Toward A Classification of Compact Homogeneous Complex Finsler Manifolds

1. Introduction

Let M be a complex manifold, h be an Hermitian metric. For a compact complex manifold, there is always some Hermitian metric h by the partition of the unity argument. If h is an Hermitian metric and G is a compact Lie group acting on M biholomorphically, then by taking average on G, we can always assume that h is invariant under G.

A compact complex homogeneous space with an invariant Hermitian structure was classified by H. C. Wang in [13], see also [8]. In fact, they classified the compact complex homogeneous spaces with compact Lie groups. In particular, an Hermitian manifold is a Riemannian manifold. The identity component of the Riemannian isometric group for a compact Riemannian manifold is a compact Lie group. So is the identity component of the Hermitian isometric group for a compact Hermitian manifold. Moreover, the isotropic group of a homogeneous Hermitian manifold is a subgroup of U(n) with n = ${dim}_C$M for any Lie group of complex automorphisms which keeps an Hermitian metric invariant.

Therefore, we have:

Proposition 1.1. If M = G/H is a compact homogeneous Riemannian manifold with G connected, then G is a subgroup of a compact Lie group. In particular, both G and H are reductive with compact semisimple parts.

We then have (see [8] Theorem B):

Proposition 1.2. Any compact Hermitian homogeneous manifold is a complex torus bundle over a rational (therefore simply connected) projective homogeneous space.

One could also see [7] page 66, Remark for a detail understanding of this fibration.

There is also a similar fibration [12] closely related to Proposition 1.2 for any general compact complex homogeneous space:

Proposition 1.3. Let M = G/H be a compact complex homogeneous s pace such that

G is a complex Lie group, H a complex Lie subgroup and M = G/H is the complex quotient. Then there is a complex fibration

G/H → G/N

such that

N =${Norm}_G$($H_0$)

and G/N is a rational projective homogeneous space.-

Here, $H_0$ is the identity component of H and

${Norm}_G$($H_0$) = {$g$∈ G|$g{H}_0{g}^{-1}$⊂$H_0$}

is the normalizer of $H_0$ in G.

With these results at hand, now we come to a generalization of the compact homgeneous Hermitian manifolds. At a point p, a nonnegative function F from the complex tangent space $T_p^{1.0}$M to the real numbers define a complex (pseudo-convex) Finster metric [1] (p.84-85, Definitions 2.3.1, 2.3.5) if

1. F(v) ≥ 0 and = 0 iff v = 0;

2. F(cv) = |c|F(v) for any c ∈C;

3. ∂$\overline{\partial }$($F^2$) > 0 on F = 1.

The third condition is also called the pseudo-convex property. Therefore, sometime, we also call a complex Finler metric a pseudo-convex complex Finsler metric. This is equivalent to requiring that the domain

${\mathrm{\Omega }}_F$= {v ∈ $C^n$|F(v)<1}

is strongly pseudoconvex. This condition seems is necessary if we need to define a meaningful connection and to do some kind of geometry like parallel transformations, geodesics, torsion, curvature, etc.. We shall also address the Kähler-like conditions in the next section, which were, are, and will be in the center of the complex Finsler geometry, especially for the relation between the first variation of the length and the geodesics.

A complex manifold is complex Finsler [1] if there is smooth function F on $T_p^{1.0}$M −M such that on each fiber $T_p^{1.0}$M, F comes from a pseudo-convex Finsler metric.

The pseudo-convex condition gives an Hermitian metrics on $T_p^{1.0}$M −M. Again, this is necessary to the so called the Chern-Finsler connection in general even without the relation between the first variation of the lenghth and the geodesics.

Similar to the Hermitian case, if G is a compact subgroup of biholomor-phic automorphisms, by average, we can always make the complex Finsler metric to be invariant under G.

On the other hand, one want to know that if the manifold is compact, a group of biholomorphic automorphisms which keep the complex Finsler structure invariant can be chosen to be compact.

The first difficulty comes from the question whether the big group G is compact, where G keeps F and the complex structure invariant.

Main Theorem 1. The isometric group is compact if the manifold is.

This is good since in the pseudo-Kähler case, or complex manifold with an invariant volume case (including quasi-Hermitian case), one might not expect that the group to be compact. See [?, 6, 7] and the references therein for examples.

The second (difficulty) question is whether the isotropic group H is a subgroup of U(n).

This is the question Professor C. P. Zhong (from Xiamen University) asked us in the Chinese Annual National Conference of Several Complex Variables August 2026. Fortunately, in the efforts to give some kind of answer we were able to overcome the difficulty with some knowledge from another project we were working on recently and obtained:

Main Theorem 2. The isotropic subgroup is always a subgroup of U(n).

