On Submanifolds and Deformations in Poisson Geometry

This thesis concerns specific classes of submanifolds in Poisson geometry. The emphasis lies on normal form statements, and we present an application in deformation theory. The results are divided into three themes. We first study coisotropic submanifolds in log-symplectic manifolds. We provide a normal form around coisotropic submanifolds transverse to the degeneracy locus, and we prove a reduction statement for coisotropic submanifolds transverse to the symplectic leaves. Next, we address Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We establish a normal form around such Lagrangians, which we use to study their deformations. On the algebraic side, we show that the deformations correspond with Maurer-Cartan elements of a suitable DGLA. On the geometric side, we discuss when small deformations of the Lagrangian are constrained to the singular locus, and we find criteria for unobstructedness of first order deformations. We also address equivalences of deformations and we prove a rigidity result. At last, we consider a class of submanifolds in arbitrary Poisson manifolds, which are defined by imposing a suitable constant rank condition. We show that their local Poisson saturation is smooth, and we give a normal form for the induced Poisson structure. This result extends some normal form theorems around distinguished types of submanifolds in symplectic and Poisson geometry. As an application, we prove a uniqueness statement concerning coisotropic embeddings of Dirac manifolds into Poisson manifolds.

Jaar van publicatie:2021

Toegankelijkheid:Open