Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Abstract

Let X be a complex manifold with strongly pseudoconvex boundary M . If \psi is a defining function for M , then -\log\psi is plurisubharmonic on a neighborhood of M in X , and the (real) 2-form \sigma = i \partial \bar \partial (-\log \psi) is a symplectic structure on the complement of M in a neighborhood in X of M ; it blows up along M . The Poisson structure obtained by inverting \sigma extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure near M is completely determined up to isomorphism by the contact structure on M . In addition, when -\log\psi is plurisubharmonic throughout X , and X is compact, bidifferential operators constructed by Engliš for the Berezin–Toeplitz deformation quantization of X are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on M , along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.