b84c503
THE ORDINARY DOUBLE POINT GAP IN EVERY DIMENSION
A plain-language companion to the proof
(original: The-ordinary-double-point-gap-in-every-dimension)
HOW TO USE THIS NOTE
The original is a research paper. Every line means something to a
specialist, but the paper is dense and the thread of the argument is
easy to lose. This note reorganises it for a reader with high-school
mathematics who wants to know what is proved and why. The mathematics
is the paper's; what is added is explanation, signposts, and worked
checks. Anything marked CHECK is arithmetic or calculus you can do
yourself with a pencil.
The honesty rules of the note:
- a statement is either proved here, proved in another paper, or
assumed; each is labelled.
- when a step is technical, the note says what the step accomplishes
and leaves the epsilons to the original.
- the base cases n = 2, 3, 4 come from earlier work; the new content
is the induction step n >= 5.
==== 1. THE SHORT VERSION ==========================================
Every singular klt complex n-fold germ has normalised volume at most
2 (n-1)^n, with equality only at an ordinary double point. It holds
for every n >= 2. Smooth points have value n^n and are the unique
maximum; the ordinary double point is the unique second-highest value.
That is the whole theorem.
==== 2. SINGULARITIES AND THE ORDINARY DOUBLE POINT ===============
A complex variety of dimension n is a shape that near a typical point
looks like a piece of C^n. The point is smooth if that is literally so
in some neighbourhood, and singular otherwise.
Mildness of a singularity comes in degrees. The mildest condition used
here is "klt" (Kawamata log terminal). It is technical; for this note
two facts about klt suffice: (i) klt points are exactly the ones on
which normalised volume is well behaved (a minimiser exists, volume is
lower-semicontinuous in families, quotient and finite-cover
constructions preserve klt), and (ii) many natural operations take klt
to klt. The paper works with "zero boundary": no log pair is involved,
just a variety X and its canonical divisor K_X.
The simplest singular klt germ is the ordinary double point (ODP). It
is the solution set, near the origin, of a nondegenerate quadratic
z_0^2 + z_1^2 + ... + z_n^2 = 0 in C^{n+1}.
"Nondegenerate" means the quadratic form has full rank. The holomorphic
Morse lemma then says any hypersurface whose defining equation has a
nondegenerate quadratic part at the point is analytically the standard
one above. The ODP is a cone over a smooth quadric Q^{n-1} in P^n,
with a single singular point at the tip.
For n = 2, the ODP is the A_1 surface singularity, the quotient of C^2
by (+/- the identity). For n = 3 it is the cone over a smooth quadric
surface in P^3. In every dimension it is the simplest isolated singular
hypersurface.
The theorem is NOT about isolated singularities, hypersurfaces, or
quotients. It is about every singular klt germ. That generality is
what makes the result interesting and hard.
==== 3. NORMALISED VOLUME =========================================
The idea is to measure how many holomorphic functions at x vanish to
"order at least t", as t grows.
Fix a normal complex germ (x, X) of dimension n, with local ring
O = O_{X,x} and maximal ideal m = m_x.
- For a real t >= 0, the "valuation ideals" of a ruler v are
a_t(v) = { f in O : v(f) >= t },
an m-primary ideal.
- The "length" length(O / a_t(v)) is its complex dimension. For a
valuation v the count grows like a constant times t^n:
length( O / a_t(v) ) = vol(v) * t^n / n! + (lower order in t).
Define vol(v) to be that leading coefficient:
vol(v) = lim_{t -> infinity} n! * length( O / a_t(v) ) / t^n.
- The "log discrepancy" A_X(v) is a correction that sees how the
divisor of v meets the canonical divisor K_X. On a smooth variety
with v the ordinary vanishing order, A_X(v) = n.
- Define
nvol(x, X) = inf over v centered at x of A_X(v)^n * vol(v).
The infimum is finite and positive. It is invariant under analytic
isomorphism of germs, so it depends only on the completed local
ring.
Two anchor computations:
(smooth) x smooth: the ordinary order valuation has A = n and
vol = 1, so nvol = n^n. This is the maximum.
(ODP) x an ODP: blow up the tip. The exceptional divisor is a
smooth quadric Q^{n-1} of multiplicity 2, with log
discrepancy n-1 in the standard normalisation. The
valuation that orders along this exceptional divisor has
discrepancy n-1 and volume 2, so
nvol(0, ODP) = 2 (n-1)^n.
Define the "density" d(x, X) = nvol(x, X) / n^n. It lies in (0, 1].
It equals 1 exactly at smooth points. Every singular point has d < 1,
and the theorem says that for a klt point the deficiency is at least a
fixed amount:
d(x, X) <= p_n, p_n = 2 ((n-1)/n)^n.
The gap is real. Smooth means d = 1; singular klt means d <= p_n, with
equality only at the ODP.
==== 4. THE THEOREM ===============================================
THEOREM (ordinary double point gap; Theorem 1.1 of the paper).
Let n >= 2. Let x be a singular closed point of a normal complex
algebraic n-fold X. Suppose K_X is Q-Cartier and X is klt near x, with
zero boundary. Then
nvol(x, X) <= 2 (n-1)^n.
Equality holds if and only if the analytic germ of X at x is
( { z_0^2 + ... + z_n^2 = 0 } in C^{n+1} , 0 ).
In density form: d(x, X) <= p_n = 2 ((n-1)/n)^n, with equality iff the
germ is an analytic ODP.
Numerical remarks:
- n = 2: nvol <= 2. Known (Li-Liu); equality is the A_1 surface.
- n = 3: nvol <= 16. Known (Liu-Xu).
- n = 4: nvol <= 162. Proved in a companion paper (the fourfold
theorem); this paper uses it as its base case for induction.
- n >= 5: proved here.
GLOBAL COROLLARY (Corollary 1.2 of the paper). Let V be an n-dimensional
K-semistable Q-Fano variety with n >= 2.
- If V is singular, then (-K_V)^n <= 2 ((n^2-1)/n)^n < 2 n^n.
- If V is smooth and V is not P^n, then (-K_V)^n <= 2 n^n, with
equality exactly at a smooth quadric Q^n in P^{n+1} or at
P^1 x P^{n-1}. (These coincide for n = 2, where Q^2 = P^1 x P^1.)
The singular bound is strictly smaller than the smooth one. The paper
does not claim the singular bound is optimal; it only needs it to be
strictly below 2 n^n.
==== 5. WHY THE NUMBER 2 (n-1)^n IS THE RIGHT ONE =================
Two anchor values. A smooth point has nvol = n^n, the champion. The
smallest genuinely singular shape is the ODP, and Section 3 computed
nvol(0, ODP) = 2 (n-1)^n.
So 2 (n-1)^n is achieved by an honest singularity. The gap conjecture
of Spotti and Sun says that nothing strictly between n^n and
2(n-1)^n can occur among klt germs; this paper proves it.
The ratio of the two values is exactly the density p_n:
p_n = nvol(ODP) / nvol(smooth) = 2 ((n-1)/n)^n = 2 (1 - 1/n)^n.
CHECK: p_n is increasing in n, with values
p_2 = 1/2, p_3 = 16/27, p_4 = 81/128, p_5 = 2048/3125, ...
and limit p_n -> 2/e = 0.7358... as n -> infinity. To see monotonicity,
put g(t) = t log(1 - 1/t) for t > 1. Then
g'(t) = log(1 - 1/t) + 1/(t-1) = u - log(1+u),
with u = 1/(t-1) > 0. This is positive because log(1+u) < u for u > 0.
Hence p_k = 2 exp(g(k)) is increasing in k.
So the worst singularity's deficiency 1 - p_n shrinks from 1/2 in
dimension 2 to about 0.264 in high dimensions. The gap is always
present, and it is largest in dimension 2.
==== 6. THE PROOF AT A GLANCE =====================================
The proof is an induction on n. Base cases n = 2 and n = 3 come from
known theorems, and n = 4 from the companion paper. So assume n = N+1
>= 5 and suppose the theorem is known in all dimensions below n.
The induction step runs as follows.
