Same number, same shape?

0:00The μ-constant problem

Take a polynomial in three complex variables with an isolated critical point at the origin, like x² + y² + z². Its zero set is a surface with one singular point. Nearby level sets are smooth, and inside a small ball each one, the Milnor fiber, looks like a bouquet of two-spheres. The number of spheres is the Milnor number, μ. Algebraically, it is the dimension of the local ring modulo the three partial derivatives: one for x² + y² + z², and six for x² + y³ + z⁴. Now let the polynomial move in a family with a parameter t. If the Milnor number stays constant, is the family topologically trivial: is there, for each t, a homeomorphism of space, varying continuously with t, that turns each function into the first one? Lê and Ramanujam proved in 1976 that constant Milnor number gives constant topological type whenever the hypersurface dimension is not two, and Timourian and King upgraded this to triviality of the family. For surfaces in three-space, where their cobordism argument does not give a product, the question stayed open: the μ-constant problem.

1:17The claim

Result 064 is one OpenAI preprint, 37 pages, dated September 24. It claims a positive answer for surfaces: every holomorphic one-parameter family of isolated surface singularities in complex three-space with constant Milnor number is topologically right-trivial. After shrinking, there are homeomorphisms φ_t of the ambient space, fixing the origin and varying continuously with t, together with their inverses, such that f_t ∘ φ_t = f_0. So the zero sets and all nearby level sets move together. Constant Milnor number is weaker than Whitney regularity, a stronger form of equisingularity. By Teissier and by Briançon and Speder, Whitney regularity means that the whole sequence of sectional Milnor numbers is constant. In the Briançon–Speder family, z⁵ + t·y⁶z + x·y⁷ + x¹⁵, μ stays 364, but the Milnor number of a generic plane section drops from 28 to 26 once t is nonzero. A proof must handle such families. Fernández de Bobadilla and Pełka had already shown that the multiplicity stays constant.

2:36A logarithmic invariant

Here is the plan. If the central surface is a double point, or log canonical, the paper reduces to known tools: plane curves in the first case, and multiplicity at most three, constant multiplicity and Teissier’s criterion in the second. The main case is a surface that is not log canonical. There the paper uses Wahl’s logarithmic invariant beta: minus the self-intersection of the positive part of the sum of the canonical divisor and all exceptional curves on a resolution. If beta is constant, Okuma’s theorem gives a simultaneous resolution after a finite base change. So the numerical task is to show that constant μ forces constant beta. To compare beta across the family, the paper compactifies to projective surfaces and runs semistable reduction and the threefold log minimal model program. The special fiber becomes one main component plus extra surfaces over the singular point. Constant Milnor number makes the Euler characteristics cancel: zero equals the sum of the Euler characteristics of the extra open surfaces, plus a correction that is never negative, by Greuel and Steenbrink. With nefness from the model, Langer’s orbifold Bogomolov–Miyaoka–Yau inequality bounds each extra log canonical square between zero and three times that Euler characteristic. Together these force every extra square to vanish, so beta is constant.

4:05From resolution to homeomorphism

With beta constant, Okuma’s theorem gives, after a base change t = uᵉ, a simultaneous resolution whose exceptional curves stay reduced. An Euler characteristic count shows that they keep normal crossings, since smoothing a node would lower it. On these resolved tubes, the paper lifts the parameter direction to complete vector fields. The flows descend to the punctured zero sets and extend to the nearby nonzero levels, preserving the function. Properness of the tubes and reverse transport give continuity at the origin, for the maps and for their inverses. An explicit correction along the angular seam of the base change brings the trivialization back to the original disk. Parusiński had built trivializing vector fields for linear families, f + t·g, by a gradient method. An appendix explains why a continuous gradient lift alone is not enough: the trajectories need control too.

5:01Where it stands

The preprint is not yet peer reviewed, and it has not been revised. There is no Lean formalization, and it cites no other paper from the release, though one of the Zariski counterexample papers of episode 059 mentions it for context. Its inputs are substantial established theorems: Okuma’s simultaneous resolution, the threefold minimal model program, Langer’s inequality, Greuel and Steenbrink, nearby cycles, and Fernández de Bobadilla and Pełka. We found no public assessment. If it holds, the μ-constant problem is settled for surfaces: in three variables, a constant Milnor number means the function itself does not change topologically along the family, even when finer invariants, like the Milnor number of a plane section, jump.