|Dept.||Imperial College, London|
We develop a theory of complex Kuranishi structures on projective schemes. These 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 projective moduli spaces M of stable sheaves on Calabi-Yau 4-folds. Borisov-Joyce produced a real virtual homology cycle on M using real derived differential geometry. In the prequel to this work we constructed an algebraic virtual cycle on M. We prove the 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.