Status of global spherical shell conjecture for minimal complex surfaces?

Question

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?

Answer 1

Teleman has shown that the global spherical shell conjecture holds for $b_2 = 1$, $b_2 = 2$, and $b_2 = 3$ in the following papers respectively:

1. Teleman, Andrei, Donaldson theory on non-Kählerian surfaces and class VII surfaces with $b_2 = 1$, Invent. Math. 162, No. 3, 493-521 (2005). ZBL1093.32006.

2. Teleman, Andrei, Instantons and curves on class VII surfaces, Ann. Math. (2) 172, No. 3, 1749-1804 (2010). ZBL1231.14028.

3. Teleman, Andrei, Donaldson theory in non-Kählerian geometry, in "Modern Geometry: A Celebration of the Work of Simon Donaldson”, Proceedings of Symposia in Pure Mathematics, Vol. 99, 363-392 (2018).