Table of contents

Let be the category of crystalline representations of the Galois group of the field of fractions of the ring of Witt vectors of an algebraically closed field of characteristic . The author describes the subfactors annihilated by multiplication by  of the representations from arising from filtered modules of filtration length , and proves a generalization of the Shafarevich conjecture that there do not exist abelian schemes over : if  is a smooth proper scheme over the ring of integers of the field  (respectively , , ), then the Hodge numbers of the complex manifold  satisfy for and (respectively ).

Bibliography: 17 titles.

This paper studies the categorical version of the concept of mutations of an exceptional set, as used in the theory of vector bundles. The basic object of study is a triangulated category with a family of subcategories satisfying the so-called admissibility condition. A natural notion arising here is that of a Serre functor, effecting a certain duality in the triangulated category.

Bibliography: 16 titles.

When working with a set of linear functions and the configration of hyperplanes they define, one encounters a matrix of multidimensional hypergeometric integrals. We give a formula which expresses the determinant of this matrix in terms of critical values of the linear functions restricted to the configuration.

Bibliography: 26 titles.

The author studies the spectral properties of the operator A = id/dt in the space whose domain of definition is the kernel of some functional that is bounded in but not bounded in Necessary and sufficient conditions are given under which the operators ±iA generate C₀-semigroups, and criteria for the similarity of A with a dissipative operator are proved. The results are used to study the basis properties of families of exponentials and to solve S. G. Krein's problem on the description of generators of semigroups in terms of their dissipative extensions. The solvability of integral equations of Delsarte type for mean periodic extensions of functions is also proved.

Bibliography: 32 titles.

The author obtains a classification, to within a rigid isotopy (i.e. an isotopy in the class of algebraic surfaces), of algebraic surfaces of degree four in with isolated singularities which have at least one nonsimple singular point (i.e. a singular point different from A_p, D_p, E₆, E₇, and E₈) . Tables: 4. Figures: 6.

Bibliography: 10 titles.

Complete solutions of the commutation equations of ordinary differential operators are obtained, to which there corresponds a three-dimensional vector bundle of common eigenfunctions over an elliptic curve. The deformation of the commuting pair by the Kadomtsev–Petviashvili equation is studied. The finite-zone solutions of the Kadomtsev–Petviashvili equation of rank 3 and genus 1 are explicitly expressed in terms of functional parameters satisfying a Boussinesq-type system of two evolution equations.

Bibliography: 40 titles.

Let Z be a del Pezzo surface with log-terminal singularities and a minimal resolution of singularities. Then the Picard number of Y satisfies where e is the maximal multiplicity of the singularities of Z and N a certain function of e.

Bibliography: 18 titles.