Infinitesimal deformation quantization of complex analytic spaces

For the physical aspects of deformation quantization we refer to the expository and survey papers [4] and [10]. Our objective is to initiate a version of global theory of quantization deformation in the category of complex analytic spaces in the same lines as the theory of (commutative) deformation. The goal of inifinitesimal theory is to do few steps towards construction of a star-product in the structure sheaf OX of holomorphic functions on a complex analytic space X, occasionally with singularity. A formal power series with representing a star-product can have a chance to converge on holomorphic functions for, at least, a sequence of values of the parameter. This phenomenon is described in Berezin’s global theory, [2], [3]. We shall see strong similarity between commutative and skew-commutative deformations, in particular, any Poisson bracket and any Kodaira-Spencer class (commutative deformation) are locally in ”one flacon”, that is in the same Hochschild cohomology space. On the global level, extension of an arbitrary infinitesimal quantization meets obstructions which are elements of the Čech cohomology of the sheaves of the analytic Hochschild cohomology in the same way as infinitesimal deformations of complex analytic spaces do. The construction of analytic Hochschild (co)homology is given in terms of analytic tensor products and bounded linear mappings of analytic algebras. We prove here the