The Kähler Ricci flow on Fano manifolds (I)

Abstract

We study the evolution of pluri-anticanonical line bundles K -νM 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 21(M)=1 or c 12 = 3. Combined with the work of [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 [Tian90]. © 2012 European Mathematical Society.