Asymptotic Properties of Extremal Kähler Metrics of Poincaré Type

Extremal Kähler metrics and K-stability

Note on Poincaré Type Kähler Metrics and Futaki Characters

A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric onX\D with cusp singularity alongD. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application ...

On the Kähler Classes of Extremal Metrics

Let (M,J) be a compact complex manifold, and let E ⊂ H1,1(M,R) be the set of all cohomology classes which can be represented by Kähler forms of extremal Kähler metrics, in the sense of Calabi [3]. Then E is an open subset. ∗Supported in part by NSF grant DMS 92-04093. †Supported in part by NSF grant DMS 90-22140.

Extremal Almost-kähler Metrics

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain example...

Kähler Geometry of Toric Varieties and Extremal Metrics

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope ∆ ⊂ Rn. Recently Guillemin gave an explicit combinatorial way of constructing “toric” Kähler metrics on X, using only data on ∆. In this paper, differential geometric properties of these metrics are investigated using Guillemin’s construction. In particular, a nice combinatorial formula for the...