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 K\"{a}hler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost K\"{a}hler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for K\"{a}hler-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 K\"{a}hler-Ricci soliton on its smooth part.