Abstract
Let S be a compact complex surface in class VII0
+ containing a cycle of rational curves C = ∑Dj
. 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 Ai
of A is a chain of rational curves which intersects a curve Dj
of the cycle and for each curve Dj
of the cycle there at most one chain which meets Dj
. 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.
Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey
