Normal forms in Poisson geometry

Ioan Marcut

Published 2013-01-19, updated 2013-01-23Version 2

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.