Kähler-Ricci flow for deformed complex structures

Abstract

Let ( M , J 0 ) (M,J_0) be a Fano manifold which admits a Kähler-Ricci soliton, we analyze the behavior of Kähler-Ricci flow near this soliton as we deform the complex structure J 0 J_0 . First, we will establish an inequality of Lojasiewicz’s type for Perelman’s entropy along the Kähler-Ricci flow. Then we prove the convergence of Kähler-Ricci flow when the complex structure associated to the initial value lies in the kernel Z Z or negative part of the second variation operator of Perelman’s entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons in the moduli space of complex structures near J 0 J_0 , and we show that the kernel Z Z corresponds to the local moduli space of Fano manifolds which are modified K K -semistable. We also prove a uniqueness theorem for Kähler-Ricci solitons.