The Lefschetz hyperplane section theorem asserts that a complex affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to give an explicit description of attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real numbers. The cells and attaching maps are described in combinatorial terms of chambers. We also discuss the cellular chain complex with coefficients in a local system and a presentation for the fundamental group associated to the minimal CW-decomposition for the complement.

We give a necessary and sufficient condition for a collection of two-point sets to have the uniqueness property for meromorphic functions.

Take a pair of two disjoint nonpolar compact subsets A and B of the complex plane C = Ĉ\{∞}, the complex sphere less the point at infinity, with connected complement Ĉ\(A ∪ B) and a simple arc γ in Ĉ\(A ∪ B). We form the two sheeted covering surface Ĉ_γ of Ĉ by pasting Ĉ\γ with another copy Ĉ\γ crosswise along γ. Embed A and B in Ĉ_γ either in the same sheet or in the different sheets and consider the variational 2-capacity cap(A, Ĉ_γ\B) of A contained in the open subset Ĉ_γ\B of Ĉ_γ. Concerning the relation between the above capacity and the variational 2-capacity cap(A, Ĉ\B) of A contained in the open subset Ĉ\B of Ĉ, we will establish the following capacity inequality for the two sheeted cover and its base:

0 < cap (A, Ĉ_γ\B) < 2 · cap(A, Ĉ\B),

where the bound 2 in the above inequality is the best possible in the sense that, for any 0 < τ < 2, there is a triple of A, B, and γ such that cap(A, Ĉ_γ\B) > τ · cap(A, Ĉ\B), where A and B may in the same sheet or in the different sheets.

We prove that any Kähler manifold admitting a flat complex conformal connection is a Bochner-Kähler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kähler manifolds under consideration.

Let Mⁿ (n ≥ 2) be a n-dimensional oriented closed submanifolds with parallel mean curvature in S^{n + p} (1), denote by S, the norm square of the second fundamental form of M. H is the constant mean curvature of M. We prove that if ∫_M S^n/2 ≤ A(n), where A(n) is a positive universal constant, then M must be a totally umbilical hypersurface in the sphere S^{n + 1}.

Let Ω and Π be two simply connected domains in the complex plane C, which are not equal to the whole plane C, and let A(Ω, Π) denote the set of functions f : Ω → Π analytic in Ω. Define the quantities C_n (Ω, Π) by

$C_{n}(\Omega,\Pi):=\sup\limits_{f\in A(\Omega,\Pi)}\sup\limits_{z\in \Omega} \frac{|f^{(n)}(z)|\lambda_{\Pi}(f(z))}{n!(\lambda_{\Omega}(z))^{n}},\;\; n\in \mathbb{N}$

where λ_Ω and λ_Π are the densities of the Poincaré metric in Ω and Π, respectively. We derive sharp upper bounds for |f⁽ⁿ⁾(z)| (z $\in$ Ω) and C_n(Ω, Π) if 2 ≤ n ≤ 8 and Ω is a convex domain. The detailed equality condition of the estimate on |f⁽ⁿ⁾(z)| is also given.

We prove growth estimates for logarithmic derivatives of functions in the Nevanlinna class. Blaschke products with radially restricted zero sequences will be of particular interest. Our results are sharper in a certain sense than the corresponding estimates in [2] obtained for meromorphic functions in the unit disc.

In this paper, we prove a Chevalley type restriction theorem for a proper complex equifocal submanifold. The proof is performed by showing the same type restriction theorem for an infinite dimensional proper anti-Kaehlerian isoparametric submanifold and using it.