We describe the set of points over which a dominant polynomial map $f=(f_1,...,f_n):{ℂ}^n\to {ℂ}^n$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $({\prod }_{i=1}^{n}deg{f}_i-\mu \left(f\right))/\left(mi{n}_{i=1,...,n}deg{f}_i\right)$.

We prove that the symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains.

Generalizations of the theorem of Forelli to holomorphic mappings into complex spaces are given.

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f:ℂ{\mathbb{P}}^k\to ℂ{\mathbb{P}}^k$, when k >1. Classical theory describes U(f) as the complement in $ℂ{\mathbb{P}}^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ{\mathbb{P}}^k$, we give a definition of linking number between closed loops in $ℂ{\mathbb{P}}^k\setminus suppS$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁\left(ℂ{\mathbb{P}}^k\setminus suppS\right)$. As an application, we use these linking...