The normalized-volume gap in dimension four

1. Reduction to a cone with the same volume

Stable degeneration produces $C=\operatorname{Spec}\operatorname{gr}_v\mathcal O_{X,x}$ and a minimizing Reeb vector $\xi_*$. Finite generation, preservation of volume and log discrepancy, and minimization over all centered valuations are essential: minimization only within a fixed Reeb cone would not suffice. Lemma 2.1 lifts homogeneous generators and uses the Izumi estimate to obtain $\operatorname{edim}(x,X)\le\operatorname{edim}(o,C)$.

[XZ25, Theorem 1.2, p. 2] [LX18, Theorem 1.3, p. 3; Lemma 4.10, p. 35] [Li18, Theorem 3.1, p. 6]

A singular point on the punctured cone would have an orbit closure making the singular locus positive-dimensional at the vertex, contradicting $V\le4096/27<162$. The finite-degree formula applies to the index-one cover of degree $d$ and gives $dV\le256$, hence $d=1$. Cohen–Macaulayness of klt singularities and graded Nakayama make $C$ Gorenstein. For primitive integral gradings approaching the minimizing ray, write $r>0$ and $h$ for the discrepancy and volume of the degree valuation: $r^4h\ge V$ and $r^4h\to V$.

Proposition 2.2; Lemma 2.1 · pp. 5–6 [XZ21, Theorem 1.3, p. 1] [Liu22, Corollary 2.18, p. 9]

2. The two estimates meet at a terminal section

Proposition 3.4 gives $A_C(F)\ge3$ for every primitive prime divisor centered at the vertex. Proposition 5.1 gives $\operatorname{lct}_o(C;\mathfrak m_o)>3/2$. On a log resolution of the maximal ideal, write $\mathfrak m_o\mathcal O_Y=\mathcal O_Y(-\sum b_iE_i)$ and $A_i=A_C(E_i)$. The ordinary discrepancy along a general section $H$ is $A_i-1-b_i$. As a Cartier section of a Gorenstein Cohen–Macaulay fourfold, $H$ is Gorenstein and $S_2$; Bertini gives smoothness away from the vertex, hence $R_1$ and normality.

If $b_i=1$, then $A_i-b_i\ge2$; if $b_i\ge2$, then $A_i-b_i>b_i/2\ge1$. Every coefficient is positive, so $H$ is a terminal Gorenstein threefold germ. The threshold estimate alone is not being used to infer terminality. Reid's index-one classification makes a singular $H$ an isolated cDV point, hence an analytic hypersurface. Including the smooth case, $\operatorname{edim}(o,H)\le4$, and therefore

Proposition 6.1 · pp. 17–18 [Reid83, Theorem (1.1), print p. 138 / PDF p. 5; Section (0.5), print p. 134 / PDF p. 3]

3. Upper bound, necessity for equality, and sharpness

A singular normal fourfold germ with embedding dimension at most five has local ring equal to a five-dimensional regular local ring modulo a height-one prime. A regular local ring is a UFD, so this ideal is principal. Thus the original $X$ is an algebraic local hypersurface, hence lci; showing only that the degeneration cone is a hypersurface would not suffice.

Liu's lci theorem now yields $V\le2(4-1)^4=162$. Its equality classification says that $V=162$ forces the original germ to be an ODP. Conversely, one needs the actual ODP volume to be 162: merely exhibiting one valuation of normalized volume 162 would not establish the infimum. For the exceptional divisor $E$ of the ordinary blow-up of the ODP vertex, $A_X(E)=3$ and $\operatorname{vol}(E)=2$, giving the candidate value $3^4\cdot2=162$. The manuscript invokes the established lci equality theorem and this ODP calculation at this point, and uses invariance under isomorphism of completed local rings to pass to the analytic characterization. This also establishes sharpness of the constant.

Proof of Theorem 1.1 · p. 18 [Liu22, Theorem 2.12, p. 6] [Liu22, Lemma 2.15, p. 7]

1. Test valuations against the minimum over all valuations

The grading extraction in Lemma 3.1 has $K_{\widetilde C}=\pi^*K_C+(r-1)E$ and an lc pair $(\widetilde C,E)$. On the smooth slice cover with stabilizer $\mu_m$, assign weights $(1,\alpha,\alpha,\alpha)$ to $(u,z_1,z_2,z_3)$ and restrict the resulting valuation to the function field. For rational $\alpha>0$, this $v_\alpha$ is proportional to a divisorial valuation and $A_C(v_\alpha)=r+3\alpha$.

The image of functions of degree $j$ is bounded by the minimum of $\dim S_j$ and the dimension of slice jets in the matching character. The latter has leading term $\ell^3/(6m)$: forgetting the character and hence $m$ would lose the required estimate. Integrating over degrees gives

Lemma 3.1 · p. 7 Lemma 3.3 · p. 8

2. Contradiction to approximate minimality, then integrality

If $m/r\ge2/5$, then $Q=mhr^3\ge324/5>64$. Set $\alpha=r\delta$ and divide the bound by $r^4h$. The derivative at zero of $(1+3\delta)^4/(1+\delta Q^{1/3})^3$ is $12-3Q^{1/3}<0$. A fixed small positive rational $\delta$ gives a ratio uniformly below one, contradicting $V/(r^4h)\to1$. The cutoff limit is taken separately for each valuation, so uniformity of the jet-error constants in the grading is unnecessary.

