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Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

1
Faculty of Mechanics and Mathematics, Moscow State University, GSP-1, Moscow 119991, Russia
2
ICMAT (CSIC-UAM-UC3M-UCM), Dept. of Algebra, Geometry and Topology, Complutense University of Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
3
Instituto de Matemática Interdisciplinar (IMI), Dept. of Algebra, Geometry and Topology, Complutense University of Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 902; https://doi.org/10.3390/sym11070902
Received: 6 June 2019 / Revised: 5 July 2019 / Accepted: 9 July 2019 / Published: 11 July 2019
(This article belongs to the Special Issue Mirror Symmetry and Algebraic Geometry)
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PDF [279 KB, uploaded 11 July 2019]

Abstract

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space. View Full-Text
Keywords: actions of finite groups; complex quasi-projective varieties; Grothendieck rings; λ-structure; power structure; Macdonald-type equations actions of finite groups; complex quasi-projective varieties; Grothendieck rings; λ-structure; power structure; Macdonald-type equations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Gusein-Zade, S.M.; Luengo, I.; Melle-Hernández, A. Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups. Symmetry 2019, 11, 902.

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