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Search: MSC category 14J27 ( Elliptic surfaces )

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1. CJM 2003 (vol 55 pp. 609)

Moraru, Ruxandra
Integrable Systems Associated to a Hopf Surface
A Hopf surface is the quotient of the complex surface $\mathbb{C}^2 \setminus \{0\}$ 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 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 $G \colon \mathcal{M}^n \rightarrow \mathbb{P}^{2n+1}$ 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 K\"ahler, it is an elliptic fibration that does not admit a section.

Categories:14J60, 14D21, 14H70, 14J27

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