This actually can be also obtained by Main Theorem 1, since any compact subgroup of GL(n,C) is conjugate to a subgroup of U(n).

This is also good, since even in the pseudo-Kähler case, one can not expect that the isotropic subgroup is in U(n), which is compact, even the isotropic representation is still faithful. That made the classification of the compact homogeneous pseudo-Kähler spaces very difficult.

And third, we came to the homogeneous case and obtained:

Corollary 1.1. All the compact homogeneous complex Finsler manifolds, as complex homogeneous spaces, are those of compact homogeneous Hermitian manifolds.

Remark 1.1. Although we are able to classify the complex manifolds, it is not very clear how to check the properties of all the possible complex Finsler metrics for us. For example, are they Kähler, weakly Kähler but not Kähler, or not weakly Kähler? To give some concrete examples for complex Finsler manifolds is the purpose of the classification. We shall give some discussions in the next section. Also, the Main Theorems still apply to the open manifolds, or in particular, with an action of a reductive Lie group.

However, the situation might be more complicated.

2. Methods

This is a study to get many examples of complex Finsler manifolds. Of course, the easiest way to get them is to classify the homogeneous ones. Therefore, one of way to obtain them is to consider the cosets of Lie groups. Therefore, first, we used the basic Lie theory to get the work down.

The classifications of compact complex homogeneous spaces or with a reductive groups were well-down for many special cases, e.g., Kähler, pseudo-Kähler, symplectic (invariant or not), with invariant volume, etc., but not very well taught to a general mathematicians. Therefore, we introduce some of them in this article and also refer them to [6,7,8,12,13], and the references therein. .

Of course, the next thing we need is the theory for complex Finsler geometry. We got them from [1], Chapter 2, also from [3].

Under the condition of the smooth strictly pseudo-convexity, one actually can get a Hermitian metric on the complex projective tangent bundle and the Chern connection on the complex Finsler manifold. From the Chern connection one get the geodesics. That make the isotropic representation faithful. Once it is faithful, we can get the compact ness of the isotropic subgroup from the invariant Hermitian metric on Projective fibers.

This implies with a standard result of the Lie theory guaranteed that the isotropic subgroup is a subgroup of U(n).

In this article, we also use several results from Several Complex Variables. That also give elegant proofs that the isotropic subgroup is a subgroup of U(n), by translate it into isotropic subgroup for invariant Kähler metrics. The existence of the Bergman metrics for bounded smooth strictly pseudo-convex domains is used in this article. Another two results from Several Complex Variables which we use are the existence and the uniqueness of the Kähler Einstein metrics on bounded smooth strictly pseudo-convex domains.

No artificial intelligence (AI) has been used in this article, although it could be.

3. Proofs

To any Hermitian metric is associated a unique complex linear connection such that the metric tensor is parallel: the Chern connection.

One observes that F defines a Hermitian metric on the holomorphic tangent bundle $T^{1.0}$M. We write G = $F^2$ for convenience. Theorem 2.3.2 in [1] page 87 to 89 gives a unique connection D (determined by F) which is also called Chern connection. It is a generalization of the Chern connection from the Hermitian case, although it is much more complicated. However, as in the Riemannian case, we can define that a vector field X along a curve α(t) is parallel if $D_{{\mathsf{\alpha }}^{\prime }\left(\mathrm{t}\right)}$X = 0 and α(t) is a geodesic if $D_{{\mathsf{\alpha }}^{\prime }\left(\mathrm{t}\right)}$α’(t) = 0.

Lemma 2.1. Given a point and a vector direction at that point, one obtains an unique geodesic.

This is because the geodesic equation is a first order system of equations

Z’= A(Z)

which is regarded as orbits of the vector field A and can be solved.

Here, we notice that the geodesics exists without the different kinds of Kähler-like conditions. There is no need to have any relation between the first variation of the length and the geodesics.

Theorem 2.1. The isotropic representation is faithful.

Proof of Theorem 2.1.: This is similar to the Riemannian case. Once we have an invariant connection, given a point and a vector direction at that point, one obtain an unique geodesic. The kernel of the isotropic representation keep all the geodesics through the given point invariant [1]. Therefore, it fixes a neighborhood of that point. It must be the identity.

Q E. D.

Proof of the Main Theorem 1.: Now, the isotropic subgroup acts linearly on

P($T_p^{1.0}$M)

with the induced invariant Hermitian metric on

$P_{S_F}$=$S_F$F/$S^1$

where

$S_F$= {v ∈ $T_p^{1.0}$M|F(v)=1}.

