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
Full-text PDF
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 $p$.
References
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- Olivier Haution, Involutions of varieties and Rost’s degree formula, J. Reine Angew. Math. 745 (2018), 231–252. MR 3881477, DOI https://doi.org/10.1515/crelle-2016-0003
- Alexander S. Merkurjev, Rost’s degree formula, notes of mini-course in Lens, 2001, http://www.math.ucla.edu/~merkurev/papers/lens.dvi.
- Markus Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393. MR 1418952
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- Philippe Gille and Patrick Polo (eds.), Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 7, Société Mathématique de France, Paris, 2011 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64]; A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; Revised and annotated edition of the 1970 French original. MR 2867621
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References
- M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR 0232406, DOI https://doi.org/10.2307/1970721
- Alexander Ritchie Boisvert, A new definition of the Steenrod operations in algebraic geometry, Thesis (Ph.D.)–University of California, Los Angeles, 2007, ProQuest LLC, Ann Arbor, MI. MR 2710781
- N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, Paris, 1970 (French). MR 0274237
- William Browder, Pulling back fixed points, Invent. Math. 87 (1987), no. 2, 331–342. MR 870731, DOI https://doi.org/10.1007/BF01389418
- P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
- P. E. Conner and E. E. Floyd, Maps of odd period, Ann. of Math. (2) 84 (1966), 132–156. MR 0203738, DOI https://doi.org/10.2307/1970515
- A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222 (French). MR 217084
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181
- John Ewing and Robert Stong, Group actions having one fixed point, Math. Z. 191 (1986), no. 1, 159–164. MR 812609, DOI https://doi.org/10.1007/BF01163616
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- Olivier Haution, Involutions of varieties and Rost’s degree formula, J. Reine Angew. Math. 745 (2018), 231–252. MR 3881477, DOI https://doi.org/10.1515/crelle-2016-0003
- Alexander S. Merkurjev, Rost’s degree formula, notes of mini-course in Lens, 2001, http://www.math.ucla.edu/~merkurev/papers/lens.dvi.
- Markus Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393. MR 1418952
- Markus Rost, Notes on the degree formula, preprint, 2001, https://www.math.uni-bielefeld.de/~rost/degree-formula.html.
- Jean-Pierre Serre, Représentations linéaires des groupes finis, Third revised edition, Hermann, Paris, 1978 (French). MR 543841
- Philippe Gille and Patrick Polo (eds.), Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 7, Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; revised and annotated edition of the 1970 French original, Société Mathématique de France, Paris, 2011 (French). MR 2867621
- William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
Additional Information
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.
