In this paper, we prove that every proper holomorphic self-map of a smoothly bounded pseudoconvex circular or Hartogs domain of finite type in ${\bf C}^2$ is biholomorphic.

We investigate the uniform perfectness on a boundary point of a hyperbolic open set and distortion of a holomorphic function from the unit disk $\Delta$ into a hyperbolic domain with a uniformly perfect boundary point, especially of a universal covering map of such a domain from $\Delta$, and we obtain similar results to celebrated Koebe's Theorems on univalent functions.

Let $X$ be a complex $n$-dimensional reduced analytic space with isolated singular point $x_{0}$, and with a strongly plurisubharmonic function $\rho:X \to [0,\infty)$ such that $\rho(x_{0}) = 0$. A smooth Kähler form on $X\setminus\{x_{0}\}$ is then defined by ${\bf i}\partial\bar{\partial}\rho$. The associated metric is assumed to have $L^{n}_{\rm{loc}}$-curvature, to admit the Sobolev inequality and to have suitable volume growth near $x_{0}$. Let $E \to X\setminus\{x_{0}\}$ be a Hermitian-holomorphic vector bundle, and $\xi$ a smooth $(0,1)$-form with coefficients in $E$. The main result of this article states that if $\xi$ and the curvature of $E$ are both $L^{n}_{\rm{loc}}$, then the equation $\bar{\partial}u = \xi$ has a smooth solution on a punctured neighbourhood of $x_{0}$. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a $CR$-hypersurface, are discussed in the final section.

We show that a (Spin$^q$-style) twistor space admits a canonical Spin structure. The adiabatic limits of $\eta$-invariants of the associated Dirac operator and of an intrinsically twisted Dirac operator are then investigated.

The aim of this paper is to determine when there exists a quasicontinuous Sobolev function $u \in W^{1,p}({\bf R}^n;\mu)$ whose trace $u|_{{\mathbf R}^{n-1}}$ is the characteristic function of a bounded set $E \subset {\bf R}^{n-1}$, where $d\mu(x) = |x_n|^\alpha dx$ with $-1< \alpha < p-1$. As application we discuss the existence of harmonic measures for weighted $p$-Laplacians in the unit ball.

We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.

We give an asymptotically sharp lower bound for the slope $\lambda (f)$ of a fibration $f:S\rightarrow B$, where $S$ is a surface and $B$ is a curve, if there exists an involution on the general fibre $F$ of $f$. We also construct a new lower bound of $\lambda (f)$ depending increasingly on the irregularity of $S$; as an application of this new bound we have a criteria to control the existence of other fibrations on $S$.

Let $K/k$ be a Galois extension of a number field of degree $n$ and $p$ a prime number which does not divide $n$. The study of the $p$-rank of the ideal class group of $K$ by using those of intermediate fields of $K/k$ has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of $\mathbb{Q}$.

Let $f:N\to P$ be a holomorphic map between $n$-dimensional complex manifolds which has only fold singularities. Such a map is called a holomorphic fold map. In the complex 2-jet space $J^2(n, n; \mathbf{C})$, let $\Omega^{10}$ denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that $\Omega^{10}$ is homotopy equivalent to SU$(n+1)$. By using this result we prove that if the tangent bundles $TN$ and $TP$ are equipped with SU$(n)$-structures in addition, then a holomorphic fold map $f$ canonically determines the homotopy class of an SU$(n+1)$-bundle map of $TN\oplus\theta_N$ to $TP\oplus\theta_P$, where $\theta_N$ and $\theta_P$ are the trivial line bundles.

We slightly modify the definitions of $q$-Hurwitz $\zeta$-functions and $q$-$L$-series constructed by J. Satoh. By using these modified functions, we give some relations for the ordinary Dirichlet $L$-series. Especially we give an elementary proof of Katsurada's formula on the values of Dirichlet $L$-series at positive integers.