Step 1 (cone reduction). A minimiser of normalised volume at the
singular germ degenerates by "stable degeneration" to a K-polystable
Fano cone C. The new cone has the same normalised volume at its
vertex as the original germ, and its embedding dimension dominates
the germ's. The cone has a smooth puncture and a Gorenstein
coordinate ring. So the problem becomes: control the volume of an
isolated cone singularity built from a smooth projective base.
Step 2 (shell). Approximate the cone's Reeb vector -- its preferred
real one-parameter grading -- by a primitive integral grading.
Quotient the smooth puncture by that one-parameter group. The
quotient B is a smooth projective orbifold with an ample
"weight-one" line bundle L and -K_B = r L. The largest stabilizer
order of B is m_max.
Step 3 (case split). Compare m_max to r/N.
(a) Small stabilizers, m_max <= r/N. A minimal orbifold rational
curve on B has deformations; taking the symmetric product of
its infinitesimal translations gives a nonzero symmetric
tensor of negative weight on the cone's tangent bundle. A
Ricci-flat cone metric on the smooth puncture then bounds the
weight from below. The two combine to force every stabilizer
to be trivial, r to be N or N+1, and the quotient to be
(P^N, O(1)) or (Q^N, O(1)). Hence C is smooth affine space
or the ODP.
(b) Large stabilizers, m_max >= r/N. Choose a point of maximal
stabilizer and build the semigroup of (original degree,
leading Taylor exponent) pairs. Two independent inputs
control it:
- an interior-point theorem puts the canonical lattice
point c_0 = (r, 1, 1, ..., 1) inside the cone of the
semigroup, using two moment inequalities and an
exponential expectation bound;
- an adjoint-lifting theorem fills the semigroup exactly,
using a weighted blowup, multiplier ideals and vanishing.
Conclusion: the semigroup is the entire lattice cone, a
finitely generated saturated affine semigroup.
Step 4 (two filtrations). Filter the cone's coordinate ring first
by total Taylor order and then by a "second" order. The second Rees
algebra has a singular central fiber G with a stabilizer of order
h >= 2 at a chosen point. A horizontal slice through that point
gives a singular klt germ Y of the same dimension n. On Y a
character class accounts for only 1/h of each valuation colength.
That strict fraction is the whole source of strictness in the bound.
Step 5 (bootstrap). The slice satisfies a clean inequality
d(o, C) <= (p_n + M_n) / 2,
where M_n is the supremum of densities over ALL singular klt
n-fold germs -- an unknown free to define. If M_n were strictly
bigger than p_n, taking germs that approach M_n from above makes
the inequality fail. Hence M_n <= p_n, which is the upper bound.
Step 6 (equality). If a cone has d(o, C) = p_n, every inequality in
Steps 3(b) and 4 is an equality. In particular h = 2, the slice's
minimiser has A = N L and volume 2, and the distinguished element v
used to construct the second filtration has w(v) <= 2. An integral
rounding then gives N m_max <= r for every sufficiently far
grading, so the cone falls into the small-stabilizer case and is
an ODP.
Step 7 (back to the germ). Embedding dimension edim(X, x) <=
edim(C, o) = n + 1, so the singular germ X is a hypersurface. A
weighted hypersurface test then forces its quadratic part to be
nondegenerate. Hence the germ is an analytic ODP.
That is the arc. The next several sections unpack each step.
==== 7. STEP 1. REDUCE THE GERM TO A CONE =========================
Fix a singular klt germ (x, X) with d(x, X) >= p_n. This hypothesis
is the running assumption of the induction step: the final proof will
show it forces ODP, because everything with d > p_n is impossible.
The first reduction turns the problem about (x, X) into a problem
about a cone.
Take a minimiser v* of normalised volume at x. Such a minimiser
exists (Blum; Xu; Xu-Zhuang). It is unique up to positive scaling,
quasi-monomial, has finite discrepancy, and its associated graded
ring is finitely generated.
That associated graded ring defines a Fano cone C = Spec S. Two
passages make it K-polystable:
(i) "Stable degeneration" (Li-Xu) passes from (x, X) to a
K-semistable Fano cone with the same minimising valuation,
discrepancy and volume.
(ii) A K-polystable degeneration (Li-Wang-Xu) passes from
K-semistable to K-polystable, preserving the same quantities.
Both passages are equivariant flat degenerations, and both preserve
the Reeb-normalised volume and the Reeb discrepancy of the chosen
minimiser. The upshot is
d(o, C) = d(x, X),
with the same normalised volume at the vertex o of C.
Also, both passages are compatible with the embedding dimension
(the lifting-generators lemma: lifts of a graded minimal generator
set span the maximal ideal modulo its square), so
edim(X, x) <= edim(C, o).
Because X is singular, C is singular at its vertex o.
The cone's puncture is smooth. Here is why. Choose any P on the
puncture. Some positive-integral grading contracts P to the vertex,
so by lower semicontinuity (Blum-Liu)
d(o, C) <= d(P, C).
Choose a hypersurface slice Y_P through P transverse to the grading
orbit. Because the grading action has finite stabilizer at P, Y_P is
klt. If P were singular on C, then Y_P would be a singular germ of
dimension N = n-1, and by the induction hypothesis plus a smooth-
factor lemma (applied to the product C ~ Y_P x C^1, plus etale
invariance of normalised volume) one would get
d(P, C) <= p_{n-1} < p_n <= d(o, C),
contradicting the lower semicontinuity inequality. Hence the puncture
C minus {o} is smooth.
Finally the coordinate ring S is Gorenstein. The canonical index q is
the smallest positive integer with q K_C Cartier at o. The cyclic
index cover of degree q is quasi-etale, crepant, and klt, and by a
tangency argument has a single reduced point over o. The finite-
degree formula then gives d(o, C) <= 1/q. But d(o, C) >= p_n > 1/2,
so q = 1. Klt graded rings are Cohen-Macaulay, so the canonical
module is rank one; a suitable homogeneous generator of its fiber at
the vertex extends to the whole ring because the graded support of
the cokernel would otherwise be a positive-grading-invariant closed
set containing the vertex. So the canonical module is principal,
S is Gorenstein, and it has one homogeneous canonical generator.
Conclusion after Step 1: (C, o) is a normal Gorenstein Fano cone
of dimension n. The vertex o is singular, the puncture C minus {o}
is smooth, and
d(o, C) = d(x, X) >= p_n.
All later steps work with this cone. The original germ returns only
at the very end (Step 7).
==== 8. STEP 2. THE PUNCTURE QUOTIENT IS A SMOOTH ORBIFOLD ========
The Reeb vector of a Fano cone is a real one-parameter linear
grading xi on S: every positive-degree homogeneous function has
strictly positive xi-weight. Normalise so that A_C(xi) = n. Because
-K_C is a positive multiple of the cone class, such xi exists and
is unique up to scaling. Its weights are usually irrational.
The grading torus T of S has a preferred lattice. Approximate the
class of xi by primitive positive integral cocharacters zeta_i:
zeta_i = (r_i / n) xi + eps_i,
with eps_i -> 0 in the real cocharacter lattice and r_i = A_C(zeta_i)
a positive integer called the canonical weight. If the ray of xi is
rational, take its exact primitive generator (constant sequence);
otherwise r_i -> infinity. The canonical-weight condition matches the
paper's Lemma 2.6.
For each zeta_i collapse the orbit structure on the puncture:
B = [ (C minus {o}) / G_m ].
This is a smooth proper Deligne-Mumford (orbifold) stack whose coarse
space is the projective base Proj S. Its stabilizers are finite cyclic
groups. A weight-one line bundle L is fixed once and for all, with
S_j = H^0(B, j L), -K_B = r L.
At a point of order m a smooth transverse uniformizer carries a
faithful mu_m-action, and L has a faithful fiber character. Let
m_max be the largest stabilizer order on B, and r = r_i the canonical
weight of the grading.
The orbifold structure is real: passing to the coarse scheme Proj S
would forget the cyclic character data that govern rational curves,
the semigroup colength and the final volume bound. So keep B and L.
==== 9. STEP 3a. SMALL STABILIZERS ================================
Assume m_max <= r/N. (The complementary case m_max >= r/N is
Section 10.)
