The integrable case of Adler–van Moerbeke. Discriminant set and bifurcation diagram

The Adler–van Moerbeke integrable case of the Euler equations on the Lie algebra so(4) is investigated. For the L–A pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler–van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler–van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on so(4).     

Syst��mes hamiltoniens compl��tement int��grables

This article is dedicated to one of the greatest mathematicians of our time: V.I. Arnold, who died suddenly Thursday, June 3, 2010 in France. Integrable hamiltonian systems are nonlinear ordinary differential equations described by a hamiltonian function and possessing sufficiently many independent constants of motion in involution. The regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following purely differential geometric fact: a compact and connected n-dimensional manifold on which there exist n vector fields which commute and are independent at every point is diffeomorphic to an n-dimensional real torus and each vector field will define a linear flow there. We make a careful study of the connection with the concept of completely integrable systems and we apply the methods to several problems.     

On the Weak Kowalevski–Painlevé Property for Hyperelliptically Separable Systems

We consider so called hyperelliptically separable systems (h.s.s.) arising in various physical problems, whose generic invariant manifolds can be completed either to hyperelliptic Jacobians or to their nonlinear subvarieties (strata) or their finite coverings. In the case of strata the algebraic geometrical structure of such systems has much in common with that of algebraic completely integrable systems (a.c.i.s.). Using this property we study formal singular solutions of a.c.i.s. and h.s.s., which may contain fractional powers of time. We give estimates for the number and leading behavior of their principal and lower balances both for a generic and for the so called physical direction of the flow. This can be regarded as an useful extension of the Kowalevski–Painlevé integrability test. We also prove that when the system is h.s. but not a.c.i., its generic solutions are single-valued on an infinitely sheeted ramified covering of the complex time plane. Some model examples are considered, such as the hierarchy of integrable generalizations of the Henon–Heiles and the Neumann systems.     

Hamiltonian torus actions on symplectic orbifolds and toric varieties

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.

     

The generalized Kupershmidt deformation for constructing new discrete integrable systems

It is known that the KdV6 equation can be represented as the Kupershmidt deformation of the KdV equation. We propose a generalized Kupershmidt deformation for constructing new discrete integrable systems starting from the bi-Hamiltonian structure of a discrete integrable system. We consider the Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies and obtain Lax representations for these new deformed systems. The generalized Kupershmidt deformation provides a new way to construct discrete integrable systems.     

Quaternionic pryms and Hodge classes

Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n−1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds.     

On diffusion in high-dimensional Hamiltonian systems

The purpose of this paper is to construct examples of diffusion for ε-Hamiltonian perturbations of completely integrable Hamiltonian systems in 2d-dimensional phase space, with d large.In the first part of the paper, simple and explicit examples are constructed illustrating absence of ‘long-time’ stability for size ε Hamiltonian perturbations of quasi-convex integrable systems already when the dimension 2d of phase space becomes as large as . We first produce the example in Gevrey class and then a real analytic one, with some additional work.In the second part, we consider again ε-Hamiltonian perturbations of completely integrable Hamiltonian system in 2d-dimensional space with ε-small but not too small, |ε|>exp(-d), with d the number of degrees of freedom assumed large. It is shown that for a class of analytic time-periodic perturbations, there exist linearly diffusing trajectories. The underlying idea for both examples is similar and consists in coupling a fixed degree of freedom with a large number of them. The procedure and analytical details are however significantly different. As mentioned, the construction in Part I is totally elementary while Part II is more involved, relying in particular on the theory of normally hyperbolic invariant manifolds, methods of generating functions, Aubry-Mather theory, and Mather's variational methods.Part I is due to Bourgain and Part II due to Kaloshin.     

On the cases of Kirchhoff and Chaplygin of the Kirchhoff equations

It is proven that the completely integrable general Kirchhoff case of the Kirchhoff equations for B ?? 0 is not an algebraic complete integrable system. Similar analytic behavior of the general solution of the Chaplygin case is detected. Four-dimensional analogues of the Kirchhoff and the Chaplygin cases are defined on e(4) with the standard Lie-Poisson bracket.     

Geometrical equivalence of groups

The notion of geometrical equivalence of two algebras, which is basic for this paper, is introduced in [5], [6]. It is motivated in the framework of universal algebraic geometry, in which algebraic varieties are considered in arbitrary varieties of algebras. Universal algebraic geometry (as well as classic algebraic geometry) studies systems of equations and its geometric images, i.e., algebraic varieties, consisting of solutions of equations. Geometrical equivalence of algebras means, in some sense, equal possibilities for solving systems of equations.

In this paper we consider results about geometrical equivalence of algebras, and special attention is paied on groups (abelian and nilpotent).     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

Quantum integrable systems and differential Galois theory

This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.     

Foliations in deformation spaces of local G-shtukas

We study local G-shtukas with level structure over a base scheme whose Newton polygons are constant on the base. We show that after a finite base change and after passing to an étale covering, such a local G-shtuka is isogenous to a completely slope divisible one, generalizing corresponding results for p-divisible groups by Oort and Zink. As an application we establish a product structure up to finite surjective morphism on the closed Newton stratum of the universal deformation of a local G-shtuka, similarly to Oort?s foliations for p-divisible groups and abelian varieties. This also yields bounds on the dimensions of affine Deligne–Lusztig varieties and proves equidimensionality of affine Deligne–Lusztig varieties in the affine Grassmannian.     

Algebraic and abelian solutions to the projective translation equation

Let \({{\mathrm x}=(x,y)}\). A projective two-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) \({(1-z)\phi({\mathrm x})=\phi(\phi({\mathrm x}z)(1-z)/z)}\), \({\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}}\). Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field \({\varpi(x,y)\bullet \varrho(x,y)}\) which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, such as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows us to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are described in terms of abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR), and the orbits of these flows can be transformed into algebraic curves \({x^{A}(x-y)^{B}y^{C}\equiv{\mathrm{const.}}}\) (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines \({y\equiv{\mathrm{const.}}}\) (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.     

Integrable systems and group actions

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.     

Support problem for the intermediate Jacobians of l-adic representations

We consider the support problem of Erdös in the context of l-adic representations of the absolute Galois group of a number field. Main applications of the results of the paper concern Galois cohomology of the Tate module of abelian varieties with real and complex multiplications, the algebraic K-theory groups of number fields and the integral homology of the general linear group of rings of integers. We answer the question of Corrales-Rodrigáñez and Schoof concerning the support problem for higher dimensional abelian varieties.     

Noncommutative integrable systems on <Emphasis Type="Italic">b</Emphasis>-symplectic manifolds

In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Rational points of the group of components¶of a Néron model

The structure of component groups of Néron models has been investigated on several occasions. Here we admit non-separably closed residue fields and are interested in the subgroup of rational points or, in other terms, in the subgroup of geometrically connected components of a Néron model. We consider Néron models of abelian varieties and of algebraic tori and give detailed computations in the case of Jacobians of curves. Received: 4 May 1998 / Revised version: 12 October 1998