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Reductivity of the automorphism group of K-polystable Fano varieties

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Throughout, we work over an algebraically closed field k of characteristic 0.

Abstract

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and \(\Theta \)-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

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Notes

  1. Since the first version of the current paper was written, this expectation was proved in [43].

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Acknowledgements

JA and CX thank Xiaowei Wang, and CX thanks Jun Yu for helpful conversations. We thank the referees for suggestions on revising the paper. Much of the work on this paper was completed while the authors enjoyed the hospitality of the MSRI, which is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    Jarod Alper

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 48112, USA

    Harold Blum

  3. Malott Hall, Mathematics Department, Cornell University, Ithaca, NY, 14853, USA

    Daniel Halpern-Leistner

  4. MIT, Cambridge, 02139, USA

    Chenyang Xu

  5. Beijing International Center for Mathematical Research, Beijing, 100871, China

    Chenyang Xu

Authors
  1. Jarod Alper
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  2. Harold Blum
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  3. Daniel Halpern-Leistner
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  4. Chenyang Xu
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Correspondence to Chenyang Xu.

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JA was partially supported by NSF Grant DMS-1801976. HB was partially supported by NSF Grant DMS-1803102. DHL was partially supported by NSF Grant DMS-1762669. CX was partially supported by a Chern Professorship of the MSRI (NSF No. DMS-1440140), by NSFC Grant No. 11425101 (2015-2018) and by NSF Grant DMS-1901849.

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Alper, J., Blum, H., Halpern-Leistner, D. et al. Reductivity of the automorphism group of K-polystable Fano varieties. Invent. math. 222, 995–1032 (2020). https://doi.org/10.1007/s00222-020-00987-2

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  • DOI: https://doi.org/10.1007/s00222-020-00987-2

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