Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Deformations of Fano Manifolds

Huai-Dong Cao, Xiaofeng Sun, Shing-Tung Yau, Yingying Zhang

Abstract

In this paper we provide new necessary and sufficient conditions for the existence of Kähler-Einstein metrics on small deformations of a Fano Kähler-Einstein manifold. We also show that the Weil-Petersson metric can be approximated by the Ricci curvatures of the canonical $L^2$ metrics on the direct image bundles. In addition, we describe the plurisubharmonicity of the energy functional of harmonic maps on the Kuranishi space of the deformation of compact Kähler-Einstein manifolds of general type.

On Deformations of Fano Manifolds

Abstract

In this paper we provide new necessary and sufficient conditions for the existence of Kähler-Einstein metrics on small deformations of a Fano Kähler-Einstein manifold. We also show that the Weil-Petersson metric can be approximated by the Ricci curvatures of the canonical metrics on the direct image bundles. In addition, we describe the plurisubharmonicity of the energy functional of harmonic maps on the Kuranishi space of the deformation of compact Kähler-Einstein manifolds of general type.

Paper Structure

This paper contains 5 sections, 15 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The Kuranishi Gauge
  3. Small Deformation of Fano Kähler-Einstein Manifolds
  4. Curvature of the $L^2$ Metrics on Direct Image Sheaves
  5. Plurisubharmonicity of Energy of Harmonic Maps

Key Result

Theorem 1.1

Let $\left ( X_0,\omega_0\right )$ be a Fano Kähler-Einstein manifold and let $\left ( {\mathfrak X},B,\pi\right )$, with $X_t=\pi^{-1}(t)$, be the Kuranishi family of $X_0$ with respect to $\omega_0$. Then the following statements are equivalent:

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.2
  • Remark 4
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • proof
  • ...and 21 more