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Journal of the London Mathematical Society 2000 62(2):357-369; doi:10.1112/S0024610700001253
© 2000 by London Mathematical Society
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© The London Mathematical Society

The Moduli Problem for Integrable Systems: The Example of a Geodesic Flow on SO(4)

Peter Bueken and Pol Vanhaecke

Department of Mathematics, Katholieke Universiteit Leuven Celestijnenlaan 200 B, B-3001 Heverlee (Leuven), Belgium, Peter.Bueken{at}wis.kuleuven.ac.be
Mathématiques, SP2MI, Université de Poitiers Boulevard 3 – Téléport 2 – BP 179, 86960 Futuroscope Cedex, France, Pol.Vanhaecke{at}mathlabo.univ-poitiers.fr

Received 4 June 1999. Revision received 7 October 1999.

The moduli problem for (algebraic completely) integrable systems is introduced. This problem consists in constructing a moduli space of affine algebraic varieties and explicitly describing a map which associates to a generic affine variety, which appears as a level set of the first integrals of the system (or, equivalently, a generic affine variety which is preserved by the flows of the integrable vector fields), a point in this moduli space. As an illustration, the example of an integrable geodesic flow on SO(4) is worked out. In this case, the generic invariant variety is an affine part of the Jacobian of a Riemann surface of genus 2. The construction relies heavily on the fact that these affine parts have the additional property of being 4:1 unramified covers of Abelian surfaces of type (1,4).


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