Department of Mathematics
Ben Gurion University
Prof. Amnon
Yekutieli
Lecture:
Deformation
Quantization in Algebraic Geometry
Abstract:
We study deformation quantization of the structure sheaf O_X of a
smooth algebraic variety X in characteristic 0. The universal
deformation formula of Kontsevich gives rise to an L_infinity
quasi-isomorphism between the pullbacks of the DG Lie algebras
T_{poly,X} and D_{poly,X} to the bundle of formal coordinate systems of
X. Using simplicial sections we obtain an induced twisted L_infinity
quasi-isomorphism between the mixed resolutions Mix(T_{poly,X}) and
Mix(D_{poly,X}). If certain cohomologies vanish (e.g.\ if X is
D-affine) it follows that there is a canonical function from the set of
gauge equivalence classes of formal Poisson structures on X to the set
of gauge equivalence classes of deformation quantizations of O_X. This
is the quantization map. When X is affine the quantization map is in
fact bijective. This is an algebro-geometric analogue of Kontsevich's
celebrated result.
Eprint: math.AG/0310399
Lecture notes: postscript
Presentation: acrobat
updated 25 June 2006