Let X be a compact complex manifold and L → X a positive holomorphic line bundle. Assume that Aut(X,L)/C× is discrete, where Aut(X,L) is the group of holomorphic automorphisms of the pair (X,L). Donaldson [D] has recently proved that if X admits a metric ω ∈ c1(L) of constant scalar curvature, then (X,L) is Hilbert-Mumford stable for k sufficiently large. Since Kähler-Einstein metrics have cons...

We obtain a vanishing theorem for the half-kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the half-kernel of a spinc Dirac operator twisted by a line bundle with curvature of a mixed ...

The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain YII(r, p;K) is obtained in this paper when the parameter K equals p 2 + 1 p+1 . The estimate of holomorphic sectional curvature under this metric is also given which intervenes between −2K and − 2K p and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalen...

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature −1.

Let X be a compact complex manifold and L → X a positive holomorphic line bundle. Assume that Aut(X,L)/C× is discrete, where Aut(X,L) is the group of holomorphic automorphisms of the pair (X,L). Donaldson [D] has recently proved that if X admits a metric ω ∈ c1(L) of constant scalar curvature, then (X,L) is Hilbert-Mumford stable for k sufficiently large. Since Kähler-Einstein metrics have cons...

Let (M, g) be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that M is holomorphically covered by a pseudoconvex domain in C which is homeomorphic to R, provided (M, g) has uniformly faster than linear average quadratic curvature decay.

In this paper we shall prove two theorems about extending holo-morphic mappings between complex manifolds. Both results involve extending such mappings across pseudo-concave boundaries. The first is a removable singularities statement for meromorphic mappings into compact K~ihler manifolds. The precise result and several illustrative examples are given in Section 1. The second theorem is a Hart...