Skip to main content
eScholarship
Open Access Publications from the University of California

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

  • Author(s): Leichtnam, Eric
  • Tang, Xiang
  • Weinstein, Alan
  • et al.
Abstract

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.

Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you.

Main Content
Current View