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.

v2 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