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The Modified Cusp Kähler–Ricci Flow and Soliton

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Abstract

In this paper, we study the limiting flow of conical Kähler–Ricci flow modified by a holomorphic vector field on a compact Kähler manifold M which carries an effective \({\mathbb {R}}\)-ample divisor D with simple normal crossing support. By smooth approximation, we prove the existence and uniqueness of the flow with cusp singularity when the twisted canonical bundle \(K_M+D\) is ample. At last, we show that the flow converges to a soliton-type metric.

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Acknowledgements

The author would like to thank the referees for their careful reading of the manuscript. The author would like to thank Dr. Jia Wei Liu for introducing this topic to him, as well as to Dr. Xishen Jin for conversation. The author is supported by the Natural Science Foundation of Universities of Anhui Province [Grant Number K120431039]. The author is partially supported by National Natural Science Foundation of China [Grant Numbers 11625106, 11801535, 11721101]. The research is partially supported by the project Analysis and Geometry on Bundle of Ministry of Science and Technology of the People’s Republic of China [Grant Number SQ2020YFA070080].

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    Pan Zhang

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Zhang, P. The Modified Cusp Kähler–Ricci Flow and Soliton. J Geom Anal 31, 10402–10435 (2021). https://doi.org/10.1007/s12220-021-00650-z

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