Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

  • Eric Leichtnam

    Institut Mathématique de Jussieu, Paris, France
  • Xiang Tang

    Washington University, St. Louis, United States
  • Alan Weinstein

    University of California, Berkeley, United States

Abstract

Let be a complex manifold with strongly pseudoconvex boundary . If is a defining function for , then is plurisubharmonic on a neighborhood of in , and the (real) 2-form is a symplectic structure on the complement of in a neighborhood in of ; it blows up along .

The Poisson structure obtained by inverting extends smoothly across and determines a contact structure on which is the same as the one induced by the complex structure. When is compact, the Poisson structure near is completely determined up to isomorphism by the contact structure on . In addition, when is plurisubharmonic throughout , and is compact, bidifferential operators constructed by Engliš for the Berezin–Toeplitz deformation quantization of are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on , along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.

Cite this article

Eric Leichtnam, Xiang Tang, Alan Weinstein, Poisson geometry and deformation quantization near a strictly pseudoconvex boundary. J. Eur. Math. Soc. 9 (2007), no. 4, pp. 681–704

DOI 10.4171/JEMS/93