Table of contents

In this paper the question is studied of the possibility of obtaining a converse of the well-known theorem of Scheffer on homotopic representation of a given map of a compact connected group into a locally compact Abelian group by a (unique) homomorphism. Namely, several conditions are found under which the existence of a representing homomorphism (its uniqueness) implies that the group is Abelian.

The Weil pairing of two elements of the torsion of the Jacobian of an algebraic curve can be expressed in terms of the product of the local Hilbert symbols of two special idèles associated with the torsion elements of the Jacobian. On the other hand, Arbarello, De Concini, and Kac have constructed a central extension of the group of idèles on an algebraic curve in which the commutator is also equal up to a sign to the product of all the local Hilbert symbols of two idèles. The aim of the paper is to explain this similarity. It turns out that there exists a close connection between the Poincaré biextension over the square of the Jacobian defining the Weil pairing and the central extension constructed by Arbarello, de Concini, and Kac. The latter is a quotient of a certain biextension associated with the central extension.

The operator cross ratio, which is meaningful, in particular, for the infinite-dimensional Sato Grassmannian is defined and investigated. Its homological interpretation is presented. A matrix and operator analogue of the Schwartzian differential operator is introduced and its relation to linear Hamiltonian systems and Riccati's equation is established. The aim of these constructions is application to the KP-hierarchy (the Kadomtsev-Petviashvili hierarchy).

A special solution of Abel's ordinary differential equation of the first kind is considered, which describes the behaviour of a broad spectrum of solutions of partial differential equations with a small parameter in the neighbourhood of cusp points of their slowly varying equilibrium positions. The existence of this special solution is demonstrated; an asymptotic formula for it as , is constructed and substantiated.

Precise (in order) estimates of the Kolmogorov widths in the space , , of the classes and  and also of the trigonometric widths of the classes in  for   and  satisfying certain relations are obtained.

The genericity of the embeddability of lattice actions in flows with multidimensional time is studied. In particular, questions of de la Rue and de Sam Lazaro on the genericity of the embeddability of an action of a 2-lattice in a flow and the embeddability of a transformation in injective flow actions with multidimensional time are answered. It is also shown that a generic transformation has a set of roots of continuum cardinality in an arbitrary prescribed massive set.

A criterion for the non-singularity of a complete intersection of two fibrewise quadrics in   is obtained. The following addition to Alexeev's theorem on the rationality of standard Del Pezzo fibrations of degree 4 over   is deduced as a consequence: each fibration of this kind with topological Euler characteristic is proved to be rational.