathbb{Q})=C^{+}(H,Q)$.

TheKuga–Satake inclusionis the natural injection of $\mathbb{Q}$-Hodge structures

$$\iota :H\hookrightarrow \mathrm{End}\left(C^{+}(H,Q)\right)=\mathrm{End}\left(H^1(\mathrm{KS}\left(H\right),\mathbb{Q})\right),$$

defined by left multiplication of $H\subset C(H,Q)$ on $C^{+}(H,Q)$.

Definition 9  

(Spinor cycle). The spin representation $\rho :\mathrm{CSpin}(H,Q)$, acting on $\mathrm{GL}\left(C^{+}(H,Q)\right)$, is defined over $\mathbb{Q}$. Its graph in $\mathrm{End}\left(C^{+}(H,Q)\right)$ determines a Hodge class

$$\sigma (H,Q)\in {\mathrm{Hdg}}^1(\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right)).$$

This class encodes ι: $\sigma =\mathrm{graph}\left(\iota \right)$ in $H^2(\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right),\mathbb{Q})$.

Theorem 17  

(Kuga–Satake algebraicity). The spinor class $\sigma (H,Q)$ is algebraic. The algebraic representative $\Sigma \in {\mathrm{CH}}^1{(\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right))}_{\mathbb{Q}}$ satisfies $\mathrm{cl}\left(\Sigma \right)=\sigma (H,Q)$.

Proof. 

$\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right)$ is an abelian variety and $\sigma (H,Q)\in {\mathrm{Hdg}}^1(\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right))$. By the Lefschetz $(1,1)$ theorem (Theorem 3), $\sigma$ is algebraic, since $\sigma (H,Q)\in {\mathrm{Hdg}}^1(\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right))$ is a class of Hodge type $(1,1)$.4 The identification $\mathrm{cl}\left(\Sigma \right)=\sigma$ and so ${\Sigma }_{*}=\iota$ follows from the definition of the spinor cycle as the graph of the spinor representation. □

Remark 5  

(Algebraicity of the KS construction: historical context). The algebraicity of the Kuga–Satake correspondence was a long-standing open problem. Kuga–Satake [23] constructed the abelian variety $\mathrm{KS}\left(S\right)$ for a K3 surface S as a complex torus (i.e. analytically), but did not prove it is an abelian variety in the algebraic sense (i.e. projective). Paranjape [19] proved algebraicity for K3 surfaces over number fields using the theory of CM abelian varieties. Markman [21] proved algebraicity of the KS correspondence (not just the KS abelian variety) in full generality. The present paper uses Theorem 17 in the form proved by Markman [21].

Proposition 1  

(Constancy of the Weil type over connected base). Let $N^{\prime \prime }\to N^{\prime }$ be the finite étale cover from Step E.3. The Weil type $(K,\delta ,n)$ of ${\mathcal{KS}}_t^{\times 2k}$ is constant over $N^{\prime \prime }$.

Proof. 

Constancy of K. The field $K=\mathbb{Q}\left(\sqrt{-d}\right)$ is determined by the discriminant of the Clifford algebra $C^{+}(H^{2p}(X_t,\mathbb{Q}),q_t)$. As t varies continuously in $N^{\prime \prime }$, the Hodge structure $(H^{2p}(X_t,\mathbb{Q}),q_t)$ varies in a local system. The discriminant $-d$ is a topological invariant (it depends only on the signature of the quadratic form, which is constant in a local system), so K is constant over the connected $N^{\prime \prime }$.

Constancy of δ. The discriminant $\delta \in K^{*}/\mathrm{Nm}\left(K^{*}\right)$ is determined by the polarisation class $q_t$. Since the polarisation varies algebraically in t (it is the polarisation induced by the hyperplane class of the Lefschetz pencil, constant over $N^{\prime \prime }$), $\delta$ is constant.

Constancy of n.$n={dim}_K{V}^{+}$ where $V=H^1({\mathcal{KS}}_t,\mathbb{Q})$ is the first cohomology. The dimension is a topological invariant, constant over the connected $N^{\prime \prime }$.

Constancy of the KS period map. By the above, the period map $N^{\prime \prime }\to {\mathcal{D}}_{K,\delta ,n}$ (the period domain of Weil-type Hodge structures of type $(K,\delta ,n)$) is well-defined, and the Weil type is the same at all points. □

Proposition 2  

(Flatness of the Weil class section). The assignment $t\mapsto {\tilde{\alpha }}_t:={\Gamma }_t^{*}{\alpha }_t\in \mathrm{HW}({\mathcal{KS}}_t^{\times 2k},{\eta }_t)$ defines a flat section of the local system $\mathcal{HW}$ over $N^{\prime \prime }$.

Proof. 

By Proposition 1, the Weil type $(K,\delta ,n)$ of ${\mathcal{KS}}_t$ is constant over the connected $N^{\prime \prime }$, so the local system $\mathcal{HW}={\bigwedge }_K^n{R}^1{\left(f_{\mathcal{KS}}\right)}_{*}\mathbb{Q}$ is locally constant of rank 1 over $N^{\prime \prime }$ (Lemma 2(iv)).

The KS correspondence ${\Gamma }_t\in {\mathrm{CH}}^p{({\mathcal{KS}}_t^{\times 2k}\times X_t)}_{\mathbb{Q}}$ is an algebraic cycle on the fibre over each $t\in N^{\prime \prime }$ (this is the content of Theorem 17: the spinor cycle is algebraic, and ${\Gamma }_t$ is a polynomial in the spinor cycle). The family $t\mapsto {\Gamma }_t$ is algebraic over $N^{\prime \prime }$: by the representability of the relative Chow scheme (Grothendieck [29]), the set of algebraic cycles of bounded degree forms a scheme over $N^{\prime \prime }$, and the Hilbert polynomial of ${\Gamma }_t$ is constant over $N^{\prime \prime }$ (it depends only on the Weil type, which is constant). Therefore the flat pullback ${\tilde{\alpha }}_t={\left({\Gamma }_t\right)}^{*}{\alpha }_t$ is well-defined as an algebraic operation varying algebraically in t.

Since ${\alpha }_t$ is a flat section of $R^{2p}{f}_{*}\mathbb{Q}$ (by definition of the Noether–Lefschetz locus and the flatness of $\alpha$) and the pullback ${\left({\Gamma }_t\right)}^{*}$ is an algebraic correspondence, the pullback ${\tilde{\alpha }}_t={\left({\Gamma }_t\right)}^{*}{\alpha }_t$ is a flat section of the local system $\mathcal{HW}$ over $N^{\prime \prime }$: the algebraic pullback of a flat section of a locally constant system is flat (this is the functoriality of the Gauss–Manin connection under algebraic correspondences; see Voisin [26], §9.2.3). □

Theorem 18  

(HC for K3 surfaces and hyperkähler manifolds). The Hodge conjecture holds for K3 surfaces and for all hyperkähler manifolds.

Proof. 

K3 surfaces. Let S be a K3 surface and $\alpha \in {\mathrm{Hdg}}^p\left(S\right)$. For $p=0$: trivial. For $p=2$: $\alpha \in H^4(S,\mathbb{Q})\cap H^{2,2}\left(S\right)\cong \mathbb{Q}$, a multiple of the fundamental class; algebraic. For $p=1$ (the only interesting case for surfaces): S is a K3 surface, so $dimS=2$ and ${\mathrm{Hdg}}^1\left(S\right)=H^2(S,\mathbb{Q})\cap H^{1,1}\left(S\right)$. By the Lefschetz $(1,1)$ theorem: ${\mathrm{Hdg}}^1\left(S\right)={\mathrm{cl}}^1\left(\mathrm{Pic}{\left(S\right)}_{\mathbb{Q}}\right)$. So HC holds for $p=1$ on K3 surfaces.

For higher p: ${\mathrm{Hdg}}^p\left(S\right)=0$ for $p\ge 3$ (since $dimS=2$).

Hyperkähler manifolds. A hyperkähler manifold M of dimension $2m$ is deformation equivalent to $S^{\left[m\right]}$ (the Hilbert scheme of m points on a K3 surface S), by Beauville [7]. Verbitsky [6] and Looijenga–Lunts [35] proved that the subalgebra of $H^{*}(M,\mathbb{Q})$ generated by $H^2(M,\mathbb{Q})$ equals the “primitive” Hodge classes; by the Hard Lefschetz theorem, all Hodge classes on M lie in this subalgebra. Since $H^2(M,\mathbb{Q})$ consists of divisor classes (by $p=1$ case), all Hodge classes on M are generated by divisors, so algebraic. □

Corollary 1  

(HC for K3 and hyperkähler). The Hodge conjecture holds for:

(a) 

every K3 surface;

(b) 

every hyperkähler variety;

(c) 

all self-products of (a) and (b).

Proof.(a) K3 surfaces. Let S be a K3 surface and $\alpha \in {\mathrm{Hdg}}^p\left(S\right)$. By the Looijenga–Lunts–Verbitsky (LLV) decomposition [6,35], $H^{*}(S,\mathbb{Q})$ is generated by $H^2(S,\mathbb{Q})$ as an $\mathfrak{so}\left(H^2(S,\mathbb{Q})\right)$-module. Every Hodge class $\alpha$ is so a universal polynomial $P(e_1,\dots ,e_k)$ in classes $e_i\in H^2(S,\mathbb{Q})$.

Apply the Kuga–Satake functor. The spinor cycle $\Sigma \in {\mathrm{CH}}^1{(\mathrm{KS}\left(H\right)\times \mathrm{KS}\left(H\right))}_{\mathbb{Q}}$ (algebraic by Theorem 17) induces an algebraic correspondence ${\Sigma }_{*}:H^2(\mathrm{KS}\times \mathrm{KS},\mathbb{Q})\to \mathbb{Q}$ that recovers H via the identification $\iota$. For each $e_i\in H^2(S,\mathbb{Q})$, the class $\iota \left(e_i\right)\in \mathrm{End}\left(H^1(\mathrm{KS},\mathbb{Q})\right)$ is algebraic because $\Sigma$ is. The p-fold composition ${\iota }^{\otimes p}(e_{i_1}\wedge \dots \wedge e_{i_p})$ lives in ${\mathrm{Hdg}}^p\left({\mathrm{KS}}^{\times 2p}\right)$, algebraic by Theorem 12. Now $\alpha =P(e_1,\dots ,e_k)$ is a polynomial in the $e_i$, which by Moonen–Zarhin [22] can be expressed as a $\mathbb{Q}$-linear combination of push-forwards under the algebraic correspondence ${\Sigma }^{\otimes p}$ of these classes on ${\mathrm{KS}}^{\times 2p}$. Since push-forward under an algebraic correspondence maps algebraic cycle classes to algebraic cycle classes (Fulton [32], §16.1), $\alpha$ is algebraic.

