McLean's question: does the contact structure of the link characterize smoothness for all normal isolated singularities?

For a normal isolated complex singularity, the link - the intersection with a small sphere - carries a natural contact structure induced by the ambient complex structure. Mumford proved that a normal surface singularity is smooth exactly when its link is a sphere, and McLean proved that in complex dimension three smoothness is characterized by the link being contactomorphic to the standard contact sphere; de Fernex and Tu extended this to all dimensions for singularities whose normalized blow-up exceptional divisor has a generically reduced irreducible component, and posed two further conjectures (a CR-deformation criterion and a contact-form degeneration criterion). The CR-deformation part of their program has since been completed by de Fernex in corrected form (a normal isolated singularity is smooth iff the semirigid CR deformation equivalence class of its link is that of the standard CR sphere), while the central contact-theoretic question remains open in general: it is unknown, in complex dimension at least four and without the reduced-component hypothesis, whether a singular normal isolated singularity can have a link contactomorphic to the standard contact sphere. By McLean's reduction the equality case of Shokurov's minimal-log-discrepancy conjecture in dimension at least four would imply an affirmative answer, and an affirmative answer would settle that equality case for the class of singularities whose link is contactomorphic to the standard contact sphere; the equality case is open even for isolated Fano cone singularities (the vertices of affine cones over Fano varieties, i.e. singularities given by the spectrum of a finitely generated positively graded algebra whose projective base has ample anti-canonical class) as of 2025. The problem posed is to settle this characterization question in full generality, by proof or by counterexample.

Background

Let $X\subset\mathbb{C}^N$ be an $n$-dimensional complex analytic variety with an isolated singularity at the origin $0$; "normal" means the local rings are integrally closed, and "isolated" means $X$ is a complex manifold at every point of a punctured neighborhood of $0$. For small $\epsilon>0$ the link $L_{X,\epsilon}=X\cap S^{2N-1}_\epsilon$ is a compact real manifold of dimension $2n-1$ whose diffeomorphism type does not depend on $\epsilon$; it is the boundary of a small tubular neighborhood of the singularity. A classical theorem of Mumford states that a normal isolated surface singularity is smooth if and only if its link is diffeomorphic to a sphere. In higher dimensions the topological statement fails: Brieskorn produced singularities whose links are nevertheless spheres, so the bare topology of the link cannot detect smoothness.

The link, however, carries finer structure. Since $X$ is a complex manifold along the link, the complex structure $J$ on the regular part of $X$ restricts to a hyperplane field $\xi=T(L_X)\cap J(T(L_X))$ on the link, where $L_X$ is any link of $X$ at $0$; this field is maximally non-integrable, making the link a contact manifold (a contact structure is a hyperplane field on an odd-dimensional manifold that is locally the kernel of a 1-form $\alpha$ with $\alpha\wedge(d\alpha)^{n-1}$ nowhere zero). A contactomorphism is a diffeomorphism pulling one contact structure back to the other. When $X$ is smooth at $0$, the link is contactomorphic to the standard contact sphere $(S^{2n-1},\xi_{\mathrm{st}})$, the link of $\mathbb{C}^n$ at the origin; and the contactomorphism class of the link is an analytic invariant of the singularity, as recalled by de Fernex and Tu. This raises the question, formulated by McLean, of whether the contact structure of the link characterizes smooth points among all normal isolated singularities.

McLean answered his question affirmatively in complex dimension three. His argument shows more: if the link of a normal isolated singularity is contactomorphic to the standard contact sphere, then $X$ is numerically $\mathbb{Q}$-Gorenstein - meaning that a positive multiple of the canonical class is $\mathbb{Q}$-Cartier up to numerical equivalence, where $\mathbb{Q}$-Cartier means some multiple of the canonical divisor is defined by a local equation - and then, through a dynamical analysis of the Reeb vector fields (for a contact form $\alpha$ with $\xi=\ker\alpha$, the Reeb field is the unique vector field $R$ with $\alpha(R)=1$ and $\iota_R d\alpha=0$; its closed orbits are the Reeb orbits) and the Conley–Zehnder index (a Floer-theoretic index attached to each closed Reeb orbit), that $X$ is actually $\mathbb{Q}$-Gorenstein with minimal log discrepancy equal to $n$. The minimal log discrepancy $\operatorname{mld}_x(X)$ of a normal $\mathbb{Q}$-Gorenstein variety at a point $x$ is the infimum over all prime divisors $E$ appearing on birational models $f:Y\to X$ above $x$ of the log discrepancies $a_E=1+k_E$, where $K_Y=f^*K_X+\sum k_EE$; a smooth point has $\operatorname{mld}=\dim X$, and small values measure how bad the singularity is. In dimension three, the results of Shokurov (dimension two, unpublished) and Ambro (dimension three) show that an isolated $\mathbb{Q}$-Gorenstein singularity with $\operatorname{mld}=\dim X$ must be smooth - this is the low-dimensional case of a conjecture of Shokurov, which in general asserts that an isolated $\mathbb{Q}$-Gorenstein klt (Kawamata log terminal, meaning that on a log resolution all discrepancies exceed $-1$) singularity with $\operatorname{mld}_x(X)=\dim X$ is smooth at $x$, a statement of interest in the minimal model program because of its connection to the conjecture on termination of flips. In higher dimensions this step is open, which is precisely where the general contact question stalls.

