Family 048 — Free vector fields on a singular surface

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Family 048 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

On a smooth surface, a coordinate chart supplies two coordinate vector fields. Every other vector field on the chart can be written uniquely as a function times the first plus a function times the second. Could the converse hold: if vector fields admit such a free local basis, must the surface be smooth?

The Lipman–Zariski conjecture predicts yes in characteristic zero. It is a proposed way to recognize nonsingularity through differentiation, without first finding coordinates or checking a Jacobian matrix.

Family 048 contains one manuscript claiming a counterexample over $\mathbb C$. It constructs a normal affine surface with a singular point, while its entire module of algebraic vector fields is free of rank two. Freeness is asserted on the whole affine surface, including at the singularity, not merely on its smooth part.

The construction first takes place in complex-analytic geometry and then passes to an algebraic surface. It is therefore an existence construction through neighborhoods, contraction, and algebraization, rather than a short hypersurface equation in three variables.

2. Background and essential definitions

Vector fields as derivations

If $A$ is the ring of polynomial functions on an affine variety, an algebraic vector field is a derivation

$$D:A\longrightarrow A, \qquad D(fg)=fD(g)+gD(f),$$

which is linear over $\mathbb C$. The product rule captures the familiar behavior of directional differentiation.

All derivations form an $A$-module, denoted $\operatorname{Der}_{\mathbb C}(A)$. Multiplying a vector field by a function gives another vector field. On the affine plane,

$$\operatorname{Der}_{\mathbb C}(\mathbb C[x,y]) =\mathbb C[x,y]\,\partial_x\oplus\mathbb C[x,y]\,\partial_y.$$

A module is free of rank two if it has two generators and every element has a unique expression as their linear combination with coefficients in $A$. The associated tangent sheaf records the corresponding vector fields on each open set. Local freeness allows a basis that depends on the open set; global freeness supplies one basis everywhere on the affine variety.

Smoothness and the pointwise tangent space

A local ring at a point is regular if its maximal ideal can be generated by as many elements as the local dimension. For a complex algebraic variety, regularity is equivalent to smoothness at that point.

The Zariski tangent space at a point is dual to $\mathfrak m/\mathfrak m^2$, where $\mathfrak m$ is the maximal ideal. At a singular point its dimension can exceed the dimension of the variety.

This pointwise tangent space must be distinguished from a free module of derivations on a neighborhood. At singularities, taking the dual of the module of differentials and then taking a fiber need not reproduce the dual of the fiber of differentials. Thus one cannot dismiss the conjecture by saying that a free rank-two derivation module automatically makes the pointwise tangent space two-dimensional. That implication is part of the subtlety being investigated.

Normality and resolutions

An integral ring is normal if it is integrally closed in its fraction field. Geometrically, normality rules out some elementary defects, such as branches that should have been separated, but it still permits isolated singularities on surfaces.

A resolution replaces a singular point by a configuration of smooth curves in a smooth surface. These exceptional curves record information hidden at the point. Their genera, intersections, and the behavior of differential forms along them help determine the severity of the singularity.

The manuscript builds the neighborhood of one such curve first and then contracts the curve to produce the singular point.

3. Development history and previous partial results

Lipman’s 1965 work established that local freeness of the derivation module forces normality. The conjecture therefore asks whether free vector fields also rule out the normal singularities that remain.

In positive characteristic, the corresponding assertion can fail because differentiation loses information: for example, the derivative of $z^p$ vanishes in characteristic $p$. Those examples do not settle the complex case, where ordinary differentiation behaves differently.

Strong positive results cover several broad classes in characteristic zero. Hochster treated positively graded algebras with degree-zero part equal to the field. Scheja–Storch handled hypersurfaces, and Källström’s corrected complete-intersection result handles the more general locally complete-intersection setting. Van Straten–Steenbrink proved the isolated-singularity case in dimension at least three using extension of differential forms, leaving surfaces as a distinct difficulty.

The extension viewpoint also connects the problem to birational singularities. Greb–Kebekus–Kovács–Peternell proved the conjecture for Kawamata log terminal spaces. Druel and Graf–Kovács established the log-canonical case, a larger class of mild singularities measured by discrepancies on resolutions.

For surfaces, Graf obtained results in low geometric genus. Bergner–Graf proved smoothness when

$$p_g-g-b\le2,$$

where $p_g$ is the geometric genus of the local singularity, $g$ the sum of the genera of exceptional curves, and $b$ the first Betti number of the resolution’s dual graph. Here the local geometric genus measures a cohomological obstruction on a resolution; it is not the geometric genus of a compact surface from the classification theory.

Another useful distinction is the Lie bracket. Jörder proved smoothness when a tangent basis can be chosen with pairwise commuting vector fields. Arbitrary free vector fields need not commute, whereas coordinate vector fields do.

M1 places its example outside these established classes. The neighborhood is deliberately not a positively graded cone, the singularity is more severe than log canonical, and no commuting tangent frame is claimed.

4. What the new manuscripts claim

Theorem 1.1 asserts a finitely generated normal integral complex algebra $A$ of dimension two and a maximal ideal $\mathfrak m$ such that

$$\operatorname{Der}_{\mathbb C}(A)\simeq A^{\oplus2}, \qquad A_{\mathfrak m}\text{ is not regular}.$$

The affine surface $X=\operatorname{Spec}A$ can be chosen smooth away from the marked singular point. The claim is therefore a counterexample with an isolated normal surface singularity and a globally free tangent sheaf.

