Bergman Kernels and algebraic structure of limit space for a sequence of almost Kähler-Ricci solitons

Abstract

In this paper, we give a lower bound of Bergman kernels for a sequence of almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost Kähler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive scalar curvature. As an application of our result, we prove that the Gromov-Hausdorff limit of sequence is homomorphic to a log terminal $Q$-Fano variety which admits a Kähler-Ricci soliton on its smooth part.

Section 4 was rewritten, Lemma 8.3 was newly added. Keywords: Kähler-Einstein metrics, almost Kähler-Ricci solitons, Ricci flow, $\bar\partial$-equation