We prove a rigidity theorem in Poisson geometry around compact Poisson
submanifolds, using the Nash-Moser fast convergence method. In the case of
one-point submanifolds (fixed points), this immediately implies a stronger
version of Conn's linearization theorem, also proving that Conn's theorem is,
indeed, just a manifestation of a rigidity phenomenon; similarly, in the case
of arbitrary symplectic leaves, it gives a stronger version of the local normal
form theorem; another interesting case corresponds to spheres inside duals of
compact semisimple Lie algebras, our result can be used to fully compute the
resulting Poisson moduli space.Comment: 43 pages, v3: published versio
