Integrable Systems Associated to a Hopf Surface

A Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on 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 . 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 Kähler, it is an elliptic fibration that does not admit a section.