Counting sheaves on Calabi-Yau 4-folds, I

TL;DR

This paper constructs algebraic virtual cycles for sheaves on Calabi-Yau 4-folds, extending Borisov-Joyce's work with new localization techniques and K-theoretic refinements, enabling computability and noncompact case analysis.

Contribution

It introduces an algebraic virtual cycle construction for Calabi-Yau 4-folds, including localization formulas and K-theoretic refinements, advancing the computational framework.

Findings

Established a localization formula for Euler classes on isotropic cones.

Extended invariants to noncompact Calabi-Yau 4-folds with compact fixed loci.

Developed K-theoretic square root Euler classes and their localizations.

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.