We classify generic germs of contracting holomorphic mappings which factorize through blowing-ups, under the relation of conjugation by invertible germs of mappings. As for Hopf surfaces, this is the key to the study of compact complex surfaces with $b_1=1$ and $b₂>0$ which contain a global spherical shell. We study automorphisms and deformations and we show that these generic surfaces are endowed with a holomorphic foliation which is unique and stable under any deformation.

We study compact complex manifolds covered by a domain in $n$-dimensional projective space whose complement $E$ is non-empty with $(2n-2)$-dimensional Hausdorff measure zero. Such manifolds only exist for $n\ge 3$. They do not belong to the class $𝒞$, so they are neither Kähler nor Moishezon, their Kodaira dimension is $-\infty$, their fundamental groups are generalized Kleinian groups, and they are rationally chain connected. We also consider the two main classes of known 3-dimensional examples: Blanchard manifolds, for which...

A compact complex space $X$ is called complex-symmetric with respect to a subgroup $G$ of the group ${\mathrm{Aut}}_0\left(X\right)$, if each point of $X$ is isolated fixed point of an involutive automorphism of $G$. It follows that $G$ is almost $G^0$-homogeneous. After some examples we classify normal complex-symmetric varieties with $G^0$ reductive. It turns out that $X$ is a product of a Hermitian symmetric space and a compact torus embedding satisfying some additional conditions. In the smooth case these torus embeddings are classified using...