Deformation quantization of algebraic varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semi-formal deformations are Noetherian and have polynomial growth. We propose constructions of semi-formal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson-Lie groups can be semi-formally quantized.