Normalized volume gap in dimension four

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Normalized volume gap in dimension four

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Take a single number, normalized volume at least 162 at a singular klt fourfold point, and follow how it is turned into an equation that pins the point down as an ordinary double point.

The numbers to keep in mind

Normalized volume is the infimum of A(v)⁴ · vol(v) over valuations centered at the point. Smooth points reach the maximum n^n = 256 [0:17].
The Spotti–Sun conjecture says the runner-up is the ordinary double point, at 2(n−1)^n. In dimension four that is 2·3⁴ = 162, and the theorem proves that bound, with equality only at the ODP.
The difficulty: a general klt point has no equations to compute with. So the proof turns the numerical hypothesis into a local-ring constraint: embedding dimension at most five [1:07].

The chain, link by link

Cone [1:33]. Stable degeneration gives a graded cone with the same volume, and its embedding dimension bounds the germ's. Non-isolated points are capped at 4096/27 ≈ 151.7. A degree-d index-one cover would push the volume past 256, so the cone is isolated and Gorenstein.
Stabilizer bound [2:19]. Counting jets on a transverse slice shows that a large stabilizer would beat the minimizing valuation. So m < r/(5/2), and every primitive divisor over the vertex has A ≥ 3 (an integer, because the cone is Gorenstein).
Hilbert series [2:47]. With the canonical weight at 5/2, Stanley duality turns the Hilbert coefficients into a signed measure. Its nonzero frequencies lie outside (−1, 1), and a band-limited test acts as a low-pass filter: Poisson summation leaves only a polynomial integral. That integral forces a function of weight in (0, 5/3).
Threshold [3:42]. If lct ≤ 3/2, a boundary built from those functions has weight below 3/2 · 5/3 = 5/2, which leads to a contradiction on a minimal lc center. So lct > 3/2.
Terminal section [4:15]. A general hyperplane section has discrepancy A − 1 − b along each divisor. For b = 1, A ≥ 3 is enough. For b = 2 and A = 3 the value is 0, so the threshold is needed: it forces A ≥ 4. Both bounds are needed.
Hypersurface [5:09]. By Reid, the section is cDV, so edim ≤ 4, and the cone has edim ≤ 5. That makes the germ one equation in five-space, and Liu's lci theorem gives ≤ 162, with equality only at the ODP.

When you face a volume-gap question, don't hunt for a better valuation. Squeeze structure out of the number until an equation appears [5:36].

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