The Kähler Ricci flow on Fano manifolds (I)

Abstract

We study the evolution of pluri-anticanonical line bundles K_M^{-\nu} along the Kähler Ricci flow on a Fano manifold M . Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of M . For example, the Kähler Ricci flow on M converges when M is a Fano surface satisfying c_1^2(M)=1 or c_1^2(M)=3 . Combined with the works in [CW1] and [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 in [Tian90].