JournalsjemsVol. 9 , No. 4DOI 10.4171/jems/93

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
Poisson geometry and deformation quantization near a strictly pseudoconvex boundary cover

Abstract

Let XX be a complex manifold with strongly pseudoconvex boundary MM. If ψ\psi is a defining function for MM, then logψ-\log\psi is plurisubharmonic on a neighborhood of MM in XX, and the (real) 2-form σ=i\del\delbar(logψ)\sigma = i \del \delbar(-\log \psi) is a symplectic structure on the complement of MM in a neighborhood in XX of MM; it blows up along MM. The Poisson structure obtained by inverting σ\sigma extends smoothly across MM and determines a contact structure on MM which is the same as the one induced by the complex structure. When MM is compact, the Poisson structure near MM is completely determined up to isomorphism by the contact structure on MM. In addition, when logψ-\log\psi is plurisubharmonic throughout XX, and XX is compact, bidifferential operators constructed by Engli\v{s} for the Berezin-Toeplitz deformation quantization of XX are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on MM, along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.