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

TL;DR

This paper introduces a Weil-Petersson type metric on the space of Fano Kähler-Ricci solitons, characterizes its independence, and explores its properties and deformations, advancing understanding of the geometric structure of these solitons.

Contribution

It defines a new Weil-Petersson type metric for Fano Kähler-Ricci solitons and analyzes its independence, Kähler property, and deformation behavior, providing foundational insights.

Findings

The Weil-Petersson metric is Kähler on the Kuranishi space.

A necessary and sufficient condition for metric independence is established.

First and second order deformations relate directly to the Weil-Petersson metric.

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 K\"ahler-Ricci solitons and show that, essentially, the first effective term in deforming Kaehler-Ricci solitons leads to the Weil-Petersson metric.