Counting sheaves on Calabi-Yau 4-folds


Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We constructed an algebraic virtual cycle. A key step is a localisation of Edidin-Graham’s square root Euler class for $SO(2n,\mathbb{C})$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We also develop a theory of complex Kuranishi structures on projective schemes which are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to the moduli spaces to prove the two virtual cycles coincide in homology after inverting 2 in the coefficients. In particular, when Borisov-Joyce’s real virtual dimension is odd, their virtual cycle is torsion. This is a joint work with Richard Thomas.