1 · The essentials
The manuscript claims to construct a complex surface with a singularity—a place where the surface is not smooth. Yet its directions of motion can be described by two independent choices that work across the whole surface. The surprising point is that this description remains available despite the singularity. The claimed example contradicts the characteristic-zero Lipman–Zariski conjecture, which would rule out this combination under the stated mathematical conditions. The supplied abstract announces the construction but does not explain how it works. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a surface with a flawed spot, but with a two-direction navigation system that still works everywhere. The analogy captures the contrast between a singular surface and a uniformly describable system of directions; it does not explain the mathematical construction.
Why it matters
The claimed example separates two conditions that the Lipman–Zariski conjecture links: having a free tangent sheaf and being smooth. Its stated scope is particularly specific: a normal affine complex surface, in characteristic zero. If the construction is valid, it supplies a counterexample within that scope.
2 · Going further
The precise claim is the existence of a singular normal affine complex surface with tangent sheaf . Here denotes the sheaf of regular functions, so this isomorphism says that the tangent sheaf is free of rank two. The surface is nevertheless singular.
The abstract presents this combination as a counterexample to the Lipman–Zariski conjecture in characteristic zero. The relevant contrast is between the algebraic regularity of the tangent sheaf and the geometric smoothness of the surface: the claimed construction has the former without the latter, even with normality imposed.
The only argument-level information supplied is that the manuscript constructs such a surface. No defining equations, construction method, or checks of normality, singularity, and freeness are included in the abstract, so its mechanism cannot be explained from the supplied text. Only one manuscript is provided; no relationship to companion manuscripts is described.
Useful notions
- Complex algebraic surfaces
- Affine varieties and regular functions
- Normality and singularities
- Tangent sheaves and freeness
What was known before / what is new
Before
The source identifies the Lipman–Zariski conjecture in characteristic zero as the statement being challenged. It gives no account of earlier results, special cases, or the conjecture’s history.
New
The manuscript claims a singular normal affine complex surface whose tangent sheaf is free of rank two. It presents this construction as a counterexample to the characteristic-zero Lipman–Zariski conjecture.
Limits and assumptions
The claim concerns a normal affine complex surface with a free rank-two tangent sheaf; the source does not state results for other dimensions or characteristics. The supplied abstract contains no construction details or supporting argument, so it does not permit assessment of whether the example satisfies all the stated conditions. The main result is not formalized in Lean and has not been formally verified. No remaining open problems are specified in the supplied text.
Mini glossary
- Complex algebraic surface —
- An algebraic space of complex dimension two, described using polynomial equations over the complex numbers.
- Singularity —
- A point where an algebraic space is not smooth.
- Normality —
- An algebraic condition requiring local rings to be integrally closed in their fields of fractions.
- Tangent sheaf —
- The sheaf describing local algebraic vector fields, or derivations of regular functions.
- Free of rank two —
- A sheaf is free of rank two when it is isomorphic to two copies of the sheaf of regular functions.
- Characteristic zero —
- A field has characteristic zero when no positive number of copies of its multiplicative identity sums to zero.
Keywords : Lipman–Zariski conjecture · Tangent sheaf · Singular surface
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A singular normal affine surface with free tangent sheaf
23/09/2026We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.