Blowup behavior of the Kahler-Ricci flow on Fano manifolds

TL;DR

This paper investigates the blowup behavior of the normalized Kahler-Ricci flow on Fano manifolds lacking Kahler-Einstein metrics, providing estimates and convergence results for the flow and related methods.

Contribution

It establishes new estimates for the Kahler potential and volume form convergence away from certain subschemes, extending to Aubin's continuity method.

Findings

Volume forms converge to zero away from a multiplier ideal subscheme.

Kahler potential estimates are obtained outside the subscheme.

Results apply to both Kahler-Ricci flow and Aubin's continuity method.

Abstract

We study the blowup behavior at infinity of the normalized Kahler-Ricci flow on a Fano manifold which does not admit Kahler-Einstein metrics. We prove an estimate for the Kahler potential away from a multiplier ideal subscheme, which implies that the volume forms along the flow converge to zero locally uniformly away from the same set. Similar results are also proved for Aubin's continuity method.