Holomorphic Frobenius Actions for DQ-Modules

Abstract

Given a complex manifold endowed with a ${\mathbb{C}}^{\times }$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the ${\mathbb{C}}^{\times }$-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension-three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.