Power structure over the Grothendieck ring of maps

TL;DR

This paper explores new examples of power and lambda structures over Grothendieck rings, focusing on maps of complex quasi-projective varieties, and demonstrates their effectiveness and applications to Hilbert-Chow morphisms.

Contribution

It introduces novel lambda and power structures over Grothendieck rings of maps, expanding the understanding of their algebraic properties and applications.

Findings

Two natural lambda-structures lead to the same power structure

The power structure is shown to be effective

Applications to equations involving Hilbert-Chow morphisms

Abstract

A power structure over a ring is a method to give sense to expressions of the form $(1+a_1t+a_2t^2+\ldots)^m$, 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 of varieties. 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.