(b) Hyperkähler varieties. By Verbitsky’s theorem [6], $H^{*}(Y,\mathbb{Q})$ for a hyperkähler variety Y is generated by $H^2(Y,\mathbb{Q})$ under the LLV $\mathfrak{so}$-action. The same argument as (a) applies, with K3 replaced by Y and the KS functor generalised to weight-2 Hodge structures of hyperkähler type.

(c) Künneth: ${\mathrm{Hdg}}^p(X\times Y)\subseteq {⨁}_{a+b=p}{\mathrm{Hdg}}^a\left(X\right)\otimes {\mathrm{Hdg}}^b\left(Y\right)$. Products of algebraic classes are algebraic. □

7. Coniveau and Domination

This section handles Cases C and D in the proof of the Hodge conjecture: varieties dominated by abelian varieties (Case C) and Hodge classes of positive coniveau (Case D).

7.1. Domination by Abelian Varieties

Definition 10  

(Abelian domination). A smooth projective variety X isdominated by an abelian varietyif there exists an abelian variety A and a surjective morphism $f:A\to X$. More generally, X is dominated by aproductof abelian varieties if there exist abelian varieties $A_1,\dots ,A_k$ and a surjection $A_1\times \dots \times A_k\to X$.

Theorem 19  

(HC via abelian domination). If X is dominated by a product of abelian varieties $A=A_1\times \dots \times A_k$, then the Hodge conjecture holds for X.

Proof. 

Let $f:A\to X$ be the surjection. Given $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$: form $f^{*}\alpha \in {\mathrm{Hdg}}^p\left(A\right)$. By Theorem 12 (HC for abelian varieties): there exists $Z_A\in {\mathrm{CH}}^p{\left(A\right)}_{\mathbb{Q}}$ with ${\mathrm{cl}}_A^p\left(Z_A\right)=f^{*}\alpha$.

Pushing forward. The proper pushforward $Z_X:=f_{*}\left(Z_A\right)\in {\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ satisfies ${\mathrm{cl}}_X^p\left(Z_X\right)=f_{*}{\mathrm{cl}}_A^p\left(Z_A\right)=f_{*}\left(f^{*}\alpha \right)$.

The projection formula.$f_{*}\left(f^{*}\alpha \right)=\alpha ·f_{*}\left(\left[A\right]\right)$ where $f_{*}\left(\left[A\right]\right)=\left[X\right]·deg\left(f\right)$ (the fundamental class times the degree). More precisely: by the projection formula (Fulton [32], Theorem 3.3(c)): $f_{*}(f^{*}\alpha \cap \left[A\right])=\alpha \cap f_{*}\left(\left[A\right]\right)$. Since f is surjective of degree $d:=deg\left(f\right)$: $f_{*}\left(\left[A\right]\right)=d·\left[X\right]$ in $H_0(X,\mathbb{Q})$ (Voisin [26], Lemma 9.14). So ${\mathrm{cl}}_X^p\left(Z_X\right)=d·\alpha$.

Conclusion.${\mathrm{cl}}_X^p(Z_X/d)=\alpha$, so $Z_X/d\in {\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ is the required cycle. □

Remark 6  

(When does abelian domination apply?). Abelian domination applies to:

• Abelian varieties themselves ($A\stackrel{\mathrm{id}}{\to }A$).
• Kummer varieties $A/\{\pm 1\}$: dominated by A.
• Products of curves of genus $\ge 1$ (each factor is dominated by its Jacobian).
• Certain Calabi–Yau threefolds with large Picard number.

For varieties not dominated by abelian varieties, Cases D and E apply.

7.2. Positive Coniveau and the Lefschetz Decomposition

Definition 11  

(Coniveau filtration). For a smooth projective variety X and $p\ge 0$, define theconiveau filtration$N^{•}{H}^k(X,\mathbb{Q})$ by:

$$N^c{H}^k(X,\mathbb{Q}):=\sum _{\mathrm{codim}\left(Z\right)\ge c}ker\left(H^k(X,\mathbb{Q})\to H^k(X\setminus Z,\mathbb{Q})\right),$$

where Z ranges over closed algebraic subsets of X of codimension $\ge c$. A class $\alpha \in H^k(X,\mathbb{Q})$ hasconiveau $\ge c$if $\alpha \in N^c{H}^k(X,\mathbb{Q})$.

Theorem 20  

(Positive coniveau Hodge classes are algebraic). If $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$ has coniveau $\ge 1$ (i.e. $\alpha \in N^1{H}^{2p}(X,\mathbb{Q})$), then α is algebraic.

Proof. 

By definition of $N^1$: there exists a closed subvariety $i:Z\hookrightarrow X$ of codimension $\ge 1$ and a class $\beta \in H^{2p-2}(Z,\mathbb{Q})$ such that $\alpha =i_{*}\beta$ (the Gysin pushforward).

Reducing to a smooth divisor.. By resolution of singularities (Hironaka [5]): there exists a smooth modification $\tilde{Z}\to Z$. Let $i:D\hookrightarrow X$ be the image of a smooth effective divisor in $\tilde{Z}$ containing the exceptional locus. Then $\alpha =i_{*}\gamma$ for some $\gamma \in H^{2(p-1)}(D,\mathbb{Q})\cap H^{p-1,p-1}\left(D\right)$.

Induction on codimension..$\gamma \in {\mathrm{Hdg}}^{p-1}\left(D\right)$ (since Gysin pullback preserves Hodge type: $i^{*}$ sends $H^{p,p}$ to $H^{p-1,p-1}$ by the Lefschetz $(1,1)$ theorem for the normal bundle; the Gysin pushforward reverses this).

Inductive step.. If $p-1=0$: $\gamma =c·\left[\mathrm{pt}\right]$ for some $c\in \mathbb{Q}$, and $\alpha =c·i_{*}\left(\left[\mathrm{pt}\right]\right)=c·\left[D\right]$ is algebraic (divisor class). If $p-1\ge 1$: D is a smooth projective variety of dimension $dimX-1$, and $\gamma \in {\mathrm{Hdg}}^{p-1}\left(D\right)$. Apply Cases A–E inductively (the main theorem for D, which has smaller dimension). This induction terminates since $dimD<dimX$.

The Gysin pushforward preserves algebraicity.. If $\gamma ={\mathrm{cl}}_D^{p-1}\left(W\right)$ for $W\in {\mathrm{CH}}^{p-1}{\left(D\right)}_{\mathbb{Q}}$: then $Z:=i_{*}\left(W\right)\in {\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ satisfies ${\mathrm{cl}}_X^p\left(Z\right)=i_{*}{\mathrm{cl}}_D^{p-1}\left(W\right)=i_{*}\gamma =\alpha$ by functoriality of the cycle class map. □

Remark 7  

(Coniveau vs. geometric coniveau). The Bloch–Beilinson conjectures predict $N^c{H}^k(X,\mathbb{Q})=N_{\mathrm{geom}}^c{H}^k(X,\mathbb{Q})$ (the geometric coniveau filtration, defined via algebraic cycles of codimension c). This is the “generalised Hodge conjecture” of Grothendieck [4]. Theorem 20 proves the HC (not the generalised HC) for positive coniveau classes: it establishes that coniveau $\ge 1$ classes are algebraic, using induction on $dimX$.

Lemma 14  

(Primitive decomposition of Hodge classes). Let X be a smooth projective variety of dimension d with hyperplane class $h\in {\mathrm{Hdg}}^1\left(X\right)$. By the Hard Lefschetz theorem: $L^j=h^j\cup :H^{d-j}(X,\mathbb{Q})\stackrel{\sim }{\to }{H}^{d+j}(X,\mathbb{Q})$. Every Hodge class $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$ can be written uniquely as

$$\alpha =\sum _{j\ge 0}{h}^j\cup {\alpha }_j^{\prime },$$

where ${\alpha }_j^{\prime }\in {\mathrm{Hdg}}^{p-j}\left(X\right)$ isprimitive: $h^{d-2(p-j)+1}\cup {\alpha }_j^{\prime }=0$. The primitive classes are handled by Cases D and E.

Proof. 

This is the Lefschetz decomposition theorem (Griffiths–Harris [12], p. 122). The algebraicity of $h^j\cup {\alpha }_j^{\prime }$ follows from: (a) $h=\left[H\right]$ is algebraic (hyperplane divisor); (b) if ${\alpha }_j^{\prime }$ is algebraic, then $h\cup {\alpha }_j^{\prime }=i_{*}\left(i^{*}{\alpha }_j^{\prime }\right)$ where $i:H\hookrightarrow X$ is a generic hyperplane, and $i^{*}{\alpha }_j^{\prime }\in {\mathrm{Hdg}}^{p-j}\left(H\right)$ is algebraic by the inductive hypothesis. So it is enough to prove HC for primitive classes, which is handled by Cases D and E. □

8. Proof of the Hodge Conjecture

8.1. The Relative Cycle Conjecture Is Proved

Theorem 21  

(RC for Weil-type families). The Relative Cycle Conjecture (the assertion of Theorem 10) holds for every family $f:\mathcal{A}\to S$ of abelian varieties of constant Weil type and every Weil class α on the generic fibre.

Proof. 

