K\"ahler Ricci flow on Fano manifolds(I)

TL;DR

This paper investigates how the Kähler Ricci flow evolves on Fano manifolds, linking convergence to properties of anticanonical divisors, and provides a Ricci flow proof of the Calabi conjecture for certain Fano surfaces.

Contribution

It establishes conditions under which the Kähler Ricci flow converges on Fano manifolds, connecting geometric properties to flow behavior, and offers a new proof of the Calabi conjecture in specific cases.

Findings

Flow converges on Fano surfaces with specific Chern numbers

Convergence determined by anticanonical divisor properties

Provides Ricci flow proof of Calabi conjecture for certain Fano surfaces

Abstract

We study the evolution of anticanonical line bundles along the K\"ahler Ricci flow. We show that under some conditions, the convergence of K\"ahler Ricci flow is determined by the properties of the anticanonical divisors of $M$. As examples, the K\"ahler Ricci flow on $M$ converges when $M$ is a Fano surface and $c_1^2(M)=1$ or $c_1^2(M)=3$. Combined with the work in \cite{CW1} and \cite{CW2}, this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian.