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Integrable systems associated to a Hopf surface. (English) Zbl 1058.37043

Moduli spaces \({\mathcal M}^{n}\) of stable \(\text{SL}(2,{\mathbb C})\)-bundles on a Hopf surface \({\mathcal H}\) are studied from the point of view of non-Kählerian elliptic fibrations. More precisely, a map \(G:{\mathcal M}^{n}\rightarrow P^{2n+1}\) 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, is defined. The main result is that \(G\) yields an algebraically completely integrable Hamiltonian system with respect to a given Poisson structure on \({\mathcal M}^{n}\) and an explicit description of the fibres of this integrable system is given.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53D30 Symplectic structures of moduli spaces
14H70 Relationships between algebraic curves and integrable systems
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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