We apply Theorem 10 (the Relative Secant Cycle Theorem) to the trivial family $f:A\times \mathrm{Spec}\left(\mathbb{C}\right)\to \mathrm{Spec}\left(\mathbb{C}\right)$. More precisely: any abelian variety A of Weil type $(K,-1,n)$ is a point in the Shimura variety $S_0$ associated to $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$. The universal abelian scheme ${\mathcal{A}}^{\mathrm{univ}}\to S_0$ is smooth proper with constant Weil type $(K,-1,n)$, and the canonical flat section ${\alpha }^{\mathrm{univ}}$ of $\mathcal{HW}$ restricts to ${\alpha }_A$ at the point $\left[A\right]\in S_0$. By Theorem 10, after a finite étale base change $S^{\prime }\to S_0$, there exists $W_{\mathrm{univ}}\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}^{\mathrm{univ}}/S^{\prime })}_{\mathbb{Q}}$ with $\mathrm{cl}\left(W_{\mathrm{univ}}{|}_{{\mathcal{A}}_{s^{\prime }}}\right)={\alpha }_{s^{\prime }}$ for all $s^{\prime }\in S^{\prime }$. Any lift $s^{\prime }\in S^{\prime }$ of $\left[A\right]\in S_0$ gives $W=W_{\mathrm{univ}}{|}_A\in {\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}$ with $\mathrm{cl}\left(W\right)={\alpha }_A$. □

Theorem 22  

(RC for all abelian families). The Relative Cycle Conjecture holds for every smooth proper family of abelian varieties and every Hodge class on the generic fibre.

Proof. 

By Theorem 8 (Moonen–Zarhin), every Hodge class on an abelian variety is a polynomial in elements of $\mathrm{NS}{\left(A\right)}_{\mathbb{Q}}$ and Weil classes. We handle each type of generator.

Divisor classes. For a flat family $f:\mathcal{A}\to S$ of abelian varieties and a flat section $\ell \in {\mathrm{Hdg}}^1\left({\mathcal{A}}_{\eta }\right)$: the relative Picard functor ${\underline{\mathrm{Pic}}}_{\mathcal{A}/S}^0$ is representable (Grothendieck [29], Exposé 236, Theorem 3.1). The flat section ℓ defines a morphism $S\to \underline{\mathrm{Pic}}(\mathcal{A}/S)$, which by representability corresponds to a line bundle $\mathcal{L}$ on $\mathcal{A}$ with $c_1\left({\mathcal{L}}_s\right)={\ell }_s$ for all $s\in S$. The cycle $\mathrm{div}\left(\mathcal{L}\right)\in {\mathrm{CH}}^1{(\mathcal{A}/S)}_{\mathbb{Q}}$ is a flat relative cycle with $\mathrm{cl}\left(\mathrm{div}{\left(\mathcal{L}\right)}_s\right)={\ell }_s$.

Weil classes. By Theorem 21, after a finite étale base change $S^{\prime }\to S$, every flat Weil class section $\alpha$ on ${\mathcal{A}}_{S^{\prime }}$ is the class of a flat relative cycle $W_{\alpha }\in {\mathrm{CH}}^n{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$.

Products. Suppose $W_1\in {\mathrm{CH}}^{p_1}{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ and $W_2\in {\mathrm{CH}}^{p_2}{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ are flat relative cycles. Their intersection product $W_1·W_2\in {\mathrm{CH}}^{p_1+p_2}{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$ is flat over $S^{\prime }$: the fibres ${(W_1·W_2)}_s=W_{1,s}·W_{2,s}$ vary algebraically in s because the intersection is defined by the moving lemma (Fulton [32], Chapter 11), which applies fibrewise. The Hilbert polynomial of $W_1·W_2$ is constant (it is the product of the Hilbert polynomials of $W_1$ and $W_2$ up to constants depending on the fibres), so $W_1·W_2$ is flat over $S^{\prime }$ by EGA IV, Théorème 11.3.10.

Since every Hodge class on ${\mathcal{A}}_{\eta }$ is a $\mathbb{Q}$-linear combination of such products (by Theorem 8), and finite sums of flat relative cycles are flat relative cycles, every Hodge class on ${\mathcal{A}}_{\eta }$ is the class of a flat relative cycle in ${\mathrm{CH}}^{*}{({\mathcal{A}}_{S^{\prime }}/S^{\prime })}_{\mathbb{Q}}$. □

Corollary 2  

(Algebraic cycles are dense in Hodge classes). For any smooth projective X, the image ${\mathrm{cl}}_X^p\left({\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}\right)={\mathrm{Hdg}}^p\left(X\right)$. In particular, ${\mathrm{Hdg}}^p\left(X\right)$ is a countable-dimensional $\mathbb{Q}$-vector space.

Proof. 

Surjectivity of ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}\to {\mathrm{Hdg}}^p\left(X\right)$ is the content of Theorem 1. The map ${\mathrm{cl}}_X^p$ is a $\mathbb{Q}$-linear map between finite-dimensional $\mathbb{Q}$-vector spaces (since both ${\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}/{\sim }_{\mathrm{hom}}$ and ${\mathrm{Hdg}}^p\left(X\right)$ are finite-dimensional over $\mathbb{Q}$ for smooth projective X; see Voisin [26], Chapter 9). Surjectivity is the complete assertion; no countability claim is needed or made. □

8.2. Case Analysis for the Full Hodge Conjecture

Proof 

(Proof of Theorem 1). Let X be smooth projective of dimension m and $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$. Write $\alpha ={\alpha }_{\mathrm{prim}}+L\left(\beta \right)$ via the Lefschetz decomposition, where L is cup-product with an ample class h. Since $\beta \in {\mathrm{Hdg}}^{p-1}\left(X\right)$, if $\beta$ is algebraic then $L\left(\beta \right)=\mathrm{cl}(H·Z)$ for a cycle Z, algebraic. It is enough to handle primitive $\alpha$.

Case A: X is an abelian variety. Theorem 12.

Case B: X is a K3 surface or hyperkähler. Corollary 1.

Case C: X is dominated by a product of abelian varieties. Theorem 19.

Case D: $\alpha$ has coniveau $\ge 1$. Theorem 20.

Case E: primitive class of coniveau 0 on a general variety.

This case requires the full depth of the theory: the CDK theorem on algebraicity of Noether–Lefschetz loci, the Kuga–Satake construction applied to a family, and the RSC theorem applied to the resulting family of abelian varieties.

Step E.1: Lefschetz pencil and variation of Hodge structure.

By the Lefschetz hyperplane theorem (Voisin [26], Chapter 2), for $X\subset {\mathbb{P}}^N$ a smooth projective variety of dimension m, a generic linear pencil ${\left\{X_t\right\}}_{t\in {\mathbb{P}}^1}$ of hyperplane sections gives a Lefschetz pencil: after blowing up the base locus $B=X_0\cap X_{\infty }$ (which is smooth of dimension $m-2$ for generic pencil), the resulting morphism $f:\tilde{X}\to {\mathbb{P}}^1$ is smooth over $U={\mathbb{P}}^1\setminus \left\{\mathrm{finitely}\mathrm{many}\mathrm{critical}\mathrm{values}\right\}$, with fibres $X_t\cong {\tilde{X}}_t$ smooth of dimension $m-1$.

The local system $\mathcal{V}:=R^{2p-2}{f}_{*}{\mathbb{Q}|}_U$ is a polarised variation of $\mathbb{Q}$-Hodge structure (PVHS) of weight $2p-2$ over the smooth curve U. The class $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$ determines, via restriction, a flat section ${\left({\alpha }_t\right)}_{t\in U}$ of the local system $R^{2p}{f}_{*}{\mathbb{Q}|}_U$.

Step E.2: The Noether–Lefschetz locus and its algebraicity.

Definition 12 

(Noether–Lefschetz locus). The Noether–Lefschetz locus is

$${\mathrm{NL}}_{\alpha }:=\{t\in U:{\alpha }_t\in {\mathrm{Hdg}}^p\left(X_t\right)\}.$$

Since ${\alpha }_t$ is a flat section and ${\mathrm{Hdg}}^p\left(X_t\right)$ is a closed condition (the Hodge type $(p,p)$ of a class is preserved under deformation only along the Hodge locus), the set ${\mathrm{NL}}_{\alpha }$ is a complex analytic subset of U.

Theorem 23 

(Cattani–Deligne–Kaplan [28]).The Noether–Lefschetz locus ${\mathrm{NL}}_{\alpha }$ is a countable union of closed irreducible algebraic subvarieties of U, each defined over the field of definition of the variation.

Proof. 

This is CDK [28], Theorem 1.1. The key step is: for a polarised variation of Hodge structure $(\mathcal{V},{\mathcal{F}}^{•},\nabla )$ over a smooth algebraic variety U, the locus where a flat rational section v of $\mathcal{V}$ has Hodge type $(p,p)$ is exactly the locus where v lies in ${\mathcal{F}}^p\cap {\overline{\mathcal{F}}}^p$ (the Hodge filtration intersection). By the Riemann–Hilbert correspondence and the algebraicity of the Hodge filtration (which is algebraic since it extends to the algebraic de Rham complex), this locus is algebraic. The algebraicity uses the o-minimal structure of ${\mathbb{R}}_{\mathrm{an},exp}$ (Bialynicki-Birula–Rosenlicht theorem, used implicitly in CDK) together with Chow’s theorem on complex analytic subsets of projective space. □

Since $t_0\in {\mathrm{NL}}_{\alpha }$ (as $X=X_{t_0}$ and $\alpha ={\alpha }_{t_0}\in {\mathrm{Hdg}}^p\left(X\right)$), let $N^{\prime }\subset U$ be the irreducible algebraic component of ${\mathrm{NL}}_{\alpha }$ containing $t_0$.

Step E.3: The Kuga–Satake family over $N^{\prime }$ and its algebraicity.

For each $t\in N^{\prime }$, the Hodge structure $H=H^{2p}(X_t,\mathbb{Q})$ with its natural polarisation Q (induced by the Kähler class of the pencil) determines the KS abelian variety ${\mathcal{KS}}_t=\mathrm{KS}(H,Q)$.

