Integrable Systems Associated to a Hopf Surface

[摘要] A Hopf surface is the quotient of the complex surface $mathbb{C}^2setminus {0}$ by an infinite cyclic group of dilations of$mathbb{C}^2$. In this paper, we study the moduli spaces$mathcal{M}^n$ of stable $SL (2,mathbb{C})$-bundles on a Hopfsurface $mathcal{H}$, from the point of view of symplectic geometry.An important point is that the surface $mathcal{H}$ is an ellipticfibration, which implies that a vector bundle on $mathcal{H}$ can beconsidered as a family of vector bundles over an elliptic curve. Wedefine a map $G colon mathcal{M}^n ightarrow mathbb{P}^{2n+1}$that associates to every bundle on $mathcal{H}$ a divisor, called thegraph of the bundle, which encodes the isomorphism class of the bundleover each elliptic curve. We then prove that the map $G$ is analgebraically completely integrable Hamiltonian system, with respectto a given Poisson structure on $mathcal{M}^n$. We also give anexplicit description of the fibres of the integrable system. Thisexample is interesting for several reasons; in particular, since theHopf surface is not K"ahler, it is an elliptic fibration that doesnot admit a section.

[学科分类] 数学（综合）