Given a non-singular holomorphic foliation $ℱ$ on a compact manifold $M$ we analyze the relationship between the versal spaces $K$ and $K^{\mathrm{tr}}$ of deformations of $ℱ$ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of $ℱ$ parametrized by an analytic space $K^f$ isomorphic to ${\pi }^{-1}\left(0\right)\times \Sigma$ where $\Sigma$ is smooth and $\pi$ : $K\to K^{\mathrm{tr}}$ is the forgetful map. The map $\pi$ is shown to be an epimorphism in two situations: (i) if $H^2(M,{\Theta }_{ℱ}^f)=0$, where ${\Theta }_{ℱ}^f$ is the sheaf of...

In this paper we show that a 1-convex (i.e., strongly pseudoconvex) manifold $X$, with 1- dimensional exceptional set $S$ and finitely generated second homology group $H_2(X,\mathbb{Z})$, is embeddable in ${ℂ}^m\times ℂ{\mathbb{P}}_n$ if and only if $X$ is Kähler, and this case occurs only when $S$ does not contain any effective curve which is a boundary.

In this paper, we show that if $M_1$ and $M_2$ are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if $f$ is a holomorphic mapping defined near a neighborhood of $M_1$ so that $f(M_1)\subset M_2$, then $f$ is also algebraic. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which allows our main result to be stated for mappings...