Dynamics of automorphisms on compact Kähler manifolds

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.     

A note on compact Kähler manifolds with nef tangent bundles

In this paper, we study the compact Kähler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one. LetX be a compact Kähler manifold with nef tangent bundle and semiample anti-canonical bundle. We prove that κ(?K _X )=1 if and only if there exists a finite étale coverY→X such thatY??¹×A, whereA is a complex torus. As a consequence, we are able to improve upon a result of T. Fujiwara [3, 4].     

An Ohsawa-Takegoshi theorem on compact Khler manifolds

We prove two extension theorems of Ohsawa-Takegoshi type on compact Khler manifolds.In our proof,there are many complications arising from the regularization process of quasi-psh functions on compact Khler manifolds,and unfortunately we only obtain particular cases of the expected result.We remark that the two special cases we proved are natural,since they occur in many situations.We hope that the new techniques we develop here will allow us to obtain the general extension result of Ohsawa-Takegoshi type on compact Khler manifolds in a near future.     

Ricci flow on Kähler manifolds

In this Note, we announce the result that if M is a Kähler–Einstein manifold with positive scalar curvature, if the initial metric has nonnegative bisectional curvature, and the curvature is positive somewhere, then the Kähler–Ricci flow converges to a Kähler–Einstein metric with constant bisectional curvature.     

Kähler magnetic fields on Kähler manifolds of negative curvature

On a Kähler manifold we have natural uniform magnetic fields which are constant multiples of the Kähler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kähler magnetic fields and show some similarities between trajectories and geodesics on Kähler manifolds of negative curvature.     

Trees and tensors on Kähler manifolds

We present an organized method to convert between partial derivatives of metrics (functions) and covariant derivatives of curvature tensors (functions) on Kähler manifolds. Basically, it reduces the highly recursive computation in tensor calculus to the enumeration of certain trees with external legs.     

Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds

For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the ?²-homology of a sphere.     

A theorem of Tits type for compact Kähler manifolds

We prove a theorem of Tits type about automorphism groups for compact Kähler manifolds, which has been conjectured in the paper [9].     

Symplectic Dolbeault operators on Kähler manifolds

For a Kähler manifold $M$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar{\partial }$ and $\bar{\partial }^*$ , arise from Dirac operators on the canonical complex spinors on $M$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$ . We give a thorough analysis of these operators on $\mathbb{C } P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.     

Domains of definition of Monge-Ampère operators on compact Kähler manifolds

Let (X, ω) be a compact Kähler manifold. We introduce and study the largest set DMA(X, ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set PSH(X, ω) of all ω-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all ω-psh functions. As a consequence, any ω-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy.     

Eigenvalues of the complex Laplacian on compact non-Kähler manifolds

We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.     

Harmonic maps from compact Kähler manifolds to exceptional hyperbolic spaces

It has been conjectured that a lattice in a noncompact group of real rank one, other than SU(1,n), cannot be isomorphic to the fundamental group of a compact Kähler manifold; moreover, it is known to be true for SO(1,n). In this note it is shown that this conjecture also holds for the case of uniform lattices in F_4(?20), the group of isometries of the Cayley hyperbolic plane. The result is a consequence of a classification theorem for harmonic maps between Kähler and Cayley hyperbolic manifolds.     

The Bergman metric on complete Kähler manifolds

We use the existence of a bounded uniformly Hölder continuous plurisubharmonic exhaustion function to characterize the Bergman completeness of a complete Kähler manifold. As an application, we proved that any simply-connected complete Kähler manifold with sectional curvature bounded above by a negative constant is Bergman complete. Mathematics Subject Classification (2000):32H10Supported by NSFC grant no. 10271089     

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

Isospectral nearly Kähler manifolds

We give a systematic way to construct almost conjugate pairs of finite subgroups of \(\mathrm {Spin}(2n+1)\) and \({{\mathrm{Pin}}}(n)\) for \(n\in {\mathbb {N}}\) sufficiently large. As a geometric application, we give an infinite family of pairs \(M_1^{d_n}\) and \(M_2^{d_n}\) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions \(d_n>6\). We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.