N. Kurnosov

C. Spicer

The Global Spherical Shell conjecture is proved for a class of complex surfaces with two singular holomorphic foliations, a key open case.

The Global Spherical Shell conjecture says that every minimal class VII surface with positive second Betti number is a Kato surface. We prove it for surfaces with second Betti number 3 carrying two distinct singular holomorphic foliations and show that such surfaces are Inoue-Hirzebruch. This proof combines Teleman’s theorem on a cycle of rational curves with a study of the tangency divisor between the two foliations.

Let S be a minimal compact complex surface of class VII with b2(S)=3. We prove that if S carries two distinct singular holomorphic foliations, then S contains a global spherical shell. Moreover, it is an Inoue-Hirzebruch surface. The proof is based on Teleman's theorem on the existence of a cycle of rational curves and results of Dloussky and is inspired by work of Brunella.

From Foliations to Kato

Class VII surfaces with b₂ = 3 and two foliations are Kato