The Global Spherical Shell conjecture

The explanation

Take the complex plane of two variables with the origin removed. Shrink every point by half, z ↦ z/2, and treat a point and its half as the same point. What remains is a primary Hopf surface, a compact quotient of the punctured plane by a contractionSource 1. One piece of it is the shell of points with size between 1/2 and 1. Gluing its inner sphere to its outer sphere gives the whole surface. As a smooth shape it is a circle times a three-sphere.

That surface has first Betti number 1: it has one independent loop that cannot shrink. Its second Betti number b₂ is 0. Now blow up points on it, replacing each point by a sphere of directions. Each blowup raises b₂ by one. After b blowups the Euler characteristic is b. A circle times a three-sphere has Euler characteristic 0, and each blowup adds 1.

Such surfaces belong to class VII. A compact complex surface is of class VII when it has first Betti number 1 and Kodaira dimension −∞Source 2. These surfaces are never Kähler, so many standard tools do not apply. The case b₂ = 0 is classified: such minimal surfaces are Hopf or Inoue surfacesSource 3. Minimal means there is no sphere of self-intersection −1 that could be blown down.

A global spherical shell is a copy of a thin neighbourhood of the unit three-sphere, sitting inside the surface. Its complement must be connectedSource 4. Kato introduced this structure in 1977Source 5. In the Hopf example the round sphere of size 1 is such a shell. Nakamura’s classification program proposed that every minimal class VII surface with b₂ > 0 contains oneSource 6.

This paper claims to prove that conjectureSource 7. Every connected minimal class VII surface with positive second Betti number contains a global spherical shellSource 7. By Kato’s earlier work, this fixes the shape of such a surface up to smooth deformation. The surface deforms to a primary Hopf surface blown up b₂ times, and its fundamental group is the whole numbersSource 8. As a smooth shape it is a circle times a three-sphere, with b₂ reversed projective planes attachedSource 8.

Earlier work reduced the conjecture to finding curves. Enoki’s theorem handles surfaces with a nonzero effective divisor of square zeroSource 9. Otherwise, Dloussky, Oeljeklaus and Toma showed that b₂ rational curves force a shellSource 10. A rational curve is a curve whose smoothing is a sphere. So the task became producing enough rational curvesSource 11. Teleman proved the conjecture for b₂ = 1 using gauge theorySource 12. He also found a cycle of curves when b₂ = 2Source 12.

The obstacle is that the surface may show no curves at first. The paper works on the infinite cyclic cover, which unrolls the one essential loop into an infinite chain of copies. There it builds holomorphic sections and takes their zero sets. The danger is that a zero set might have open, non-compact pieces that never close upSource 13. The paper rules these out with estimates on long thin cylindersSource 14. It then counts the resulting compact rational curves downstairs, and finds exactly b₂ of themSource 15. The criterion of Dloussky, Oeljeklaus and Toma then gives the shellSource 15.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not treat the case of second Betti number zero, which was already classified, or non-minimal surfaces.