A characteristic-zero counterexample to Lipman–Zariski

The explanation

Take the surface given by xy = z². At the origin it has a sharp point, like the tip of a cone. Mathematicians call such a point singular. Everywhere else the surface is smooth. A vector field on the surface is a rule that attaches a small arrow at each point. The arrow must point along the surface, never off it.

Test one candidate. The field x∂x − y∂y pushes x up and y down at matching rates. Apply it to xy − z². The result is x·y − y·x, which is 0. So this field stays on the surface. Now try ∂z, which pushes straight up in z. Applied to xy − z², it gives −2z. That is not zero over the ordinary numbers, so ∂z leaves the surface.

Change the rules of arithmetic and the picture shifts. In arithmetic modulo 2, the number 2 equals 0. Then −2z is 0, and ∂z stays on the surface too. The paper cites this kind of example. Take any field of characteristic p. There the surface xy − z^p = 0 has two fields that generate all the others. They are x∂x − y∂y and ∂z. This happens even though the surface has a singular pointSource 1.

The tangent sheaf of a surface is the collection of all its vector fields. It is free of rank two when two fields act like a coordinate system. Every other field is then a unique combination of those two, with polynomial coefficients. A smooth surface always has this property near each point. The Lipman–Zariski conjecture asks for the converse over the complex numbers. A complex variety with locally free tangent sheaf should be nonsingularSource 2. Variety here means a shape cut out by polynomial equations.

Much was known before. Lipman showed that such freeness forces normality, a mild condition that rules out the worst kinds of singular pointsSource 3. Many cases of the conjecture were then proved. Hochster settled the graded case, which includes cones like xy = z²Source 4. Van Straten and Steenbrink proved it for isolated singular points in dimension at least threeSource 5. That left surfaces as a separate problemSource 6. Greb, Kebekus, Kovács and Peternell proved it for Kawamata log terminal spaces, a standard class of mild singularitiesSource 7. Others proved the log canonical case, a wider classSource 8. Graf, and then Bergner and Graf, proved it for surfaces whose singular point has small numerical invariantsSource 9,Source 10.

This paper claims to disprove the conjectureSource 11. It builds a normal affine complex surface with a free tangent sheaf of rank two. The surface still has a singular pointSource 12. Affine means the surface is cut out by polynomial equations in ordinary complex space. The freeness holds on the whole surface, and the surface is smooth away from that one pointSource 13.

The obstacle is that positive results cover every easy place to look. Hochster’s theorem rules out a graded coneSource 14. The earlier theorems also rule out mild singular points and small invariants. So the paper starts from a different shape. Blow up the singular point to look at it closely, and it becomes a curve. Here that curve has genus 21, meaning it is a surface with 21 holes. Its self-intersection is −10Source 15. The paper builds the neighbourhood of this curve by hand. It starts from a simple model and changes how the local pieces are glued together. This freedom supplies the two needed vector fieldsSource 16.

The numbers show how far the example sits from the known cases. The genus formula gives the curve’s canonical degree as 2 × 21 − 2 = 40. The paper’s line bundle L has degree 10, and four copies give exactly 40. The geometric genus p_g, a count of independent holomorphic two-forms near the point, is at least 33. The Bergner–Graf criterion proves smoothness when p_g − g − b is at most 2. Here g is 21, and b is 0, since one curve forms no loop. So the quantity is at least 33 − 21 − 0 = 12Source 17. That places the example outside the criteria just describedSource 18.

The construction first gives an isolated analytic singular point, defined by convergent power series rather than polynomialsSource 19. A further step turns it into an affine algebraic surface with the same local structureSource 19. The paper also notes a side result. A related question asked whether repeatedly blowing up at the module of vector fields resolves every singular point. For this surface, every such step does nothing, so the answer is noSource 20,Source 21.

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 give a graded or cone-shaped example, and it does not settle any positive case of the conjecture.