Deformations of coisotropic submanifolds in locally conformal symplectic manifolds

TL;DR

This paper investigates the deformation theory of coisotropic submanifolds within locally conformal symplectic manifolds, establishing governing equations, moduli spaces, and the influence of an $L_$-structure, while confirming certain obstructions persist.

Contribution

It introduces a new $L_$-structure governing deformations and extends the deformation theory of coisotropic submanifolds to locally conformal symplectic manifolds.

Findings

Deformation equations for coisotropic submanifolds are derived.

The deformation problem is governed by an $L_$-structure, a deformation of homotopy Lie algebroids.

Obstructions to deformations are confirmed to persist in this setting.

Abstract

In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. Secondly, we prove that the formal deformation problem is governed by an $L_\infty$-structure which is a $\frak b$-deformation of strong homotopy Lie algebroids introduced in Oh and Park (2005) in the symplectic context. Then we study deformations of locally conformal symplectic structures and their moduli space, and the corresponding bulk deformations of coisotropic submanifolds. Finally we revisit Zambon's obstructed infinitesimal deformation (Zambon, 2002) in this enlarged context and prove that it is still obstructed.