The moduli space of Fano manifolds with K\"ahler-Ricci solitons

TL;DR

This paper constructs a canonical moduli space for Fano manifolds with K"ahler-Ricci solitons, extending previous work on K"ahler-Einstein metrics, and introduces a moment map framework with complex analytic charts.

Contribution

It develops a complex analytic moduli space for Fano manifolds with K"ahler-Ricci solitons, generalizing existing moduli spaces and employing a moment map approach.

Findings

Constructed a Hausdorff complex analytic moduli space.

Established a moment map picture for K"ahler-Ricci solitons.

Provided complex analytic charts on the space of solitons.

Abstract

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with K\"ahler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with K\"ahler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for K\"ahler-Ricci solitons, and give complex analytic charts on the topological space consisting of K\"ahler-Ricci solitons, by studying differential geometric aspects of this infinite dimensional moment map. Some stacky words and arguments on Gromov-Hausdorff convergence help to glue them together in the holomorphic manner.