The $\gamma$-support as a micro-support

TL;DR

This paper establishes a fundamental link between the $\gamma$-support of Lagrangian submanifolds and their sheaf-theoretic micro-support, providing new insights into symplectic topology and sheaf quantization.

Contribution

It proves the equivalence of $\gamma$-support and reduced micro-support for elements in the completion of Lagrangian submanifolds, connecting spectral invariants with sheaf theory.

Findings

$\gamma$-support equals reduced micro-support for Lagrangians.

Characterization of Vichery subdifferential via $\gamma$-support.

Application to spectral distance and sheaf quantization.

Abstract

We prove that for any element $L$ in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the $\gamma$-support of $L$ coincides with the reduced micro-support of its sheaf quantization. As an application, we give a characterization of the Vichery subdifferential in terms of $\gamma$-support.