Abstract
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.
Highlights
Let S be a compact complex surface in class VII+0 containing a cycle of rational curves C = Dj
With deformation theory, the following result which show with theorem 0.3 that the maximal divisor looks like the one of a Kato surface
The following theorem shows that the class of the maximal divisor of a surface with a cycle of rational curves is of the same type as the class of a cycle
Summary
One suppose that S contains a cycle of s rational curves, 1 ≤ s ≤ n. The following theorem shows that the class of the maximal divisor of a surface with a cycle of rational curves is of the same type as the class of a cycle. Let S be a compact complex surface in class VII+0 endowed with exactly one cycle C = D0 + · · · + Dαi + · · · + Dαs−1 of s rational curves, 1 ≤ s < n, of class −eIC. By lemma 18, there is a deformation which smoothes a singular point of the cycle, say Dα0 ∩ Dα1, self-intersection does not change and the number of curves decreases by one, the deformed surface in not minimal (recall the formula b2(S) = −C2 + (C) for minimal surfaces) with exactly one exceptional curve of the first kind of class eα.
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