Non K\"ahlerian surfaces with a cycle of rational curves

TL;DR

This paper classifies certain non-Kähler complex surfaces with cycles of rational curves, showing their structure resembles Kato surfaces and applying topological and deformation theory techniques.

Contribution

It provides a topological classification of class VII$_0^+$ surfaces with rational curve cycles, linking their structure to Kato surfaces and Inoue-Hirzebruch surfaces.

Findings

If multiple connected components exist, the surface is an Inoue-Hirzebruch surface.

Connected components of the divisor are chains of rational curves intersecting the cycle.

A twisted logarithmic 1-form has a trivial vanishing divisor.

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.