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Higher index Fano varieties with finitely many birational automorphisms

Part of: Birational geometry Special varieties

Published online by Cambridge University Press:  02 December 2022

Nathan Chen  and
David Stapleton
Nathan Chen
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA [email protected]
David Stapleton
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA [email protected]
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Abstract

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

Keywords

Fano varietieshypersurfacesbirational automorphismsrigidity

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations
Secondary: 14E05: Rational and birational maps 14M20: Rational and unirational varieties
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 11 , November 2022 , pp. 2033 - 2045
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The first author's research was partially supported by an NSF postdoctoral fellowship, DMS-2103099. The second author's research is partially supported by the NSF FRG grant number 1952399.

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