The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds

TL;DR

This paper proves conjectures on the rationality and pole structure of generating functions for Pandharipande-Thomas invariants of projective complex 3-manifolds with superpositive curve classes, using wall-crossing theory.

Contribution

It establishes the Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds by applying wall-crossing techniques in abelian categories.

Findings

Proves rationality of generating functions for superpositive classes.

Identifies the pole structure of Pandharipande-Thomas invariants.

Extends previous conjectures to a broader class of curve classes.

Abstract

Let $X$ be a projective complex 3-manifold. An effective curve class $\beta\in H_2(X,\mathbb Z)$ is called positive if $c_1(X)\cdot\beta>0$, and superpositive if all the effective summands of $\beta$ are positive. If $X$ is Fano then all curve classes are superpositive. In arXiv:2111.04694 the second author developed a theory of enumerative invariants in abelian categories and wall-crossing formulae. We use this theory to prove conjectures by Pandharipande and Thomas on the rationality and poles of generating functions of Pandharipande-Thomas invariants of $X$ with descendent insertions, for superpositive curve classes.