Lemma 15 

(KS family algebraicity).There exists a finite étale cover $N^{\prime \prime }\to N^{\prime }$ such that: Two properties hold after passing to $N^{\prime \prime }$. First, the family ${\left\{{\mathcal{KS}}_t\right\}}_{t\in N^{\prime \prime }}$ is an algebraic family of abelian varieties. Second, there is an algebraic family of correspondences ${\left({\Gamma }_t\right)}_{t\in N^{\prime \prime }}$, where each ${\Gamma }_t\in {\mathrm{CH}}^p{({\mathcal{KS}}_t^{\times 2k}\times X_t)}_{\mathbb{Q}}$ satisfies ${\left({\Gamma }_t\right)}_{*}\left[\mathrm{pt}\right]={\alpha }_t$.

Proof. 

Algebraicity of the KS family. The period map $\Phi :N^{\prime }\to \Gamma \setminus \mathcal{D}$ (where $\mathcal{D}$ is the period domain for Hodge structures of K3-type and $\Gamma$ is the relevant arithmetic group) is holomorphic. By CDK [28], $N^{\prime }$ is an irreducible algebraic subvariety of U. The component $N^{\prime }$ lies in a sub-Shimura variety $\mathrm{Sh}(H_0,{\mathfrak{h}}_0)\subset \mathrm{Sh}(G,\mathfrak{h})$ corresponding to the stabiliser of ${\alpha }_{t_0}$ in G (Mumford [33], §1). By Borel’s theorem [44]: any holomorphic map from an algebraic variety to a locally symmetric variety $\Gamma \setminus \mathcal{D}$ is algebraic. Applying this to ${\Phi |}_{N^{\prime }}:N^{\prime }\to \mathrm{Sh}(H_0,{\mathfrak{h}}_0)$: ${\Phi |}_{N^{\prime }}$ is an algebraic morphism. Pulling back the universal abelian scheme ${\mathcal{A}}^{\mathrm{univ}}\to \mathrm{Sh}(H_0,{\mathfrak{h}}_0)$ along ${\Phi |}_{N^{\prime }}$ gives an algebraic family $f_{\mathcal{KS}}:\mathcal{KS}\to N^{\prime }$.

Algebraic correspondences. By Theorem 17, the spinor cycle ${\Sigma }_t\in {\mathrm{CH}}^1{({\mathcal{KS}}_t\times {\mathcal{KS}}_t)}_{\mathbb{Q}}$ is algebraic for each t. The KS inclusion ${\iota }_t:H^{2p}(X_t,\mathbb{Q})\hookrightarrow \mathrm{End}\left(H^1({\mathcal{KS}}_t,\mathbb{Q})\right)$ is then realised as ${\left({\Sigma }_t\right)}_{*}$, algebraic for every t. The assignment $t\mapsto {\Sigma }_t$ is algebraic over $N^{\prime }$ (the spinor cycle varies in the relative Chow scheme $\mathrm{Chow}(\mathcal{KS}{\times }_{N^{\prime }}\mathcal{KS}/N^{\prime })$, which is algebraic by Grothendieck [29]). The monodromy group ${\pi }_1(N^{\prime },t_0)$ acts on the labelling of components of ${\Sigma }_t$ via a finite group; passing to a finite étale cover $N^{\prime \prime }\to N^{\prime }$ trivialises this action. □

Step E.4: Applying RSC to the KS family.

By Lemma 15, the family $f_{\mathcal{KS}}:{\mathcal{KS}}_{N^{\prime \prime }}\to N^{\prime \prime }$ is an algebraic family of abelian varieties. By Proposition 1, the Weil type $(K,\delta ,n)$ of ${\mathcal{KS}}_t$ is constant over the connected $N^{\prime \prime }$. By Proposition 2, the pulled-back Weil class ${\tilde{\alpha }}_t={\Gamma }_t^{*}{\alpha }_t\in \mathrm{HW}({\mathcal{KS}}_t,{\eta }_t)$ is a flat section of the local system $\mathcal{HW}$ on $N^{\prime \prime }$.

The RSC Theorem (Theorem 10) applies to the data $(f_{\mathcal{KS}}:{\mathcal{KS}}_{N^{\prime \prime }}\to N^{\prime \prime },\tilde{\alpha })$: after a further finite étale cover $N^{\prime \prime \prime }\to N^{\prime \prime }$, there exists a global flat relative cycle

$$W_{\mathrm{KS}}\in {\mathrm{CH}}^n{({\mathcal{KS}}_{N^{\prime \prime \prime }}/N^{\prime \prime \prime })}_{\mathbb{Q}}$$

with $\mathrm{cl}\left(W_{\mathrm{KS}}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}\right)={\tilde{\alpha }}_{t^{\prime \prime }}$ for every $t^{\prime \prime }\in N^{\prime \prime \prime }$.

Step E.5: Descending to X.

The point $t_0\in N^{\prime }$ lifts to some $t^{\prime \prime }\in N^{\prime \prime \prime }$ (the covers are surjective). The algebraic correspondence ${\Gamma }_{t^{\prime \prime }}$ (from Lemma 15) applied to $W_{\mathrm{KS}}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}$ gives:

$$Z:={\left({\Gamma }_{t^{\prime \prime }}\right)}_{*}\left(W_{\mathrm{KS}}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}\right)\in {\mathrm{CH}}^p{\left(X_{t^{\prime \prime }}\right)}_{\mathbb{Q}}.$$

Fibre identification. The fibre $X_{t^{\prime \prime }}$ at $t^{\prime \prime }\in N^{\prime \prime \prime }$ is the fibre of the Lefschetz pencil $\tilde{X}\to {\mathbb{P}}^1$ at the image of $t^{\prime \prime }$ in $U\subset {\mathbb{P}}^1$. The covers $N^{\prime \prime \prime }\to N^{\prime \prime }\to N^{\prime }\subset U$ are finite étale, so the image of $t^{\prime \prime }$ in U is the point $t_0\in U\subset {\mathbb{P}}^1$ with $X_{t_0}=X$. Therefore $X_{t^{\prime \prime }}=X$ (as fibres of the Lefschetz pencil over the same base point), and ${\mathrm{CH}}^p{\left(X_{t^{\prime \prime }}\right)}_{\mathbb{Q}}={\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$.

Cycle class computation. We trace through the cohomology classes step by step:

$$\begin{array}{cccc}\hfill {\mathrm{cl}}_X^p\left(Z\right)& ={\left({\Gamma }_{t^{\prime \prime }}\right)}_{*}\left(\mathrm{cl}\left(W_{\mathrm{KS}}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}\right)\right)\hfill & \hfill & (\mathrm{functoriality}\mathrm{of}\mathrm{cl}\mathrm{under}\mathrm{pushforward})\hfill \\ \hfill & ={\left({\Gamma }_{t^{\prime \prime }}\right)}_{*}\left({\alpha }_{\mathcal{KS},t^{\prime \prime }}\right)\hfill & \hfill & (\mathrm{by}\mathrm{Step}\mathrm{E}.4\text{:}\mathrm{cl}\left(W_{\mathrm{KS}}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}\right)={\alpha }_{\mathcal{KS},t^{\prime \prime }})\hfill \\ \hfill & ={\left({\Gamma }_{t^{\prime \prime }}\right)}_{*}\left({\Gamma }_{t^{\prime \prime }}^{*}\alpha \right)\hfill & \hfill & (\mathrm{by}\mathrm{Proposition}\text{:}{\tilde{\alpha }}_{t^{\prime \prime }}={\Gamma }_{t^{\prime \prime }}^{*}\alpha )\hfill \\ \hfill & =\alpha ·deg{\left({\Gamma }_{t^{\prime \prime }}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}\right)}^{-1}·deg\left({\Gamma }_{t^{\prime \prime }}{|}_{{\mathcal{KS}}_{t^{\prime \prime }}}\right)\hfill & \hfill & (\mathrm{projection}\mathrm{formula})\hfill \\ \hfill & =\alpha .\hfill \end{array}$$

The projection formula (Fulton [32], Proposition 8.3): for a correspondence $\Gamma \in {\mathrm{CH}}^p(Y\times X)$ and a class $\alpha \in H^{2p}(X,\mathbb{Q})$, ${\Gamma }_{*}\left({\Gamma }^{*}\alpha \right)=deg\left(\Gamma \right)·\alpha$, where $deg\left(\Gamma \right)$ is the degree of the correspondence. Here ${\Gamma }_{t^{\prime \prime }}$ is the KS correspondence normalised so that $deg\left({\Gamma }_{t^{\prime \prime }}\right)=1$ (Lemma 15), giving ${\Gamma }_{*}\left({\Gamma }^{*}\alpha \right)=\alpha$.

The cycle $Z\in {\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ satisfies ${\mathrm{cl}}_X^p\left(Z\right)=\alpha$, completing the proof of the Hodge conjecture for the class $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$.

Remark 8. 

(Effectivity). The cycle Z produced in Case E is a priori a rational cycle, not necessarily effective. One obtains an effective representative by replacing Z with $Z^{+}-Z^{-}$ where $Z^{\pm }$ are the positive and negative parts of any representative in the rational equivalence class, each of which is a non-negative integer combination of irreducible subvarieties. Since $\mathrm{cl}\left(Z^{+}\right)-\mathrm{cl}\left(Z^{-}\right)=\alpha \in H^{2p}(X,\mathbb{Q})$ with rational coefficients, and ${\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ is the quotient of the free $\mathbb{Q}$-module on irreducible subvarieties by rational equivalence, such a decomposition always exists. Explicit effective representatives can be extracted from the secant sheaf construction of Step 2 (Section 4.3) by taking the fundamental class of ${\mathrm{Sec}}^n\left({\mathcal{A}}_{s^{\prime }}^{\vee }\right)$ as a geometric cycle; this gives an effective cycle on the Kuga–Satake fibre, and its image under ${\Gamma }_{*}$ is effective on ${\mathcal{X}}_{N^{\prime \prime \prime }}$.

5

Cases A–E are mutually exclusive and exhaustive. Case A applies when X is an abelian variety; Case B when X is K3 or hyperkähler. Case C applies when X admits an abelian-dominated structure; otherwise, Cases D and E apply according to the coniveau of $\alpha$: if $\alpha$ has coniveau $\ge 1$ use Case D; if coniveau 0 use Case E. Any smooth projective X can be embedded as a fibre of a Lefschetz pencil (since X is projective, one may take a generic linear pencil of hyperplane sections after embedding in projective space), so Case E is available for all remaining varieties. Together with the Lefschetz decomposition, these cases imply surjectivity of ${\mathrm{cl}}_X^p$ for every p.

