Poisson geometry and deformation quantization near a strictly pseudoconvex boundary
نویسندگان
چکیده
Let X be a complex manifold with strongly pseudoconvex boundary M . If ψ is a defining function for M , then − logψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form σ = i∂∂(− logψ) 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 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 − logψ is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Englǐs 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. ∗Research partially supported by NSF Grant DMS-0204100 MSC2000 Subject Classification Numbers: 32T15 (Primary); 53D10, 53D17
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