The Lipman–Zariski conjecture for normal surface singularities

The Lipman–Zariski conjecture predicts that a complex algebraic variety or complex analytic space whose tangent sheaf (the sheaf of Kähler derivations) is locally free must be smooth. Classical reductions due to Lipman, Becker, and Flenner make the conjecture equivalent to its restriction to normal surface singularities: if the module of derivations of a two-dimensional normal local ring over a field of characteristic zero is free, must the ring be regular? This surviving case is precisely the obstruction that, as de Fernex observed in his work on the complex Plateau problem, blocks the extension of the characterization of smooth fillings of strongly pseudoconvex Calabi–Yau CR manifolds from real dimension at least $5$ to dimension $3$. Despite substantial partial results — log canonical, locally complete intersection, toric, terminal, and low-genus singularities — the question remains open as of 2026, and the conjecture is known to fail in positive characteristic, so characteristic-zero methods are essential.

Background

On a smooth complex algebraic variety the sheaf of Kähler differentials $\Omega^1_X$ is locally free, and hence so is its dual, the tangent sheaf $T_X=\operatorname{Hom}_{\mathcal O_X}(\Omega^1_X,\mathcal O_X)$, which is the sheaf of holomorphic derivations $\operatorname{Der}_{\mathbb C}(\mathcal O_X)$. The Lipman–Zariski conjecture, posed as a question by Lipman in 1965, predicts the converse: a complex algebraic variety or complex space $X$ whose tangent sheaf is locally free must be smooth. Lipman proved that freeness of $T_X$ already forces $X$ to be normal (normality means that the local rings are integrally closed; equivalently, the variety has no codimension-one singularities); Becker showed that a counterexample, if one exists, may be taken to have isolated singularities; and Flenner, building on work of van Straten and Steenbrink, settled the case where the singular locus has codimension at least $3$. Since the singular locus of a normal variety has codimension at least $2$, and a normal surface therefore has only isolated singularities, these three reductions show that the entire conjecture is equivalent to its restriction to normal surface singularities. Partial results cover proper subclasses: homogeneous complete intersections (Hochster), locally complete intersection singularities (Källström), log terminal and more generally log canonical singularities (Greb–Kebekus–Kovács–Peternell; Druel; Graf–Kovács; log canonical and klt are mild singularity classes from the minimal model program), toric varieties (Tipler), terminal singularities (Bongiorno), and singularities of low genus: Graf proved the conjecture for normal surface singularities of geometric genus $p_g\le 1$ (the geometric genus is $\dim R^1f_*\mathcal O_Y$ for a resolution $f:Y\to X$) and for $p_g=2$ when the link (the 3-manifold bounding the singularity) is not a rational homology sphere, and Bergner–Graf extended this to singularities with $p_g-g-b\le 2$, where $g$ is the sum of the genera of the exceptional curves of a resolution and $b$ the first Betti number of the resolution dual graph. Biswas–Gurjar–Kolte verified the conjecture for several classes of affine and projective surfaces. In positive characteristic the conjecture is false (Graf), so any proof must use characteristic-zero input in an essential way. The conjecture also blocks a question from a neighboring field. A CR (Cauchy–Riemann) manifold is a real manifold whose complexified tangent bundle carries a distinguished subbundle $T^{1,0}$ behaving like the holomorphic directions of a real hypersurface in complex space; it is strongly pseudoconvex when its Levi form is positive definite. By a theorem of Harvey and Lawson, a compact strongly pseudoconvex CR manifold $M$ of dimension $2n-1$ contained in the boundary of a strongly pseudoconvex bounded domain $\Omega\subset\mathbb C^N$ bounds a canonical normal Stein space $V\subset\Omega$ with boundary regularity, called the Stein filling (a Stein space is the analytic analogue of an affine variety). The complex Plateau problem, in the formulation studied by Yau, asks when this filling $V$ is smooth. Yau characterized smooth fillings for hypersurface boundaries with $n\ge 3$ by the vanishing of suitable Kohn–Rossi cohomology groups, the cohomology of the tangential Cauchy–Riemann complex on $M$ introduced by Kohn and Rossi. Du, Gao and Yau later introduced a numerical invariant $g^{(\wedge^p 1)}(M)$, defined as the dimension of the cokernel of the natural map from the $p$-th exterior power of the space of global holomorphic cross sections of the dual of the holomorphic tangent bundle of $M$ to the space of global holomorphic cross sections of its $p$-th exterior power; Yau conjectured that this invariant is positive for every singular normal surface singularity, a conjecture proved for rational surface singularities by Du and Gao. de Fernex proved (Theorem A) that for a strongly pseudoconvex compact Calabi–Yau CR manifold $M$ of dimension $2n-1\ge 5$ (Calabi–Yau meaning that the canonical line bundle of the CR structure is trivial) the Stein filling $V$ has smooth normalization if and only if $g^{(\wedge^n 1)}(M)=0$, and (Theorem B) that in the hypersurface case the filling itself is smooth if and only if $g^{(\wedge^n 1)}(M)=0$, for all $n\ge 2$. The final step of the proof of Theorem A is exactly an instance of the Lipman–Zariski conjecture, supplied for $n\ge 3$ by the theorem of van Straten and Steenbrink for isolated singularities of dimension at least $3$; de Fernex states explicitly that the proof would extend to $n=2$, the case of Calabi–Yau CR 3-manifolds, if the conjecture were known for all normal surface singularities. In the hypersurface case the needed locally-complete-intersection instance was available from Källström, which is why Theorem B already covers $n=2$.

Problem Statement

Let $(X,0)$ be a normal complex surface singularity, that is, a germ of a two-dimensional normal complex analytic space; equivalently, by Artin approximation, one may work with the local ring at the isolated singular point of a normal algebraic surface over a field of characteristic zero. Let $\Omega^1_X$ denote the sheaf of Kähler differentials and let $T_X=\operatorname{Hom}_{\mathcal O_X}(\Omega^1_X,\mathcal O_X)$ be the tangent sheaf, i.e., the sheaf of holomorphic derivations $\operatorname{Der}_{\mathbb C}(\mathcal O_X)$. Question (Lipman–Zariski conjecture for normal surface singularities): if $T_{X,0}$ is a free $\mathcal O_{X,0}$-module, must $(X,0)$ be smooth, i.e., $\mathcal O_{X,0}\cong\mathbb C\{z_1,z_2\}$? By the reductions of Lipman (freeness implies normality), Becker (a counterexample may be assumed to have isolated singularities), and Flenner (singular locus of codimension at least $3$ is settled), an affirmative answer for all normal surface singularities is equivalent to the full Lipman–Zariski conjecture: every complex algebraic variety, or complex space, with locally free tangent sheaf is smooth. A negative answer would consist of an explicit singular normal surface germ with free tangent sheaf. Derived consequence, as recorded in the source: an affirmative answer would remove the dimensional restriction in Theorem A of de Fernex, extending to $n=2$ the statement that for every strongly pseudoconvex compact Calabi–Yau CR manifold $M$ of dimension $2n-1$ contained in the boundary of a strongly pseudoconvex bounded domain of some $\mathbb C^N$, the Stein filling $V$ has smooth normalization if and only if the invariant $g^{(\wedge^n 1)}(M)$ vanishes; the case $n=2$ concerns Calabi–Yau CR 3-manifolds.

References

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