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Birational superrigidity and slope stability of Fano manifolds

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Abstract

We prove a relation between birational superrigidity of Fano manifold and its slope stability in the sense of Ross and Thomas (J Alg Geom 16:201–205, 2007).

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Notes

  1. This modification is due to a technical issue. For any polarized variety we can find a class of test configurations which are \(\mathbb{G }_m\)-equivariantly isomorphic to trivial test configuration, away from closed subschemes of codimension at least two. Note that those normalizations are the trivial test configurations. The Donaldson–Futaki invariant of the normalization of these “pathological” test configurations automatically vanish. The modification of the definition excluded those class by considering only normal test configurations.

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Acknowledgments

We are grateful to Professors Shigefumi Mori, Constantin Shramov and Alexander Pukhlikov for helpful comments. We would like to thank Doctors Jesus Martinez-Garcia and Julius Ross for their careful reading of the draft versions and Professor Ivan Cheltsov for his interest in this work. We also would like to thank the referee for her/his efforts to point out many errors. The first author is partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 21-3748) and the Grant-in-Aid for JSPS fellows (DC1). The second author is partially supported by the Grand-in-Aid for Scientific Research (KAKENHI No. 23-2053) and the Grand-in-Aid for JSPS fellows (PD).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, Japan

    Yuji Odaka

  2. Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, Japan

    Takuzo Okada

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  1. Yuji Odaka
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  2. Takuzo Okada
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Correspondence to Takuzo Okada.

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Odaka, Y., Okada, T. Birational superrigidity and slope stability of Fano manifolds. Math. Z. 275, 1109–1119 (2013). https://doi.org/10.1007/s00209-013-1172-7

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