Cobordism invariants of the moduli space of stable pairs

TL;DR

This paper constructs virtual cobordism classes for schemes with perfect obstruction theories, studies cobordism invariants of moduli spaces of stable pairs, and proves rationality of partition functions in specific cases.

Contribution

It introduces the virtual cobordism class for schemes with perfect obstruction theories and analyzes cobordism invariants of stable pairs moduli spaces.

Findings

Virtual cobordism class construction for schemes with perfect obstruction theory.

Proof of rationality of partition functions for nonsingular projective toric 3-folds.

Relation of Chern numbers of virtual classes to integrals of virtual tangent bundle Chern classes.

Abstract

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q <-> q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.