On reduced stable pair invariants

TL;DR

This paper proves the equivalence of two definitions of reduced stable pair invariants for the product of a K3 surface and an elliptic curve, with implications for deformation invariance and DT/PT correspondence.

Contribution

It establishes that the Behrend function weighted Euler characteristic approach naturally corresponds to the virtual class degree definition for these invariants.

Findings

The two definitions of reduced stable pair invariants are shown to agree.

The approach via the Behrend function arises naturally as the degree of a virtual class.

Applications include deformation invariance, rationality, and DT/PT correspondence for reduced invariants.

Abstract

Let $X = S \times E$ be the product of a K3 surface $S$ and an elliptic curve $E$. Reduced stable pair invariants of $X$ can be defined via (1) cutting down the reduced virtual class with incidence conditions or (2) the Behrend function weighted Euler characteristic of the quotient of the moduli space by the translation action of $E$. We show that (2) arises naturally as the degree of a virtual class, and that the invariants (1) and (2) agree. This has applications to deformation invariance, rationality and a DT/PT correspondence for reduced invariants of $S \times E$.