Deformations of coisotropic submanifolds in Jacobi manifolds

In this thesis, we investigate deformation theory and moduli theory of coisotropic submanifolds in Jacobi manifolds. Originally introduced by Kirillov as local Lie algebras with one dimensional fibers, Jacobi manifolds encompass, unifying and generalizing, locally conformal symplectic manifolds, locally conformal Poisson manifolds, and non-necessarily coorientable contact manifolds. We attach an L-infinity-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), and Le-Oh (locally conformal symplectic case). As a completely new case we also associate an L-infinity-algebra with any coisotropic submanifold in a contact manifold. The L-infinity-algebra of a coisotropic submanifold S controls the formal coisotropic deformation problem of S, even under Hamiltonian equivalence, and provides criteria both for the obstructedness and for the unobstructedness at the formal level. Additionally we prove that if a certain condition ("fiberwise entireness") is satisfied then the L-infinity-algebra controls the non-formal coisotropic deformation problem, even under Hamiltonian equivalence. We associate a BFV-complex with any coisotropic submanifold in a Jacobi manifold. Our construction extends an analogous construction by Schatz in the Poisson setting, and in particular it also applies in the locally conformal symplectic/Poisson setting and the contact setting. Unlike the L-infinity-algebra, we prove that, with no need of any restrictive hypothesis, the BFV-complex of a coisotropic submanifold S controls the non-formal coisotropic deformation problem of S, even under both Hamiltonian equivalence and Jacobi equivalence. Notwithstanding the differences there is a close relation between the approaches to the coisotropic deformation problem via L-infinity-algebra and via BFV-complex. Indeed both the L-infinity-algebra and the BFV-complex of a coisotropic submanifold S provide a cohomological reduction of S. Moreover they are L-infinity quasi-isomorphic and so they encode equally well the moduli space of formal coisotropic deformations of S under Hamiltonian equivalence. In addition we exhibit two examples of coisotropic submanifolds in the contact setting whose coisotropic deformation problem is obstructed at the formal level. Further we provide a conceptual explanation of this phenomenon both in terms of the L-infinity-algebra and in terms of the BFV-complex.