Non-linear microlocal cut-off functors

TL;DR

The paper introduces non-linear microlocal cut-off functors $L_U,L_Z,R_Z,R_U$ on sheaf categories associated to a conic closed set $Z\subset T^*M$, linking them via split Verdier sequences and establishing a non-linear microlocal cut-off lemma that ties microsupport to wrapping. It unifies these functors with Kashiwara–Schapira’s linear cut-off theory and employs Guillermou–Kashiwara–Schapira quantization to derive a wrapping formula for the corresponding kernels, enabling explicit microlocal control. Two Künneth formulas are proved to remove isotropic restrictions, yielding a functor classification for product categories of microsupport-constrained sheaves and enabling a kernel-level tensorial description ${K_{X\times Z}}\simeq K_X\boxtimes K_Z$. The work extends to pair constructions and Tamarkin-type categories, with applications to equivariant and cyclic settings, and provides a robust ∞-categorical framework for microlocal sheaf theory beyond compact generation assumptions.

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 Künneth formulas and a functor classification result for categories of sheaves with microsupport conditions.

Theorems & Definitions (52)

• Theorem 1: Non-linear microlocal cut-off lemma, \ref{['prop: non-linear microlocal cut-off lemma']} below.
• Theorem 2
• Proposition 1
• Corollary 1
• Remark 1
• Definition 1
• Lemma 1
• Remark 2
• Proposition 2: KS90
• Corollary 2: guillermouviterbo_gammasupport