Duality and kernels in microlocal geometry

TL;DR

This paper explores the dualizability and duality of sheaves with isotropic singular supports in microlocal geometry, providing a classification of functors and relating categorical dualities through the wrap-once functor.

Contribution

It introduces a classification of colimit-preserving functors via sheaf kernels and relates Verdier duality to the wrap-once functor in microlocal sheaf theory.

Findings

Classification of functors by convolutions of sheaf kernels

Relation between Verdier duality and the wrap-once functor

Extension of Verdier duality to all compact objects under certain conditions

Abstract

We study the dualizability of sheaves on manifolds with isotropic singular supports $\operatorname{Sh}_\Lambda(M)$ and microsheaves with isotropic supports $\operatorname{\mu sh}_\Lambda(\Lambda)$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports $\operatorname{Sh}_\Lambda^b(M)_0$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects $\operatorname{Sh}_\Lambda^c(M)_0$ when the wrap-once functor is an equivalence, for instance, when $\Lambda$ is a full Legendrian stop or a swappable Legendrian stop.