*-Quantizations of Fourier-Mukai transforms

TL;DR

This paper investigates how Fourier-Mukai transforms between complex manifolds can be deformed through quantizations, establishing a unique correspondence that preserves their categorical equivalences in a formal quantization setting.

Contribution

It introduces a method to deform Fourier-Mukai transforms compatibly with quantizations of complex manifolds, extending the theory to stacks of algebroids.

Findings

Constructs a unique quantization of Y from a given quantization of X.

Ensures the Fourier-Mukai transform deforms to an equivalence of derived categories.

Extends deformation theory of Fourier-Mukai transforms to complex analytic and stack frameworks.

Abstract

We study deformations of Fourier-Mukai transforms in general complex analytic settings. We start with two complex manifolds X and Y together with a coherent Fourier-Mukai kernel P on their product. Suppose that P implements an equivalence between the coherent derived categories of X and Y. Given an arbitrary formal quantization of X we construct a unique quantization of Y such that the Fourier-Mukai transform deforms to an equivalence of the derived categories of the quantizations. Here quantizations are understood in the framework of stacks of algebroids.