On the projection of exact Lagrangians in locally conformally symplectic geometry

TL;DR

This paper explores the construction and properties of exact Lagrangians in locally conformally symplectic cotangent bundles, linking contact and lcs geometries and examining obstructions to classical theorems.

Contribution

It provides new examples of exact Lagrangians in lcs structures and investigates conditions for homotopy equivalence, connecting contact geometry with lcs structures.

Findings

Constructed examples of exact Lagrangians in lcs cotangent bundles.

Identified conditions for homotopy equivalence with the zero section.

Analyzed obstructions from Legendrians to classical theorems.

Abstract

In this paper, we construct examples of exact Lagrangians (of "locally conformally symplectic" type) in cotangent bundles of closed manifolds with locally conformally symplectic (lcs) structures and give conditions under which the projection induces a simple homotopy equivalence between an exact Lagrangian and the $0$-section of the cotangent bundle. This line of questioning leads us to investigate the links between the contact geometry of jet spaces and the lcs geometry of cotangent bundles. Among other things, we will study essential Liouville chords, which seem to be the lcs equivalent to Reeb chords. We will also see how Legendrians in jet spaces are an obstruction to the straightforward adaptation of the Abouzaid-Kragh theorem to lcs geometry.