Counting invariant of perverse coherent sheaves and its wall-crossing
Kentaro Nagao, Hiraku Nakajima
TL;DR
The paper develops a comprehensive framework for counting invariants of stable perverse coherent systems on the resolved conifold, unifying DT, PT and Szendroi-type invariants through a perverse-coherent-system perspective. By modeling perverse objects via a noncommutative crepant resolution with framing and a quiver-with-potential description, it derives stability conditions, constructs moduli spaces, and establishes a detailed wall-crossing theory across chambers. The main results include an explicit wall classification, a universal wall-crossing formula, and product-type relations among DT, PT and noncommutative DT invariants, together with a flop-invariant picture and a Behrend-weighted interpretation. The work also provides alternative tilting-based descriptions and combinatorial realizations of fixed points via pyramid partitions, offering both structural insight and calculable generating functions for conifold invariants. These contributions advance the understanding of how perverse coherent systems encode DT-type counting across geometric transitions and their noncommutative avatars.
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.