Symmetric powers of null motivic Euler characteristic

TL;DR

This paper explores a conjecture linking trivial motivic Euler characteristics of varieties to their symmetric powers, proposing a compatible power structure on the Grothendieck--Witt ring and implications for Hilbert schemes.

Contribution

It introduces a conjecture about motivic Euler characteristics and demonstrates how, if true, it leads to a compatible power structure on the Grothendieck--Witt ring and enriches G"ottsche's formula.

Findings

Conjecture relating trivial motivic Euler characteristic to symmetric powers.

Conditional construction of a compatible power structure on the Grothendieck--Witt ring.

Implications for G"ottsche's formula for Hilbert schemes.

Abstract

Let k be a field of characteristic not 2. We conjecture that if X is a quasi-projective k-variety with trivial motivic Euler characteristic, then Sym$^n$X has trivial motivic Euler characteristic for all n. Conditional on this conjecture, we show that the Grothendieck--Witt ring admits a power structure that is compatible with the motivic Euler characteristic and the power structure on the Grothendieck ring of varieties. We then discuss how these conditional results would imply an enrichment of G\"ottsche's formula for the Euler characteristics of Hilbert schemes.