NRVE-Acc: ACC for Normalized Volumes

• NRVE-Acc is the ACC for local normalized volumes of klt singularities, characterizing their accumulation behavior under DCC-constrained boundary coefficients.
• It leverages δ-plt blow-ups and Lipschitz estimates to ensure uniform control of volumes in analytically bounded families.
• Dimension-specific results confirm ACC for surfaces and discreteness for threefold terminals, highlighting its impact on singularity classification.

NRVE-Acc (Ascending Chain Condition for Local (Normalized) Volumes)

The Ascending Chain Condition (ACC) for local (normalized) volumes, abbreviated here as NRVE-Acc (Editor's term), concerns the structure and accumulation behavior of the set of normalized volumes attached to Kawamata log terminal (klt) singularities. The central theme is the ACC property for the local volumes $\operatorname{vol}(x,(X,\Delta))$ under the constraint that the coefficients of $\Delta$ are drawn from a set $I$ obeying the Descending Chain Condition (DCC). Substantial progress was made by Han, Liu, and Qi, who established the ACC conjecture for analytically bounded families of klt germs, verified the existence of uniform $\delta$-plt blow-ups under quantitative hypotheses, and obtained unconditional results in dimension 2 and for three-dimensional terminal singularities (Han et al., 2020).

1. Definitions and the Folklore ACC Conjecture

Let $(X, \Delta)$ be an $n$-dimensional klt germ at a closed point $x \in X$, where $X$ is normal, $K_X+\Delta$ is $\mathbb{R}$-Cartier, and every prime divisor $E$ over $X$ satisfies

$$A_{X,\Delta}(E) = 1 + \operatorname{coeff}_E(K_Y - p^*(K_X+\Delta)) > 0$$

for some log resolution $p\!: Y \to X$. The coefficients of $\Delta = \sum a_i D_i$ are assumed to be in $I \subset [0,1]$, which is DCC or finite.

A real valuation $v$ on $K(X)$ centered at $x$ has a log discrepancy $A_{X, \Delta}(v) \in \mathbb{R}_{>0}$ and a volume

$$\operatorname{vol}_x(v) = \lim_{m \to \infty} \frac{\ell(\mathcal{O}_{X,x}/\mathfrak{a}_m(v))}{m^n/n!}, \quad \mathfrak{a}_m(v) = \{ f \mid v(f) \geq m \}.$$

The normalized volume of $v$ is defined by

$$\widehat{\operatorname{vol}}_{x,(X,\Delta)}(v) = \begin{cases} A_{X,\Delta}(v)^n\;\operatorname{vol}_x(v) & \text{if }A_{X,\Delta}(v)<\infty,\ +\infty & \text{otherwise}. \end{cases}$$

The local (normalized) volume of the germ is then

$$\operatorname{vol}(x,(X,\Delta)) = \inf_v \widehat{\operatorname{vol}}_{x,(X,\Delta)}(v).$$

ACC conjecture for local volumes: For fixed $n$ and DCC set $I \subset [0,1]$, consider combinations

$$\operatorname{Vol}_{n,I} = \{ \operatorname{vol}(x,(X,\Delta)) \},$$

where $(X,\Delta)$ runs over all $n$-dimensional klt germs with boundary coefficients $a_i \in I$. The conjecture posits:

1. If $I$ is finite, the only accumulation point of $\operatorname{Vol}_{n,I}$ is $0$.
2. If $I$ satisfies DCC, then $\operatorname{Vol}_{n,I}$ satisfies ACC.

2. Analytically Bounded Families and Main ACC Theorem

An analytically bounded $\mathbb{Q}$-Gorenstein family consists of a flat family

$$\pi\!: \mathcal{X} \to B, \quad \Delta \subset \mathcal{X}$$

where $\mathcal{X}$ is normal, $K_{\mathcal{X}/B} + \Delta$ is $\mathbb{R}$-Cartier, every fiber $(X_b,\Delta_b)$ is klt of fixed dimension $n$, and locally along the section the germs lie in a bounded analytic class.

