Premium
The Moduli Problem for Integrable Systems: The Example of a Geodesic Flow on SO(4)
Author(s) -
Bueken Peter,
Vanhaecke Pol
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001253
Subject(s) - moduli space , mathematics , integrable system , pure mathematics , affine transformation , moduli of algebraic curves , geodesic , algebraic variety , algebra over a field , mathematical analysis , algebraic number
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).