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THE MODULI PROBLEM FOR INTEGRABLE SYSTEMS: THE EXAMPLE OF A GEODESIC FLOW ON SO(4)

Published online by Cambridge University Press:  08 January 2001

PETER BUEKEN  and
POL VANHAECKE
PETER BUEKEN
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Heverlee (Leuven), Belgium; [email protected]
POL VANHAECKE
Affiliation:
Mathématiques, SP2MI, Université de Poitiers, Boulevard 3 – Téléport 2 – BP 179, 86960 Futuroscope Cedex, France; [email protected]
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Abstract

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).

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 2 , October 2000 , pp. 357 - 369
Copyright
The London Mathematical Society 2000

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