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

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.