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Deformation quantizations with separation of variables on a Kähler manifold. (English) Zbl 0866.58037
A deformation quantization on a symplectic manifold $$M$$ is an associative algebra structure on the space $$C^\infty(M)[[v]]$$ of formal power series such that the algebra multiplication * is a deformation of the ordinary multiplication of functions on $$M$$ and the *-commutator is a deformation of the Poisson bracket. In this paper the author considers deformation quantizations on Kähler manifolds that satisfy the following separation of variables property. For each open set $$U\subseteq M$$ the *-multiplication from the left by a holomorphic function and from the right by an antiholomorphic function coincides with the ordinary multiplication. He shows that these quantizations are in 1-1 correspondence with the formal deformations of the original Kähler metric.
Reviewer: V.Perlick (Berlin)

##### MSC:
 53D50 Geometric quantization 81S10 Geometry and quantization, symplectic methods 53B35 Local differential geometry of Hermitian and Kählerian structures
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##### References:
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