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Integrable Systems Associated to a Hopf Surface


Published:20030601
Printed: Jun 2003
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Abstract
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.
MSC Classifications: 
14J60, 14D21, 14H70, 14J27 show english descriptions
Vector bundles on surfaces and higherdimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) [See also 32L25, 81Txx] Relationships with integrable systems Elliptic surfaces
14J60  Vector bundles on surfaces and higherdimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14D21  Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) [See also 32L25, 81Txx] 14H70  Relationships with integrable systems 14J27  Elliptic surfaces
