Counting sheaves on Calabi–Yau 4-folds, I

Abstract

Borisov and 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 localization of Edidin and Graham’s square root Euler class for $\mathrm{SO}(2n,\mathbb{C})$-bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization 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 localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.