Topological triviality of μ-constant surface singularities

The explanation

Take the function f(x, y, z) = x² + y² + z³ of three complex variables. Its zero set is a surface with one bad point at the origin, called a singularity. There all three partial derivatives, 2x, 2y and 3z², vanish together. The Milnor number μ measures how degenerate that point is. It counts the monomials left after setting x, y and z² to zero. Only 1 and z survive, so μ = 2. Cut a nearby smooth level, such as f = 0.01, inside a small ball. Milnor showed it has the shape of μ spheres joined at one pointSource 1.

Now let the function move with a parameter t. The family x² + y² + z³ + tz² does not keep μ fixed. For t ≠ 0 its z-derivative is z(3z + 2t), which vanishes only simply at z = 0. So μ drops to 1 there. The family x² + y² + z³ + tz⁴ keeps μ = 2 for every t. Near the origin its z-derivative is z² times 3 + 4tz, a factor close to 3.

The μ-constant problem asks whether a constant μ forces the topology to stay the same. The precise form is topological right-trivialitySource 2. A homeomorphism is a continuous map with a continuous inverse. Right-triviality asks for homeomorphisms φ_t fixing the origin with f_t(φ_t(x)) = f_0(x). The maps and their inverses must also vary continuously with t together.

Lê Dũng Tráng and C. P. Ramanujam proved constant topological type in 1976 for hypersurfaces of every dimension except twoSource 3. Their method links nearby smooth levels through a cobordism, a space whose boundary is the two shapes. In higher dimensions, the h-cobordism theorem turns that link into a product. Complex dimension two is exactly where that step failsSource 4. Outside dimension two, Timourian and King later proved full triviality of the functions themselvesSource 5. Teissier, Briançon and Speder tied a stronger condition, Whitney regularity, to constancy of a longer list of Milnor numbers. Briançon and Speder also found families that are topologically trivial but not Whitney regularSource 6. So a proof from μ alone cannot lean on Whitney regularity. In 2024, Fernández de Bobadilla and Pełka proved that constant μ keeps the multiplicity constantSource 7.

This paper claims a positive answer in the surface caseSource 8,Source 9. The dependence on t may be any holomorphic one, meaning given by convergent power seriesSource 10. The main obstacle is numerical. Some singularities belong to a milder class called log canonical. For the others, the paper must show that Wahl's invariant β stays constant, using μ aloneSource 11. That invariant is read from a resolution, which replaces the bad point by smooth curves.

The proof rests on a cancellation of Euler characteristicsSource 12. An Euler characteristic is a count like vertices minus edges plus faces. Constant μ keeps one such count fixed on the surfaces of the familySource 13. A comparison then writes zero as a sum of counts for extra surfaces plus a correction term. A theorem of Greuel and Steenbrink makes that correction never negativeSource 14. Langer's form of the Bogomolov–Miyaoka–Yau inequality supplies the opposite signSource 15. So every extra square term must vanish, and β stays constantSource 16.

Okuma's theorem then gives a resolution of all the surfaces at once, after a finite cover of the diskSource 17. The paper lifts the motion in t to these resolved surfaces. The lifts push down to the zero sets and extend to nearby levels, keeping f fixedSource 18. Log-canonical singularities are handled by a separate argumentSource 19.

How sure is this? There is no Lean scope note for this result, so no computer proof checker has tested it. The paper claims to prove the full surface case. No specialist review has been published.

What it doesn't show. This does not give smooth or Whitney-regular coordinate changes, and it says nothing about families with several parameters.