Holomorphic Frobenius actions for DQ-modules

TL;DR

This paper establishes an equivalence between categories of F-equivariant DQ-modules and modules over invariant sections, under certain conditions, and applies this to prove a conjecture in microdifferential modules.

Contribution

It introduces a new equivalence of categories for DQ-modules with Frobenius actions on complex manifolds with free and proper $C^ imes$-actions.

Findings

Proves category equivalence under free and proper actions.

Derives the codimension three conjecture for formal microdifferential modules.

Connects DQ-modules with invariant sections to geometric symplectic structures.

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.