Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Abstract

Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$. If $\psi$ is a defining function for $M$, then $-\log \psi$ is plurisubharmonic on a neighborhood of $M$ in $X$, and the (real) 2-form $\sigma =i\partial \bar{\partial }(-\log \psi )$ is a symplectic structure on the complement of $M$ in a neighborhood in $X$ of $M$; 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 $-\log \psi$ is plurisubharmonic throughout $X$, and $X$ is compact, bidifferential operators constructed by Engliš 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.