Shokurov's conjecture on the maximal minimal log discrepancy of singularities

Shokurov's conjecture predicts that the maximal possible minimal log discrepancy of a singularity is attained only at smooth points: if a normal, numerically Q-Gorenstein complex germ $(A, 0)$ of complex dimension $n$ has minimal discrepancy $\mathrm{md}(A, 0) = n - 1$ (equivalently, minimal log discrepancy equal to $n$), then $A$ is smooth at $0$. Formulated in Shokurov's work on the geometry of log flips and highlighted by McLean's theorem identifying the minimal discrepancy of an isolated singularity with a contact-dynamical invariant of its link (half the minimal mean Conley-Zehnder index of Reeb orbits), the conjecture is known only in dimension at most three, for toric germs, and for special classes such as germs whose normalized blow-up has a generically reduced exceptional component. The general case remains open.

Background

A complex algebraic or analytic germ $(A, 0)$ of complex dimension $n$ is a small neighborhood of a point $0$ in a complex variety, considered up to biholomorphism; it is normal if its singular locus has codimension at least two and its local rings are integrally closed. Birational geometry measures how singular such a germ is by discrepancies: if $f : Y \to A$ is a resolution and $K_Y = f^*K_A + \sum a_E E$ expresses the canonical divisor of $Y$ in terms of the pullback of the canonical divisor $K_A$ (this needs $K_A$ Q-Cartier, i.e. some multiple of it a Cartier divisor; the minimal adequate substitute, used in the conjecture below, is the numerically Q-Gorenstein condition of Boucksom-de Fernex-Favre-Urbinati, meaning $K_A$ is Q-Cartier up to numerical equivalence), the coefficients $a_E$ attached to the exceptional divisors $E$ are the discrepancies. The minimal discrepancy $\mathrm{md}(A, 0)$ is the infimum of the $a_E$ over all prime divisors $E$ over $A$ whose center is the closed point $0$ (that is, divisors $E$ on some proper birational model $f : Y \to A$ with $f(E) = 0$); the minimal log discrepancy is $\mathrm{mld}(A, 0) = \mathrm{md}(A, 0) + 1$. A smooth point has $\mathrm{md} = n - 1$, the largest value expected to occur; the classes of canonical ($\mathrm{md} \ge 0$) and terminal ($\mathrm{md} > 0$) singularities form the basic hierarchy of the minimal model program, and these numbers control termination of flips (the property that the birational surgery steps of the minimal model program cannot continue indefinitely), inversion of adjunction, and boundedness of singularities.

Shokurov's conjecture, which appeared in his 'Letters of a Bi-Rationalist IV' on the geometry of log flips, asserts that the maximal value characterizes smoothness: a normal numerically Q-Gorenstein germ with $\mathrm{md}(A, 0) = n - 1$ is smooth at $0$. In the original, stronger pair form, for a boundary $B$ (an effective Q-divisor with coefficients in $[0,1]$) one expects $\mathrm{mld}(X, B, P) \le \operatorname{codim} P$ for every point $P$, with equality only when $P$ is a nonsingular point of $X$ and $B = 0$ near $P$. The conjecture is known in complex dimension at most three, through the classification of terminal threefold singularities and the computation of their minimal discrepancies; Shokurov also noted the toric case, and de Fernex-Tu later proved it for germs whose normalized blow-up at the point has an exceptional divisor with a generically reduced irreducible component. In general it remains open and is regarded as a core problem of singularity theory, with consequences for termination of flips.

