Extension of functors for algebras of formal deformation

TL;DR

This paper develops a method to extend functors for modules over algebras of formal deformation on complex manifolds, enabling new applications in microlocal analysis and deformation quantization.

Contribution

It provides explicit conditions and constructions for extending functors to coherent modules over deformed algebras, preserving exactness and enabling advanced analysis tools.

Findings

Extended functors include inverse image, Fourier transform, and microlocalization.

Established a non-characteristic Cauchy-Kowalewskaia-Kashiwara theorem.

Proved comparison theorems for regular holonomic modules.

Abstract

Suppose we are given complex manifolds $X$ and $Y$ together with substacks $\mathcal{S}$ and $\mathcal{S}'$ of modules over algebras of formal deformation $\mathcal{A}$ on $X$ and $\mathcal{A}'$ on $Y$, respectively. Suppose also we are given a functor $\Phi$ from the category of open subsets of $X$ to the category of open subsets of $Y$ together with a functor $F$ of prestacks from $\mathcal{S}$ to $\mathcal{S}'\circ\Phi$. Then we give conditions for the existence of a canonical functor, extension of $F$ to the category of coherent ${\sha}$-modules such that the cohomology associated to the action of the formal parameter $\hbar$ takes values in $\mathcal{S}$. We give an explicit construction and prove that when the initial functor $F$ is exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialization and microlocalization, nearby and vanishing cycles in the framework of $\shd[[\hbar]]$-modules. We also obtain a Cauchy-Kowalewskaia-Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic $\shd[[\hbar]]$-modules and a coherency criterion for proper direct images of good $\shd[[\hbar]]$-modules.