Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Leichtnam, Eric, Tang, Xiang, and Weinstein, Alan. "Poisson geometry and deformation quantization near a strictly pseudoconvex boundary." Journal of the European Mathematical Society 009.4 (2007): 681-704. <http://eudml.org/doc/277513>.

@article{Leichtnam2007,
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 $\sigma =i\partial \overline\{\partial \}(−\log \psi )$ 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 \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.},
author = {Leichtnam, Eric, Tang, Xiang, Weinstein, Alan},
journal = {Journal of the European Mathematical Society},
keywords = {Poisson structure; pseudoconvexity; plurisubharmonic function; contact structure; Lie algebroid},
language = {eng},
number = {4},
pages = {681-704},
publisher = {European Mathematical Society Publishing House},
title = {Poisson geometry and deformation quantization near a strictly pseudoconvex boundary},
url = {http://eudml.org/doc/277513},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Leichtnam, Eric
AU - Tang, Xiang
AU - Weinstein, Alan
TI - Poisson geometry and deformation quantization near a strictly pseudoconvex boundary
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 4
SP - 681
EP - 704
AB - 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 $\sigma =i\partial \overline{\partial }(−\log \psi )$ 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 \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.
LA - eng
KW - Poisson structure; pseudoconvexity; plurisubharmonic function; contact structure; Lie algebroid
UR - http://eudml.org/doc/277513
ER -