The construction is as following: First, there is a map p from $S_F$ to

$P_{S_F}$= P($T_p^{1.0}$M).

This gives the skew two form ∂$\overline{\partial }$(G) on $T_{S_F}$. By the real dimension of $T_{S_F}$ to be 2n − 1, it has a real 1-dimensional kernel, which is at the J(${\partial }_l$) with l being the length of the tangent vectors. By our construction J(${\partial }_l$) has the same direction of the $S^1$ action.

Therefore,

$p_{\prime }$(∂$\overline{\partial }$(G)|$T_{S_F}$)

is meaningful and gives an Hermitian metric on

T($P_{S_F}$) = T(P($T_p^{1.0}$M)).

Therefore, the image group $H_1$ on

$P_{S_F}$=$S_F$/$S^1$

can be a subgroup of

PSU(n),

and can be compact.

Let π be the homomorphism from the isotropic subgroup to PSU(n). The kernel of π is in $C^{\prime }$ and keep $S_F$ invariant, which is $S^1$.

This implies that the isotropic group H is an $S^1$ fiber bundle over $H_1$ and can be chosen to be compact.

By M being compact, we get that the isometric group is compact. This is our Main

Theorem 1.

Q. E. D.

Proposition 2.1. There is a complete Bergman metric on any bounded pseudo-convex domain with a $C^1$ smooth boundary.

This comes from [10].

Proposition 2.2. There is a complete Kähler Einstein metric with a Einstein constant −1 on any bounded pseudo-convex domain with a $C^5$ smooth boundary.

This comes from [4].

We also have the uniqueness of the Kähler Einstein metrics:

Proposition 2.3. The complete Kähler Einstein metric with an Einstein constant −1 is unique on any bounded pseudo-convex domain.

This comes from [4] and [9]. Actually in [9], they proved a Schwarz Lemma which can be applied to prove the uniqueness of a Kähler Einstein metric with an Einstein constant −1.

In fact, in [5], we gave a self-contained proof for the special case in which the domain in $C^n$ can be unbounded and the metric is Kähler Einstein metric with an Einstein constant −1. This implies that the Kähler Einstein metric is invariant under the action of the holomorphic automorphism. With an earlier result of the non-existence of certain kind of invariant Kähler Einstein metric in [2], we proved that against the general belief that any pseudo-convex domain in $C^n$ existing a complete Kähler Einstein metric we found many nontrivial (or a product type) unbounded pseudo-convex domains which do not admit any complete Kähler Einstein metric. Moreover, in that class of unbounded domains, the ones which admit complete Kähler Einstein metrics are those on the boundary of the class and those which admit no complete Kähler Einstein metric are exactly those in the interior of that class. This is somehow different from the general view point that those which admit complete Kähler Einstein metric are certain ”stable” ones. This is also similar to some situations in the compact cases which we dealt with in several earlier papers.

We also notice that according to [11] the uniqueness for complete Kähler Einstein metric with an Einstein constant 0 does not hold even for $C^n$.

Theorem 2.2. There are invariant Kähler metrics on

${\mathrm{\Omega }}_{p,F}$= {v ∈ $T_p^{1.0}$M|F(v) < 1}

under the automorphism group.

Proof of Theorem 2.2： ${\mathrm{\Omega }}_{p,F}$ is a smooth bounded strongly pseudo-convex domain. Then by Propositions 2.1 to 2.3, it admits both a complete Bergman metric and a complete Kähler Einstein metric. The Bergman metric is always invariant under the automorphism group. By Proposition 2.3, the complete Kähler Einstein metric is also invariant under the automorphism group.

Q. E. D.

Proof of the Main Theorem 2.: Then, the isotropic representation fixes the zero point of ${\mathrm{\Omega }}_{p,F}$ and is in the isotropic subgroup of the invariant Kähler metrics at the zero point, which also have faithful representations at the zero point. Therefore, it is a subgroup of U(n). This proves our Main Theorem 2.

Q. E. D.

Proof of The Corollary: It is obvious then both the big group G and the isotropic group H for a compact homogeneous Finsler Manifold M =G/H are compact. And one can easily construct an invariant Hermitian metric on M by a standard complex norm on $T_p^{1.0}$M.

Q. E. D.

Moreover, we can easily obtain all the possible complex Finsler metrics on M by constructing an invariant complex Finsler metric on $T_p^{1.0}$M.

