If a Lie group G acts on a symplectic manifold (M, ω) and preserves the symplectic form ω, then in some cases there may exist a moment map ([C]) Φ from M to the dual of the Lie algebra. When the action is (locally) free, the preimage of a point in the dual of the Lie algebra modulo the isotropy group of this point will still be a symplectic manifold (orbifold). This process is called the Marsde...

Let X be a quasiprojective manifold given by the complement of a divisor D with normal crossings in a smooth projective manifold X. Using a natural compactification of X by a manifold with corners e X, we describe the full asymptotic behavior at infinity of certain complete Kähler metrics of finite volume on X. When these metrics evolve according to the Ricci flow, we prove that such asymptotic...

We consider compact Kähler manifolds acted on by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||μ|| is constant if and only if the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K. The authors do not know whether the compactnes...

Let M be a close complex manifold and T M its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then M is a complex homogeneous manifold. It implies that every irreducible close Kähler manifold with ample tangent bundle is isomorphic to the projective space. This provides an alternative proof of Hartshore's conjecture in algebrai...

We prove the existence of a (unique) S-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a realanalytic Kähler manifold X, extending the metric on X. In the important paper [3], Calabi proved existence of Ricci-flat Kähler metrics on two classes of manifolds: a) cotangent bundles of projective spaces; b) canonical bundles of Kähler-Einstein man...

Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a constant scalar curvature (resp. an extremal) Kähler metric, possibly of Poincar...

Locally conformally Kähler (LCK) manifolds with potential are those which admit a Kähler covering with a proper, automorphic, global potential. Existence of a potential can be characterized cohomologically as vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manif...

This paper is the continuation of the paper [6] of our study of the Moduli space of polarized Calabi-Yau threefold. A polarized Calabi-Yau manifold is a pair (X,ω) of a compact algebraic manifold X with zero first Chern class and a Kähler form ω ∈ H(X,Z). The form ω is called a polarization. Let U be the universal deformation space of (X,ω). U is smooth by a theorem of Tian [12]. By [15], we ma...

In this paper we describe the Einstein-Kähler metric for the Cartan-Hartogs domains which are the special case of the Hua domains. First of all, we reduce the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w). This differential equation can be solved to give an implicit function in X. Secondly, for some cases, we obtained explicit fo...

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D ⊂ TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M, g). If Z also admits a second complex contact structure D̃ 6= D, then Z = C...