Deformation quantization and perverse sheaves

Abstract

Kashiwara, Polesello, Schapira and D'Agnolo defined canonical deformation quantizations of a holomorphic symplectic manifold and a holomorphic Lagrangian submanifold equipped with an orientation data. The goal of this paper is to use deformation quantization modules to construct a Fukaya-like category of holomorphic Lagrangians, resolving a conjecture of Joyce. Our main result describes the RHom complex between two such deformation quantization modules associated to a pair of Lagrangian submanifolds in terms of the derived geometry of the Lagrangian intersection. Namely, we identify the RHom complex with the DT sheaf associated to the d-critical structure on the Lagrangian intersection. Via a Riemann-Hilbert correspondence this describes the RHom complex in the category of microsheaves of sheaf quantizations of conic holomorphic Lagrangians.