Complex Kuranishi structures and counting sheaves on Calabi-Yau 4-folds, II

TL;DR

This paper develops a theory of complex Kuranishi structures on projective schemes, establishing their equivalence to weak perfect obstruction theories and applying them to moduli spaces of stable sheaves on Calabi-Yau 4-folds, linking algebraic and real derived geometry approaches.

Contribution

It introduces a new framework for complex Kuranishi structures that unify algebraic and differential geometric methods on Calabi-Yau 4-fold moduli spaces.

Findings

The algebraic and real virtual cycles coincide in homology after inverting 2.

Borisov-Joyce's real virtual cycle is 2-torsion when the virtual dimension is odd.

The developed theory provides a flexible yet rigid structure for moduli space analysis.

Abstract

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. Using real derived differential geometry, Borisov-Joyce produced a virtual homology cycle on M. In the prequel to this paper we constructed an algebraic virtual cycle on M. We prove the cycles coincide in homology after inverting 2 in the coefficients. And when Borisov-Joyce's real virtual dimension is odd, their virtual cycle is 2-torsion.