Corrigendum: The affine part of the Picard scheme

We give a corrected version of Theorem 3, Lemma 4, and Proposition 9 in the above-mentioned paper, which are incorrect as stated (as was pointed out by O. Gabber).

Systèmes dynamiques algébriquement complètement intégrables et géométrie

Résumé In this paper I present the basic ideas and properties of the complex algebraic completely integrable dynamical systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several examples of algebraic completely integrable systems : Kowalewski’s top, geodesic flow on SO(4), Hénon-Heiles system, Garnier potential, two coupled nonlinear Schrödinger equations and Yang-Mills system.

The affine part of the Picard scheme

We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field.

THE YANG–MILLS EQUATIONS AND THE INTERSECTION OF QUARTICS IN PROJECTIVE 4-SPACE ℂℙ4

In this paper, we discuss an interesting interaction between complex algebraic geometry and dynamics: the integrability of the Yang–Mills system for a field with gauge group SU(2) and the intersection of quartics in projective 4-space ℂℙ4. Using Enriques classification of algebraic surfaces and dynamics, we show that these two quartics intesect in the affine part of an abelian surface and it follows that the system of differential equations is algebraically completely integrable.

Affine Parts of Algebraic Theories II

This paper concerns relative complexity of an algebraic theory T and its affine part A, primarily for theories TR of modules over a ring R. TR, AR and R itself are all, or none, finitely generated or finitely related. The minimum number of relations is the same for T R and AR. The minimum number of generators is a very crude invariant for these theories, being 1 for AR if it is finite, and 2 for TR if it is finite (and 1 ≠ 0 in R). The minimum arity of generators is barely less crude: 2 for TR} and 2 or 3 for AR (1 ≠ 0). AR is generated by binary operations if and only if R admits no homomorphism onto Z2.

Siegel domains over self-dual cones and their automorphisms

The Lie algebra gr of all infinitesimal automorphisms of a Siegel domain in terms of polynomial vector fields was investigated by Kaup, Matsushima and Ochiai [6]. It was proved in [6] that gr is a graded Lie algebra; gr = g-1 + g-1/2 + g0 + g1/2 + g1 and the Lie subalgebra ga of all infinitesimal affine automorphisms is given by the graded subalgebra; ga = g-1 + g-1/2 + g0. Nakajima [9] proved without the assumption of homogeneity that the non-affine parts g1/2 and g1 can be determined from the affine part ga.