Commuting Families in Skew Fields and Quantization of Beauville’s Fibration

We construct commuting families in fraction fields of symmetric powers of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a K3 surface S, they correspond to lagrangian fibrations introduced by Beauville. When S is the canonical cone of an algebraic curve C, we construct commuting families of differential operators on symmetric powers of C, quantizing the Beauville systems. Introduction. In [1], Beauville introduced lagrangian fibrations associated with a K3 surface S. These fibrations have the form S [g] → P(H(S,L)), where S [g] is the Hilbert scheme of g points of S, equipped with a symplectic structure introduced in [10], and L is a line bundle on S. Later, the authors of [5] explained that these systems are natural deformations of the ”separated” (in the sense of [7]) versions of Hitchin’s integrable systems, more precisely, of their description in terms of spectral curves (already present in [8]). Beauville’s systems can be generalized to surfaces with a Poisson structure (see [4]). When S is the canonical cone Cone(C) of an algebraic curve C, then this system coincides with the separated version of Hitchin’s systems. A quantization of Hitchin’s system was proposed in [3]. It seems interesting to construct quantizations of Beauville’s systems. In this paper, we construct generalizations of the birational version of Beauville’s construction (Theorem 2.1) and a quantum analogue of this construction (Theorem 1.1). We show that to obtain a quantization of Beauville’s fibration, it would be sufficient to have quantizations of function fields of K3 surfaces. Such a quantization is not known explicitly, in general. However, in the case of the canonical cone of an algebraic curve C, an explicit quantization is known (see [2, 6]). In Section 4, we construct the quantized Beauville systems explicitly in this case, and we show that in some cases these systems correspond to commuting families of rational differential operators on symmetric powers of C. We make these operators explicit in the case of a rational curve with marked points. Finally, in Appendix A, we discuss the relation of Theorems 1.1 and 2.1 with the formal non-commutative geometry of [9].