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Openness of uniformly valuative stability on the Kähler cone of projective manifolds

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Abstract

Assume that a projective variety is uniformly valuatively stable with respect to a polarization. We show that the projective variety is uniformly valuatively stable with respect to any polarization sufficiently close to the original polarization. The definition of uniformly valuatively stability in this paper is stronger than that given by Dervan and Legendre (Valuative stability of polarised varieties, arXiv:2010.04023, 2020). We also define the valuative stability for the transcendental Kähler classes. Our openness result can be extended to the Kähler cone of projective manifolds.

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Acknowledgements

I would like to thank my supervisor Akito Futaki for constant help, his guidance and teaching over many years. I am grateful to Jian Xiao for telling me the differentiability result of Witt Nyström [41] and Kewei Zhang for helpful discussions. I wish to thank the anonymous referees for their many helpful comments.

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  1. Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Shuangqing, Beijing, 100084, China

    Yaxiong Liu

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Liu, Y. Openness of uniformly valuative stability on the Kähler cone of projective manifolds. Math. Z. 303, 52 (2023). https://doi.org/10.1007/s00209-023-03209-6

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