K\"ahler-Ricci flow for deformed complex structures

TL;DR

This paper studies the behavior of the K"ahler-Ricci flow on Fano manifolds near a soliton, establishing convergence results, solving the Yau-Tian-Donaldson conjecture locally, and proving uniqueness of K"ahler-Ricci solitons.

Contribution

It proves convergence of the K"ahler-Ricci flow under complex structure deformations, solves the Yau-Tian-Donaldson conjecture locally, and establishes the uniqueness of K"ahler-Ricci solitons.

Findings

Convergence of K"ahler-Ricci flow near a soliton under certain conditions

Solution to the Yau-Tian-Donaldson conjecture in a local moduli space

Uniqueness theorem for K"ahler-Ricci solitons

Abstract

Let $(M,J_0)$ be a Fano manifold which admits a K\"ahler-Ricci soliton, we analyze the behavior of the K\"ahler-Ricci flow near this soliton as we deform the complex structure $J_0$. First, we will establish an inequality of Lojasiewicz's type for Perelman's entropy along the K\"ahler-Ricci flow. Then we prove the convergence of K\"ahler-Ricci flow when the complex structure associated to the initial value lies in the kernel $Z$ or negative part of the second variation operator of Perelman's entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of K\"ahler-Ricci solitons in the moduli space of complex structures near $J_0$, and we show that the kernel $Z$ corresponds to the local moduli space of Fano manifolds which are modified $K$-semistable. We also prove an uniqueness theorem for K\"ahler-Ricci solitons.