Kähler-Ricci flow for deformed complex structures

Tian, Gang; Zhang, Liang; Zhu, Xiaohua

doi:10.1090/tran/8821

Kähler-Ricci flow for deformed complex structures
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by Gang Tian, Liang Zhang and Xiaohua Zhu HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 1999-2046 Request permission

Abstract:

Let $(M,J_0)$ 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 $J_0$. 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 $Z$ 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 $J_0$, and we show that the kernel $Z$ corresponds to the local moduli space of Fano manifolds which are modified $K$-semistable. We also prove a uniqueness theorem for Kähler-Ricci solitons.

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