Toshiki Mabuchi

UNIQUENESS OF EXTREMAL KÄHLER METRICS FOR AN INTEGRAL KÄHLER CLASS

For an integral Kahler class on a compact connected complex manifold, an extremal Kahler metric, if any, in the class is unique up to the action of Aut0(M). This generalizes a recent result of Donaldson (see [4] for cases of metrics of constant scalar curvature) and that of Chen [3] for c1(M)≤0.

An obstruction to asymptotic semistability and approximate critical metrics

In this paper, we consider an obstruction to asymptotic Chow-semistability of a polarized Kaehler algebraic manifold. Even when a linear algebraic group of positive dimension acts nontrivially and holomorphically on a polarized Kaehler algebraic manifold with constant scalar curvature, the vanishing of the obstruction allows us to generalize Donaldsons construction of approximate solutions for equations of balanced metrics.

Multiplier Hermitian structures on Kähler manifolds

The main purpose of this paper is to make a systematic study of a special type of conformally Kahler manifolds, called multiplier Hermitian manifolds, which we often encounter in the study of Hamiltonian holomorphic group actions on Kahler manifolds. In particular, we obtain a multiplier Hermitian analogue of Myers’ Theorem on diameter bounds with an application (see [M5]) to the uniquness up to biholomorphisms of the “Kahler-Einstein metrics” in the sense of [M1] on a given Fano manifold with nonvanishing Futaki character.

An Algebraic Character associated with the Poisson Brackets

This chapter focuses on an algebraic character associated with the Poisson brackets. If N is a connected compact Kahler manifold and Aut( N ) is the group of holomorphic automorphisms of N , then if c 1 R c 1 R = 0, then the celebrated solution of Calabis conjecture by Aubin and Yau asserts that N always admits an Einstein–Kahler metric. In the case c 1 R > 0, however, the existence problem is still open, and a couple of obstructions to the existence are known.

Asymptotics of polybalanced metrics under relative stability constraints

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds (M , L), a series of weighted balanced metrics !m , m 1, called polybalanced metrics, are obtained from complete linear systems jLm j on M . Then the asymptotic behavior of the weights as m !1 will be studied.

Strong K-stability and Asymptotic Chow-stability

For a polarized algebraic manifold (X,L), let T be an algebraic torus in the group Aut(X) of all holomorphic automorphisms of X. Then strong relative K-stability (cf. [6]) will be shown to imply asymptotic relative Chow-stability. In particular, by taking T to be trivial, we see that asymptotic Chow-stability follows from strong K-stability.

The Yau-Tian-Donaldson Conjecture for General Polarizations, I

In this paper, some \(C^0\) boundedness property (BP) is introduced for balanced metrics on a polarized algebraic manifold (X, L). Then by assuming that (X, L) is strongly K-stable in the sense of [8], we shall show that the balanced metrics have (BP). In a subsequent paper [10], this property (BP) plays a very important role in the study of the Yau-Tian-Donaldson conjecture for general polarizations.

The Donaldson–Futaki Invariant for Sequences of Test Configurations

In this paper, given a polarized algebraic manifold (X,L), we define the Donaldson–Futaki invariant \(F_1\left(\{\mu_{i}\}\right)\) for a sequence \(\{\mu_{i}\}\) of test configurations for (X,L) of exponents lisatisfying

An ℓ-th root of a test configuration of exponent ℓ

l_i\rightarrow\;\infty,\quad \mathrm{as} \ j\rightarrow\;\infty.