The Kodaira dimension of diffeomorphic Kähler threefolds

We provide infinitely many examples of pairs of diffeomorphic, non-simply-connected Kähler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non-deformation equivalent, diffeomorphic Kähler threefolds.

     

Convexity of moment polytopes of algebraic varieties

We consider the situation of a compact irreducible subvariety of a smooth compact complex variety equipped with a Kähler form preserved by a torus action. We study the image of that subvariety under the moment map of the Kähler form.

     

Convergence of Kähler-Ricci flow

In this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold which admits a Kähler-Ricci soliton.

     

Almost Hermitian structures induced from a Kähler structure which has constant holomorphic sectional curvature

We obtain a non-Kähler almost Hermitian manifold of constant holomorphic sectional curvature by changing the almost complex structure in a Kähler manifold of constant holomorphic sectional curvature.

     

On the Albanese maps of compact Kähler manifolds

We show that compact Kähler manifolds of dimension up to four have a surjective connected Albanese map.

     

Pseudo-holomorphic curves in complex Grassmann manifolds

It is proved that the Kähler angle of the pseudo-holomorphic sphere of constant curvature in complex Grassmannians is constant. At the same time we also prove several pinching theorems for the curvature and the Kähler angle of the pseudo-holomorphic spheres in complex Grassmannians with non-degenerate associated harmonic sequence.

     

Central Kähler metrics

The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.

     

Symplectic 4-manifolds with Hermitian Weyl tensor

It is proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kähler.     

-metrics, projective flatness and families of polarized abelian varieties

We compute the curvature of the -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.

     

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.     

Functorial Hodge identities and quantization

By a uniform abstract procedure, we obtain integrated forms of the classical Hodge identities for Riemannian, Kähler and hyper-Kähler manifolds, as well as of the analogous identities for metrics of arbitrary signature. These identities depend only on the type of geometry and, for each of the three types of geometry, define a multiplicative functor from the corresponding category of real, graded, flat vector bundles to the category of infinite-dimensional -projective representations of an algebraic structure. We define new multiplicative numerical invariants of closed Kähler and hyper-Kähler manifolds which are invariant under deformations of the metric.

     

Ricci flatness of asymptotically locally Euclidean metrics

In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.

     

Bochner-Kähler metrics

A Kähler metric is said to be Bochner-Kähler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least . In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kähler metrics in complex dimension has real dimension and a recipe for an explicit formula for any Bochner-Kähler metric will be given.

It is shown that any Bochner-Kähler metric in complex dimension  has local (real) cohomogeneity at most . The Bochner-Kähler metrics that can be `analytically continued' to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kähler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kähler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kähler metric.

The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kähler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.

     

On the critical points of the functionals in Kähler geometry

We prove that a Kähler metric in the anticanonical class, that is a critical point of the functional and has nonnegative Ricci curvature, is necessarily Kähler-Einstein. This partially answers a question of X.X. Chen.

     

Polar and coisotropic actions on Kähler manifolds

The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

     

Embeddability of some strongly pseudoconvex CR manifolds

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are boundaries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kähler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

     

A uniqueness result of Kähler Ricci flow with an application

In this paper, we will study the problem of uniqueness of Kähler Ricci flow on some complete noncompact Kähler manifolds and the convergence of the flow on with the initial metric constructed by Wu and Zheng.

     

Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

     

On cohomology of ACH Kähler manifolds

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.

     

A K-energy characterization of extremal Kähler metrics

We show that for any polarized compact Kähler manifold, the extremal Kähler metrics that represent the given cohomology class can be characterized as critical points of a suitably defined K-energy functional.