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A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

Huai-Dong Cao, Xiaofeng Sun, Yingying Zhang

Abstract

In this paper we define a Weil-Petersson type metric on the space of shrinking Kaehler-Ricci solitons and prove a necessary and sufficient condition on when it is independent of the choices of Kaehler-Ricci soliton metrics. We also show that the Weil-Petersson metric is Kaehler when it defines a metric on the Kuranishi space of small deformations of Fano Kaehler-Ricci solitons. Finally, we establish the first and second order deformation of Fano Kähler-Ricci solitons and show that, essentially, the first effective term in deforming Kaehler-Ricci solitons leads to the Weil-Petersson metric.

A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

Abstract

In this paper we define a Weil-Petersson type metric on the space of shrinking Kaehler-Ricci solitons and prove a necessary and sufficient condition on when it is independent of the choices of Kaehler-Ricci soliton metrics. We also show that the Weil-Petersson metric is Kaehler when it defines a metric on the Kuranishi space of small deformations of Fano Kaehler-Ricci solitons. Finally, we establish the first and second order deformation of Fano Kähler-Ricci solitons and show that, essentially, the first effective term in deforming Kaehler-Ricci solitons leads to the Weil-Petersson metric.
Paper Structure (5 sections, 13 theorems, 70 equations)

This paper contains 5 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. The Twisted Kuranishi-Divergence Gauge
  3. The Weil-Petersson Metric
  4. The Weil-Petersson metric on the space of Fano Kähler-Einstein manifolds
  5. Weil-Petersson type metric on the space of Fano Kähler-Ricci solitons

Key Result

Theorem 2.1

Let $\left ( M, \omega_g\right )$ be a Fano manifold with Ricci potential $f$. Then $\lambda_1\equiv \lambda_1 (\Delta_f)\geq 1$. Moreover, if $H^0\left ( M, T^{1,0}M\right )\ne 0$ then $\lambda_1=1$ and $H^0\left ( M, T^{1,0}M\right )\cong \Lambda_f^1.$ In particular, are linear isomorphisms with $\nabla^{1,0}\circ \text{div}_f=-Id$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.1
  • proof
  • Theorem 3.3
  • proof
  • ...and 9 more