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.
Categories: math.DG