Counting invariant of perverse coherent sheaves and its wall-crossing

TL;DR

This paper studies moduli spaces of stable perverse coherent systems on Calabi-Yau 3-folds, computes their invariants across different stability chambers, and relates them to known enumerative invariants like Donaldson-Thomas and Pandharipande-Thomas.

Contribution

It introduces a new framework for counting invariants of perverse coherent sheaves on crepant resolutions and explicitly determines wall-crossing behavior for the resolved conifold.

Findings

All walls in the stability space are explicitly determined.

Generating functions of invariants are computed for all chambers.

Invariants specialize to known DT, PT, and Szendroi invariants.

Abstract

We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi-Yau 3-folds and consider their Donaldson-Thomas type counting invariants. The stability depends on the choice of a component (= a chamber) in the complement of finitely many lines (= walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi-Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson-Thomas, Pandharipande-Thomas and Szendroi invariants.