The K\"ahler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations

TL;DR

This paper links the K"ahler-Ricci flow to bounds on Donaldson-Futaki invariants, providing new insights into Fano manifolds' stability and extending previous finiteness results.

Contribution

It establishes a lower bound for the Donaldson-Futaki invariant via the K"ahler-Ricci flow, generalizing finiteness of Futaki invariants to all Fano manifolds.

Findings

Lower bound for Donaldson-Futaki invariant in terms of Ricci lower bounds

Generalization of Futaki invariants finiteness to all Fano manifolds

Discussion of relation to Hisamoto's inequality for the $H$-functional

Abstract

We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the K\"ahler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on K\"ahler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the $H$-functional.