The analytic precursor has one exceptional curve of genus 21 and self-intersection $-10$. M1 reports canonical discrepancy $-5$ and local geometric genus at least 33. Thus, for this resolution,

$$g=21,\qquad b=0,\qquad p_g-g-b\ge12.$$

These values locate the proposed example beyond the earlier numerical smoothness criterion. The discrepancy is the coefficient measuring the change of the canonical divisor along the exceptional curve; log-canonical singularities require it to be at least $-1$.

Two further conclusions follow in the manuscript. First, the smooth locus $U=X_{\mathrm{reg}}$ has logarithmic Kodaira dimension two, by applying an earlier theorem of Biswas–Gurjar–Kolte. Logarithmic Kodaira dimension measures the growth of canonical forms allowed logarithmic poles along a boundary compactifying $U$.

Second, repeatedly blowing up at the module of derivations does not resolve this singularity. The relevant blowup is designed to make the module locally free. Since it already is free, the operation is the identity; normalization also changes nothing because the surface is normal. This addresses the specific derivation-module procedure, not resolution of singularities by arbitrary blowups.

5. Why the advance matters

Importance: 5/5. If correct, the construction would disprove a longstanding intrinsic criterion for smoothness in characteristic zero, in a dimension central to the remaining difficulty.

Vector fields are among the most basic local geometric objects. A free tangent sheaf looks formally like the tangent bundle of a smooth surface, so a counterexample would show that this module alone can miss singular geometry. The issue is not a shortage of vector fields but the failure of their algebraic organization to force coordinate directions.

The example would also clarify the reach of existing theorems. Their additional hypotheses—mild singularities, low resolution genus, grading, complete intersections, or commuting vector fields—carry essential information. The proposed counterexample does not negate those results; it would show why a general converse requires more than freeness.

6. The main idea

Start with a curve whose neighborhood can be changed

M1 chooses a smooth curve $C$ of genus 21 and a degree-ten line bundle $L$ satisfying

$$L^{\otimes4}\simeq K_C.$$

The dual line bundle $L^{-1}$ has negative degree. Its zero section gives a starting neighborhood with normal bundle $L^{-1}$, but the manuscript modifies the holomorphic gluing of that neighborhood. This changes more than the normal bundle records.

In local base and fiber coordinates $(z,t)$, the construction uses the meromorphic two-form

$$\sigma=t^{-5}\,dt\wedge dz.$$

The aim is to build two vector fields whose determinant vanishes to exactly the required order along $C$, while remaining nonzero everywhere off $C$. They then form a tangent frame on the punctured neighborhood.

Match local corrections with global residue constraints

The leading coefficients of the fields have six common zeros on $C$. Correcting the determinant at those points requires local choices, but arbitrary choices need not glue around the whole curve.

An auxiliary elliptic curve and a trace pairing provide meromorphic data whose residues satisfy the needed global cancellation identities. The construction then extends the vector fields and their desired determinant through finitely many Taylor orders. Choosing determinant coefficients separately retains freedom to remove the compatibility obstructions.

The manuscript emphasizes a finite construction. It does not rely on asserting that an uncontrolled infinite sequence of neighborhood changes converges. Cohomological vanishing and formal-functions arguments are used to lift the finite data to actual holomorphic fields after shrinking the neighborhood.

Contract, then pass from analytic to algebraic geometry

Grauert’s contraction criterion collapses the negative curve to a point. The vector fields give a frame away from that point, and M1 uses normality to extend tangent freeness across it. Adjunction records that the resulting point is nonetheless singular.

The volume form dual to the frame has a fifth-order pole along the exceptional curve. The construction therefore does not assume the holomorphic extension of forms that powers many earlier positive results.

Finally, approximation and finite determinacy produce an algebraic local ring with the same completed local ring as the analytic singularity. Completion records all formal infinitesimal orders around the point. The manuscript keeps the equations and their relations together, then uses a Jacobian presentation and faithful flatness to descend freeness of derivations. Shrinking to one affine neighborhood gives the globally free module asserted in Theorem 1.1.

7. Scope, limitations, and open questions

The claimed surface is affine and noncompact. It does not contradict the compact classification result identifying compact complex surfaces with globally free rank-two tangent sheaf as complex tori.

Its tangent frame cannot be chosen commuting near the singular point, by the earlier commuting-frame theorem. A reader should therefore not interpret the two generators as the partial derivatives of a hidden coordinate chart.

The example also lies outside the known complete-intersection, graded-cone, and log-canonical affirmative settings. Its construction does not provide a universal classification of singularities with free tangent sheaf or a minimal possible value of the local geometric genus.

The derivation-blowup consequence is restricted to that prescribed operation. General resolution methods remain available in characteristic zero. A natural further question is which additional information about vector-field brackets or differential forms restores an effective smoothness criterion.

8. Sources and further reading

M1’s Theorem 1.1 gives the algebraic endpoint. Sections 2–5 construct the analytic germ, Remark 5.5 records its resolution invariants, and Section 6 explains algebraization and the two further consequences.

The primary papers linked in Section 3 describe the earlier affirmative cases. Optional background on derivations and regular local rings helps connect the statement to ordinary differentiation and the algebraic definition of a smooth point.

Source manuscripts

• M1. OpenAI, A singular normal affine surface with free tangent sheaf (September 23, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.