Deformations of Lagrangian submanifolds in log-symplectic manifolds

TL;DR

This paper studies how Lagrangian submanifolds within the singular locus of log-symplectic manifolds can be deformed, providing a normal form, algebraic framework, and criteria for unobstructedness and equivalence of such deformations.

Contribution

It introduces a normal form for log-symplectic structures around Lagrangians and links deformation theory to a DGLA, offering new insights into deformation obstructions and equivalences.

Findings

Deformation problem governed by a DGLA.

Criteria for unobstructed first-order deformations.

Gauge equivalence matches Hamiltonian isotopies.

Abstract

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.