arXiv:1403.3493v2 [math.AG] 18 Feb 2015 QUANTIZATION OF LINE BUNDLES ON LAGRANGIAN SUBVARIETIES VLADIMIR BARANOVSKY, VICTOR GINZBURG, DMITRY KALEDIN, AND JEREMY PECHARICH To Volodya Drinfeld on the occasion of his 60th Birthday ABSTRACT. We apply the technique of formal geometry to give a nec- essary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed de- formation quantization of the structure sheaf of an algebraic symplectic variety. CONTENTS 1. Introduction 1 2. Basic constructions 3 3. Comparison of Harish-Chandra pairs 8 4. Harish-Chandra Torsors 12 5. Torsors associated with a quantization 17 6. Proof of the main result 21 References 24 1. I NTRODUCTION 1.1. Let X be a smooth algebraic symplectic variety over a field k of char- acteristic zero. Let ω denote the symplectic 2-form and {−, −} the associ- ated Poisson bracket on O X , the structure sheaf of X. A formal quantiza- tion of X is, by definition, a sheaf O  on X (in the Zariski topology) of flat associative k[[]]-algebras (which is complete and separated in the linear topology of a k[[]]-module), equipped with an isomorphism O  /O  ∼ = O X and such that 1  (ab − ba) mod  = {a mod ,b mod } for all a, b ∈O  . We will be interested in the problem of quantization of O X -modules. Thus, given a coherent O X -module L and a formal quantization O  of X, we are looking for O  -modules L  , flat over k[[]], such that L  /L  ∼ = L. A necessary condition for the existence of such an L  is provided by the fundamental integrability of characteristics theorem, due to Gabber [Ga]. It says that if L admits a flat deformation to an O  / 3 O  -module then the (smooth locus of) every irreducible component of Supp L, the support of the coherent sheaf L, must be a coisotropic subvariety of X. 1