Poisson geometry and deformation quantization near a strictly pseudoconvex boundary
- Author(s): Leichtnam, Eric
- Tang, Xiang
- Weinstein, Alan
- et al.
Let X be a complex manifold with strongly pseudoconvex boundary M. If 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 derivative partial derivative- log psi) is a symplectic structure on the complement of M in a neighborhood of M in X; 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 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. 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.