Moduli spaces of stable pairs

TL;DR

This paper constructs a moduli space for stable pairs on smooth projective varieties, generalizing previous notions, and explores their deformation, obstruction theories, and virtual fundamental classes.

Contribution

It introduces a generalized framework for stable pairs, extending prior models by allowing a fixed coherent sheaf, and analyzes their deformation and obstruction theories.

Findings

Constructed a moduli space of stable pairs.

Described deformation and obstruction theories.

Proved existence of virtual fundamental class in special cases.

Abstract

We construct a moduli space of stable pairs over a smooth projective variety, parametrizing morphisms from a fixed coherent sheaf to a varying sheaf of fixed topological type, subject to a stability condition. This generalizes the notion used by Pandharipande and Thomas, following Le Potier, where the fixed sheaf is the structure sheaf of the variety. We then describe the relevant deformation and obstruction theories. We also show the existence of the virtual fundamental class in special cases.