Derived sheaves in locally conformally symplectic geometry

TL;DR

This paper employs derived sheaves to analyze rigidity in locally conformally symplectic manifolds, introducing a quantization method and proving a non-squeezing theorem analogous to contact geometry results.

Contribution

It defines a new sheaf-theoretic quantization for locally conformally symplectic isotopies and establishes a non-squeezing theorem in this setting.

Findings

Sheaf-theoretic proof of the Chantraine-Murphy theorem.

Introduction of asymptotic Betti numbers for sheaves.

Derivation of a non-squeezing theorem in locally conformally symplectic geometry.

Abstract

In this paper, we use derived sheaves to study rigidity phenomena in the cotangent bundles of manifolds endowed with some locally conformally symplectic ($\frak{lcs}$) structure. Taking inspiration from the work of Guillermou, Kashiwara and Shapira, we define a quantization for ``$\frak{lcs}$'' Hamiltonian isotopies, as well as new quantities: the asymptotic Betti numbers of a sheaf. We then show that those quantities are ``well behaved'' with respect of said quantization and use this to give a sheaf-theoretical proof of the Chantraine-Murphy theorem. We also consider the quantization in light of the Tamarkin morphism and the displacement energy of sheaves. This allows us to derive a non-squeezing theorem for $\frak{lcs}$ geometry that is similar, although not identical, to the one recently proven by Bertelson, Chakravarthy, and Sandon. Indeed, the result shown in this paper is more in line with the contact non-squeezing theorem shown by Eliashberg, Kim and Polterovich in 2006.