A class VII surface is a compact complex surface $M$ such that $b_1(M)=1$ and $kd(M)=-\infty$. Class VII surfaces with vanishing second Betti number have been classified by Bogomolov (and are either Hopf surfaces or Inoue surfaces).

The situation is somewhat more complicated for surfaces with positive second Betti number. Conjecturally, all such surfaces admit a global spherical shell (a neighbourhood of $S^3\subset \mathbb{C}^2/\{0\}$ holomorphically embedded into $M$ so that the complement is connected). Dloussky, Oeljeklaus and Toma have proved that if a minimal complex surface $M$ with $b_2(M)>0$ admits a global spherical shell iff it contains $b_2(M)$ rational curves. Results of Nakamura & Teleman imply existence of global spherical shell for minimal surfaces with $b_2(M)=1$.

My question is: what progress was made on the conjecture since Teleman?