Which closed contact manifolds are links of isolated complex singularities, and how much of the singularity does the contact link determine?

Every isolated complex singularity has a link, a closed strongly pseudoconvex Cauchy–Riemann (CR) manifold whose underlying hyperplane field is a canonical contact structure, and this makes singularity theory and contact topology confront each other: one may ask exactly which closed contact manifolds arise this way, and how much of the analytic type of the singularity is encoded in the contact structure of the link. Both directions of this dictionary are open in general. On the characterization side, the only known tools are obstructions: McLean proved that the growth rate of wrapped Floer cohomology of any Liouville filling whose boundary is such a link is bounded by the complex dimension, forcing for instance the base of a unit cotangent bundle of a simply connected closed manifold to be rationally elliptic if the bundle is such a link; no nontrivial sufficient conditions are known. On the rigidity side, Scherk proved that the full CR structure of the link determines the singularity up to analytic equivalence, while at the contact level only coarse analytic invariants (minimal discrepancy, multiplicity, and log canonical threshold — standard numerical measures of how severe a singularity is) are known to be link invariants, and the statement that a normal isolated singularity with link contactomorphic to the standard contact sphere must be smooth — equivalently, uniqueness of the Milnor filling of the standard contact sphere up to normalization — is proven in complex dimension at most 3 and for a large class in all dimensions, but in full generality is conditional on Shokurov's conjecture that maximal minimal log discrepancy characterizes smooth points, which remains open.

Background

Let $X\subset\mathbb{C}^N$ be a complex analytic variety with an isolated singularity at the origin, of complex dimension $n$. Its link $L_X$ is the intersection of $X$ with a small sphere $S^{2N-1}_\varepsilon$ centered at the origin; for $\varepsilon$ small this is a closed smooth manifold of real dimension $2n-1$, well defined up to diffeomorphism, and it inherits from the complex structure a canonical contact structure $\xi_X=TL_X\cap J_0(TL_X)$, where $J_0$ is the multiplication by $i$ on the tangent bundle. Here a contact structure on a $(2n-1)$-manifold is a hyperplane field locally given as the kernel of a $1$-form $\alpha$ with $\alpha\wedge(d\alpha)^{n-1}$ vanishing nowhere; a contactomorphism is a diffeomorphism pulling one contact structure back to the other. The link also carries a strongly pseudoconvex Cauchy–Riemann (CR) structure (a complex hyperplane field on the complexified tangent bundle defined by the tangential Cauchy–Riemann equations along $L_X$, locally the boundary structure of a strictly pseudoconvex domain), of which the contact structure is the underlying real part. A closed contact manifold contactomorphic to the link of some isolated complex singularity is called Milnor fillable, and the singularity (or the Milnor fiber of a smoothing, i.e. a nearby smooth fiber of a deformation) is called a Milnor filling of it.

Two classical facts frame the problem. Mumford proved in 1961 that a normal surface singularity is smooth if and only if its link is diffeomorphic to $S^3$; in higher dimensions this fails, since links of isolated singularities can even be exotic spheres. Scherk proved in 1986 a rigidity theorem in the CR category: if the links of two normal isolated singularities, with their induced strongly pseudoconvex CR structures, are CR isomorphic, then the two germs are analytically equivalent. So the full CR structure of the link determines the singularity — but the CR structure depends on the embedding and on the radius of the cutting sphere, and is therefore not an invariant of the singularity, whereas the contactomorphism class of $(L_X,\xi_X)$ is. This raises the rigidity question of how much analytic information survives at the contact level: the central test case is whether smoothness among normal isolated singularities is characterized by the link being contactomorphic to the standard contact sphere $(S^{2n-1},\xi_{\mathrm{std}})$, the sphere in $\mathbb{C}^n$ with its standard contact structure of complex tangencies $\xi_{\mathrm{std}}=TS^{2n-1}\cap J_0(TS^{2n-1})$, defined as for links above.

On the characterization side, the main tool so far is Floer-theoretic. A Liouville domain is a compact exact symplectic manifold $(M,\omega=d\lambda)$, of real dimension $2n$, whose Liouville vector field (defined by $\iota_V\omega=\lambda$) points strictly outward along the boundary; links of isolated hypersurface singularities bound such domains. Wrapped Floer cohomology $HW(L_1,L_2)$ is a version of Lagrangian Floer cohomology, defined for suitable exact (admissible) Lagrangian submanifolds of the completion of $M$, that counts Hamiltonian-perturbed intersection points with orbits escaping to infinity. Its growth rate $\Gamma(L_1,L_2;\mathbb{K})$ is the polynomial rate at which the ranks of its action-filtered pieces grow. McLean proved that if $\partial M$ is contactomorphic to the link of an isolated complex singularity of complex dimension $n$, then $\Gamma(L_1,L_2;\mathbb{K})\le n$ for all transverse admissible Lagrangians and all coefficient fields. This is a necessary condition for a contact manifold to be a link; combined with computations of growth rates in unit cotangent bundles, it implies for example that if the unit cotangent bundle $ST^*Q$ of a simply connected closed manifold $Q$ is contactomorphic to such a link, then $Q$ must be rationally elliptic (its rational homotopy must be finite-dimensional), yielding strong bounds on the Betti numbers of $Q$.

