Power structure over the Grothendieck ring of maps

  • S. M. Gusein-Zade
  • I. Luengo
  • A. Melle-Hernández
Article
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Abstract

A power structure over a ring is a method to give sense to expressions of the form \((1+a_1t+a_2t^2+\cdots )^m\), where \(a_i\), \(i=1, 2,\ldots \), and m are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of \(\lambda \)-and power structures over some Grothendieck rings. The main example is for the Grothendieck ring of maps of complex quasi-projective varieties. We describe two natural \(\lambda \)-structures on it which lead to the same power structure. We show that this power structure is effective. In the terms of this power structure we write some equations containing classes of Hilbert–Chow morphisms. We describe some generalizations of this construction for maps of varieties with some additional structures.

Keywords

Lambda-structure Power structure Complex quasi-projective varieties Maps Grothendieck ring 

Mathematics Subject Classification

14A10 18F30 55M35 

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Mechanics, GSP-1Moscow State UniversityMoscowRussia
  2. 2.Department of Algebra, Geometry and Topology, ICMAT (CSIC-UAM-UC3M-UCM)Complutense University of MadridMadridSpain
  3. 3.Department of Algebra, Geometry and Topology, Instituto de Matemática Interdisciplinar (IMI)Complutense University of MadridMadridSpain

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