Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Let X be a complex manifold with strongly pseudoconvex boundary M. If ψ is a defining function for M, then -log ψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form σ = i∂∂̄(- log ψ) is a symplectic structure on the complement of M in a neighborhood of M in ψ; it blows up along M. The Poisson structure obtained by inverting σ 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 ψ 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.