TL;DR
This paper introduces a family of additive endomorphisms on Grothendieck rings of varieties and motives, analogous to Adams operations, and explores their relations to power structures and configuration space formulas.
Contribution
It constructs new Adams-like operations on Grothendieck rings and relates them to existing power structures and formulas in algebraic geometry.
Findings
Defined additive endomorphisms similar to Adams operations
Connected these operations to power structures in algebraic geometry
Discussed interpretation of Getzler's formula for configuration spaces
Abstract
We construct a family of additive endomorphisms of the Grothendieck ring of quasiprojective varieties and the Grothendieck ring of Chow motives similar to the Adams operations in the K-theory. The speciality of the -structure on the Grothendieck ring of motives (proved by F. Heinloth) gives a set of natural equations for these operations. We discuss this construction in a general setting and relate it to the concept of power structures introduced by S. Gusein-Zade, I. Luengo and A. Melle-Hernandez. Some interpretation of the E. Getzler's formula for the equivariant Hodge-Deligne polynomial of the configuration spaces is also discussed.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology