The Global Spherical Shell conjecture for minimal class VII surfaces with positive second Betti number

Theorem 1.1: every minimal class VII surface with $b_2>0$ contains a global spherical shell. Corollaries: by Kato, each such surface deforms to blown-up primary Hopf surfaces, has $\pi_1\cong\mathbb Z$ and is diffeomorphic to $(S^1\times S^3)\#b_2\overline{\mathbb{CP}}^2$; and aspherical compact complex surfaces satisfy $\chi\ge\tfrac95|\sigma|$. The proof builds divisors on the infinite cyclic cover and shows all their components are compact rational curves, then applies Dloussky-Oeljeklaus-Toma. It does not give a complete biholomorphic classification of the resulting Kato surfaces beyond what Kato's theory provides.

Theorem 1.1: every minimal class VII surface with $b_2>0$ contains a global spherical shell. Corollaries: by Kato, each such surface deforms to blown-up primary Hopf surfaces, has $\pi_1\cong\mathbb Z$ and is diffeomorphic to $(S^1\times S^3)\#b_2\overline{\mathbb{CP}}^2$; and aspherical compact complex surfaces satisfy $\chi\ge\tfrac95|\sigma|$. The proof builds divisors on the infinite cyclic cover and shows all their components are compact rational curves, then applies Dloussky-Oeljeklaus-Toma. It does not give a complete biholomorphic classification of the resulting Kato surfaces beyond what Kato's theory provides.