The global spherical shell conjecture for class VII_0 surfaces

The conjecture says that every minimal class $\mathrm{VII}_0$ surface with $b_2>0$ contains a global spherical shell, equivalently is a Kato surface. The general classification remains unresolved.

Known results

• The case $b_2=1$ is classified: every minimal class $\mathrm{VII}$ surface with $b_2=1$ has a global spherical shell (2007).
• Having $b_2$ rational curves, a numerically pluri-anticanonical divisor, or a cycle of curves gives sufficient conditions for a global spherical shell (2007).
• Dloussky, Oeljeklaus, and Toma proved that exactly $b_2$ rational curves imply a global spherical shell.
• Surfaces with $b_2=0$ are Hopf or Inoue surfaces; known class $\mathrm{VII}_0^+$ examples are Kato surfaces.

10 September, year not given: special-case proof claim

A report claims that the conjecture has been proved for a class of complex surfaces carrying two singular holomorphic foliations, described as a key open case. This is progress toward, not a resolution of, the full conjecture; the claim has no proof details or independent verification in the retrieved source.

Current status (as of September 2026): The $b_2=1$ case and several geometric subclasses are settled, while the global spherical shell conjecture for arbitrary class $\mathrm{VII}_0$ surfaces with $b_2>0$ remains open; the reported foliated-subclass result is unverified.