Integrable Systems Associated to a Hopf Surface

A Hopf surface is the quotient of the complex surface C^2 \ {0} by an infinite cyclic group of dilations of C^2. In this paper, we study the moduli spaces M^n of stable SL(2,C)-bundles on a Hopf surface H , from the point of view of symplectic geometry. An important point is that the surface H is an elliptic fibration, which implies that a vector bundle on H can be considered as a family of vector bundles over an elliptic curve. We define a map G : M^n rightarrow P^ (2n+1) that associates to every bundle on H a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on M^n. We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not K?hler, it is an elliptic fibration that does not admit a section.