Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension $(1,1)$ for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions $({ℂ}^2,0)\to ({ℂ}^2,0)$.

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...

If E is a closed subset of locally finite Hausdorff (2n-2)-measure on an n-dimensional complex manifold Ω and all the points of E are nonremovable for a meromorphic mapping of Ω E into a compact Kähler manifold, then E is a pure (n-1)-dimensional complex analytic subset of Ω.

The purpose of this paper is twofold. The first is to weaken or omit the condition $dim{f}^{-1}\left(H_i\cap H_j\right)\le m-2$ for i ≠ j in some previous uniqueness theorems for meromorphic mappings. The second is to decrease the number q of hyperplanes $H_j$ such that f(z) = g(z) on ${\bigcup }_{j=1}^q{f}^{-1}\left(H_j\right)$, where f,g are meromorphic mappings.

We study the topological invariant ϕ of Kwieciński and Tworzewski, particularly beyond the case of mappings with smooth targets. We derive a lower bound for ϕ of a general mapping, which is similarly effective as the upper bound given by Kwieciński and Tworzewski. Some classes of mappings are identified for which the exact value of ϕ can be computed. Also, we prove that the variation of ϕ on the source space of a mapping with a smooth target is semicontinuous in the Zariski topology.