Deformation Quantization of Algebraic Varieties

By M. Kontsevich. 2001

69 citations recorded by the source corpus.

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 semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal 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 semiformally quantized.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3cd6a8224ee37afa104c0394bc3815b0af37b77f

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Citations in the corpus

Double affine Hecke algebras of rank 1 and affine cubic surfaces (2003, methodology): “We will need the following corollary of Kontsevich’s quantization theory [11, 12 ].”
Topological strings on noncommutative manifolds (2003, background): “: the star-product can be made associative only if θis a Poisson bi-vector. M. Kontsevich showed that there are no other obstructions to deforming the algebra O(U) up to an arbitrarily high order in ~[5, 8]. Actually, the above requirements do not determine higher-order terms in the star-product uni”
Electric-Magnetic Duality And The Geometric Langlands Program (2006, background): “For a general discussion of such matters, see [138].”
Relative formality theorem and quantisation of coisotropic submanifolds (2005, background): “als over conﬁguration spaces as in [K1]. One can then deduce the existence of a morphism of Q-manifolds U: T(A)[1] →D(A)[1] for the algebra Aof functions on any supermanifold along the lines of [K1], [K2] or [D]. The explicit contruction goes as follows. To ﬁx notations, let E→M be the graded vector”
On Flatness and Completion for Infinitely Generated Modules over Noetherian Rings (2009, background): “ed, it is not clear if its completion Mc is itself complete. (We do not know a counterexample.) An R-module Mis called m-adically free if M∼= Nb for some free R-module N. (This is a new deﬁnition. In [Ko] such a module is called topologically free.) In this paper we study the structure of m-adically”