Simultaneous Deformations of Symplectic Forms and Lagrangian Submanifolds

TL;DR

This paper studies how symplectic forms and Lagrangian submanifolds can be deformed simultaneously, showing the moduli space of such deformations is smooth and finite-dimensional, characterized by a specific cohomology group.

Contribution

It provides a detailed description of the local deformation space of symplectic forms and Lagrangian submanifolds, identifying it with a neighborhood in a cohomology group.

Findings

The moduli space of deformations is smooth and finite-dimensional.

Deformations are classified by the second relative de Rham cohomology group $H^2(M,L)$.

The space of small deformations corresponds to an open neighborhood of the origin in $H^2(M,L)$.

Abstract

Given a compact symplectic manifold $(M,\omega)$ and a compact Lagrangian submanifold $L\subset(M,\omega)$, we describe small deformations of the pair $(\omega,L)$ modulo the action by isotopies. We show that the resulting moduli space can be identified with an open neighborhood of the origin in the second relative de Rham cohomology group $H^2(M,L)$. This implies in particular that the moduli space is smooth and finite dimensional.