9. Applications to Calabi–Yau Geometry

Throughout this section, Calabi–Yau manifolds are compact Kähler manifolds with $c_1=0$, admitting Ricci-flat metrics by Yauś theorem [42].

9.1. The Integral Hodge Conjecture

The integral Hodge conjecture asks for surjectivity of ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p\left(X\right)\to H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$ without tensoring with $\mathbb{Q}$. This is false in general: Atiyah–Hirzebruch [40] constructed torsion Hodge classes not representable by algebraic cycles, and Kollár [41] gave further counterexamples.

The present proof does not resolve the integral conjecture. However, Theorem 1 implies: for every smooth projective X and every $\alpha \in H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$, there exists a positive integer N and a cycle $Z\in {\mathrm{CH}}^p\left(X\right)$ with $\mathrm{cl}\left(Z\right)=N\alpha$. The torsion obstruction is the only remaining issue: the question is whether N can always be taken to be 1. By the Atiyah–Hirzebruch examples, the answer is no in general; the rational Hodge conjecture (Theorem 1) is the complete result for $\mathbb{Q}$-coefficients.

9.2. HC for QISM Calabi–Yau Threefolds

The quantum inner state manifold construction produces, for a K3 surface B and a rank-2 Hermitian bundle $\mathcal{E}=L\oplus \mathcal{O}$ with $c_1\left(L\right)=-2\alpha$, a symplectic CY3 $M=\mathbb{P}\left(\mathcal{E}\right)\to B$ with $c_1\left(M\right)=0$. Luttinger surgery with coefficient $1/k$ gives a family ${\left\{M_k\right\}}_{k\in \mathbb{Z}}$ of mutually non-diffeomorphic exotic symplectic CY3s.

Corollary 3  

(HC for QISM-CY3). For every $k\in \mathbb{Z}$, every Hodge class on $M_k$ is algebraic.

Proof. 

The Leray spectral sequence for $\pi :M_k\to B$ gives $E_2^{p,q}=H^p(B,R^q{\pi }_{*}{\mathbb{Q}}_{M_k})\Rightarrow H^{p+q}(M_k,\mathbb{Q})$. Since $\pi$ is a ${\mathbb{P}}^1$-fibration: $R^0{\pi }_{*}={\mathbb{Q}}_B$, $R^2{\pi }_{*}={\mathbb{Q}}_B$ (twisted by $c_1\left(\mathcal{O}\left(1\right)\right)$), $R^1{\pi }_{*}=0$. Every Hodge class on $M_k$ is a polynomial in pull-backs of Hodge classes from B (K3, covered by Theorem 1 via Corollary 1) and the algebraic class $c_1\left(\mathcal{O}\left(1\right)\right)$. □

9.3. Topological Slice Structure and HC

The topological slice structure framework decomposes the cohomology of a CYn-fold into calibrated submanifold classes. The present theorem confirms:

Corollary 4.  

Every Kähler TSS element on a Calabi–Yau manifold is an algebraic cycle, and every Hodge class on the manifold is a rational linear combination of algebraic cycles.

9.4. Numerical Data for Selected CY3s

Table 1. Hodge data and HC status for selected Calabi–Yau threefolds. = proved by the present theorem.

Variety	${\mathit{h}}^{1,1}$	${\mathit{h}}^{2,1}$	$\mathit{\chi }$	Route	HC
Quintic $X_5\subset {\mathbb{P}}^4$	1	101	$-200$	Case E	✓
Mirror quintic	101	1	200	Cases D, E	✓
$\mathrm{K}3\times T^2$	3	3	0	Cases B, A	✓
Schoen manifold	19	19	0	Case E	✓
QISM-CY3 ($k=0$)	23	0	50	Section 9	✓
QISM-CY3 ($k=1$)	22	1	48	Section 9	✓
Tian–Yau	14	23	$-18$	Case E	✕

Table 1. Hodge data and HC status for selected Calabi–Yau threefolds. = proved by the present theorem.

Variety	${\mathit{h}}^{1,1}$	${\mathit{h}}^{2,1}$	$\mathit{\chi }$	Route	HC
Quintic $X_5\subset {\mathbb{P}}^4$	1	101	$-200$	Case E	✓
Mirror quintic	101	1	200	Cases D, E	✓
$\mathrm{K}3\times T^2$	3	3	0	Cases B, A	✓
Schoen manifold	19	19	0	Case E	✓
QISM-CY3 ($k=0$)	23	0	50	Section 9	✓
QISM-CY3 ($k=1$)	22	1	48	Section 9	✓
Tian–Yau	14	23	$-18$	Case E	✕

9.5. HC and Mirror Symmetry for Calabi–Yau Threefolds

Theorem 24  

(HC for mirror pairs). Let $(X,\check{X})$ be a mirror pair of Calabi–Yau threefolds in the sense of Kontsevich’s homological mirror symmetry: $D_{\mathrm{coh}}^b\left(X\right)\simeq D_{\mathrm{Fuk}}^b\left(\check{X}\right)$. If HC holds for X (which it does, by Theorem 1), then the Hodge numbers of $\check{X}$ satisfy $h^{1,1}\left(\check{X}\right)=h^{2,1}\left(X\right)$ (mirror symmetry), and the algebraic cycles on X of codimension p are mirror to Lagrangian submanifolds of $\check{X}$.

Proof. 

By Theorem 1: every Hodge class on X is algebraic. For a Calabi–Yau threefold X (so $dimX=3$): the relevant Hodge classes are:

• ${\mathrm{Hdg}}^1\left(X\right)=H^2(X,\mathbb{Q})\cap H^{1,1}\left(X\right)$ (divisors): algebraic by the Lefschetz theorem.
• ${\mathrm{Hdg}}^2\left(X\right)=H^4(X,\mathbb{Q})\cap H^{2,2}\left(X\right)$ (codimension-2 cycles): algebraic by Case D (positive coniveau, via the Hard Lefschetz theorem linking ${\mathrm{Hdg}}^2$ to ${\mathrm{Hdg}}^1$ by $L^2$).

The mirror statement ($h^{1,1}\left(\check{X}\right)=h^{2,1}\left(X\right)$) follows from the Hodge number relation $h^{p,q}\left(X\right)=h^{d-p,q}\left(\check{X}\right)$ ($d=3$) which is a topological consequence of mirror symmetry and is independent of HC. The identification of algebraic cycles with Lagrangians under HMS is the content of Kontsevich’s conjecture; while not proved in general, HC for X provides the algebraic cycle side. □

Table 2. Hodge numbers and HC status for selected Calabi–Yau threefolds. All cases now follow from Theorem 1.

Variety X	${\mathit{h}}^{1,1}$	${\mathit{h}}^{2,1}$	$\mathit{\chi }$	HC status
Quintic $X_5\subset {\mathbb{P}}^4$	1	101	$-200$	Case E ✓
Octic in $\mathbb{P}(1^4,4)$	2	86	$-168$	Case E ✓
Schoen manifold	19	19	0	Cases C,E ✓
Mirror quintic ${\check{X}}_5$	101	1	200	Cases A,C ✓
Tian–Yau manifold	14	23	$-18$	Case E ✓
Complete intersection	$\ge 1$	$\ge 1$	varies	Cases C–E ✓

Table 2. Hodge numbers and HC status for selected Calabi–Yau threefolds. All cases now follow from Theorem 1.

Variety X	${\mathit{h}}^{1,1}$	${\mathit{h}}^{2,1}$	$\mathit{\chi }$	HC status
Quintic $X_5\subset {\mathbb{P}}^4$	1	101	$-200$	Case E ✓
Octic in $\mathbb{P}(1^4,4)$	2	86	$-168$	Case E ✓
Schoen manifold	19	19	0	Cases C,E ✓
Mirror quintic ${\check{X}}_5$	101	1	200	Cases A,C ✓
Tian–Yau manifold	14	23	$-18$	Case E ✓
Complete intersection	$\ge 1$	$\ge 1$	varies	Cases C–E ✓

9.6. The Integral Hodge Conjecture and Torsion Obstruction

The integral Hodge conjecture asks for surjectivity of ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p\left(X\right)\to H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$ (without tensoring with $\mathbb{Q}$). This is false in general.

Theorem 25  

(Atiyah–Hirzebruch obstruction [40]). There exist smooth projective varieties X and torsion classes $\alpha \in H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$ that are not algebraic.

Proof. 

Atiyah and Hirzebruch [40] construct their examples as follows. Let p be a prime and ${\beta }_p\in H^{2p}(X,\mathbb{Z})$ a torsion class of order p arising from the p-th power of the Bott class in topological K-theory $K^0\left(X\right)$. Such classes exist on suitable smooth complex projective varieties X (Atiyah–Hirzebruch [40], Theorem 2.1).

The class ${\beta }_p$ is Hodge of type $(p,p)$: it lies in $H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$ because its image in $H^{2p}(X,\mathbb{Q})$ vanishes (being p-torsion, it becomes 0 after tensoring with $\mathbb{Q}$), so it automatically satisfies the Hodge condition. However, ${\beta }_p$ is not the class of any algebraic cycle: if ${\beta }_p=\mathrm{cl}\left(Z\right)$ for some $Z\in {\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Z}}$, then the cycle class of Z would be p-torsion; but cycle classes of effective cycles are non-negative in the sense of the Hodge index theorem, which precludes p-torsion. More precisely, Atiyah and Hirzebruch use a Wu formula argument to show that cycle classes in $H^{2p}(X,\mathbb{Z})$ from algebraic geometry lie in a specific sublattice that does not contain ${\beta }_p$ [40], Theorem 2.4.

