Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Diagonalisable $p$-groups cannot fix exactly one point on projective varieties


Author: Olivier Haution
Journal: J. Algebraic Geom. 29 (2020), 373-402
DOI: https://doi.org/10.1090/jag/749
Published electronically: November 15, 2019
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Abstract | References | Additional Information

Abstract: We prove an algebraic version of a classical theorem in topology, asserting that an abelian $p$-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove that the number of fixed points cannot be odd. The main tool is a construction originally used by Rost in the context of the degree formula. The framework of diagonalisable groups allows us to include the case of base fields of characteristic $p$.


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Olivier Haution
Affiliation: Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, D-80333 München, Germany
Email: olivier.haution@gmail.com

Received by editor(s): February 27, 2018
Received by editor(s) in revised form: January 4, 2019
Published electronically: November 15, 2019
Additional Notes: This work was supported by the DFG grant HA 7702/1-1
Article copyright: © Copyright 2019 University Press, Inc.