Том 52, № 4 (2018)

In this paper we discuss some problems of the deformation theory of zero-dimensional singularities, which are closely related to the study of properties of differential forms and the Poincaré–de Rham complex. We also investigate the cotangent homology and cohomology of zerodimensional singularities, compute the basic analytic invariants for certain types of such singularities, and examine in detail some interesting examples and applications.

It is proved that, for n = 8, 9, 10, the natural algebra of automorphic forms of the group O⁺_2,n(ℤ) acting on the n-dimensional symmetric domain of type IV is free, and the weights of generators are found. This extends results obtained in the author’s previous paper for n ≤ 7. On the other hand, as proved in a recent joint paper of the author and O. V. Shvartsman, similar algebras of automorphic forms cannot be free for n > 10.

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring freely generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler characteristic for V-manifolds and for cell complexes of the described type.

A plane algebraic curve whose Newton polygon contains d integer points is completely determined by d points in the plane through which it passes. Its coefficients regarded as functions of sets of coordinates of these points commute with respect to the Poisson bracket corresponding to the pair of coordinates of any of these points. This observation was made by Babelon and Talon in 2002. A result more general in some respects and less general in others was obtained by Enriquez and Rubtsov in 2003. As a particular case, we obtain that the coefficients of the Lagrange interpolation polynomial commute with respect to a Poisson bracket on the set of interpolation data. We prove a general assertion in the framework of the method of separation of variables which explains all these facts. This assertion is as follows: Any (nondegenerate) system of n smooth functions in n+2 variables generates an integrable system with n degrees of freedom. In addition to those mentioned above, the examples include a version of the Hermite interpolation polynomial and systems related to Weierstrass models of curves (= miniversal deformations of singularities). The integrable system related to the Lagrange interpolation polynomial has recently arisen as a reduction of rank-2 Hitchin systems (and, thereby, it gives particular solutions of such systems; see the author’s paper in Doklady Mathematics), and it is also closely related to the integrable systems on universal bundles of symmetric powers of curves introduced by Buchstaber and Mikhailov in 2017.

The Thoma simplex Ω is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are probability measures on Ω depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit as it goes to 0, the z-measures turn into the Poisson–Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space Ω.