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Poisson geometry and deformation quantization near a strictly pseudoconvex boundary. (English) Zbl 1146.53072
Summary: 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 \(\sigma = i \delta \deltabar(-\log \psi)\) 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 C. L. Epstein, R. B. Melrose and G. A. Mendoza [Acta Math. 167, No. 1–2, 1–106 (1991; Zbl 0758.32010)] concerning manifolds with contact boundary.

MSC:
53D55 Deformation quantization, star products
32U99 Pluripotential theory
53D17 Poisson manifolds; Poisson groupoids and algebroids
Citations:
Zbl 0758.32010
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