The Kato-surface characterization conjecture for class VII surfaces

A class VII surface is a compact complex surface with first Betti number one and no nonconstant meromorphic functions. A surface is minimal if it contains no exceptional curves of the first kind, and $b_2$ denotes its second Betti number. A cycle of rational curves is a connected configuration of rational curves arranged in a cycle. A Kato surface is a minimal class VII surface with positive $b_2$ that admits a global spherical shell.

Kato-surface characterization conjecture. Any minimal class VII surface with $b_2>0$ containing a cycle of rational curves is a Kato surface.

Together with the cycle conjecture, this characterization would imply the global spherical shell conjecture. The paper presents it as the second, more accessible step toward the classification of class VII surfaces, and no resolution is supplied here.