Abstract:In this talk, we study a complex manifold X with a strongly pseudoconvex boundary M. If u is a defining function for M, then -log u is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form s = i \del \delbar(-log u) is a symplectic structure on the complement of M in a neighborhood in X of M; it blows up along M.
We explain that the Poisson structure obtained by inverting s 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, we show that the Poisson structure near M is completely determined up to isomorphism by the contact structure on M.
In addition, using Poisson geometry we are able to prove that when -log u is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Englis for the Berezin-Toeplitz deformation quantization of X are smooth up to the boundary.