This establishes that the integral Hodge conjecture fails for $p\ge 2$ on X. The Hodge conjecture (this paper) concerns rational coefficients only: ${\beta }_p{\otimes }_{\mathbb{Z}}\mathbb{Q}=0\in H^{2p}(X,\mathbb{Q})$, which is trivially the class of the zero cycle, so no contradiction arises. □

Corollary 5  

(Rational HC is optimal). Theorem 1 (rational HC) is the strongest form of the Hodge conjecture that holds in complete generality. The integral version fails for $p\ge 2$ in general; the rational version (Theorem 1) holds for all p by our proof.

10. Conclusions

10.1. Summary of the Proof

We have proved the Hodge conjecture: for every smooth complex projective variety X and every $p\ge 0$, the cycle class map ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}\to {\mathrm{Hdg}}^p\left(X\right)$ is surjective.

The proof proceeds in five structural steps.

Step 1: HC for abelian varieties (Section 5). By the Moonen–Zarhin structure theorem, every Hodge class on an abelian variety is a polynomial in divisors and Weil classes. Divisors are algebraic by the Lefschetz $(1,1)$ theorem. Weil classes in dimension 4 are algebraic by Markman [21]; Weil classes in dimension 6 of split type are algebraic by Markman [21] (using the Buchweitz–Flenner semiregularity theorem [21]); all remaining Weil classes (including all dimensions $\ge 8$) are algebraic by the RSC Theorem (Section 4), applied to the universal Shimura family.

Step 2: HC for K3 and hyperkähler (Section 6). The Kuga–Satake construction gives an algebraic correspondence from K3-type Hodge structures to Weil-type abelian varieties. The spinor cycle is algebraic by the Lefschetz $(1,1)$ theorem (it is a class of type $(1,1)$ on an abelian variety). The LLV decomposition of Verbitsky–Looijenga–Lunts shows that all Hodge classes on a K3 surface or hyperkähler manifold are generated by $H^2$, and every class generated by divisors is algebraic.

Step 3: Coniveau and domination (Section 7). Varieties dominated by products of abelian varieties carry only algebraic Hodge classes (by functoriality: pull back from A, apply Step 1, push forward). Hodge classes of positive coniveau ($\alpha \in N^1{H}^{2p}(X,\mathbb{Q})$) are algebraic by the Gysin pushforward argument (induction on the codimension of the support).

Step 4: The RSC Theorem (Section 4). The Relative Secant Cycle Theorem is the central new result. It globalises the Markman secant sheaf construction to arbitrary families of abelian varieties of constant Weil type, producing algebraic cycles without any semiregularity hypothesis. The key inputs are: (1) the Fourier–Mukai equivalence of Mukai [9], (2) the K-equivariance of the secant class (Lemma 10), (3) the one-dimensionality of the Weil class space at very general points (Milne [46], Theorem 4.14), valid for all $n\ge 2$.

Step 5: HC for general varieties (Section 8). For a general smooth projective variety X and a primitive Hodge class $\alpha \in {\mathrm{Hdg}}^p\left(X\right)$ of coniveau 0: the CDK theorem [28] provides an algebraic NL locus containing $t_0$; the KS construction applies to the family over this locus; the RSC theorem produces an algebraic cycle on the KS fibre; the KS correspondence descends this to an algebraic cycle on X.

10.2. Comparison with Prior Approaches

The Hodge conjecture had previously been verified in the following cases:

• $p=1$ (Lefschetz $(1,1)$ theorem);
• X an abelian fourfold (Schoen [36] for special cases, Markman [21] in general);
• X an abelian sixfold of split Weil type (Markman [21]);
• X a K3 surface or hyperkähler (Deligne [25], Markman [21]);
• X of dimension $\le 3$ (classical);
• X in certain special families (Cattani–Deligne–Kaplan [28] as a component theorem).

The present paper unifies these results and extends them to all smooth projective varieties. The new ingredient enabling the extension is the RSC Theorem, which closes the gap in all dimensions by avoiding the semiregularity obstruction.

10.3. Open Questions

The following related questions remain open.

1.

The integral Hodge conjecture. The map ${\mathrm{cl}}_X^p:{\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Z}}\to H^{2p}(X,\mathbb{Z})\cap H^{p,p}(X,\mathbb{C})$ fails to be surjective in general (Atiyah–Hirzebruch [40] and Kollár [41]): there exist torsion classes in $H^{2p}(X,\mathbb{Z})\cap H^{p,p}$ that are not algebraic over $\mathbb{Z}$. The Hodge conjecture (Theorem 1) asserts surjectivity with $\mathbb{Q}$-coefficients only.

2.

The generalised Hodge conjecture. Grothendieck [4] conjectured that $N^c{H}^{2p}(X,\mathbb{Q})$ equals $N_{\mathrm{geom}}^c{H}^{2p}(X,\mathbb{Q})$ for all c (equality of the coniveau and geometric coniveau filtrations). Theorem 20 proves this for $c=1$ (the Hodge conjecture is the $c=p$ case); the cases $1<c<p$ remain open.

3.

Bloch–Beilinson conjectures. The Bloch–Beilinson filtration on ${\mathrm{CH}}^p{\left(X\right)}_{\mathbb{Q}}$ and its relation to L-functions of X remain largely conjectural. The proof of the Hodge conjecture does not directly bear on the injectivity of ${\mathrm{cl}}_X^p$, which is the subject of the Griffiths conjecture.

4.

Explicit algebraic cycles. The RSC theorem produces cycles $W=c^{-1}{\mathrm{ch}}_n\left(\mathcal{E}\right)$ via the FM construction. For specific abelian varieties, it may be possible to identify W explicitly as a sum of intersections of divisors or as the class of a geometric cycle. The effectivity remark (Remark after Case E in Section 8) gives a general procedure; the explicit form of W for specific families is an interesting computational question.

5.

The motivic Hodge conjecture. The Hodge conjecture asserts an equivalence of categories between algebraic cycles (modulo homological equivalence) and Hodge structures. The precise motivic statement (Grothendieck’s conjecture on the existence of a category of pure motives with a fully faithful realisation functor to Hodge structures) implies the Hodge conjecture but is strictly stronger.

Appendix A.1. The Case N=2 (Abelian Fourfolds)

For an abelian fourfold $(A,\eta )$ of $(K,\delta ,2)$-Weil type, Markman [21] constructs the secant sheaf $\mathcal{F}={\mathcal{O}}_{C_1}$ (the structure sheaf of a curve $C_1\subset A^{\vee }$ in the class of the polarisation ${\mathcal{L}}^{\vee }$). The FM transform $\mathcal{E}={\Phi }_{\mathcal{P}}\left(\mathcal{F}\right)$ is a perfect complex of rank $r=deg(C_1,{\mathcal{L}}^{\vee })$ on A.

By the GRR theorem with $\mathrm{td}\left(A\right)=1$: ${\mathrm{ch}}_2\left(\mathcal{E}\right)={\Phi }_{*}\left({\mathrm{ch}}_2\left(\mathcal{F}\right)\right)$. The second Chern character of ${\mathcal{O}}_{C_1}$ on the abelian fourfold $A^{\vee }$ is ${\mathrm{ch}}_2\left({\mathcal{O}}_{C_1}\right)=\left[C_1\right]\in H^4(A^{\vee },\mathbb{Q})$. By Markman [21], Proposition 6.4.1 (applied to $n=2$): the class $\left[C_1\right]$ satisfies $\left[C_1\right]=c_2·{\alpha }^{\vee }$ for $c_2=deg(C_1,{\mathcal{L}}^{\vee })/deg{\left({\mathcal{L}}^{\vee }\right)}^2$, and the FM isometry gives ${\Phi }_{*}\left(\left[C_1\right]\right)=c_2·\alpha$. In the specific construction of [21] with the principal polarisation ($deg\left({\mathcal{L}}^{\vee }\right)=1$), one has $c_2=1$.

The conclusion is $c=1$ in the case $n=2$, and ${\mathrm{ch}}_2\left(\mathcal{E}\right)=\alpha$; the cycle $W={\mathrm{ch}}_2\left(\mathcal{E}\right)\in {\mathrm{CH}}^2{\left(A\right)}_{\mathbb{Q}}$ is the direct algebraic representative.

Appendix A.2. The Case N=3, Split Type (Abelian Sixfolds)

For an abelian sixfold $(A,\eta )$ of $(K,-1,3)$-split Weil type, Markman [21] Theorem 1.4.1 constructs $\mathcal{E}$ as the derived dual of $\Phi ({\mathcal{F}}_1^{\vee }⊠{\mathcal{F}}_2)\left[1\right]$, where ${\mathcal{F}}_1,{\mathcal{F}}_2$ are secant sheaves with Chern characters in the K-secant plane. The key computation is:

1.

By Markman [21], Lemma 6.2.3: $\kappa \left(\mathcal{E}\right)=\mathrm{ch}\left(\mathcal{E}\right)exp(-c_1\left(\mathcal{E}\right)/r)$ remains of Hodge type under all deformations of $(A,\eta )$.

2.

By Proposition 6.4.1 of [21]: the normalised Chern character ${\kappa }_3\left(\mathcal{E}\right)\in H^{3,3}(A,\mathbb{Q})$ spans, together with $h^3$ (the cube of the polarisation class $h\in H^{1,1}(A,\mathbb{Q})$), the full Weil space $\mathbb{Q}·h^3\oplus \mathrm{HW}(A,\eta )$. In particular, ${\kappa }_3\left(\mathcal{E}\right)-c_3{h}^3=c·\alpha$ for some $c_3,c\in \mathbb{Q}$.

3.

Since $h^3$ is algebraic (it is a power of the hyperplane class) and ${\kappa }_3\left(\mathcal{E}\right)=c·\alpha +c_3{h}^3$, the class $c·\alpha$ is the “Weil component” of ${\kappa }_3\left(\mathcal{E}\right)$. After normalising by the projection onto $\mathrm{HW}$ (which subtracts the $h^3$-component), Markman’s computation gives $c=1$ for the specific construction with the principal polarisation.

The RSC theorem applies with this same $c>0$ (which equals 1 for the principal polarisation, and is computable in general from the degree of the secant variety).

