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BY 4.0 license Open Access Published by De Gruyter Open Access July 16, 2021

Non Kählerian surfaces with a cycle of rational curves

  • Georges Dloussky EMAIL logo
From the journal Complex Manifolds


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.

MSC 2010: 32J15; 32J27


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Received: 2021-03-22
Accepted: 2021-06-20
Published Online: 2021-07-16
Published in Print: 2021-01-01

© 2021 Georges Dloussky, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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