Local Cartierness of $mE$ gives $\operatorname{ord}_F(E)\ge1/m$. The lc pair then yields $A_C(F)\ge r\operatorname{ord}_F(E)>5/2$. Since $K_C$ is Cartier, this is an integer, hence at least three. Rescaling to $y=\lambda j$, $\lambda=5/(2r)$, fixes the canonical weight at $5/2$ and gives $\lambda m<1$ at every nonvertex point.

Proposition 3.4 · p. 9

1. A Hilbert-series frequency gap forces short weights

For a normal Cohen–Macaulay graded domain $T$, use the natural weights on its canonical module to form

Lemma 4.1 uses $\lambda m<1$ to place all nonzero Hilbert-series pole frequencies outside $[-1,1]$. A Schwartz test $F$ with Fourier transform supported in $(-1,1)$ then satisfies $\int F\,d\mu_T=\int F(y)P_T(y)\,dy$, where $P_T$ encodes the polar part of $H_T(s)=\sum_y\dim T_y e^{-sy}$ at $s=0$. The manuscript derives $H_{\omega_T}(z)=(-1)^{\dim T}H_T(z^{-1})$ by dualizing a graded free resolution; we follow that internal calculation. In this identity $H_T(z)$ is the integer-graded Hilbert series, with $z=e^{-\lambda s}$.

For the four-dimensional ring $S$, canonical weight $5/2$ makes the density $a(x^3+cx)$ after shifting to $x=y+5/4$. The manuscript multiplies squares of band-limited functions by sign-changing polynomials. After deriving $c\ge-1$ and $a\ge432/625$, it shows that the absence of function weights in $(0,D)$, $D=5/3$, gives contradictory integral bounds. This is Proposition 4.3(a). Its two- and three-dimensional version (b) has an additional hypothesis: an injection $\omega_T^{[q]}\hookrightarrow T$ must lower weights by a positive amount. Lemma 4.2 then gives positive canonical-section weights and positivity of the next-to-leading coefficient of $P_T$.

Lemma 4.1 · pp. 10–11 Lemma 4.2 · pp. 11–12 Proposition 4.3 · pp. 12–14

2. A small threshold creates an lc center where all low weights vanish

Assume $\operatorname{lct}_o(C;\mathfrak m_o)\le3/2$. Choose a basis $f_i$ of the homogeneous functions with $0<y<D$ and put $d_0=\max\operatorname{wt}(f_i)<D$. Power them to a common weight $B$, obtaining an ideal $\mathfrak b$. Its threshold $t=\operatorname{lct}_o(C;\mathfrak b^{d_0/B})$ satisfies $0<t\le3/2$ and $td_0<5/2$. Averaging sufficiently many general members constructs an effective homogeneous boundary $G$ of total weight $td_0$. The pair $(C,G)$ is lc, and every $f_i$ vanishes on every lc center. The argument does not assume that $\mathfrak b$ is maximal-ideal primary.

On the grading extraction, $A_{C,G}(F)>0$ for every divisor centered exactly at $o$: the strict inequality $td_0<5/2$ makes the exceptional coefficient difference positive. Thus $o$ itself is not an lc center. A minimal center $Z$ has $1\le\dim Z\le3$. Affine subadjunction applies because $C$ is normal affine, $G$ is an effective rational boundary, $K_C+G$ is $\mathbb Q$-Cartier, and the pair is lc. The resulting klt pair $(Z,\Delta_Z)$ makes $Z$ normal with rational singularities, hence Cohen–Macaulay.

Proposition 5.1, boundary and lc centers · pp. 15–16 [FG12, Theorem 7.2, p. 11]

3. Check the nonvanishing hypothesis in each center dimension

Give $T=S/I_Z$ the induced grading. Its stabilizers equal those of the same points in $C$, so $\lambda m<1$ remains valid even if the action is ineffective. However, $T_y$ is an image of $S_y$, and all $f_i$ vanish on $Z$, so $T_y=0$ for $0<y<D$.

If $\dim Z=1$, the normal graded curve has ring $\mathbb C[z]$, and the stabilizer bound forces the weight of $z$ below one, a contradiction. If $\dim Z=2$, a surface admitting a klt boundary is $\mathbb Q$-factorial; removing the effective boundary makes $Z$ itself $\mathbb Q$-Gorenstein klt. A homogeneous generator of a reflexive canonical power has positive weight. Sending it to $1$ gives the required weight-lowering injection.

If $\dim Z=3$, then $U=Z\setminus\{o\}$ is a Cartier divisor in smooth $C\setminus\{o\}$. Choose a nonzero low-weight function $f$, put $b=\operatorname{wt}(f)<D$ and $q=\operatorname{ord}_Z(f)\ge1$. Equivariant adjunction and multiplication by $f$ give $\omega_U^{\otimes q}\to\mathcal O_U$ of degree $b-q(5/2)<0$. Reflexive extension across the vertex yields $\omega_T^{[q]}\hookrightarrow T$. No global $\mathbb Q$-Gorenstein hypothesis on $Z$ is required. In dimensions two and three, Proposition 4.3(b) now contradicts low-weight vanishing, proving the strict threshold bound.

Proposition 5.1, center dimensions · pp. 16–17 [dFH09, Proposition 7.2, p. 16; Corollary 7.15, p. 18]