4. Further Discussions

As it is well known, in the usual Hermitian geometry, the vanishing of the torsion of the Hermitian connection is equivalent to the metric being Kähler. For the Finsler cases, people consider the tangent spaces of holomorphic tangent spaces instead of just considering the tangent spaces themselves. Since the torsion of the Chern-Finsler connection has two parts: a horizontal part and a mixed part, although the vertical part is zero, the situation for a pseudoconvex Finsler metric to be Kähler is a bit subtler. There is also a radian horizontal vector field

r =${\sum }_{i=1}^n{\displaystyle \frac{v_i\mathsf{\delta }}{\mathsf{\delta }{z}_i}}$

in the parallel distribution H. In many works, there are three kinds of metric notions called respectively the strongly Kähler, Kähler and weakly Kähler according to the vanishing of some parts of the torsion of the Chern-Finsler connection (see [1]).

We denote (${\mathrm{G}}_{i\overline{\mathrm{j}}}$) := (${\partial }_i{\overline{\partial }}_j$G). By the pseudo-convex condition, we have (${\mathrm{G}}_{i\overline{\mathrm{j}}}$) > 0 on TM − M. Therefore, there is an inverse ($G^{\overline{\mathrm{i}\mathrm{j}}}$).

From now on, the lower indices of G always mean to take derivatives in linear (tangent) space, and the lower indices after a sign ”;” of G always mean to take derivatives with respect to the coordinates z.

We set

$N_k^i$=$G^{\overline{\mathrm{i}\mathrm{j}}}{\mathrm{G}}_{i\overline{\mathrm{j}};k}$.

The horizontal base vectors and vertical base covectors can be defined by

${\displaystyle \frac{\mathsf{\delta }}{\mathsf{\delta }{z}_k}}$:=${\displaystyle \frac{\partial }{\partial z_k}}$− $N_k^i{\displaystyle \frac{\partial }{\partial v_i}}$, δ${\mathrm{v}}_k$:= d${\mathrm{v}}_k$+ $N_k^i$d${\mathrm{z}}_i$.

Notice that the symbol N is not the usual Christoffel symbol which has three indices. Therefore, we prefer to use N instead of Γ as in [3].

Let

θ(X,Y ) := ${\mathrm{D}}_{\mathrm{X}}$Y −${\mathrm{D}}_{\mathrm{Y}}$X − [X,Y ]

with X,Y ∈ Γ(T(TM)) be the torsion.

Definition 2.1. (see [1,3]) Let F be a complex Finsler metric, and

r =${\sum }_{i=1}^n{\displaystyle \frac{v_i\mathsf{\delta }}{\mathsf{\delta }{z}_i}}$ ∈ Γ(H)

be the radial horizontal vector field. We say that F is

(1) strongly Kähler, if θ(X,Y ) = 0 for any X,Y ∈ H;

(2) Kähler, if θ(X,r) = 0 for any X ∈ H;

(3) weakly Kähler, if (θ(X,r),r) = 0 for any X ∈ H.

These conditions obviously are only related to H.

We also see from [1] page 94 Proposition 2.3.9 (i) that the mixed part of the torsion vanishes iff the complex Finsler metric is actually an Hermitian metric. Proposition 2.3.9 (ii) then describes the relation for the metric to be classical Hermitian Kähler. These give the reasons for the Kähler-like conditions for the pseudo-convex complex Finsler metric.

It was surprising that in [3], the authors were able to prove that (1) is the same as (2). But people are still working on finding example of a weakly Kähler which is not Kähler.

Therefore, we hope that our classification would provide enough examples for them.

However, the calculations might be still hard.

To make our classification a little bit easier for the readers, we explain some of our manifolds before the end of this article.

A rational projective homogeneous space M = G/P is a complex homogeneous space with a complex semisimple Lie group G such that P is a parabolic subgroup of G. This kind of M is always simply connected and rational, that is, one can get an open set $C^{{\mathrm{d}\mathrm{i}\mathrm{m}}_{\mathrm{C}}\mathrm{M}}$ in it. Moreover, M is homogeneous Kähler-Einstein with a positive Einstein constant.

A parabolic subgroup P of G is a closed Lie subgroup such that the Lie algebra of P contains a Cartan subalgebra of G and all the negative root vectors with respect to this Cartan algebra.

A classical example from our example is the complex two dimensional Hopf surface S = $C^2$/(2). It is a complex one dimensional torus bundle over ${CP}^1$.

5. Conclusions

In this paper we finished the classification of compact homogeneous complex Finsler manifolds as complex manifolds and also all the possible homogenous complex Finsler metrics algebraically. Part of our results can be also used for any general complex Finsler metrics (with the strictly pseudo-convex condition, which is needed for the definition of a meaningful connection). However, we are not going into the geometric properties of these manifolds and metrics. We leave the further calculations to more advanced colleagues who worked, work and will still work in complex Finsler area or they might get some help even from AI. And we just leave here with a good wish for future fruitful developments.