The rigidity side runs through birational geometry. For a normal $\mathbb{Q}$-Gorenstein singularity (one for which some multiple of the canonical divisor is Cartier) of dimension $n$, the minimal log discrepancy $\mathrm{mld}_0(X)$ is the infimum, over all prime divisors obtained in blowups of ideals supported at the singular point, of the discrepancy coefficient measuring how the canonical divisor changes under such a blowup, plus one; it quantifies how mild the singularity is, with $\mathrm{mld}_0(X)=n$ at smooth points, the maximal value, and $\mathrm{mld}_0(X)>1$ defining terminal singularities, the mildest class in the minimal model program (the birational classification program for complex varieties). McLean defined a purely contact-theoretic invariant $\mathrm{hmi}(C,\xi)$, the highest minimal symplectic field theory (SFT) index, as the supremum over contact forms of the infimum over closed Reeb orbits (orbits of the vector field dual to the contact form) of a corrected Conley–Zehnder index (a rational-valued measure of the twisting of the linearized Reeb flow around the orbit), and proved that for a normal isolated numerically $\mathbb{Q}$-Gorenstein singularity (one for which some multiple of the canonical class is numerically trivial) whose link satisfies $H^1(L_X;\mathbb{Q})=0$, one has $\mathrm{hmi}(L_X,\xi_X)=2\,\mathrm{md}(X,0)$ whenever the minimal discrepancy $\mathrm{md}=\mathrm{mld}-1$ is non-negative. Thus the minimal discrepancy is computable from the contact structure of the link. Consequently, if the link of a normal isolated numerically $\mathbb{Q}$-Gorenstein singularity with $H^1(L_X;\mathbb{Q})=0$ is contactomorphic to the standard contact sphere, then $\mathrm{md}=n-1$; the conclusion that the singularity is smooth then follows from Shokurov's conjecture that a numerically $\mathbb{Q}$-Gorenstein singularity with $\mathrm{md}=n-1$ (equivalently $\mathrm{mld}=n$) is smooth, a special case of his conjecture on the maximum of minimal log discrepancy in fixed dimension. This conjecture is known in dimension at most 3 through the classification of threefold singularities, and de Fernex and Tu proved it when the exceptional divisor of the normalized blow-up at the point has a generically reduced irreducible component, but it remains open in general.

Problem Statement

Determine the exact correspondence between the contact geometry of links and the analytic theory of isolated complex singularities, in full generality (any isolated complex singularity, any dimension), through two linked questions:

(1) Characterize the closed contact manifolds that are contactomorphic to links of isolated complex singularities. Provide necessary and sufficient conditions; the general characterization is the objective of (1), while deciding the question for natural classes such as unit cotangent bundles $ST^*Q$ of simply connected closed manifolds, where currently only the necessary condition of rational ellipticity (with its Betti number bounds) is known, is a concrete test case of it rather than a substitute for the general characterization.

(2) Determine the extent to which the contact structure of the link determines the analytic type of the singularity, given that the full CR structure already determines the germ up to analytic equivalence — namely, which analytic invariants of the germ, beyond the coarse ones already known to be link invariants, are determined by the contactomorphism class, and in particular whether smoothness is. The central open case: is it true in every dimension that a normal isolated singularity whose link is contactomorphic to the standard contact sphere $(S^{2n-1},\xi_{\mathrm{std}})$ is smooth at the point — equivalently, that the standard contact sphere has a unique Milnor filling up to normalization in all dimensions? What (2) calls for is an unconditional determination — a definite answer to this case, or a determination of which further analytic data the contact structure retains — and not only the reduction to Shokurov's conjecture. This is proven in complex dimension at most 3 and for singularities whose normalized blow-up has a generically reduced exceptional component; in general it is conditional on Shokurov's conjecture that maximal minimal log discrepancy ($\mathrm{md}=n-1$, equivalently $\mathrm{mld}=n$) characterizes smooth points, which itself remains open.

The two questions share one objective: a two-way dictionary in which the class of link contact manifolds among closed contact manifolds is characterized from the contact-geometric side, and the analytic information retained by the link's contact structure is identified from the singularity-theoretic side.

References

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