Minimal rational curve. The Li-Zhou orbifold rational-curve theorem
gives a representable nonconstant minimal curve f: P(1, l) -> B whose
only possible stack point is infinity and whose anticanonical degree
is at most n = dim B + 1: (-f^* K_B) <= n. Choose f of minimum
positive L-degree among such curves. Write f^* L = O(e) with e > 0,
and let m be the stabilizer order at f(infinity). Representability
forces l | m.
Two inequalities.
(-f^* K_B) = r * e / l <= n (from the degree bound), and
r / m >= N (since m <= m_max <= r/N).
Divide and use l | m to see that em/l is an integer:
1 <= em / l = (r e / l) / (r / m) <= (N+1)/N < 2.
So em/l = 1, hence
e = 1, l = m, N <= r / m <= n.
Deformations. Consider the affine scheme of homogeneous maps
F: A^2(1, m) -> C with F(0, 1) fixed to a point P lifting f(infinity)
to the cone. The deformations lemma (Lemma 3.2 of the paper) says that
every component through the chosen map has dimension at least
chi = n + r/m - sum_{j=1}^n b_j'/m,
where b_j' in {1, ..., m} are the tangent weights at P, with the
trivial character encoded as m instead of 0. When r/m = N and the
tangent action at P is entirely trivial, chi = n exactly; otherwise
chi > n, and so chi >= n.
Finite evaluation. Let H be the closure of a suitable component of the
open locus of maps avoiding the vertex. The positive G_m-grading
(x, y) -> (a x, y) acts, and the resulting evaluation
ev: H -> C, F |-> F(1, 0),
is finite and surjective (Lemma 3.3 of the paper: properness of twisted
stable maps plus graded Nakayama). In particular dim H = n.
A negative-weight tensor. The additive group (x, y) -> (x, y + t x^m),
t in C, fixes the map pointwise at (0, 1) and preserves the component
H. Differentiating gives a regular vector V along ev over C minus {o},
of derivation weight -m. Taking the symmetric product of the tangent
lifts over all b = deg(ev) sheets gives a nonzero section
s in H^0( C minus {o}, Sym^b T_{C minus {o}} ),
of integral grading weight -b m. Normality of C minus {o} makes the
symmetric product well defined without assuming H is normal.
Weight bound. A Fano cone with smooth puncture admits a Ricci-flat
Kaehler cone metric (Collins-Szekelyhidi). The holomorphic Bochner
formula and subharmonicity then give (Lemma 3.5 of the paper): every
nonzero section of Sym^b T has xi-weight at least -b.
Character estimate and rounding. Let chi be any nonzero eigencomponent
of the tensor. The tensor has zeta-weight -b m, so
-b m = chi(zeta) = (r/n) chi(xi) + chi(eps)
>= -(r/n) b - c b ||eps||,
using the weight bound on the xi-weight chi(xi). The character estimate
(Lemma 3.6 of the paper) provides ||chi|| <= c b. Divide by b:
m <= r / n + c ||eps||, i.e. n m - r <= n c ||eps||.
The left side is a nonnegative integer because N <= r/m <= n forces
n m - r >= 0. Since ||eps|| -> 0,
n m = r for all sufficiently far gradings.
The rational case has eps = 0 and the same conclusion directly.
Conclusion. With n m = r, chi becomes
chi = n + r/m - sum b_j'/m = 2n - sum b_j'/m.
Because every b_j' <= m, sum b_j'/m <= n, so chi >= n. But dim H = n
and chi <= dim H, so chi = n. All equalities are tight, hence every
b_j' = m: the tangent action at P is trivial. Faithfulness then forces
m = 1. The rounding nm = r gives r = n = N+1 in the main case; in the
exceptional case of the deformation lemma, the trivial tangent action
already forces m = 1 and then r/m = N gives r = N. Either way,
m_max <= r/N <= (N+1)/N < 2, hence m_max = 1.
So the quotient B has trivial stabilizers: it is an honest smooth
projective Fano N-fold, L is an actual ample line bundle, and
-K_B = r L, r in {N, N+1}.
Kobayashi-Ochiai (in its ample-line-bundle form, also supplied by
Araujo-Druel-Kovacs) then gives
(B, L) = (P^N, O(1)) if r = N+1,
(B, L) = (Q^N, O(1)) if r = N,
with Q^N a smooth quadric. The section ring S = sum_j H^0(B, j L) has,
for its affine cone, respectively: C^{N+1} = C^n (smooth), or the
cone over a smooth quadric (= the n-dimensional ODP). This finishes
the small-stabilizer case.
==== 10. STEP 3b, PART I. LARGE STABILIZERS: AN INTERIOR POINT =====
Now assume m_max >= r/N. Choose a point of order m = m_max and look
at the Taylor expansion there. This builds a semigroup whose geometry
encodes the cone's normalised volume. This section proves the
canonical lattice point sits in its interior; the next section fills
the semigroup exactly.
Setup. At the chosen point, take smooth equivariant uniformizer
coordinates z_1, ..., z_N with characters b_1, ..., b_N in Z/m and a
local frame of L of character 1. For a section s of jL, let in(s) be
its leading Taylor monomial (lex-leading after total order). Define
Gamma = { (j, gamma) : some nonzero s in H^0(B, jL)
has in(s) = z^gamma },
Lambda = { (j, gamma) in Z x Z^N : j = sum_i b_i gamma_i (mod m) },
K = closure of the cone over Gamma.
Note: j is the ORIGINAL degree, not scaled. For integration only,
put D_0 = N j / r and s = r / (N m), so 0 < s <= 1.
Facts (Proposition 4.1 of the paper). Gamma is an additive semigroup
with one-dimensional leaves and generated group Lambda. K is a full-
dimensional closed convex cone; it contains the positive constant
ray, projects onto the exponent orthant R^N_{>=0}, and equals the
closure of its interior. It is the epigraph of a finite convex
positively homogeneous coercive function f in the scaled coordinates
(D_0, x). The lattice density in those coordinates is s. The
canonical lattice point c_0 = (r, 1, ..., 1) lies in Lambda. Every
finite compatible Taylor jet is realised by sections of all
sufficiently large degrees in each admissible class modulo m.
Test valuations. For a real delta >= 0 define
v_delta( sum_j s_j ) =
min over s_j != 0 of ( (N j / r) + delta * ord_z(s_j) ).
This is a valuation centered at o, divisorial up to scaling
(Lemma 4.2 of the paper), and
A_C(v_delta) = N + N delta,
A_C(v_delta)^n vol(v_delta)
= N^n (1+delta)^n s
* ( integral over R^N_{>=0} of e^{-f(x) - delta |x|} dx ).
Two moment inequalities. Let X = (X_1, ..., X_N) be independent
exponential variables of mean 1, put B_0(x) = f(x) - |x|, and
H = E[ e^{-B_0(X)} ]. The hypothesis nvol(o, C) >= 2 N^n and the
test at delta = 0 give
H >= 2 / s (first inequality).
A near-minimality computation (Lemma 4.3 of the paper: homogeneity of
f, uniform second moment of |x|, passage to a small rational delta)
gives
E[ (1 + B_0(X)) e^{-B_0(X)} ] >= -eta H, eta >= 0, eta -> 0
(second inequality).
The exponential expectation bound. The decisive input bounds H when
B_0 is at least a zero-sum linear function. With u = 4/3 define the
envelope function
g(y) = 1 for y <= 0,
g(y) = (1+y) e^{-y} for y > 0.
Theorem 4.4 of the paper shows
sup over N >= 1, sup over a in R^N with sum_i a_i = 0,
of (e^u/u) E[ g(u + a . X) ] = C_* := (3/4) e^{4/3} - e/3 < 2.
The supremum is approached as N -> infinity by
(2, -2/(N-1), ..., -2/(N-1)). CHECK the numerical certificate:
e > 1359/500 and e^{4/3} < 1897/500 give
C_* < (3/4)(1897/500) - (1/3)(1359/500) = 3879/2000 < 2.
Interiority. Suppose f(1, ..., 1) >= N. At the interior point
1 = (1, ..., 1) of the exponent orthant, pick a subgradient p of f.
