We first give a finiteness theorem for meromorphic mappings. Next, we give conditions under which two holomorphic mappings from a finite analytic covering space over the complex $m$-space into a smooth elliptic curve are algebraically related.

Let $(V, o)$ be a normal isolated singularity in a complex Euclidean space $(C^N, o)$. Let $M$ be the intersection of this singularity and the real hypersphere $S_{\epsilon}^{2N-1} (o)$, centered at the origin $o$ with an $\epsilon$ radius. Then, naturally, this link $M$ admits a CR structure, induced from $V$, and the deformation theory of this CR structures has been studied in [1], [2],[3]. Especially in [3], a particular subspace of the infinitesimal deformation space is found, and we propose to study the relation between this subspace and simultaneous deformation. We note that: if the canonical line bundle of the CR structure is trivial, then the infinitesimal space of the deformation of CR structures is a part of the middle dimension cohomology. And in this line, we conjecture that $Z^1$, introduced in [3], might be related to the simultaneous deformation of isolated singularity $(V, o)$ (see also [2]). We discuss this problem for $A_l$ singularities.

Introducing a new notion of the exact steepest descent path, we develop a new steepest descent method applicable to general ordinary differential equations with polynomial coefficients. Its application to the connection problem for solutions is also discussed.

We prove here a weak negativity property on Demailly's 2-jet bundles of hyperbolic (singular) fibrations on hyperbolic curves with some restrictions on the singularities of special fibres.

We show some methods of constructing hyperbolic hypersurfaces in the complex projective space, which gives a hyperbolic hypersurface of degree $2^n$ in $P^n (\mathbf{C})$ for every $n \ge 2$. Moreover, we show that there are some hyperbolic hypersurfaces of degree $d$ in $P^n (\mathbf{C})$ for every $d \ge 2 \times 6^n$ for each $n \ge 3$.

We prove that there exists no $C^m$ Levi-flat real hypersurface in $\mathbb{CP}_n$ for $n \geqslant 2$ and $m \geqslant 4$. This is an improvement of the regularity in a theorem of Y.-T. Siu who proved this result for $m \geqslant 8$.

The main theme of this article concerns the characterization problem of analytic polyhedra in $\mathbf{C}^n$ with non-compact automorphism group. In particular, we include a proof that every bounded convex analytic polyhedron in $\mathbf{C}^n$ is biholomorphic to the product of a Kobayashi hyperbolic convex cone and a bounded convex domain. Several related recent developments are also introduced.

Smooth quadric hypersuraces in $\mathbb{P}^{n+1} (\mathbb{C})$ are numerically characterised as the smooth Fano $n$-folds of length $n$, i.e., a smooth Fano $n$-fold $X$ is isomorphic to a hyperquadric if and only if the minimum of the intersection number $(C, -K_X)$ is $n$, where $C$ runs through the rational curves on $X$.

In this note, we shall discuss elimination theorems of defects of hypersurfaces or rational moving targets for a meromorphic mapping or a holomorphic curve into $\mathbf{P}^n (\mathbf{C})$ by its small deformation.

Here we discuss the intersection multiplicities of holomorphic and algebraic curves with divisors on an algebraic variety as an analogue to the abc-Conjecture. We announce some new results.

To any given Laurent polynomial $f$ on $\mathbf{C}_{*}^{n}$ we associate two natural convex functions $M_f$ and $N_f$ on $\mathbf{R}^n$. We compute the Hessian of $M_f$ and obtain an explicit formula for the volume of the Newton polytope $\Delta_f$. We also establish asymptotic formulas relating our convex functions to coherent triangulations of $\Delta_f$ and to the secondary polytope.

We give a short survey on some new (and old) results on deformations of cyclic quotient surface singularities which are mainly contained in the doctoral thesis of Stephan Brohme.

We give a general criterion for the existence of a coarse moduli space as an algebraic space.

— It is possible to determine meromorphic functions on $\mathbb{C}$ by inverse images of some sets since R. Nevanlinna. However, analogous problems to holomorphic mappings of $\mathbb{C}$ into $\mathbb{P}^n (\mathbb{C})$ are complicated. In this paper some results for such problems are given. —

We state some fundamental properties of the intersection theory for pseudo-effective line bundles defined by Tsuji, and some applications of that theory. As one of applications, we generalize Seshadri's criterion for ampleness of nef line bundles as a criterion for bigness of pseudo-effective line bundles.

We give a subadjunction theorem which relates the multi-adjoint linear system of the ambient space and the linear system of the restricted bundle on a subvariety.

We study the fixed point indices of some polynomial automorphisms of $\mathbf{C}^n$. In particular, it is shown that, for a composition of generalized Hénon maps, the sum of the fixed point indices vanishes. A consequence is that a generic polynomial automorphism of $\mathbf{C}^2$ has a saddle fixed point.

We give some observations and results on Nevanlinna theory for holomorphic curves in algebraic varieties.

In this note, we prove that the “adiabatic limit” of the equivariant analytic torsion of a Nikulin's $K3$ surface converges to the value of norm of the Borcherds $\Phi$-function at its period point after a certain renormalization.