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)


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