Homogeneity gives p . 1 = f(1) and f(x) >= p . x on the orthant, so
B_0(x) >= a . x, where a = p - 1, sum_i a_i = f(1) - N >= 0.
Replace a by a - (min coefficient), giving a zero-sum vector, at the
cost of shifting the inequality by a constant; one verifies the point
estimate (1 + (1+b)/u) e^{-b} <= (e^u/u) g(u+b) absorbs that shift.
Averaging against the measure H^{-1} e^{-f(x)} dx and using the two
moment inequalities yields
(1 - eta/u) H
<= E[ (1 + (1+B_0)/u) e^{-B_0} ]
<= (e^u/u) E[ g(u + a . X) ]
<= C_* < 2.
But H >= 2/s >= 2. For eta < (2 - C_*)/2 and s <= 1 both lines
cannot hold. Therefore f(1, ..., 1) < N, which after rescaling back
to original degree says
c_0 = (r, 1, ..., 1) lies in Int K (Theorem 4.8 of the paper).
What matters for the rest of the proof: interiority is a STRICT
inequality with a universal margin, so approximating xi by zeta_i
does not destroy it. That is the whole point of the constant C_* < 2.
==== 11. STEP 3b, PART II. LARGE STABILIZERS: ADJOINT LIFTING ====
Section 10 gave one half of Step 3b: the canonical lattice point c_0
sits in the interior of K. This section gives the other half: every
interior lattice point is realised, so Gamma is exactly the lattice
cone K intersect Lambda. Three ingredients: adjoint lifting, a
vanishing theorem, and the geometry of supporting hyperplanes at c_0.
The lifting statement. Recall Lambda (lattice) and Gamma (semigroup
of realised leading Taylor indices) from Section 10. The adjoint
lifting lemma (Lemma 5.2 of the paper) says:
if (j, alpha) in Lambda and (j, alpha) + c_0 in Int K,
then there is a section of jL with in(s) = z^alpha.
Equivalently (j, alpha) in Gamma. The shift by c_0 = (r, 1, ..., 1)
is what pays the canonical divisor in the vanishing theorem below.
Without it the vanishing hypothesis fails; with it, it holds.
Construction.
Weighted blowup. Choose positive primitive integer weights
u_1, ..., u_N matching the lex-leading order on a finite relevant
set of monomials. Let I_k be the ideal generated by monomials of
u-weight at least k, and blow up their Rees algebra:
pi: Z -> B, exceptional divisor E.
Z is a smooth stack; E = P(u_1, ..., u_N) modulo the residual cyclic
group.
Line bundles. On Z put
P_0 = pi^*(jL) - p_0 E, H_0 = pi^*(a j_0 L) - a U E,
where p_0 = u . alpha, U = u . (alpha + 1) = p_0 + sum_i u_i, and
j_0 is a rational degree chosen so that (j_0, alpha + 1) is
interior.
Positivity. The canonical identity K_Z + E = pi^*K_B + (sum_i u_i)E
and -K_B = r L give
P_0 - (K_Z + E) - (1/a) H_0 = (j + r - j_0) pi^* L.
The right side is nef and big with strictly positive coefficient.
This is the positivity that lets us extend sections from E to Z.
Multiplier membership on E. The sections generating H_0 restrict to
an ideal b on E. The character congruence for (j, alpha) and the
convexity coming from "(j, alpha) + c_0 interior" together imply
z^alpha lies in the multiplier ideal J_E(b^{1/a}). This is Howald's
monomial multiplier-ideal formula, applied chart-by-chart on the
smooth weighted-projective E: monomial x^nu lies in J(I^c) iff
nu + 1 is interior to c times the Newton polyhedron of I. The
hypothesis holds on the smooth atlas of Z, not on the coarse blowup.
Vanishing and pushdown. Resolve E and b by a log resolution
sigma: Y -> Z. On Y, Kawamata-Viehweg vanishing (Lemma 5.1 of the
paper, in its stack form) makes the map H^0(Y, ...) -> H^0(E, ...)
surjective. The multiplier membership provides a section on E whose
rational representative is the pullback of z^alpha; surjectivity
lifts it to a section of P_0 on Y. The pole estimate
(K_Y - sigma^*K_Z - sigma^*E + E' - floor(G_0/a) is effective up to
exceptional terms) shows the section is regular after pushing down
to Z. Restricting to E recovers the leading monomial z^alpha
exactly, because the weighted order built into the blowup kills the
lower-order terms.
That is Lemma 5.2. It is the sole input from multiplier ideals and
vanishing, and it explains why the canonical shift c_0 is exactly
(r, 1, ..., 1): the coordinate 1, ..., 1 is the Newton-polyhedron
shift that Howald's formula requires.
Saturation. Combine interiority (Section 10) with liftability. The
saturation theorem (Theorem 5.3 of the paper) states:
(i) K is a rational polyhedral cone;
(ii) every facet of K has a primitive nonnegative support form
eta in the dual lattice Lambda^* with eta(c_0) = 1;
(iii) Gamma = K intersect Lambda, and
Int K intersect Lambda = c_0 + Gamma.
Here is how each part is proved.
Facet forms have height one at c_0. Let x be a differentiability
point of the boundary of K in the positive orthant. Its supporting
normal is eta_0(j, x') = j - grad f . x'. Suppose either that eta_0
is irrational, or that its primitive integral generator eta has
eta(c_0) >= 2. In both cases one can find a lattice vector w in
Lambda with -eta(c_0) < eta(w) < 0. Take lattice points l_k of
Lambda approaching multiples of the boundary's tangent direction.
Then l_k + w sits just outside K, while (l_k + w) + c_0 sits just
inside Int K. Adjoint lifting would put l_k + w in Gamma, hence in
K. Contradiction. So every supporting ray is rational with height
exactly one at c_0.
Finitely many support forms. With eta(c_0) = 1, testing on a small
Euclidean ball around c_0 contained in K bounds ||eta|| <= 1/eps.
Since Lambda^* is discrete, only finitely many primitive forms fit
in any ball. Convex functions are differentiable almost everywhere,
so the finite list of forms recovered at differentiability points
defines all of K. Hence K is rational polyhedral.
Saturation. Take q in K intersect Lambda. Because c_0 is interior,
q + c_0 is interior, so adjoint lifting puts q in Gamma. Conversely,
take q in Int K intersect Lambda. Every primitive facet form eta
takes a positive integral value on q, hence at least one; because
eta(c_0) = 1, subtracting c_0 keeps q - c_0 in K intersect Lambda =
Gamma. Therefore Int K intersect Lambda = c_0 + Gamma.
Finite generation. Triangulate K into finitely many simplicial
cones with rational rays and take their primitive lattice
generators. Points of K intersect Lambda reduce, modulo the bounded
half-open parallelepiped of each simplex, to finitely many residues.
So Gamma is finitely generated. Saturation is automatic by the same
length argument.
That finishes Step 3b.
What Step 3b delivered: c_0 in Int K, and Gamma = K intersect Lambda
is a finitely generated saturated affine semigroup, normal, Cohen-
Macaulay, and Gorenstein. That is exactly the data Step 4 needs.
==== 12. STEP 4. TWO FILTRATIONS AND A STABILIZER OF ORDER TWO ====
We now have saturation: c_0 in Int K and Gamma = K intersect Lambda.
Write S for the cone's coordinate ring and n = N+1 >= 5. The task of
this section is to convert semigroup data into a singular klt germ Y
of the SAME dimension n whose normalised volume is strictly smaller
than d(o, C) by a fixed fraction. That fraction is the sole source of
strictness in the whole bound.
The plan is to filter S twice.
Filtration one: total Taylor order.
For a homogeneous section s of S_j, let T(s) be its total Taylor
order at the chosen point. Set T_t S = span of basis elements of
Taylor order at least t.
Lemma 6.1 (first filtration). The associated graded ring
bar S = gr_T S is finitely generated, normal, Cohen-Macaulay,
Gorenstein. Its canonical module has a homogeneous generator of
bidegree (r, N). There is a section v in S_m with initial form
tau^m (so T(v) = 0), such that
bar S[bar v^{-1}] = C[bar v, bar v^{-1}][u_1, ..., u_N],
T(bar v) = 0, T(u_i) = 1.