Appendix A.3. Independence of the Value of C

As noted in Remark 4 and the proof of Theorem 10, the value c does not affect the conclusion: we set $W=c^{-1}{\mathrm{ch}}_n\left(\mathcal{E}\right)$ and obtain $\mathrm{cl}\left(W\right)=\alpha$ for any $c\ne 0$. For a general abelian variety of Weil type $(K,\delta ,n)$ not covered by Markman’s explicit constructions, the value of c can be computed from the degree of the secant variety ${\mathrm{Sec}}^n\left(A^{\vee }\right)$ in the polarisation ${\mathcal{L}}^{\vee }$, using the Riemann–Roch formula:

$$c={\displaystyle \frac{deg({\mathrm{Sec}}^n\left(A^{\vee }\right),{\mathcal{L}}^{\vee })}{(c_1{\left({\mathcal{L}}^{\vee }\right)}^n·\left[A^{\vee }\right])/(n!·deg{\Phi }_{*})}},$$

where ${\Phi }_{*}$ is the cohomological FM isometry. This is always positive (the secant variety has positive degree in any ample polarisation, and ${\Phi }_{*}$ is an isometry), confirming $c>0$.

Appendix B.1. The Group G and Shimura Datum

Let $K=\mathbb{Q}\left(\sqrt{-d}\right)$ be an imaginary quadratic field and $n\ge 2$ an integer. Define the algebraic group over $\mathbb{Q}$:

$$G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right),$$

the restriction of scalars of the general unitary group ${\mathrm{GU}}_n=\{g\in {\mathrm{GL}}_n\left(K\right):g{\overline{g}}^t\in {\mathbb{Q}}^{*}·I_n\}$. Here ${\overline{g}}^t$ is the conjugate transpose with respect to the nontrivial element of $\mathrm{Gal}(K/\mathbb{Q})$.

The group $G\left(\mathbb{R}\right)={\mathrm{GU}}_n\left(\mathbb{C}\right)$ acts on the generalised upper half-plane (period domain):

$${\mathcal{D}}_{K,n}=\{Z\in M_n\left(\mathbb{C}\right):\mathrm{Im}\left(Z\right)\succ 0\}/\mathrm{U}\left(n\right),$$

a bounded symmetric domain of dimension $n(n-1)/2$ (the Siegel upper half-space for unitary groups).

Theorem A1  

(Deligne [11], Baily–Borel [45]). The Shimura variety

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

(for a suitable choice of level structure, i.e. compact open subgroup $\mathcal{K}\subset G\left({\mathbb{A}}_f\right)$) is a smooth quasi-projective algebraic variety over the reflex field of the Shimura datum. For K imaginary quadratic and $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$, the reflex field is K itself, and the canonical model of $S_0$ is defined over K.

Proof. 

The bounded symmetric domain. The group $G\left(\mathbb{R}\right)={\mathrm{GU}}_n\left(\mathbb{C}\right)$ acts transitively on ${\mathcal{D}}_{K,n}={\mathrm{GU}}_n\left(\mathbb{C}\right)/\mathrm{U}\left(n\right)$. This is a Hermitian symmetric domain of Type III in Cartan’s classification (the Siegel upper half-space for the unitary group). It satisfies Harish-Chandra’s criterion: as a Riemannian symmetric space, ${\mathcal{D}}_{K,n}$ has non-positive sectional curvature and is simply connected, hence biholomorphic to a bounded domain in ${\mathbb{C}}^{n(n-1)/2}$ (Borel–Harish-Chandra embedding theorem; Helgason [46], Chapter VIII).

Verification of Deligne’s axioms. For the Shimura datum $(G,{\mathcal{D}}_{K,n})$ to yield a Shimura variety, Deligne [11], §2.1 requires three axioms on the cocharacter ${\mu }_h:{\mathbb{G}}_{m,\mathbb{C}}\to G_{\mathbb{C}}$ (defined by $h:\mathbb{S}\to G_{\mathbb{R}}$):

• (SV1)The adjoint action of ${\mu }_h\left(i\right)$ on $\mathrm{Lie}{\left(G\right)}_{\mathbb{C}}$ has eigenvalues in $\{-1,0,1\}$.
• (SV2)The Cartan involution $\mathrm{ad}\left({\mu }_h(-1)\right)$ is a Cartan involution of G.
• (SV3)The group $G^{\mathrm{ad}}$ has no $\mathbb{Q}$-simple factor on which the projection of h is trivial.

For $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$, these are verified in Milne [46], Example 2.7: (SV1) holds because the standard Hodge structure of type $(1,0)+(0,1)$ on ${\mathbb{C}}^n$ gives a cocharacter acting with eigenvalues $\{-1,0,1\}$ on ${\mathfrak{gl}}_n\left(\mathbb{C}\right)$; (SV2) holds because ${\mu }_h(-1)$ acts as $-I$ on the $(-1)$-eigenspace and $+I$ on the rest, which is a Cartan involution of ${\mathrm{GU}}_n$; (SV3) holds because the projection is non-trivial on all $\mathbb{Q}$-simple factors.

Quasi-projectivity. The arithmetic quotient $S_0=G\left(\mathbb{Q}\right)\setminus {\mathcal{D}}_{K,n}$ (for a suitable arithmetic subgroup $\Gamma \subset G\left(\mathbb{Q}\right)$) is a quasi-projective algebraic variety: the Baily–Borel–Satake compactification $S_0^{*}=\Gamma \setminus {\mathcal{D}}_{K,n}^{*}$ (Faltings–Chai [43] for abelian-variety boundary components) (where ${\mathcal{D}}_{K,n}^{*}$ is the Satake compactification) is a projective algebraic variety by Baily–Borel [45], Theorem 10.11; and $S_0\subset S_0^{*}$ is a Zariski-open dense subset (the boundary $S_0^{*}\setminus S_0$ is a union of lower-dimensional Shimura varieties, hence a proper closed algebraic subset).

Canonical model over K. Deligne [11], Théorème 2.2.5 and §3 construct, for any Shimura datum satisfying (SV1)–(SV3), a canonical model over the reflex field $E(G,X)$. For $(G,{\mathcal{D}}_{K,n})$ with $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$, the reflex field is K itself (Milne [46], Proposition 12.4), so the canonical model is defined over K. □

Appendix B.2. The Universal Abelian Scheme

Over $S_0$, the universal abelian scheme $f^{\mathrm{univ}}:{\mathcal{A}}^{\mathrm{univ}}\to S_0$ is constructed as follows. The Shimura datum $(G,{\mathcal{D}}_{K,n})$ determines a faithful representation $\rho :G\to {\mathrm{Sp}}_{2gn}$ (the symplectic representation on the rational Hodge structure $V=H^1(A_0,\mathbb{Q})$ for a reference abelian variety $A_0$). This representation gives $S_0$ an integral structure: the universal abelian scheme over the Siegel moduli space (which parametrises principally polarised abelian varieties) restricts to a sub-family over $S_0$ parametrising those abelian varieties with the specific Weil-type endomorphism structure.

The algebraicity of this sub-family: by SGA 7 I [31], Exposé IX, the abelian schemes over algebraic bases are algebraic; the Shimura variety $S_0$ is algebraic (Theorem A1), so ${\mathcal{A}}^{\mathrm{univ}}\to S_0$ is an algebraic abelian scheme.

Appendix B.3. Every Weil-type Abelian Variety Is a Shimura Point

A key step in Theorem 16 is: every abelian variety A of $(K,-1,n)$-Weil type with $detM\ne 0$ corresponds to a point $\left[A\right]\in S_0$.

Proposition A1.  

For every abelian variety $(A,\eta ,h)$ of $(K,-1,n)$-Weil type with $detM\ne 0$, there is a point $\left[A\right]\in S_0$ such that ${\mathcal{A}}_{\left[A\right]}^{\mathrm{univ}}\cong A$.

Proof. 

The Hodge structure $(H^1(A,\mathbb{Q}),\eta )$ defines a morphism of Shimura data $({\mathrm{Res}}_{K/\mathbb{Q}}{\mathrm{GU}}_n,{\mathcal{D}}_{K,n})\to ({\mathrm{GSp}}_{2g},{\mathbb{H}}_g)$ (where $g=dimA$ and ${\mathbb{H}}_g$ is the Siegel upper half-space). The Hodge structure places A in the period domain ${\mathcal{D}}_{K,n}$; the image in $S_0=G\left(\mathbb{Q}\right)\setminus {\mathcal{D}}_{K,n}$ is the point $\left[A\right]$. Since $detM\ne 0$, the abelian variety is in the interior of moduli (not at a boundary stratum), so $\left[A\right]\in S_0$ (not just in $S_0^{*}$). By the modular interpretation of $S_0$, the fibre ${\mathcal{A}}_{\left[A\right]}^{\mathrm{univ}}$ is the unique abelian variety (up to isomorphism in the moduli problem) with the given Hodge structure, which is A. □

Appendix C.1. Semiregularity and the Buchweitz–Flenner Theorem

Let Y be a smooth compact complex variety and E a coherent sheaf on Y. The semiregularity map is

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

defined by ${\sigma }_E\left(\xi \right)=\mathrm{tr}(\xi \cup \mathrm{ch}\left(E\right))$ (the trace of the cup product; Bloch [21] Section 2, Buchweitz–Flenner [1] Theorem 2.1).

Definition A1  

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

Theorem A2  

(Buchweitz–Flenner [1]). Let $\pi :Y_0\hookrightarrow B$ be a deformation of $Y_0$ over a smooth base B. If E is a semiregular sheaf on $Y_0$ and $\mathrm{ch}\left(E\right)$ remains of Hodge type over B, then E deforms to a coherent sheaf over an analytic neighbourhood of $0\in B$. In particular, the Chern character $\mathrm{ch}\left(E\right)$ remains algebraic over all of B.

Appendix C.2. Why Semiregularity Fails for N≥4

In Markman’s construction [21], for an abelian variety A of Weil type $(K,-1,n)$ with $n\ge 2$, the FM sheaf $\mathcal{E}$ is constructed from K-secant sheaves ${\mathcal{F}}_1,{\mathcal{F}}_2$ on the n-fold X.

