Enumerative invariants and wall-crossing formulae in abelian categories
Abstract
Enumerative invariants in Algebraic Geometry 'count' -(semi)stable objects with fixed topological invariants in some geometric problem, using a virtual class in homology, for the moduli spaces of -(semi)stable objects. We get numbers by taking integrals for cohomology classes . Let be a -linear abelian category in Algebraic Geometry. There are two moduli stacks of objects in : the usual moduli stack , and the 'projective linear' moduli stack . We give the structure of a vertex algebra, and a Lie algebra. Virtual classes lie in . We develop a universal theory of enumerative invariants in such . Virtual classes are only defined when . We define invariants in for all , with when . If are stability conditions on , we prove a wall-crossing formula writing in terms of the , using the Lie bracket on . We apply our results for the representations of a quiver or quiver with relations, or coh for a curve, surface or Fano 3-fold, or a category of 'pairs' in coh for a curve or surface. This proves conjectures in Gross-Joyce-Tanaka arXiv:2005.05637.
Keywords
Cite
@article{arxiv.2111.04694,
title = {Enumerative invariants and wall-crossing formulae in abelian categories},
author = {Dominic Joyce},
journal= {arXiv preprint arXiv:2111.04694},
year = {2021}
}
Comments
302 pages