Set D = bar S / (bar v). D is an equidimensional Cohen-Macaulay
Gorenstein ring of dimension N with
Hilb_T D(z) = (1-z)^{-N}, deg omega_D = (r - m, N).
Its T-degree-zero piece is C. A bihomogeneous vector-space basis of D
lifts to sections e_i of S such that
{ v^a e_i : a >= 0 }
is a C-basis of S, compatible with original degree and with T. The
unique basis element of level a + T_i = 0 is 1.
In words: after Taylor grading and dividing by v, the ring looks like
a polynomial ring on N things, with the tautological Hilbert series.
Rees discrepancy comparison.
The bridge from "filtered ring" to "discrepancy of a valuation on C"
is one inequality.
Lemma 6.2 (Rees comparison). Let F be a nonnegative decreasing
multiplicative integral filtration on S with F_p S = S for p <= 0,
and let R_F be the extended Rees algebra inside S[lambda, lambda^{-1}].
Suppose gr_F S is Cohen-Macaulay Gorenstein with a canonical
generator of bidegree (r, Q). Then the canonical module of R_F is
generated in bidegree (r, Q-1), and on lambda != 0 the generator is
Omega = c lambda^{-Q} Omega_C wedge d lambda, c in C^*.
For a quasi-monomial valuation V centered on the Rees space over
lambda = 0, with l = V(lambda) > 0 and w = V restricted to C(C)
centered at o, one has
A_C(w) <= A_X(V) + (Q - 1) l.
The loss is exactly Q-1 times the value of lambda: that is the only
place the filtered canonical shift enters discrepancy. We use this
inequality twice, for the first and second filtrations.
Filtration two: W(v^a e_i) = a + T_i.
Assign weight W = a + T_i to the basis vector v^a e_i. This is
positive on every nonconstant homogeneous basis vector. Equivalently,
W_p S = sum_{a = 0}^{p} v^a T_{p-a} S, p >= 0.
The two definitions agree, and W is multiplicative.
Proposition 6.4 (two-filtration structure). G = gr_W S = D[bold v]
as graded algebras, with bold v a polynomial variable of original
degree m and W-degree 1. Consequently
Hilb_W G(z) = (1-z)^{-n}, deg omega_G = (r, n).
The W-Rees space is a normal Gorenstein finite-type variety, with
canonical generator bidegree (r, n-1) and top form
c lambda^{-n} Omega_C wedge d lambda on lambda != 0.
A stabilizer of order two.
Lemma 6.5 (free gradings). Let E be a finitely generated positively
graded reduced equidimensional Cohen-Macaulay C-algebra of dimension
N >= 2, with E_0 = C and Hilb E(z) = (1-z)^{-N}. If the grading
action has trivial stabilizer at every nonvertex closed point of
Spec E, then E is a polynomial ring on N generators of degree one.
Corollary 6.6 (second stabilizer). Under the running hypothesis
nvol(o, C) >= 2N^n, there is a nonvertex closed point of Spec D whose
T-grading stabilizer has order h >= 2.
Reason. If D were a polynomial ring on generators of degree 1, then
G = D[bold v] would be polynomial, lifting those generators through
the W-filtration would make S a quotient of a polynomial ring on n
generators, and a surjection from a polynomial ring of dimension n
onto a domain of dimension n is an isomorphism. Then C would be
smooth, contradicting "C is singular". So D has a nontrivial
stabilizer somewhere, of some order h >= 2.
The quadratic cone, by hand.
The cleanest example is the cone over a smooth quadric. Take
S = C[a, z_1, ..., z_N, b] / (a b - q(z)), q(z) = sum_i z_i^2,
with all generators of original degree one. Then r = N and m = 1.
At the quotient point [a:z:b] = [1:0:0], the chart a = 1 gives
b = q(z); choose v = a = 1. The bidegrees are
element a z_i b
---------------- --- ----- -----
T (Taylor) 0 1 2
W 1 1 2
orig degree 1 1 1
The Taylor graded ring bar S is S itself, and D = C[z, b] / (q(z)).
The second Rees space, in coordinates
lambda, a-bar = lambda^{-1} a, z-bar = lambda^{-1} z,
b-bar = lambda^{-2} b,
is X = { lambda a-bar b-bar - q(z-bar) = 0 }. Its action weights are
(-1, 1, 1, 2) on (lambda, a-bar, z-bar, b-bar). The point
b-bar = 1 with all other coordinates zero is fixed by the subgroup of
order 2, because the weight of b-bar is 2. The transverse slice
Y = { b-bar = 1 } = { lambda a-bar - q(z-bar) = 0 }
is again an ordinary double point of the SAME dimension n = N + 1.
Moral: the new stabilizer h = 2 is genuinely different from the
original maximal stabilizer m (here m = 1), and it does not lower the
dimension. A character class on Y will therefore account for only 1/h
of every leading colength. That is the strict fraction the bootstrap
harvests.
==== 13. STEP 5. THE SLICE AND THE BOOTSTRAP ======================
Setup. Corollary 6.6 gives a nonvertex closed point y_D of Spec D
with T-stabilizer mu_h, h >= 2. Put y = (y_D, 0) in Spec D[bold v],
viewed on the central fiber of the W-Rees space X.
The slice.
Lemma 7.1 (slice and its tests). There is a normal Gorenstein
algebraic germ y in Y of dimension n, preserved by mu_h, with a
nonzero function lambda_Y in m_Y of character -1, such that Y is klt
away from lambda_Y = 0. Every quasi-monomial valuation u centered at
y determines a quasi-monomial valuation w centered at o in C, and
with l = u(lambda_Y) > 0,
A_C(w) <= N l + A_Y(u), vol(w) <= l^{-n}.
The comparison is valid before klt at y has been established.
Idea. Take a positive-W-degree homogeneous f on X, nonzero at y, and
slice by f = f(y). The action map Y x G_m -> X onto the slice is
finite etale, so Y inherits normality and Gorenstein and lambda
restricts to lambda_Y of character -1. Extend u to Y x G_m by the
Gauss rule, then restrict to X; this gives w. The Rees comparison
(Lemma 6.2 with Q = n, the canonical shift of G) gives the
discrepancy bound. The volume bound counts a W-adapted basis and
uses Hilb_W G = (1-z)^{-n}.
The slice is klt.
Lemma 7.2. The germ y in Y is klt.
Reason. If not, some divisor E on a log resolution has A_Y(E) <= 0
with center containing y and a = ord_E(lambda_Y) > 0. Perturb along
E to quasi-monomial valuations u_eps centered at y with
A_Y(u_eps) -> A <= 0 and l_eps -> a; their restrictions w_eps are
centered at o. Since C is klt, A_C(w_eps) > 0, so Lemma 7.1 either
already gives a nonpositive discrepancy bound (immediate
contradiction with 2N^n > N^n) or gives
2N^n <= nvol(o, C) <= A_C(w_eps)^n vol(w_eps)
<= (N + A_Y(u_eps)/l_eps)^n.
The right side has limsup at most N^n, contradicting 2N^n > N^n.
The slice is singular.
Lemma 7.3. The germ y in Y is singular.
Reason. If Y were smooth at y, the etale product chart would make X
smooth at y. The class bold v in G is then a function of character 1
with nonzero differential, and D is the hypersurface cut out in
{ bold v = 0 } by lambda|_H, a nontrivial mu_h-equivariant equation
in the smooth total space. Complete locally along the OTHER family
bar S over A^1_t: it is flat with invariant parameter t and central
fiber D, and by completing at y_D one gets a mu_h-fixed formal
section. But the generic fiber is standard affine space with every
coordinate of character 1, whose mu_h-fixed locus is the origin. A
positive-T-degree function cannot vanish at the origin and be a unit
at y_D at the same time. Contradiction.
The character contribution to colength.
By Lemma 7.2 a minimizing valuation u_* of Y exists, is quasi-monomial,
and by uniqueness up to scaling is mu_h-invariant. Write
A' = A_Y(u_*), l = u_*(lambda_Y) > 0, sigma = vol(u_*).
