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Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds

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Abstract

We prove that a general complex Monge–Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti–Weinkove: the Chern–Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern–Ricci flow on compact Hermitian manifolds, namely the twisted Chern–Ricci flow.

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Notes

  1. After this paper was completed, the author learned that Xiaolan Nie proved the first statement of the conjecture for complex surfaces (cf. [26]). She also proved that the Chern–Ricci flow can be run from a bounded data. The author would like to thank Xiaolan Nie for sending her preprint.

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Acknowledgements

The author is grateful to his supervisor Vincent Guedj for support, suggestions and encouragement. The author thanks Valentino Tosatti and Ben Weinkove for their interest in this work and helpful comments. We also thank Thu Hang Nguyen, Van Hoang Nguyen and Ahmed Zeriahi for very useful discussions. The author would like to thank the referee for useful comments and suggestions. This work is supported by the Jean-Pierre Aguilar fellowship of the CFM foundation.

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  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062, Toulouse Cedex 09, France

    Tat Dat Tô

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Communicated by Ngaiming Mok.

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Tô, T.D. Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds. Math. Ann. 372, 699–741 (2018). https://doi.org/10.1007/s00208-017-1574-7

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