Simultaneous Deformations of Symplectic Forms and Lagrangian Submanifolds

2025/12/24 by Geudens, Stephane, Schaetz, Florian, Tortorella, Alfonso G.
#17B70 #53D12 #58D27 #58H15 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2512.21225

Abstract

Given a compact symplectic manifold (M,ω) and a compact Lagrangian submanifold L⊂(M,ω), we describe small deformations of the pair (ω,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 H²(M,L). This implies in particular that the moduli space is smooth and finite dimensional.

Citations

• Deformations of symplectic groupoids
• Deformations of pre-symplectic structures and the Koszul L-infty-algebra
• Equivalences of coisotropic submanifolds
• Principal bundles of embeddings and nonlinear Grassmannians
• Symplectic manifolds and their lagrangian submanifolds

Related

• Deformations of Lagrangian submanifolds in log-symplectic manifolds
• The Moser isotopy for holomorphic symplectic and C-symplectic structures
• Hamiltonian actions and Lagrangian homogeneous submanifolds
• Deformation of singular lagrangian subvarieties
• Deformations of Lagrangian NQ-submanifolds