Lemma 7.4 (equal character lengths). For every character chi of mu_h,
dim_C (R / a_t)_chi = ( sigma / (h n!) ) t^n + o(t^n),
t -> infinity.
Reason. The valuation ideals are mu_h-invariant. Multiplication by
lambda_Y^j gives injections (R / a_{t-jl})_chi -> (R / a_t)_{chi-j}.
For any two characters choose 0 <= j <= h-1 connecting them, which
shows any two character quotients differ by a bounded shift. The
total length is sigma t^n / n! + o(t^n), so h equal-summands give the
formula.
The refined estimate.
Proposition 7.5. For the minimizing slice valuation,
vol(w) <= min over 0 <= x <= 1 of
{ x^n / l^n + (1-x)^n sigma / h }.
Reason. Split the colength of S by a cutoff k into a part covered by
S/F^pS and a part covered by F^pS. The first is at most
dim(S/F^pS) ~ p^n/n!. The second injects, via restriction and
division by lambda_Y^p, into ONE character quotient (R/a_{k-pl})_{p} of
Y, whose leading size is (1 - p l / k)^n sigma/(h n!). Set
p = floor(k x / l) and pass to the limit in k.
Optimization.
Lemma 7.6. For the fixed cone and the large-stabilizer construction,
d(o, C) <= ( (N/n) l + A'/n )^n
/ ( l^{n/N} + (h/sigma)^{1/N} )^N
<= (N/n)^n + (A'/n)^n sigma / h
<= p_n / 2 + M_n / 2.
Equality in the middle (Holder) holds iff (A')^N l sigma / h = N^N.
Reason. Direct differentiation gives
min_x { x^n l^{-n} + (1-x)^n sigma/h }
= ( l^{n/N} + (h/sigma)^{1/N} )^{-N},
attained at the unique interior point l^{n/N} / (l^{n/N} +
(h/sigma)^{1/N}). The first line is Lemma 7.1 + Proposition 7.5
divided by n^n. Holder with conjugate exponents n/N and n, applied to
the vectors (l, (h/sigma)^{1/n}) and (N, A'(sigma/h)^{1/n}), gives the
second line with exactly that equality criterion. Finally y in Y is
singular klt of dimension n by Lemmas 7.2 and 7.3, so
(A'/n)^n sigma = d(y, Y) <= M_n; and h >= 2 gives sigma/h <= sigma/2.
Replacing gives the third line. CHECK (N/n)^n = p_n/2, because
p_n = 2 ((n-1)/n)^n = 2 (N/n)^n.
The upper bound.
Theorem 7.7. Assume the theorem in all dimensions below n. Then
M_n <= p_n. Equivalently, every singular boundary-zero klt n-fold
germ has nvol <= 2 (n-1)^n.
Reason. Suppose M_n > p_n. Choose singular klt germs with
d_i > p_n tending up to M_n, apply the cone reduction to each, and
take a sufficiently far grading for each cone C_i. If the small-
stabilizer hypothesis holds for some i, then C_i is smooth (giving
d_i = 1, incompatible with the singular vertex the reduction always
produces) or is the ODP (giving d_i = p_n, contradicting d_i > p_n).
So the large-stabilizer hypothesis holds, and Lemma 7.6 gives
d_i <= p_n / 2 + M_n / 2.
Letting i -> infinity gives M_n <= p_n/2 + M_n/2, i.e. M_n <= p_n,
contradicting M_n > p_n. So M_n <= p_n.
Each choice of a sufficiently fast grading is made separately for
each fixed cone. No uniformity of approximation across cones is
required. That is why defining M_n as a supremum is harmless.
==== 14. STEP 6. THE EQUALITY CASE ===============================
Now suppose the fixed cone satisfies d(o, C) = p_n. Theorem 7.7 is
available (M_n <= p_n). For every fast grading in the large-stabilizer
range take m = m_max and run the whole Step 4 / 5 construction. The
main observation is that every inequality is automatically tight.
Equality data.
Lemma 7.8. h = 2, d(y, Y) = p_n, and A' = N l. After rescaling u_*
so that l = 1, one has
A' = N, sigma = 2, A_C(w) = 2N, vol(w) = 2^{-N},
and w = (2N/n) xi. The optimal splitting is x = 1/2, and every
inequality in Proposition 7.5 is an equality at the level of leading
coefficients.
Reason. The chain of Lemma 7.6 starts at p_n and ends at most at p_n,
so each step is equality. In particular h = 2, and the last
inequality forces (A')^n sigma / h = N^n, because (A'/n)^n sigma = p_n
and p_n = 2(N/n)^n with h = 2. The Holder equality condition
(A')^N l sigma/h = N^N then gives, with h = 2,
N^N = (A')^N l sigma/2 = N^n l / A',
so A' = N l. Rescale so l = 1; then A' = N and sigma = 2. The refined
volume bound becomes vol(w) <= 2^{-N}, and the discrepancy bound is
A_C(w) <= 2N. Since w is admissible on C,
2N^n = nvol(o, C) <= A_C(w)^n vol(w) <= (2N)^n 2^{-N} = 2N^n.
Every inequality is tight, and w computes the normalised-volume
minimum of C. By uniqueness up to scaling (A_C(xi) = n), w = (2N/n) xi.
The optimal splitting fraction x = l^{n/N}/(l^{n/N}+(h/sigma)^{1/N})
= 1/(1+1) = 1/2.
The strict count.
Some extra info about the section v that built the two filtrations is
all that remains.
Lemma 7.9. w(v) <= 2.
Reason. Suppose w(v) > 2. Choose t with 1/w(v) < t < 1/2, and set
a_k = ceil(t k), p_k = floor(k/2). For large k, p_k - a_k > 0. Using
the v^a e_i basis adapted to W, let E_k be the span of basis vectors
of W-level < p_k - a_k.
Multiplication by v^{a_k} maps E_k injectively into S/F^{p_k}S. Indeed
the initial form of v is the regular variable bold v in G = D[bold v],
so multiplication raises every nonzero initial W-level by exactly a_k
and kills no initial form. Every element of v^{a_k} E_k has w-value
> k, since w is nonnegative and a_k w(v) > k.
Hence the kernel of the natural map
S/F^{p_k}S -> S/(F^{p_k}S + a_k)
has dimension at least
dim E_k = dim S/F^{p_k - a_k}S
= ( (1/2 - t)^n / n! ) k^n + o(k^n),
a strict positive leading-order loss in the first term of the exact
splitting. The second term still satisfies the character estimate
(Lemma 7.4). At x = 1/2, l = 1, sigma/h = 1, they give
vol(w) <= 2^{-n} + 2^{-n} - (1/2 - t)^n < 2^{-N},
contradicting vol(w) = 2^{-N}. So w(v) <= 2.
Rounding the maximal stabilizer.
Lemma 7.10. N m_max <= r for every sufficiently far term.
Reason. Write the fast grading zeta = (r/n) xi + eps with eps -> 0.
Fix torus eigen-generators g_1, ..., g_q of S, positive for xi. For
all far gradings their zeta-degrees lie between c_1 r and c_2 r for
fixed c_1 > 0. Also m = m_max is O(r) by Lemma 2.6. The section
v in S_m has a polynomial lift in the g_i, homogeneous for zeta. A
monomial of degree m uses at most m/(c_1 r) = O(1) factors, so its
characters lie in a fixed finite set independent of the grading. For
any character chi occurring nontrivially in v,
m = <zeta, chi>
= (r/n) <xi, chi> + <eps, chi>,
so <xi, chi> = n m / r + O(||eps||/r). The minimum over nonzero
characters of v is exactly wt_xi(v), so the same estimate holds:
w(v) = 2N m / r + O(||eps||/r). By Lemma 7.9,
N m - r <= O(||eps||) = o(1).
The left side is an integer, hence nonpositive. The rational-ray case
has eps = 0 and gives the conclusion directly.
The equality cone.
Theorem 7.11. If a singular cone from Proposition 2.5 satisfies
d(o, C) = p_n, it is the standard ordinary double point cone.
