Fedosov quantization in positive characteristic

Abstract

The so-called deformation quantization of symplectic manifolds originally ap peared in C?? symplectic geometry; the crucial breakthrough was independently made in the early 1980s by B. Fedosov and M. De Wilde-P. Lecomte. The input of a deformation quantization problem is a symplectic (or, more generally, a Poisson) manifold M; the output is a non-commutative one-parameter deformation of the algebra of functions on M. Both Fedosov and De Wilde-Lecomte provided general procedures which solve the deformation quantization problem for C?? manifolds. Recently, motivated in part by M. Kontsevich [K], there has been much interest in generalizing the deformation quantization procedures to the case of algebraic manifolds equipped with an algebraic symplectic form (see, e.g., [BK1], [Y], and an earlier paper [NT] for the holomorphic situation). In particular, in [BK1] it has been shown that under some mild assumptions on the manifold, the Fedosov quantization can be made to work in the algebraic setting. The present paper is a continuation of [BK1] (to which we refer the reader for a more complete bibliography and historical discussion). Namely, one of the most important assumptions in [BK1] (as well as in other papers on the subject) was that the field of definition for all algebraic manifolds has characteristic 0. In this paper, we study what happens in the case of positive characteristic. The most obvious new feature of the theory in positive characteristic is the presence of a large Poisson center in the sheaf Ox of functions on a Poisson manifold X: since for any two local functions f,g Ox and any Poisson bracket { ?,?} on Ox we have {fp,g} = 0, the image ?px C Ox of the Frobenius map lies in the center of any Poisson structure. This phenomenon, already observed in [BMR], allows for interesting applications (see, e.g., [BK2]) but makes the quantization procedures more involved. In this paper, we were not able to prove any meaningful results for general quantizations in positive characteristic, and we had to restrict our attention to a special class of them: the so-called Frobenius-constant quantization. Roughly speaking the precise definition is Definition 1.4 below a quantization is Frobenius-constant if the Poisson center Opx C Ox stays central in the quantized algebra Oh For quantization of this type, we were able to achieve, under mild assumptions on the manifold X, a reasonably complete classification theorem. In particular, Frobenius-constant quantizations do exist.