Non-linear microlocal cut-off functors

TL;DR

This paper introduces and analyzes non-linear microlocal cut-off functors in sheaf theory, relating them to classical microlocal functors, and establishes foundational results including a cut-off lemma, K"unneth formulas, and a classification theorem.

Contribution

It extends microlocal sheaf theory by defining non-linear cut-off functors, relating them to existing functors, and proving new structural and classification results.

Findings

Established a microlocal cut-off lemma for non-linear functors

Proved two K"unneth formulas for sheaf categories

Classified categories of sheaves with microsupport conditions

Abstract

To any conic closed set of a cotangent bundle, one can associate four functors on the category of sheaves, which are called non-linear microlocal cut-off functors. Here we explain their relation with the microlocal cut-off functor defined by Kashiwara and Schapira, and prove a microlocal cut-off lemma for non-linear microlocal cut-off functors, adapting inputs from symplectic geometry. We also prove two K\"unneth formulas and a functor classification result for categories of sheaves with microsupport conditions.