zbMATH — the first resource for mathematics

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.

53D55 Deformation quantization, star products
32U99 Pluripotential theory
53D17 Poisson manifolds; Poisson groupoids and algebroids
Zbl 0758.32010
Full Text: DOI arXiv Link
[1] Block, J.: Duality and equivalence of module categories in noncommutative geometry. Preprint math.QA/0509284 · Zbl 1201.58002
[2] Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler mani- folds and gl(N ), N \rightarrow \infty limits. Comm. Math. Phys. 165 , 281-296 (1994) · Zbl 0813.58026
[3] Calaque, D.: Formality for Lie algebroids. Comm. Math. Phys. 257 , 563-578 (2005) · Zbl 1079.53138
[4] Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Math. 1764, Springer, Berlin (2001) · Zbl 1016.53001
[5] Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Berkeley Math. Lecture Notes, Amer. Math. Soc., Providence (1999) · Zbl 1135.58300
[6] Chemla, S.: A duality property for complex Lie algebroids. Math. Z. 232 , 367-388 (1999) · Zbl 0933.32015
[7] Eliashberg, Y., Gromov, M.: Convex symplectic manifolds. In: Proc. Sympos. Pure Math. 52, Part 2, Amer. Math. Soc., 135-162 (1991) · Zbl 0742.53010
[8] Engli\check s, M.: Weighted Bergman kernels and quantization. Comm. Math. Phys. 227 , 211-241 (2002) · Zbl 1010.32002
[9] Epstein, C. L.: A relative index on the space of embeddable CR-structures, I, II. Ann. of Math. 147 , 1-59, 61-91 (1998) · Zbl 0942.32025
[10] Epstein, C. L.: Subelliptic Spinc Dirac operators, III. The Atiyah-Weinstein conjecture. Preprint math.AP/0507547; Ann. of Math., to appear
[11] Epstein, C. L., Melrose, R. B., Mendoza, G. A.: Resolvent of the Laplacian on strictly pseu- doconvex domains. Acta Math. 167 , 1-106 (1991) · Zbl 0758.32010
[12] Etayo Gordejuela, F., Santamaría, R.: The canonical connection of a bi-Lagrangian manifold. J. Phys. A 34 , 981-987 (2001) · Zbl 0990.53081
[13] Evens, S., Lu, J.-H., Weinstein, A.: Transverse measures, the modular class, and a coho- mology pairing for Lie algebroids. Quart. J. Math. 50 , 417-436 (1999) · Zbl 0968.58014
[14] Fedosov, B.: A simple geometrical construction of deformation quantization. J. Differential Geom. 40 , 213-238 (1994) · Zbl 0812.53034
[15] Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis, math.DG/0401221 (2003)
[16] Guillemin, V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35 , 85-89 (1995) · Zbl 0842.58041
[17] Hess, H.: Connections on symplectic manifolds and geometric quantization. In: Differential Geometrical Methods in Mathematical Physics (Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, 153-166 (1980) · Zbl 0464.58012
[18] Hitchin, N.: Generalized Calabi-Yau manifolds. Quart. J. Math. 54 , 281-308 (2003) · Zbl 1076.32019
[19] Karabegov, A.: Deformation quantizations with separation of variables on a Kähler manifold. Comm. Math. Phys. 180 , 745-755 (1996) · Zbl 0866.58037
[20] Karabegov, A., Schlichenmaier, M.: Identification of Berezin-Toeplitz deformation quantiza- tion. J. Reine Angew. Math. 540 , 49-76 (2001) · Zbl 0997.53067
[21] Korányi, A., Reimann, H. M.: Contact transformations as limits of symplectomorphisms. C. R. Acad. Sci. Paris Sér. I Math. 318 , 1119-1124 (1994) · Zbl 0807.53026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.