Originally introduced by Kirillov and Lichnerowicz, Jacobi manifolds encompass, unifying and generalizing, locally conformal symplectic/Poisson manifolds and (non-necessarily coorientable) contact manifolds. In this talk, in order to investigate the coisotropic deformation problem, we attach two algebraic invariants to any coisotropic submanifold in a Jacobi manifold.
The first algebraic invariant is an L_∞ algebra. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), and Le-Oh (locally conformal symplectic case), and additionally, as a completely new case, it also applies in the contact setting. The L_∞ 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. The second algebraic invariant is the BFV complex. Our construction extends an analogous construction by Schaetz in the Poisson setting, and in particular it also applies in the locally conformal symplectic/Poisson setting and the contact setting. 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, the L_∞ algebra and the BFV-complex are closely related. First, both the L_∞ algebra and the BFV-complex of a coisotropic submanifold S provide cohomological reduction of S. Second, they are L_∞ quasi-isomorphic and so they encode equally well the moduli space of formal coisotropic deformations of S under Hamiltonian equivalence.
Finally we exhibit, in the contact setting, two examples of coisotropic submanifolds that are formally obstructed and provide a conceptual explanation of these phenomena in terms of both the L_∞ algebra and the BFV-complex.