Bergman Kernels 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.2000 Mathematics Subject Classification. Primary: 53C25; Secondary: 53C55, 58J05.

The moduli space of Fano manifolds with Kähler–Ricci solitons

“…Then it follows that any sequence (X σi , ω ti (α σi )) (t i → 1) is a sequence of almost Kähler-Ricci solitons in the sense of [F. Wang, X. Zhu, Definition 5.1]. Now we can deduce our estimate from [JWZ,Cororally 1.4], [DonSun1,Lemma 3.4] and the argument after the lemma. Now we can apply the arguments in [DonSun1] to the metric family…”

Kähler–Einstein metrics along the smooth continuity method

“…We can also choose a further subsequence of t k if necessary to have a weak limit (F t k ) * ω t k → ω 1 . We have the following, see [31,Corollary 1.4]. A proof can also be given in the spirit of the proof of Proposition 13.…”

“…This case is much more similar to the work of Chen-DonaldsonSun [20], since the "current part" of the equation disappears as t → 0. The case of Kähler-Ricci solitons was also studied by Jiang-Wang-Zhu [31]. We briefly describe the argument for the sake of completeness.…”

“…We briefly describe the argument for the sake of completeness. Just as in the case T < 1, we have embeddings F t : M → P N using suitable orthonormal bases for H 0 (K −m M ) with respect to the metric ω t , for some large m. The partial C 0 -estimate is still valid, in the Kähler-Einstein case by [42] based on the method in [20], and in the soliton case due to Jiang-Wang-Zhu [31]. It follows that as before, up to increasing m and choosing a sequence t k → 1 we have the algebraic convergence F t k (M ) → W ∈ P N to a normal Q-Fano variety, homeomorphic to the Gromov-Hausdorff limit (Z, d Z ) of the sequence (M, ω t k ).…”

“…The case when T = 1 has been established by JiangWang-Zhu [31]. Here v is a holomorphic vector field, such that Im v generates a compact torus of isometries of the metric α.…”