Formal equivalence of Poisson structures around Poisson submanifolds

Let (M, {\pi} ) be a Poisson manifold. A Poisson submanifold $P \in M$ gives rise to an algebroid $AP \rightarrow P$, to which we associate certain chomology groups which control formal deformations of {\pi} around P . Assuming that these groups vanish, we prove that {\pi} is formally rigid around P , i.e. any other Poisson structure on M , with the same first order jet along P as {\pi} is formally Poisson diffeomorphic to {\pi} . When P is a symplectic leaf, we find a list of criteria which imply that these cohomological obstructions vanish. In particular we obtain a formal version of the normal form theorem for Poisson manifolds around symplectic leaves.Comment: 16 page

The Normal Form Theorem around Poisson Transversals

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.Comment: 15 pages; v2: the title was changed; v3: proof of Lemma 2 was include

Normal forms in Poisson geometry

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result generalizes Conn's theorem from fixed points to arbitrary symplectic leaves. We present two proofs of this result: a geometric one relying heavily on the theory of Lie algebroids and Lie groupoids (similar to the new proof of Conn's theorem by Crainic and Fernandes), and an analytic one using the Nash-Moser fast convergence method (more in the spirit of Conn's original proof). The analytic approach gives much more, we prove a local rigidity result (Theorem 4) around compact Poisson submanifolds, which is the first of this kind in Poisson geometry. Theorem 4 has a surprising application to the study of smooth deformation of Poisson structures: in Theorem 5 we compute the Poisson-moduli space around the Lie-Poisson sphere (i.e. the invariant unit sphere inside the linear Poisson manifold corresponding to a compact semisimple Lie algebra). This is the first such computation of a Poisson moduli space in dimension greater or equal to 3 around a degenerate (i.e. non-symplectic) Poisson structure. Other results presented in the thesis are: a new proof to the existence of symplectic realizations (Theorem 0), a normal form theorem for symplectic foliations (Theorem 1), a formal normal form/rigidity result around Poisson submanifolds (Theorem 3), and a general construction of tame homotopy operators for Lie algebroid cohomology (the Tame Vanishing Lemma). We also revisit Conn's theorem and a theorem of Hamilton on rigidity of foliations.Comment: Utrecht University PhD thesis, 200 page

Local formulas for multiplicative forms

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis-Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids, respectively.Comment: 29 pages, this is the second part of an original longer paper that was split into two parts (the first part is in arXiv:1703.04411v2 [math.DG]