The Toda lattice, Dynkin diagrams, singularities and Abelian varieties

Abstract

The periodic l+1 particle Toda lattices coresponding to extended Dynkin diagrams have l+1 polynomial constants of the motion, namely as many as there are dots in the Dynkin diagram. For most values of the constants, their intersection defines as l-dimensional affine invariant manifold which completes into a complex algebraic torus (Abelian variety) by glueing on a divisor entirely specified by the extended Dynkin diagram: Therefore the global geometry of the complex invariant tori, such as polarization, dimension of certain linear systems, divisor equivalences, etc...., is entirely given by the extended Dynkin diagram. In particular, an explicit geometric description for the case of three-particle Toda lattices is given which describes the divisors, their singularities and how the various divisors intersect