The ordinary-double-point volume gap

The explanation

Stand at a point of a flat plane and draw a small disc around it. Now stand at the tip of a double cone, like two ice-cream cones joined at their points. Near the tip, the shape is squeezed. Mathematicians have a number for how roomy a shape is near one point. It is called the normalized volume. A point of a smooth n-dimensional space scores n to the power n. A flat plane is 2-dimensional, so its points score 2² = 4.

The double cone is the simplest singular point, meaning a point where the shape is not smooth. In dimension 2, it is the set of solutions of x² + y² = z². It scores 2, half of the plane's 4. One way to see this: the cone is the plane folded onto itself by the map that sends each point to its opposite. Folding by a group of two halves the room near the tip. In dimension n, the same kind of point is called an ordinary double point. It is the zero set of a sum of n + 1 squares. Its score is 2 times (n − 1) to the power n. That gives 16 in dimension 3, against 27 for a smooth point. In dimension 4 it gives 162, against 256.

The score is built from valuations. A valuation is a rule that measures how fast functions vanish as they approach the point. For each rule, one multiplies two quantities. One is a weight called the log discrepancy, raised to the power n. The other counts how many independent functions vanish slowly under that rule. The normalized volume is the smallest such product over all rules. Chi Li introduced it to measure the size of a klt singularitySource 1. Klt is short for Kawamata log terminal. It names a broad class of mild singular points, the class where this score is defined.

The top score was known. Liu and Xu showed that n to the power n is the largest value, reached exactly at smooth pointsSource 2. The question was the runner-up. The gap conjecture says every singular klt point scores at most 2(n − 1)^n. It also says only the ordinary double point reaches that valueSource 3. An ordinary double point is the germ of a nondegenerate quadratic formSource 4. A germ is the shape of a space close to one point. Spotti and Sun proposed the conjecture in 2017, from the study of limits of Kähler–Einstein metricsSource 5. These are especially balanced ways to measure distance on a shape. The ratio of the two scores grows with the dimension, from 0.5 in dimension 2 toward 2/e, about 0.736. So the gap narrows but never closes.

Some cases were already settled. In dimension 2, the folding picture with finite groups gives the answer. Liu and Xu proved the threefold case for every klt pointSource 6. Liu proved it in all dimensions for local complete intersections, points cut out by the right number of equationsSource 7. Moraga and Süß treated toric points, built from simple monomial equations. In general, though, a singular point can need many equations and have no special symmetry.

This family of two papers claims the full conjecture. The main paper claims the bound in every dimension, for klt points with zero boundarySource 8. It also claims equality only at an analytic ordinary double pointSource 9. The singular point need not be isolated from other singular pointsSource 10. The companion paper claims the case of dimension 4, where the bound is 162Source 11. The main paper starts from dimensions 2, 3 and 4, and then climbs one dimension at a timeSource 12.

The obstacle is that the hypothesis is only a number. The conjecture covers every singular klt point, not just those defined by one equation. So the proof must get structure out of a large score aloneSource 13. The paper first replaces the point by a simpler cone with the same score. The large score then forces that cone to be smooth away from its tip, with an especially simple canonical form. These facts come out of the argument, not out of extra assumptionsSource 14.

The local result also limits global shapes. A Fano variety is a shape that curves positively, like a sphere, and has a volume number of its own. The gap gives restrictions on K-semistable Fano varieties, a balanced class of such shapesSource 15. The paper states a smaller volume bound for singular ones. It does not claim that this bound is the best possibleSource 16.

How sure is this? No Lean formalization is listed for this result. Its proof is a long written argument that leans on many earlier theorems. No specialist review has been published.

What it doesn't show. This does not give the third-largest value, or classify points with scores just below the gap.