Generating functions of stable pair invariants via wall-crossings in derived categories

TL;DR

This paper explores how wall-crossing formulas for limit stable objects in derived categories can be used to prove the rationality of generating functions for stable pair invariants on Calabi-Yau 3-folds, advancing understanding in enumerative geometry.

Contribution

It demonstrates that wall-crossing formulas for limit stable objects resolve the rationality conjecture of Pandharipande-Thomas invariants' generating functions.

Findings

Wall-crossing formulas explicitly relate to generating functions.

Limit stability approximates Bridgeland stability at large volume.

The rationality conjecture is proven using these methods.

Abstract

The notion of limit stability on Calabi-Yau 3-folds is introduced by the author to construct an approximation of Bridgeland-Douglas stability conditions at the large volume limit. It has also turned out that the wall-crossing phenomena of limit stable objects seem relevant to the rationality conjecture of the generating functions of Pandharipande-Thomas invariants. In this article, we shall make it clear how wall-crossing formula of the counting invariants of limit stable objects solves the above conjecture.