The conjecture also has an unexpected symplectic-dynamical face, discovered by McLean. The link $L_A$ of an isolated singularity is the intersection of $A$ with a small sphere around $0$; it is a smooth manifold of real dimension $2n - 1$ carrying a canonical contact structure $\xi_A$ (a maximally non-integrable hyperplane field). A contact manifold contactomorphic to some link $L_A$ is said to be Milnor fillable by $A$. The prototype result is Mumford's theorem that a normal surface singularity whose link is a sphere is smooth. The Reeb vector field of a contact form (the unique vector field transverse to the contact hyperplanes whose flow preserves the contact form) has closed trajectories called Reeb orbits; each orbit carries a Conley-Zehnder index measuring the winding of the linearized flow, whose averaged version is the mean index, and $\mathrm{hmi}(L_A, \xi_A)$ denotes half the smallest mean index over Reeb orbits. McLean proved that for a normal isolated numerically Q-Gorenstein singularity with $H^1(L_A; \mathbb{Q}) = 0$, one has $\mathrm{hmi}(L_A, \xi_A) = 2\,\mathrm{md}(A, 0)$ when $\mathrm{md}(A, 0) \ge 0$ (and $\mathrm{hmi} < 0$ when $\mathrm{md} < 0$), identifying the minimal discrepancy as a purely contact-dynamical invariant of the link. Since the standard contact sphere corresponds to a smooth point, the conjecture would imply, via this identity, that the standard contact sphere of every odd dimension has a unique Milnor filling up to normalization — a statement McLean established unconditionally only for the standard contact 5-sphere (the complex-dimension-three case, its Corollary 1.2), precisely because only there is the conjecture known. The conjecture thus sits at the interface between the birational geometry of singularities and contact topology: it says exactly how much of a singularity the contact geometry of its link remembers.

Problem Statement

Let $(A, 0)$ be a normal, numerically Q-Gorenstein germ of a complex analytic space or complex algebraic variety, of complex dimension $n$, with minimal discrepancy $\mathrm{md}(A, 0)$ (equivalently, minimal log discrepancy $\mathrm{mld}(A, 0) = \mathrm{md}(A, 0) + 1$). Shokurov's conjecture, stated as Conjecture 1.3 of McLean's paper (Reference 2), asserts that if $\mathrm{md}(A, 0) = n - 1$ (equivalently $\mathrm{mld}(A, 0) = n$), then $A$ is smooth at $0$:

$$\mathrm{md}(A, 0) = n - 1 \;\Longrightarrow\; A \text{ is smooth at } 0.$$

Prove or refute this statement in full generality: no isolation, hypersurface, locally complete intersection, Gorenstein, or dimension restriction may be assumed. Shokurov's original formulation ('Letters of a Bi-Rationalist IV', Conjecture 2) is the strictly stronger pair statement: for every point $P$ of a pair $(X, B)$ with $B$ a boundary, $\mathrm{mld}(X, B, P) \le \operatorname{codim} P$, and equality holds only when $P$ is a nonsingular point of $X$ with $B = 0$ near $P$ (with $\mathrm{mld}(X, B, P)$ defined analogously, by discrepancies of the pair along prime divisors whose center is the closure of $P$). The goal of this problem is the closed-point, $B = 0$ case written above (Conjecture 1.3 of McLean's paper); the full pair form is retained as context, not as the completion requirement. Via McLean's identity $\mathrm{hmi}(L_A, \xi_A) = 2\,\mathrm{md}(A, 0)$ for isolated singularities with $H^1(L_A; \mathbb{Q}) = 0$, a proof in all dimensions would also imply that the standard contact sphere of every odd dimension greater than one has a unique Milnor filling up to normalization; this is a one-way consequence (Corollary 1.4 of McLean's paper), and no converse implication is known. An answer must resolve the discrepancy statement itself, not merely one of its known special cases or this contact-topological consequence.

References

1. V. V. Shokurov, Letters of a Bi-Rationalist IV: Geometry of log flips, arXiv:math/0206004, https://arxiv.org/abs/math/0206004 (published in Algebraic Geometry, De Gruyter, 2002, pp. 313-328, DOI 10.1515/9783110198072.313, https://doi.org/10.1515/9783110198072.313)
2. Mark McLean, Reeb orbits and the minimal discrepancy of an isolated singularity, Inventiones mathematicae 204 (2016), 505-594, DOI 10.1007/s00222-015-0620-x, https://doi.org/10.1007/s00222-015-0620-x (preprint arXiv:1404.1857, https://arxiv.org/abs/1404.1857)
3. Sébastien Boucksom, Tommaso de Fernex, Charles Favre, Stefano Urbinati, Valuation spaces and multiplier ideals on singular varieties, arXiv:1307.0227, https://arxiv.org/abs/1307.0227 (published as a chapter in Recent Advances in Algebraic Geometry, London Mathematical Society Lecture Note Series 417, Cambridge University Press, 2015, DOI 10.1017/CBO9781107416000.004, https://doi.org/10.1017/CBO9781107416000.004)
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