Enumerative invariants and wall-crossing formulae in abelian categories

Abstract

Enumerative invariants in Algebraic Geometry 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=a$ in some geometric problem, using a virtual class $[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ in homology, for the moduli spaces ${\mathcal{M}}_a^{\mathrm{st}}(\tau )\subseteq {\mathcal{M}}_a^{\mathrm{ss}}(\tau )$ of $\tau$-(semi)stable objects. We get numbers by taking integrals ${\int }_{[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}}{\rm Y}$ for cohomology classes ${\rm Y}$. Let $\mathcal{A}$ be a $\mathbb{C}$-linear abelian category in Algebraic Geometry. There are two moduli stacks of objects in $\mathcal{A}$: the usual moduli stack $\mathcal{M}$, and the 'projective linear' moduli stack ${\mathcal{M}}^{\mathrm{pl}}$. We give $H_{\ast }(\mathcal{M})$ the structure of a vertex algebra, and $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$ a Lie algebra. Virtual classes $[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ lie in $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$. We develop a universal theory of enumerative invariants in such $\mathcal{A}$. Virtual classes $[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ are only defined when ${\mathcal{M}}_a^{\mathrm{st}}(\tau )={\mathcal{M}}_a^{\mathrm{ss}}(\tau )$. We define invariants $[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{inv}}$ in $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$ for all $a$, with $[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{inv}}=[{\mathcal{M}}_a^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ when ${\mathcal{M}}_a^{\mathrm{st}}(\tau )={\mathcal{M}}_a^{\mathrm{ss}}(\tau )$. If $\tau ,{\tau }^{\prime }$ are stability conditions on $\mathcal{A}$, we prove a wall-crossing formula writing $[{\mathcal{M}}_a^{\mathrm{ss}}({\tau }^{\prime }){]}_{\mathrm{inv}}$ in terms of the $[{\mathcal{M}}_b^{\mathrm{ss}}(\tau ){]}_{\mathrm{inv}}$, using the Lie bracket on $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$. We apply our results for $\mathcal{A}$ the representations of a quiver or quiver with relations, or coh$(X)$ for $X$ a curve, surface or Fano 3-fold, or a category of 'pairs' in coh$(X)$ for $X$ a curve or surface. This proves conjectures in Gross-Joyce-Tanaka arXiv:2005.05637.

arXiv comment302 pages

References in corpus (34)

• Vertex Algebras and Algebraic Curves
• Curve counting via stable pairs in the derived category
• A Luna étale slice theorem for algebraic stacks
• Stable pairs on curves and surfaces
• The 3-fold vertex via stable pairs
• Gromov-Witten theory and Noether-Lefschetz theory
• Higher and derived stacks: a global overview
• Reduced classes and curve counting on surfaces I: theory
• Existence of moduli spaces for algebraic stacks
• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• Vafa-Witten invariants for projective surfaces II: semistable case
• A universal construction for moduli spaces of decorated vector bundles over curves
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds
• Donaldson-Thomas theory for Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• On the structure of instability in moduli theory
• Reduced classes and curve counting on surfaces II: calculations
• An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
• Curve counting via stable objects in derived categories of Calabi-Yau 4-folds
• Virtual fundamental classes of derived stacks I
• Tautological stable pair invariants of Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds II: Fano 3-folds
• Stable pairs with descendents on local surfaces I: the vertical component
• Poincare invariants
• Counting conics on sextic 4-folds
• Properness criteria for families of coherent analytic sheaves
• The homology of moduli stacks of complexes
• A note on the cotangent complex in derived algebraic geometry
• Virtual classes for the working mathematician
• On higher rank instantons & the monopole cobordism program
• Cohomological Hall algebras and vertex algebras
• Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds

Cited by in corpus (2)

• The Atiyah class on algebraic stacks
• Koszul algebras and Donaldson-Thomas invariants