Vol 298, No 1 (2017)

We study residue currents of the Bochner–Martinelli type using their relationship to the Mellin transforms of residue integrals. We present a structure formula for residue currents associated with dimension-reducing monomial mappings: they can be represented as sums of products of simple residue currents, principal value currents, and hypergeometric functions.

We consider the problems of C¹ approximation of functions by polynomial solutions and by solutions with localized singularities of homogeneous elliptic second-order systems of partial differential equations on compact subsets of the plane ℝ². We obtain a criterion of C¹-weak polynomial approximation which is analogous to Mergelyan’s criterion of uniform approximability of functions by polynomials in the complex variable. We also discuss the problem of uniform approximation of functions by solutions of the above-mentioned systems. Moreover, we consider the Dirichlet problem for systems that are not strongly elliptic and prove a result on the lack of solvability of such problems for any continuous boundary data in domains whose boundaries contain analytic arcs.

We present a universal closed formula in terms of theta functions for the Logcapacity of several segments on a line. The formula for two segments was obtained by N. Achieser (1930); three segments were considered by T. Falliero and A. Sebbar (2001).

A class of holomorphic self-mappings of a strip which is symmetric with respect to the real axis is studied. It is required that the mappings should boundedly deviate from the identity transformation on the real axis. Distortion theorems for this class of functions are obtained, and domains of univalence are found that arise for certain values of the parameter characterizing the deviation of the mappings from the identity transformation on the real axis.

For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let X_Σ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S² of the two-dimensional sphere. We complete X_Σ,n with the isomorphism classes of mappings that cover the sphere by the degenerations of Σ. In the case Σ = S², the topology that we define on the obtained completion \({\overline X _{\Sigma ,n}}\) coincides on \({X_{{s^2},n}}\) with the topology induced by the space of coefficients of rational functions P/Q, where P and Q are homogeneous polynomials of degree n on ℂP¹ ≌ S². We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ⊂ X_Σ,n consisting of isomorphism classes of branched coverings with all critical values being simple.

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.

Vector logarithmic-potential equilibrium problems with the Angelesco interaction matrix are considered. Solutions to two-dimensional problems in the class of measures and in the class of charges are studied. It is proved that in the case of two arbitrary real intervals, a solution to the problem in the class of charges exists and is unique. The Cauchy transforms of the components of the equilibrium charge are algebraic functions whose degree can take values 2, 3, 4, and 6 depending on the arrangement of the intervals. A constructive method for finding the vector equilibrium charge in an explicit form is presented, which is based on the uniformization of an algebraic curve. An explicit form of the vector equilibrium measure is found under some constraints on the arrangement of the intervals.

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the reflection principle, the scaling method, and the Kobayashi–Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.

The review is devoted to the interpretation of the Dirac spin geometry in terms of noncommutative geometry. In particular, we give an idea of the proof of the theorem stating that the classical Dirac geometry is a noncommutative spin geometry in the sense of Connes, as well as an idea of the proof of the converse theorem stating that any noncommutative spin geometry over the algebra of smooth functions on a smooth manifold is the Dirac spin geometry.

Extremal properties and localization of zeros of general (including nondiagonal) type I Hermite–Padé polynomials are studied for the exponential system {e^λjz}_j=0^k with arbitrary different complex numbers λ₀, λ₁,..., λ_k. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.