Reason. By Lemma 2.6, each fast primitive grading either already has
m_max <= r/N, or lies in the large-stabilizer range. In the second
case Lemma 7.10 forces N m_max <= r. So the entire sequence eventually
satisfies the small-stabilizer hypothesis. The small-stabilizer
classification (Section 9) then identifies C with the smooth affine
cone or the ODP. Since the vertex is singular, C is the ODP.
The equality case of the theorem is now reduced to the equality cone
of Step 1. The final section returns to the original germ.
==== 15. STEP 7. BACK TO THE GERM: HYPERSURFACES AND THE ODP ======
With Steps 1-6 done, the remaining work is to return from the cone
to the original germ and identify the equality case.
The hypersurface test.
Lemma 8.1. Let x in X be a klt hypersurface germ of dimension n, with
equation F in regular ambient parameters z_0, ..., z_n. For positive
rational weights w_0, ..., w_n, let d_F be the weighted order of F.
Then
nvol(x, X) <= d_F ( sum_i w_i - d_F )^n / prod_i w_i.
Reason. Scale to integer weights and pick a positive common multiple
D_1. The ideal I = (z_i^{D_1/w_i}) is m_x-primary. The ambient
weighted monomial valuation has log discrepancy sum_i w_i, gives
value d_F to F, and value D_1 to the ambient ideal. Inversion of
adjunction for a hypersurface in a smooth ambient gives
lct_X(I) <= ( sum_i w_i - d_F ) / D_1.
The weighted-order filtration on the hypersurface has associated
graded C[z]/(leading_w F), whose Hilbert series gives
e(I) = D_1^n d_F / prod_i w_i.
Testing with a valuation computing lct_X(I),
nvol(x, X) <= lct_X(I)^n e(I),
which is the stated bound.
Equality in the hypersurface test.
Proposition 8.2. Let x in X be a singular klt hypersurface germ of
dimension n >= 2. Then nvol(x, X) <= 2 (n-1)^n. If equality holds,
the quadratic part of a local equation is nondegenerate, and X at x
is an analytic ODP.
Reason. Let q = mult_x(F) >= 2. Equal weights 1 give
nvol <= q (n+1-q)^n,
and klt implies q <= n. The function b(t) = t(n+1-t)^n is strictly
decreasing for 1 < t < n+1, so any q > 2 already gives a strict bound
below 2(n-1)^n. Suppose q = 2. If the quadratic part is degenerate,
drop one direction to weight 1-eps and keep the rest at weight 1,
with 0 < eps < 1/3 rational. Every term of ordinary degree >= 3 has
weight > 2, so d_F = 2, and the bound is
2 (n - 1 - eps)^n / (1 - eps).
Its logarithmic derivative at eps = 0 is -n/(n-1) + 1 = -1/(n-1) < 0,
so its value is strictly less than 2(n-1)^n for small positive eps.
Hence equality forces a nondegenerate quadratic part; the holomorphic
Morse lemma then makes the germ an analytic ODP.
Completion of the main theorem.
Theorem 1.1 follows by induction. The cases n = 2, 3, 4 are known
inputs (Section 17). Assume n = N+1 >= 5 and that the theorem holds
in all dimensions below n.
Upper bound. Theorem 7.7 gives M_n <= p_n. In particular, every
singular boundary-zero klt n-fold germ satisfies
nvol(x, X) <= 2 (n-1)^n.
Equality case. Suppose d(x, X) = p_n for a singular germ. Apply
Proposition 2.5 (cone reduction), producing a K-polystable cone C
with d(o, C) = p_n and edim(X, x) <= edim(C, o). By Theorem 7.11, C
is the ODP cone, so edim(C, o) = n+1. Hence edim(X, x) = n+1, and X
is a hypersurface germ at x. (Algebraically, cut the ambient germ by
equations with independent linear parts down to dimension n+1; the
remaining ideal is a height-one prime in a regular local ring, hence
principal.) Proposition 8.2 then forces the quadratic part of a
local equation to be nondegenerate, so the analytic germ of X at x
is an ODP.
Conversely, Lemma 2.3 gives equality at every analytic ODP. That is
the full equality statement, and it closes the induction.
So Theorem 1.1 holds in every dimension n >= 2.
==== 16. THE GLOBAL COROLLARY FOR FANO VARIETIES ================
The local gap gives global bounds through a "local-to-global" volume
inequality of Liu.
Lemma (local-to-global). Let V be an n-dimensional K-semistable
Q-Fano variety. For every closed point x of V,
(-K_V)^n <= ((n+1)/n)^n nvol(x, V).
Reason. In this setting K-semistability and Ding semistability agree,
and Liu's inequality compares the anticanonical volume of V with the
local volume at any closed point x.
Corollary 1.2. Let V be an n-dimensional K-semistable Q-Fano variety,
n >= 2.
(a) If V is singular, then
(-K_V)^n <= 2 ((n^2 - 1)/n)^n < 2 n^n.
(b) If V is smooth and V is not P^n, then (-K_V)^n <= 2 n^n, with
equality exactly at a smooth quadric Q^n in P^{n+1} or at
P^1 x P^{n-1}. (These coincide for n = 2, since Q^2 = P^1 x P^1.)
Reason (a). Take a singular closed point x of V. The main theorem at
x gives nvol(x, V) <= 2 (n-1)^n. Combine with local-to-global:
(-K_V)^n <= ((n+1)/n)^n 2 (n-1)^n
= 2 ((n^2 - 1)/n)^n
= 2 n^n ((n^2 - 1)/n^2)^n
< 2 n^n,
since (n^2 - 1)/n^2 < 1.
Reason (b). The smooth case is Li-Miao's theorem: (-K_V)^n <= 2 n^n
with equality at a smooth quadric Q^n in P^{n+1} or at
P^1 x P^{n-1}. Both admit Kahler-Einstein metrics (the homogeneous
quadric metric and the scaled Fubini-Study product) and are therefore
K-polystable (Berman). Their degrees are
(-K_{Q^n})^n = (n H)^n = 2 n^n,
(-K_{P^1 x P^{n-1}})^n = (2 H_1 + n H_2)^n = 2 n^n.
The strict singular bound excludes all other equality cases.
Plain reading. Among K-semistable Q-Fanos, a singular one has
strictly smaller anticanonical volume than a smooth one, by at least
a fixed factor. The extremal smooth cases are the quadric and the
product of a line with projective space.
==== 17. WHAT IS ASSUMED: BASE CASES AND INPUTS ==================
The proof is an induction. The step n >= 5 is the new work. The base
cases are earlier theorems, and the note is careful about which is
which.
Base cases.
n = 2 (surfaces). By the quotient classification of klt surface
singularities, every klt surface germ (x, X) is analytically a
quotient C^2 / G for a finite group G acting without reflections.
For such a quotient,
nvol(0, C^2 / G) = 4 / |G|.
If the germ is singular then |G| >= 2, so nvol <= 2 = 2 (2-1)^2.
Equality requires |G| = 2 with a fixed-point-free action, i.e.
(-1, -1), the A_1 surface singularity. So n = 2 is known.
n = 3 (threefolds). Liu-Xu proved the threefold theorem for arbitrary
klt germs: nvol <= 16 = 2 (3-1)^3, with equality exactly at the ODP.
n = 4 (fourfolds). The companion paper "The normalized-volume gap in
dimension four" supplies nvol <= 162 = 2 (4-1)^4, with equality
exactly at the ODP, under the same unrestricted hypotheses (no
isolatedness, no hypersurface assumption, no smoothability). This is
the paper's sole companion input.
Slices. The induction step uses a slice of dimension N = n-1 >= 4.
That appeal is covered by the base cases plus the induction itself.
Tools used as inputs (each is cited in the paper; none is new here).
- existence, uniqueness, quasi-monomiality, finite generation of
the minimiser (Blum; Xu; Xu-Zhuang);
- lower semicontinuity of normalised volume (Blum-Liu);
- the finite-degree formula (Xu-Zhuang);
- stable degeneration and K-polystable degeneration (Li-Xu;
Li-Wang-Xu);
- Li-Zhou's minimal orbifold rational curve;
- properness of twisted stable maps (Abramovich-Olsson-Vistoli);
- existence of a Ricci-flat cone / Sasaki-Einstein metric from
K-stability (Collins-Szekelyhidi);
- Howald's monomial multiplier-ideal formula and
Kawamata-Viehweg vanishing (in Fujino's stack-friendly form);
- Kobayashi-Ochiai classification of projective space and quadric
(also in Araujo-Druel-Kovacs form);
- Ein-Mustata inversion of adjunction for hypersurfaces;
- Li-Miao (smooth Fano bound) and Berman (K-polystability from
Kahler-Einstein).
