  Let $S$ be a compact complex surface in class VII$_0^+$ containing a cycle of
rational curves $C=\sum D_j$. Let $D=C+A$ be the maximal connected divisor
containing $C$. If there is another connected component of curves $C'$ then
$C'$ is a cycle of rational curves, $A=0$ and $S$ is a Inoue-Hirzebruch
surface. If there is only one connected component $D$ then each connected
component $A_i$ of $A$ is a chain of rational curves which intersects a curve
$C_j$ of the cycle and for each curve $C_j$ of the cycle there at most one
chain which meets $C_j$. In other words, we do not prove the existence of
curves other those of the cycle $C$, but if some other curves exist the maximal
divisor looks like the maximal divisor of a Kato surface with perhaps missing
curves. The proof of this topological result is an application of Donaldson
theorem on trivialization of the intersection form and of deformation theory.
We apply this result to show that a twisted logarithmic $1$-form has a trivial
vanishing divisor.
