Counting sheaves on Calabi-Yau 4-folds, I

Abstract

Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for $SO(r,\mathbb{C})$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.

arXiv commentv2 fixed sign problem pointed out by Jørgen Rennemo and convergence problem pointed out by Andrei Okounkov. v3 referees' corrections. v4 fixed sign problem pointed out by Arkadij Bojko. References updated. 65 pages

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Cited by in corpus (17)

• $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds
• Stable maps to Looijenga pairs
• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• Gauge origami and quiver W-algebras
• Tetrahedron instantons in Donaldson-Thomas theory
• Universal structures in $\mathbb{C}$-linear enumerative invariant theories
• Cosection Localization and Vanishing for Virtual Fundamental Classes of D-Manifolds
• A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• An Overview of Crystals and Double Quiver Yangians
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
• Quantum Lefschetz without curves
• The Atiyah class on algebraic stacks
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• A Representation-Theoretic Approach to $qq$-Characters
• Virtual classes via vanishing cycles
• Simple obstructions and cone reduction