## Kähler-Ricci flow for deformed complex structures

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Gang Tian, Liang Zhang and Xiaohua Zhu
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## 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.## References

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## Additional Information

**Gang Tian**- Affiliation: School of Mathematical Sciences & BICMR, Peking University, Beijing 100871, People’s Republic of China
- MR Author ID: 220655
- Email: gtian@math.princeton.edu
**Liang Zhang**- Affiliation: School of Mathematical Sciences & BICMR, Peking University, Beijing 100871, People’s Republic of China
- Email: xhzhu@math.pku.edu.cn
**Xiaohua Zhu**- Affiliation: School of Mathematical Sciences & BICMR, Peking University, Beijing 100871, People’s Republic of China
- MR Author ID: 629360
- Email: tensor@pku.edu.cn
- Received by editor(s): January 3, 2022
- Received by editor(s) in revised form: September 8, 2022
- Published electronically: December 15, 2022
- Additional Notes: The first author and third author were partially supported by National Key R&D Program of China SQ2020YFA070059. The third author was supported by NSFC 12271009
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 1999-2046 - MSC (2020): Primary 53C25; Secondary 53C55, 32Q20, 32Q10, 58J05
- DOI: https://doi.org/10.1090/tran/8821
- MathSciNet review: 4549698