de Fernex and Tu pushed the birational side forward. Their Theorem 1.5 proves, for a normal $\mathbb{Q}$-Gorenstein variety $X$ of dimension $n$ at a point $x$ such that the exceptional divisor of the normalized blow-up of $X$ at $x$ (the blow-up of the origin in the normalization of $X$) has a generically reduced irreducible component - equivalently, the tangent cone of $X$ at $x$ is reduced at the generic point of one of its components - that $\operatorname{mld}_x(X)\le n$ with equality only at smooth points; the proof compares the normalized blow-up with the Nash blow-up (the blow-up parametrizing limits of tangent spaces, which linearizes the tangent cone). Combined with McLean's work this yields their Theorem 1.4: under the same hypothesis on the normalized blow-up, $X$ is smooth at $0$ if and only if the link is contactomorphic to the standard contact sphere, in every dimension. They state explicitly that they do not know how to remove the hypothesis on the exceptional divisor: when no component is generically reduced, the Nash blow-up mechanism has nothing to grip, and the case of singularities with nowhere-reduced normalized blow-up exceptional divisor (necessarily in complex dimension at least four, since dimensions at most three are settled) remains out of reach.

The same paper develops a second, CR-geometric thread. The link carries a Cauchy–Riemann (CR) structure $T^{1,0}$, the complex hyperplane subbundle of the complexified tangent bundle cut out by the ambient complex structure of $X$; for a normal isolated singularity the link is strongly pseudoconvex (its CR geometry is the boundary type amenable to filling by Stein spaces (Stein spaces are the complex-analytic analogue of affine varieties, the natural solution domains in several complex variables), the complex analogue of solution domains for the Dirichlet problem), and one can study the Kohn–Rossi cohomology groups, the cohomology of the tangential Cauchy–Riemann complex on the CR manifold. de Fernex and Tu introduced an equivalence relation on CR manifolds, cohomological CR deformation equivalence - deformation families along which certain Kohn–Rossi cohomological dimensions are constant, chained together - and conjectured (their Conjecture 5.14) that a normal isolated singularity is smooth if and only if this equivalence class of its link is the class of the standard CR sphere, relating the question to the complex Plateau problem of characterizing when a strongly pseudoconvex CR manifold bounds a smooth Stein filling. They also proposed a third criterion (their Conjecture 6.2) using the canonical contact form

$\theta=\frac{i}{|z|^2}\sum_i (z_i\, d\bar z_i-\bar z_i\, dz_i)$

restricted to the links $L_{X,\epsilon}$: if $L_{X,\epsilon}$ is diffeomorphic to a sphere and the contact forms $\theta|_{L_{X,\epsilon}}$ specialize to the standard contact form as $\epsilon\to 0$, then $X$ should be smooth. They note this latter criterion, while expected to be more accessible, would not by itself give a link-theoretic characterization, since the degeneration condition is not intrinsic to a single link.

Problem Statement

Let $X\subset\mathbb{C}^N$ be an $n$-dimensional complex analytic variety with a normal isolated singularity at $0$, and let $L_X$ be its link equipped with the contact structure $\xi=T(L_X)\cap J(T(L_X))$ induced by the complex structure of $X$. The problem is McLean's question in full generality: is $X$ smooth at $0$ if and only if $(L_X,\xi)$ is contactomorphic to the standard contact sphere $(S^{2n-1},\xi_{\mathrm{st}})$?

The reverse implication (smooth implies contactomorphic to the standard sphere) is standard, and the forward implication is known for $n\le 3$ (McLean) and, in all dimensions, for singularities whose normalized blow-up exceptional divisor has a generically reduced irreducible component (de Fernex and Tu, Theorem 1.4). The open case is $n\ge 4$ without that hypothesis, and the required resolution is one of the following.

1. A proof that every normal isolated singularity of every dimension whose link is contactomorphic to the standard contact sphere is smooth at the singular point.

2. A counterexample: a singular normal isolated singularity of some dimension (necessarily at least $4$) whose link is contactomorphic to the standard contact sphere.

A companion criterion proposed in the source paper, Conjecture 6.2, asks whether the degeneration of the canonical contact forms $\theta|_{L_{X,\epsilon}}$ to the standard contact form as $\epsilon\to 0$, together with $L_{X,\epsilon}\simeq S^{2n-1}$, already forces smoothness; this companion question neither implies nor is implied by the main characterization question.

References

1. Tommaso de Fernex and Yu-Chao Tu, Towards a link theoretic characterization of smoothness, Mathematical Research Letters 24 (2017), no. 5, 1239-1262, DOI 10.4310/mrl.2017.v24.n5.a1 (arXiv:1608.08510), https://arxiv.org/abs/1608.08510
2. Tommaso de Fernex, Smooth solutions to the complex Plateau problem, Journal of Differential Geometry 118 (2021), no. 2, 297-312, DOI 10.4310/jdg/1622743141 (arXiv:1801.00503), https://arxiv.org/abs/1801.00503
3. Mark McLean, Reeb orbits and the minimal discrepancy of an isolated singularity, Inventiones mathematicae 204 (2016), no. 2, 505-594, DOI 10.1007/s00222-015-0620-x (arXiv:1404.1857), https://arxiv.org/abs/1404.1857
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