We will classify $n$-dimensional real submanifolds in ${ℂ}^n$ which have a set of parabolic complex tangents of real dimension $n-1$. All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an $n$-dimensional submanifold $M$ in ${ℂ}^n$ such that its images under biholomorphisms $(z_1,\cdots ,z_n)\mapsto (r{z}_1,\cdots ,r{z}_{n-1},r^2{z}_n)$, $r\>1$, are not equivalent to $M$ via any local volume-preserving holomorphic...

Let $X\subset {{ℂ}^n}^n$ be a closed real-analytic subset and put$$𝒜:=\{z\in X\mid \exists A\subset X,\text{germ}\text{of}\text{a}\text{complex-analytic}\text{set,}z\in A,{dim}_{z}A\>0\}$$This article deals with the question of the structure of $𝒜$. In the main result a natural proof is given for the fact, that $𝒜$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited.

The space S of all non-trivial real places on a real function field K|k of trascendence degree one, endowed with a natural topology analogous to that of Dedekind and Weber's Riemann surface, is shown to be a one-dimensional k-analytic manifold, which is homeomorphic with every bounded non-singular real affine model of K|k. The ground field k is an arbitrary ordered, real-closed Cantor field (definition below). The function field K|k is thereby represented as a field of real mappings of S which might...

A stable deformation $f^t$ of a real map-germ $f:ℝⁿ,0\to {ℝ}^p,0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^t$ are real. A related notion is that of a good real perturbation $f^t$ of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of $f^t$ coincides with that of $f_C^t$. The class of map germs having an M-deformation is, in some sense, much larger than the one having a good real perturbation....

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type $\left(\mathrm{A}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for $M$ in $G_2\left({ℂ}^{m+2}\right)$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi {\phi }_1$ induced by two structure tensors $\phi$ and ${\phi }_1$. That is, this commuting shape operator is given by $\phi {\phi }_{1}A=A\phi {\phi }_1$. Using this condition, we prove that...

In this paper we give the topological classification of real primary Kodaira surfaces and we describe in detail the structure of the corresponding moduli space. Moreover, we use the notion of the orbifold fundamental group of a real variety, which was also the main tool in the classification of real hyperelliptic surfaces achieved in [10]. Our first result is that if (S,sygma) is a real primary Kodaira surface, then the differentiable tupe of the pair (S,sygma) is completely determined by the orbifold...

We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.