Integrable Systems Associated to a Hopf Surface

Moraru, Ruxandra

Canadian journal of mathematics, Tome 55 (2003) no. 3, pp. 609-635

Résumé

A Hopf surface is the quotient of the complex surface ${{\mathbb{C}}^{2}}\,\backslash \,\left\{ 0 \right\}$ by an infinite cyclic group of dilations of ${{\mathbb{C}}^{2}}$ . In this paper, we study the moduli spaces ${{\mathcal{M}}^{n}}$ of stable $\text{SL}\left( 2,\,\mathbb{C} \right)$ -bundles on a Hopf surface $\mathcal{H}$ , from the point of view of symplectic geometry. An important point is that the surface $\mathcal{H}$ is an elliptic fibration, which implies that a vector bundle on $\mathcal{H}$ can be considered as a family of vector bundles over an elliptic curve. We define a map $G:\,{{\mathcal{M}}^{n}}\,\to \,{{\mathbb{P}}^{2n+1}}$ that associates to every bundle on $\mathcal{H}$ 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 ${{\mathcal{M}}^{n}}$ . 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 Kähler, it is an elliptic fibration that does not admit a section.

Détail

DOI : 10.4153/CJM-2003-025-3

Mots-clés : 14J60, 14D21, 14H70, 14J27

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