# Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Eric Leichtnam; Xiang Tang; Alan Weinstein

Journal of the European Mathematical Society (2007)

- Volume: 009, Issue: 4, page 681-704
- ISSN: 1435-9855

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topLeichtnam, 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 -

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