Earlier related results, not used but useful for placing the theorem.
- surfaces: Li-Liu, via the quotient formula nvol = 4/|G|;
- threefolds: Liu-Xu;
- local complete intersections: Liu (K-stability of cubic
fourfolds and adjacent work);
- singular Q-Gorenstein toric germs: Moraga-Suess, giving the
sharper toric bound (16/27) n^n for n >= 3;
- Liu's threefold refinements (Gorenstein canonical
non-hypersurfaces, and classification above nvol = 9);
- global: Li-Miao (smooth Fano), Li-Miao-Zhang (general positive
Ricci).
So the new contribution is the induction step n >= 5, and inside it
the two-filtration and bootstrap machinery of Steps 4-6. Everything
else is either a theorem already in the literature or a short
computation done here.
==== 18. CHECKS YOU CAN DO BY HAND ===============================
The note sprinkled CHECKs through the argument. Here they are in one
place, so a reader can verify each without trusting the paper.
1. p_n is increasing in n, with limit 2/e. p_n = 2 (1 - 1/n)^n.
With u = 1/(n-1), g(t) = t log(1 - 1/t) has
g'(t) = u - log(1+u) > 0 because log(1+u) < u. So p_n rises
from p_2 = 1/2 toward 2/e = 0.7358.
2. C_* < 2. With u = 4/3,
C_* = (3/4) e^{4/3} - e/3
< (3/4)(1897/500) - (1/3)(1359/500)
= 3879/2000 = 1.9395 < 2.
The rational bounds e > 1359/500 and e^{4/3} < 1897/500 are the
certificates; e^{4/3} follows from the exponential series
truncated through degree eight plus a tail bound.
3. (N/n)^n = p_n/2. Since N = n-1 and p_n = 2 ((n-1)/n)^n, this is
immediate.
4. The optimization behind Lemma 7.6. For 0 < x < 1 and positive
l, c = sigma/h,
f(x) = x^n / l^n + (1-x)^n c
has a unique minimum at
x = R / (1 + R), R = l^{n/N} c^{1/N},
and the minimum value is
( l^{n/N} + (1/c)^{1/N} )^{-N}.
Numeric check: n = 3, N = 2, l = 1, c = 4 gives f(1/3) = 1/2.25
= 0.4444..., matching both sides.
5. The Holder step. With conjugate exponents n/N and n, applied to
the vectors (l, c^{1/n}) and (N, A' c^{-1/n}),
( N l + A' )^n
<= ( l^{n/N} + c^{-1/N} )^N ( N^n + (A')^n c ),
and equality holds exactly when (A')^N l c = N^N. This is the
equality criterion used in Lemma 7.8.
6. The decay of b(t) = t (n+1-t)^n. For 1 < t < n+1,
b'(t) = (n+1)(1-t)(n+1-t)^{n-1} < 0.
Hence a hypersurface of multiplicity q > 2 has nvol strictly
below 2(n-1)^n.
7. The degenerate-quadratic perturbation. The bound after giving
one direction weight 1-eps (and the rest weight 1) is
2 (n-1-eps)^n / (1-eps).
Its logarithmic derivative at eps = 0 is -n/(n-1) + 1 =
-1/(n-1) < 0, so it strictly lowers the bound.
8. The two extremal degrees in the global corollary. For a smooth
quadric Q^n in P^{n+1}, with H = O(1)|_Q and H^n = 2,
(-K_{Q^n})^n = (n H)^n = 2 n^n.
For P^1 x P^{n-1}, with -K = 2 H_1 + n H_2 and
H_1 H_2^{n-1} = 1,
(2 H_1 + n H_2)^n = 2 n^n.
Both admit Kahler-Einstein metrics, hence are K-polystable.
That is the complete set of computations the proof relies on at the
level of a careful reader. Everything else is a theorem in the cited
literature.
==== 19. GLOSSARY ================================================
Every term used in the main text, one line each.
normalised volume nvol(x, X): inf over valuations v centered at x of
A_X(v)^n vol(v). Measures the size of a klt germ. Maximum n^n at
smooth points.
density d(x, X): nvol(x, X) / n^n, in (0, 1]. Equals 1 iff smooth.
volume vol(v): leading coefficient of length(O_{X,x} / a_t(v)) as a
constant times t^n / n!. Measures how fast the valuation ideals grow.
log discrepancy A_X(v): a real number measuring how far v is from
being a divisor of the pair (X, K_X). Smooth order valuation has
A = n; the exceptional quadric of an ODP blowup has A = n-1.
klt (Kawamata log terminal): the mild condition ensuring normalised
volume is well behaved. Characterised by positive log discrepancies
of an effective boundary; with zero boundary just says the minimiser
of nvol exists, is unique up to scaling, and quasi-monomial.
Q-Cartier / Gorenstein: K_X is Q-Cartier if some positive integer
multiple is a Cartier divisor; Gorenstein if K_X itself is Cartier.
Cohen-Macaulay: a mild algebraic depth condition. Used to make
canonical modules rank one.
ODP (ordinary double point): the analytic hypersurface
z_0^2 + ... + z_n^2 = 0 in C^{n+1} near the origin. In dimension 2
this is the A_1 surface singularity, the quotient C^2 / (+/-1).
Also the cone over a smooth quadric Q^{n-1}.
Fano cone: C = Spec S for a positively graded normal Gorenstein
domain S, with a Reeb vector giving negative canonical class away
from the vertex.
Reeb vector xi: a real one-parameter linear grading on the cone with
positive weight on every nonconstant homogeneous function.
Normalised A_C(xi) = n.
K-(semi)polystable: a stability condition on Fano cones and Fano
varieties (also can be phrased as existence of a Kahler-Einstein
metric in the smooth compact case). K-semistable is the boundary;
K-polystable excludes continuous families.
orbifold, or Deligne-Mumford stack: a space locally modelled on the
quotient of a smooth scheme by a finite group. Its "stabilizer order"
at a point is the size of that finite group.
minimal orbifold rational curve: a smallest-degree map P(1, l) -> B
from a weighted projective line, used to extract characters at the
target.
semigroup Gamma, cone K: Gamma is the set of (original degree,
leading Taylor exponent) pairs realised by sections. K is the closed
real cone over Gamma. Its geometry encodes the volume.
Newton-Okounkov body: the divisor at infinity of a graded Newton
polyhedron, or equivalently a slice of K at D_0 = 1. The volume is
the (weighted) volume of the body.
extended Rees algebra R_F: sum_k lambda^{-k} F_k S, the total space
whose central fiber is the associated graded and whose general fiber
is S. Used to compare canonical shifts.
multiplier ideal J(I^c): the ideal of functions whose vanishing is
governed by c times the Newton polyhedron (monomial case, Howald),
or by c times the log canonical threshold in general.
Kawamata-Viehweg vanishing: the vanishing of higher cohomology for
divisors D with D - (K + Theta) nef and big and Theta a klt (or
simple-normal-crossing and sub-1) boundary. Used to lift sections
from the exceptional divisor.
Holder inequality: for conjugate exponents p, q,
|sum a_i b_i| <= (sum |a_i|^p)^{1/p} (sum |b_i|^q)^{1/q}. The
equality condition is a_i^p = b_i^q up to a constant, which is what
fits (A')^N l sigma/h = N^N.
==== END OF NOTE ==================================================
This note covers every result of the paper. The proofs of the many
external theorems are omitted (only their statements are needed).
For full proofs of the technical steps above, the original paper is
the intended reference; the source files are under
The-ordinary-double-point-gap-in-every-dimension/, with the main
theorem in sections/01-introduction.tex and the proof distributed
across sections/02-cone-reduction.tex through sections/08-conclusion.tex.