Enumerative invariants and wall-crossing formulae in abelian categories

Preprint

An enumerative invariant theory in Algebraic Geometry is the study of invariants which ‘count’ τ -(semi)stable objects E with fixed topological invariants JEK = α in some geometric problem, by means of a virtual class [Mssα (τ )]virt in some homology theory, for the moduli spaces Mstα (τ ) ⊆ Mssα (τ ) of τ -(semi)stable objects. We can obtain numbers by taking integrals R[ Mssα ( τ)]virt Υ for suitable universal cohomology classes Υ. Examples include Mochizuki’s invariants for coherent sheaves on surfaces [146], and Donaldson–Thomas type invariants for coherent sheaves on Calabi–Yau 3- and 4-folds and Fano 3-folds, [28, 108, 118, 152, 176]. Let A be a C-linear abelian category coming from Algebraic Geometry. There are two moduli stacks of objects E in A: the usual moduli stack M, and the ‘projective linear’ moduli stack Mpl modulo projective linear isomorphisms, that is, we quotient out by λ id E : E ! E for λ ∈ G m . Both are Artin C-stacks. Previous work by the author [106] gives H ∗ ( M ) the structure of a graded vertex algebra, and H ∗ ( Mpl) a graded Lie algebra , closely related to H ∗ ( M). Virtual classes [ Mssα ( τ )]virt lie in H ∗ ( Mpl). We develop a universal theory of enumerative invariants in such categories A, which includes and extends many cases of interest. Virtual classes [ Mssα ( τ )]virt are only defined when Mstα ( τ ) = Mssα ( τ ). We give a systematic way to define invariants [ Mssα ( τ )]inv in H ∗ ( Mpl) for all classes α ∈ C ( A), with [ Mssα ( τ )]inv = [ Mssα ( τ )]virt when Mstα ( τ ) = Mssα ( τ ). If (τ, T, 6) and (˜τ, T , ˜ 6) are two suitable (weak) stability conditions on A , we prove a wall-crossing formula which expresses [ Mssα (˜τ )]inv in terms of the [ Mssβ ( τ )]inv, using the Lie bracket on H ∗ ( Mpl). We apply our results when A is a category modC Q or modCQ/I of representations of a quiver Q or quiver with relations (Q, I), and when A = coh( X) for X a curve, surface, or Fano 3-fold, and when A is a category of ‘pairs’ ρ : V ⊗ C L ! E in coh( X) for X a curve or surface, where V is a vector space, L ! X is a fixed line bundle, and E ∈ coh( X). We also speculate on extensions of our theory to 3-Calabi–Yau categories, which would give an alternative approach to Donaldson–Thomas theory to [108, 176], and to 4-Calabi–Yau categories, which would give a theory of Donaldson–Thomas type invariants of Calabi–Yau 4-folds. Our results prove conjectures made in Gross–Joyce–Tanaka [81].

Authors: Joyce, D

• Publication status: Published
• Peer review status: Not peer reviewed
• Preprint server: arXiv
• Publication date: 2021-11-08
• DOI: 10.48550/arXiv.2111.04694
• Language: English