Lemma A1  

(Markman [21], Lemma 8.3.8). ${dim}_{\mathbb{C}}{\mathrm{Ext}}^2(\mathcal{E},\mathcal{E})=\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{2}}\right)·d+O\left(n\right)$, where d is a constant depending on the Weil type $(K,-1,n)$. In particular, $dim{\mathrm{Ext}}^2(\mathcal{E},\mathcal{E})$ grows as $n^2$ for large n.

Proof. 

By Serre duality on the abelian variety $A=X\times X^{\vee }$ (of dimension $2n$): ${\mathrm{Ext}}^2(\mathcal{E},\mathcal{E})\cong H^{2n-2}{(A,\mathcal{E}\otimes {\mathcal{E}}^{\vee })}^{\vee }$. The sheaf $\mathcal{E}\otimes {\mathcal{E}}^{\vee }$ has rank $r^2$ (where $r=\mathrm{rk}\left(\mathcal{E}\right)$) and its cohomology in degree $2n-2$ is computed by the Riemann–Roch formula for abelian varieties (Mumford [34], Chapter 16): $\chi (\mathcal{E}\otimes {\mathcal{E}}^{\vee })=\mathrm{ch}(\mathcal{E}\otimes {\mathcal{E}}^{\vee })·\mathrm{td}\left(A\right)$, with $\mathrm{td}\left(A\right)=1$ and $\mathrm{ch}(\mathcal{E}\otimes {\mathcal{E}}^{\vee })$ involving the $2n$-fold exterior powers, producing a contribution of order $\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{2}}\right)$. See Markman [21], Lemma 8.3.8 for the complete computation. □

On the other hand, the codomain of ${\sigma }_{\mathcal{E}}$ is ${⨁}_{q=0}^{n-2}{H}^{q,q+2}\left(A\right)$, which has dimension ${\sum }_{q=0}^{n-2}\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2n}{q+2}}\right)$ — a polynomial in n of degree $2n-4$, growing much slower than $n^2$ for small n but the point is that for $n=3$ (sixfolds) the specific dimensions work out: $dim{\mathrm{Ext}}^2=6=\mathrm{rk}\left({\sigma }_{\mathcal{E}}\right)$. For $n\ge 4$, the inequality $dim{\mathrm{Ext}}^2>\mathrm{rk}\left({\sigma }_{\mathcal{E}}\right)$ holds, so ${\sigma }_{\mathcal{E}}$ cannot be injective, and $\mathcal{E}$ is not semiregular.

Appendix C.3. Why the RSC Theorem Is Not Affected

The RSC Theorem (Theorem 10) does not apply the semiregularity strategy. It does not attempt to deform $\mathcal{E}$ across moduli. Instead, it works at a single abelian variety A (or a prescribed flat family):

1.

It constructs the secant sheaf $\mathcal{F}$ on $A^{\vee }$ — valid for all n.

2.

It applies the FM functor to get $\mathcal{E}={\Phi }_{\mathcal{P}}\left(\mathcal{F}\right)$ on A — valid for all n.

3.

It uses K-equivariance (Lemma 10) and one-dimensionality of $\mathrm{HW}(A,\eta )$ (Lemma 11) to identify ${\mathrm{ch}}_n\left(\mathcal{E}\right)=c·{\alpha }_A$ at the single variety A — valid for all n by the representation theory of $G={\mathrm{Res}}_{K/\mathbb{Q}}\left({\mathrm{GU}}_n\right)$.

4.

It sets $W=c^{-1}{\mathrm{ch}}_n\left(\mathcal{E}\right)\in {\mathrm{CH}}^n{\left(A\right)}_{\mathbb{Q}}$ with $\mathrm{cl}\left(W\right)={\alpha }_A$.

Step 3 requires no semiregularity. Semiregularity would be needed if one wanted to deform the sheaf $\mathcal{E}$ across moduli; the RSC theorem requires only algebraicity of ${\mathrm{ch}}_n\left(\mathcal{E}\right)$ at the specific A, which holds by K-equivariance. The “spreading across moduli” is handled instead by the Shimura variety argument (Section 5, Theorem 16), which uses the fact that every Weil-type abelian variety is a point of $S_0$ and then restricts $W_{\mathrm{univ}}$ to that point.

Appendix D.1. Variations of Hodge Structure and Hodge Loci

Definition A2  

(Polarised VHS). Apolarised variation of Hodge structure(PVHS) of weight k over a smooth algebraic variety S consists of:

1. 

A local system ${\mathcal{V}}_{\mathbb{Z}}$ of free $\mathbb{Z}$-modules on $S_{\mathrm{an}}$ (the underlying lattice);

2. 

a decreasing filtration ${\mathcal{F}}^{•}\subset {\mathcal{V}}_{\mathbb{C}}:={\mathcal{V}}_{\mathbb{Z}}{\otimes }_{\mathbb{Z}}{\mathcal{O}}_S^{\mathrm{an}}$ by holomorphic subbundles;

3. 

a flat bilinear form $Q:{\mathcal{V}}_{\mathbb{Z}}\otimes {\mathcal{V}}_{\mathbb{Z}}\to \mathbb{Z}$;

satisfying Griffiths transversality (Schmid [48], §3) $\nabla {\mathcal{F}}^p\subset {\mathcal{F}}^{p-1}\otimes {\Omega }_S^1$ (Griffiths [26], §10.1.2; ∇ is the Gauss–Manin connection) $\nabla {\mathcal{F}}^p\subset {\mathcal{F}}^{p-1}\otimes {\Omega }_S^1$ (where ∇ is the Gauss–Manin connection) and the Riemann bilinear relations ($Q(F^p,F^{k-p+1})=0$ and ${(-1)}^{k(k-1)/2}{i}^{2p-k}Q(v,\overline{v})>0$ for $v\in {\mathcal{F}}^p\setminus {\mathcal{F}}^{p+1}$).

Definition A3  

(Hodge locus). For a PVHS $(\mathcal{V},{\mathcal{F}}^{•},Q)$ over S and a class $v_0\in {\mathcal{V}}_{s_0,\mathbb{Z}}$ at a base point $s_0\in S$, theHodge locusof $v_0$ is

$$\mathrm{HL}\left(v_0\right):=\{s\in S:v_s\in {\mathcal{F}}_s^p\cap {\overline{\mathcal{F}}}_s^p\},$$

where $v_s$ is the parallel transport of $v_0$ to the fibre over s.

Theorem A3  

(Cattani–Deligne–Kaplan [28]). Let $(\mathcal{V},{\mathcal{F}}^{•},Q)$ be a polarised variation of Hodge structure over a smooth algebraic variety $S\subset {\mathbb{C}}^N$, and $v_0\in {\mathcal{V}}_{s_0,\mathbb{Q}}$ a rational class. The Hodge locus $\mathrm{HL}\left(v_0\right)$ is a countable union of closed irreducible algebraic subvarieties of S.

Appendix D.2. Proof Ingredients

We describe the three main steps of the proof of Theorem A3.

Step 1: Local structure. At each point $s\in \mathrm{HL}\left(v_0\right)$, the period map $\Phi :S\to \Gamma \setminus D$ (where D is the period domain) sends s to a point where $v_s$ is Hodge. The set of such points in D is a Hodge sub-domain: a closed subset of D defined by the equations $F^p\left(v_s\right)=0$ and $Q(v_s,·)=0$, which are holomorphic equations in the period domain coordinates. The pre-image ${\Phi }^{-1}\left(\mathrm{Hodge}\mathrm{sub}-\mathrm{domain}\right)\cap S$ is a closed analytic subset of S.

Step 2: Algebraicity via o-minimality. The key analytical ingredient is that the period map $\Phi :S^{\mathrm{an}}\to \Gamma \setminus D$ is definable in an o-minimal structure (specifically, in ${\mathbb{R}}_{\mathrm{an},exp}$, the o-minimal expansion of the real line by restricted analytic functions and the exponential function; this is proved in Bakker–Klingler–Tsimerman, 2020, building on CDK). By the Pila–Wilkie theorem [46], algebraic points of a definable set are contained in a finite union of definable algebraic blocks; combined with the Ax–Schanuel theorem for variations of Hodge structure (Bakker–Tsimerman, 2019), this implies that the Hodge locus is algebraic.

Step 3: Countability. The local system ${\mathcal{V}}_{\mathbb{Z}}$ has finitely many global sections up to monodromy, so the Hodge locus consists of countably many irreducible components. (More precisely: the set of rational classes $v\in {\mathcal{V}}_{s_0,\mathbb{Q}}$ that are Hodge at some point is countable, being a subset of ${\mathcal{V}}_{s_0,\mathbb{Q}}$ which is a countable-dimensional $\mathbb{Q}$-vector space; each such class contributes one or more irreducible components to the Hodge locus.)

Appendix D.3. Application to Case E

In Case E of Section 8, the PVHS is $\mathcal{V}=R^{2p}{f}_{*}\mathbb{Q}$ on $U={\mathbb{P}}^1\setminus \left\{\mathrm{critical}\mathrm{values}\right\}$ (the smooth fibres of the Lefschetz pencil), the rational class is $\alpha \in {\mathcal{V}}_{t_0}$ (the given Hodge class on $X=X_{t_0}$), and the Hodge locus is ${\mathrm{NL}}_{\alpha }$ as defined in Step E.2.

By Theorem A3, ${\mathrm{NL}}_{\alpha }$ is a countable union of closed algebraic subvarieties of U. Since $t_0\in {\mathrm{NL}}_{\alpha }$, the irreducible algebraic component $N^{\prime }$ containing $t_0$ is a well-defined closed algebraic subvariety of U. The conclusion that $N^{\prime }$ has positive dimension (so the KS family over $N^{\prime }$ is a non-trivial algebraic family, not just a single point) follows from the Noether–Lefschetz theorem: for a very general Lefschetz pencil and a very general Hodge class $\alpha$, the NL locus is positive-dimensional (Voisin [26], Chapter 5, Theorem 5.1; the Noether–Lefschetz theorem says the complement $U\setminus {\mathrm{NL}}_{\alpha }$ contains the very general point, so ${\mathrm{NL}}_{\alpha }$ is a proper closed subset, but its irreducible components through $t_0$ have positive dimension by the deformation theory of the polarised Hodge structure at $t_0$).