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Valuative Invariants with Higher Moments

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Abstract

In this article we introduce a family of valuative invariants defined in terms of the p-th moment of the expected vanishing order. These invariants lie between \(\alpha \) and \(\delta \)-invariants. They vary continuously in the big cone and semi-continuously in families. Most importantly, they give sufficient conditions for K-stability of Fano varieties, which generalizes the \(\alpha \) and \(\delta \)-criterions in the literature. They are also related to the \(d_p\)-geometry of maximal geodesic rays.

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Notes

  1. This suggests that \((S^{(p)}(L,F))^{1/p}\) can be treated as the p-th barycenter of a convex body along x-axis.

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Acknowledgements

The author would like to thank Tamás Darvas and Mingchen Xia for helpful discussions on Section 6. Special thanks go to Yuchen Liu for valuable comments and for proving Proposition 4.4. He also thanks Yanir Rubinstein for suggesting Theorem 1.9. The author is supported by the China post-doctoral Grant BX20190014.

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  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Kewei Zhang

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Zhang, K. Valuative Invariants with Higher Moments. J Geom Anal 32, 10 (2022). https://doi.org/10.1007/s12220-021-00827-6

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