arXiv subjects
Publications and source records attributed to Alexander Brudnyi.
We apply the methods developed in [Br1] to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin [GHS] on holomorphic L^{2} functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.
Let f_λ be a family of holomorphic functions in the unit disk, holomorphic in parameter λ\in U\subset\C^{n}. We estimate the number of zeros of f_λ in a smaller disk via some characteristic of the ideal generated by Taylor coefficients of f_λ. Our estimate is locally sharp and improve the previous estimate obtained by Roytwarf and Yomdin.
Let $M$ be a non-compact connected Riemann surface of finite type, and $R\subset\subset M$ be a relatively compact domain such that $H_{1}(M,\Z)=H_{1}(R,\Z)$. Let $\tilde R\longrightarrow R$ be a covering. We study the algebra $H^{\infty}(U)$ of bounded holomorphic functions defined in some domains $U\subset\tilde R$. Our main result is a Forelli type theorem on projections in $H^{\infty}(\Di)$.
In the paper we prove a factorization theorem for representations of fundamental groups of compact Kähler manifolds ({\em Kähler groups}) into solvable matrix groups. We apply this result to prove that the universal covering of a compact Kähler manifold with a residually solvable fundamental group is holomorphically convex.
Let M be a projective manifold, p:M_{G} --> M a regular covering over M with a free abelian transformation group G. We describe holomorphic functions on M_{G} of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary we obtain for such functions Cartwright and Liouville type theorems. Our approach brings together L_{2} cohomology technique for holomorphic vector bundles on complete Kähler manifolds and geometric properties of projective manifolds.
We study the structure of the maximal ideal space $M(H^{\infty})$ of the algebra $H^{\infty}=H^{\infty}(\Di)$ of bounded analytic functions defined on the open unit disk $\Di\subset\Co$. Based on the fact that $dim\ M(H^{\infty})=2$ we prove for $H^{\infty}$ the matrix-valued corona theorem. Our results heavily rely on the topological construction describing maximal ideal spaces of certain algebras of continuous functions defined on the covering spaces of compact manifolds.
We study traces of a class of subharmonic functions to Ahlfors regular subsets of $\Co^{n}$. In particular, we establish for the traces a generalized BMO-property and the reverse Hölder inequality.
We estimate the expected number of limit cycles situated in a neighbourhood of the origin of a planar polynomial vector field. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter. We obtain this inequality by methods of Pluripotential Theory. This inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros.
We prove local inequalities for analytic functions defined on a convex body in $\Re^{n}$ which generalize well-known classical inequalities for polynomials.
The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of polynomial, algebraic and analytic functions. The paper has much in common with an earlier paper of the author.
The paper presents a classification theorem for the class of flat connections with triangular (0,1)-components on a topologically trivial complex vector bundle over a compact Kahler manifold. As a consequence we obtain several results on the structure of Kähler groups, i.e., the fundamental groups of compact Kahler manifolds.
Let $D\subset\Co$ be a bounded domain, whose boundary $B$ consists of $k$ simple closed continuous curves and $H^{\infty}(D)$ be the algebra of bounded analytic functions on $D$. We prove the matrix-valued corona theorem for matrices with entries in $H^{\infty}(D)$.