GW/PT-correspondence for CY $4$-fold-II: Virtual Classes

This is the authors' abstract. We don't add key points for this paper.

Let B be a complex manifold and f:→ B a derived complex-analytic (not necessarily algebraic) space with an oriented relative $(-2)$-shifted symplectic structure. We suppose that the closed symplectic form is locally exact in the completed relative de Rham complex, and construct a derived smooth manifold over B, independent of additional choices. When f is proper and oriented, this gives integral virtual fundamental classes on its fibres. Our argument is heavily based on the strategy of Borisov-Joyce who already solved the algebraic version of this problem, and uses derived differential geometry. The proof also contains a relative analytic Darboux theorem a-la Brav-Bussi-Joyce, and has a natural analogue in Borel-Moore homology. We provide applications to complex-analytic enumerative geometry: in particular, to counting Pandharipande-Thomas pairs on compact and non-compact complex-analytic Calabi-Yau $4$-folds. This is part of a joint project with Felix Thimm dedicated to the proof of the GW/PT-correspondence for Calabi-Yau $4$-folds by implementing Pardon's programme in dimension $4$.