Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T02:36:05.500Z Has data issue: false hasContentIssue false
  • English
  • Français

Grothendieck Ring of Varieties with Actions of Finite Groups

Part of: (Co)homology theory Categories and geometry Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  08 March 2019

S.M. Gusein-Zade ,
I. Luengo  and
A. Melle-Hernández
S.M. Gusein-Zade
Affiliation:
Moscow State University, Faculty of Mechanics and Mathematics, GSP-1, Moscow, 119991, Russia (sabir@mccme.ru)
I. Luengo
Affiliation:
ICMAT (CSIC-UAM-UC3M-UCM), Department of Algebra, Geometry and Topology, Complutense University of Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain (iluengo@mat.ucm.es)
A. Melle-Hernández
Affiliation:
Department of Algebra, Geometry and Topology, Instituto de Matemática Interdisciplinar (IMI), Complutense University of Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain (amelle@mat.ucm.es)
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

Keywords

finite group actionscomplex quasi-projective varietiesGrothendieck ringslambda-structurepower structure

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 18F30: Grothendieck groups 55M35: Finite groups of transformations (including Smith theory)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 925 - 948
Copyright
Copyright © Edinburgh Mathematical Society 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

To the memory of Michael Atiyah

References

1Atiyah, M. and Segal, G., On equivariant Euler characteristics, J. Geom. Phys. 6(4) (1989), 671–677.Google Scholar
2Bittner, F., The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140(4) (2004), 1011–1032.Google Scholar
3Bryan, J. and Fulman, J., Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups, Ann. Comb. 2(1) (1998), 1–6.Google Scholar
4Getzler, E. and Pandharipande, R., The Betti numbers of ${\overline {\cal M}}_0,n(r,d)$, J. Algebraic Geom. 15(4) (2006), 709–732.Google Scholar
5Gusein-Zade, S. M., Luengo, I. and Melle-Hernández, A., A power structure over the Grothendieck ring of varieties, Math. Res. Lett. 11 (2004), 49–57.Google Scholar
6Gusein-Zade, S. M., Luengo, I. and Melle-Hernández, A., On the pre-λ-ring structure on the Grothendieck ring of stacks and the power structures over it, Bull. Lond. Math. Soc. 45(3) (2013), 520–528.Google Scholar
7Gusein-Zade, S. M., Luengo, I. and Melle-Hernández, A., Higher order generalized Euler characteristics and generating series, J. Geom. Phys. 95 (2015), 137–143.Google Scholar
8Gusein-Zade, S. M., Luengo, I. and Melle-Hernández, A., Equivariant versions of higher order orbifold Euler characteristics, Mosc. Math. J. 16(4) (2016), 751–765.Google Scholar
9Hirzebruch, F. and Höfer, T., On the Euler number of an orbifold, Math. Ann. 286(1–3) (1990), 255–260.Google Scholar
10Ito, Y. and Reid, M., The McKay correspondence for finite subgroups of SL(3,C), in Higher-dimensional complex varieties (Trento, 1994), (ed. Andreatta, M. and Peternell, Th.), pp. 221–240 (de Gruyter, Berlin, 1996).Google Scholar
11Knutson, D., λ-rings and the representation theory of the symmetric group, Lecture Notes in Mathematics, Volume 308 (Springer-Verlag, Berlin, 1973).Google Scholar
12Mazur, J., Rationality of motivic zeta functions for curves with finite abelian group actions, J. Pure Appl. Algebra 217(7) (2013), 1335–1349.Google Scholar
13Ramachandran, N., Zeta functions, Grothendieck groups, and the Witt ring, Bull. Sci. Math. 139(6) (2015), 599–627.Google Scholar
14Tamanoi, H., Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-theory, Algebr. Geom. Topol. 1 (2001), 115–141.Google Scholar
15Wang, W. and Zhou, J., Orbifold Hodge numbers of wreath product orbifolds, J. Geom. Phys. 38 (2001), 152–169.Google Scholar
16Zaslow, E., Topological orbifold models and quantum cohomology rings, Comm. Math. Phys. 156(2) (1993), 301–331.Google Scholar