Non Kählerian surfaces with a cycle of rational curves

Abstract: 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.