Rank 2 integrable systems of Prym varieties

Abstract

A correspondence between 1) rank 2 completely integrable systems of Jacobians of algebraic curves and 2) (holomorphically) symplectic surfaces was established in a previous paper by the first author. A more general abelian variety that occurs as a Liouville torus of integrable systems is a prym variety associated to a triple (S,W,V) consisting of a curve S, a finite group W of automorphisms of S and an integral representation V. Often W is a Weyl group of a reductive group and V is the root lattice. We establish an analogous correspondence between: i) Rank 2 integrable systems whose Liouville tori are generalized prym varieties Prym(S_u,W,V) of a family of curves S_u, u in U. ii) Varieties X of dimension 1+dim(V) with a W-action and an invariant V-valued 2-form. If V is one dimensional X is a symplectic surface. We obtain a rigidity result: When the dimension of V is at least 2, under mild additional assumptions, all the quotient curves $S_u/W$ are isomorphic to a fixed curve C. This rigidity result imposes considerable constraints on the variety X: X admits a W-invariant fibration to C and the generic fiber has an affine structure modeled after V. Examples discussed include: Hitchin systems, reduced finite dimensional coadjoint orbits of loop algebras, and principal bundles over elliptic K3 surfaces.