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Deformation quantization of algebraic varieties

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arxiv math/0106006 v1 pith:LM6E3KVO submitted 2001-06-01 math.AG math.QA

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keywords algebraicdeformationpoissonprojectivesemi-formalconditionsmanifoldsquantization
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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 semi-formal 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.

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  1. Advancements in Functorial Homological Mirror Symmetry

    hep-th 2025-02 reject novelty 4.0 of 10

    A programmatic review asserting that stability and transversality in Donaldson-Thomas degeneracy formulas correspond to abelian versus nonabelian gauging in Rozansky-Witten theory, without providing a derivation.

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