Theorem A (Han–Liu–Qi Theorem 1.2): Let $\pi\!: (\mathcal{X},\Delta) \to B$ be a $\mathbb{Q}$-Gorenstein bounded family of $n$-dimensional klt germs, and $I \subset [0,1]$ a DCC set. Then

$$\operatorname{Vol}_{B,I} = \left\{ \operatorname{vol}(x,(X,\Delta)) \mid (X,\Delta) \cong \text{fiber of }\pi,\, \operatorname{coeff}(\Delta) \subset I \right\}$$

satisfies:

• If $I$ is finite, $\operatorname{Vol}_{B,I}$ has no nonzero accumulation points.
• If $I$ is DCC, then $\operatorname{Vol}_{B,I}$ satisfies ACC.

As a consequence, any analytically bounded family of klt germs in fixed dimension affords only finitely many positive local volumes. This applies notably to smooth germs and their volumes with DCC boundary.

3. δ-plt Blow-Up Conjecture and Its Resolution

A $\delta$-plt blow-up of a klt germ $x \in (X,\Delta)$ is a projective birational morphism

$$p\!: Y \to X, \quad E = \operatorname{Exc}(p) \cong \mathbb{P}^{n-1}$$

such that $p$ is an isomorphism off $x$, $p^{-1}(x) = E$, $(Y, E + p_*^{-1} \Delta)$ is $\delta$-plt near $E$ (all discrepancies $> \delta$), and $-E$ is $p$-ample.

Conjecture B: Given $n, \varepsilon > 0$, and $\delta > 0$, there exists $d > 0$ such that any $n$-dimensional klt germ $(X,\Delta)$ with nonzero $a_i \geq \varepsilon$ and local volume $\operatorname{vol}(x,(X,\Delta)) > \delta$ admits a $d$-plt blow-up.

Theorem B (Theorem 1.7): Under analytic boundedness as in Theorem A, Conjecture B holds. Thus, in any bounded family, local volumes bounded below guarantee the existence of a $\delta$-plt blow-up with uniform parameters.

4. Dimension-Specific Results: Surfaces and Threefold Terminals

Theorem C (n=2): For klt surface germs ($n=2$), the full ACC for local volumes holds without boundedness assumptions. Explicitly, for any DCC $I \subset [0,1]$,

$$\operatorname{Vol}_{2,I} = \{ \operatorname{vol}(x,(X,\Delta)) \mid x\in (X,\Delta) \text{ klt surface, } \operatorname{coeff}(\Delta)\subset I \}$$

satisfies ACC, and for finite $I$ the only accumulation point is $0$. This follows from the explicit classification of plt blow-ups on surface germs, a parameter count for log Fano pairs on $\mathbb{P}^1$, and the Lipschitz dependence of local volumes on boundary coefficients.

Theorem D (Terminal threefolds): For terminal 3-fold singularities (no boundary), the set

$$\{ \operatorname{vol}(x,X) \mid x \in X \text{ 3-fold terminal} \}$$

is discrete in $\mathbb{R}_{>0}$—its only accumulation point is $0$. The proof reduces to the surface case by passing to the index-one cover and analyzing the analytic structure via hypersurfaces with involution.

5. Techniques: Constructibility, Semi-Continuity, and Lipschitz Estimates

Key ingredients of the analysis include:

• Minimizers: A global minimizer of $\widehat{\operatorname{vol}}_{x,(X,\Delta)}(v)$ exists, unique up to scaling, and always quasi-monomial (Blum–Xu–Zhuang). Divisorial minimizers correspond to K-semistable Kollár components (Li–Xu).
• Constructibility/Semicontinuity: In a $\mathbb{Q}$-Gorenstein family, $b \mapsto \operatorname{vol}(x_b,(X_b,\Delta_b))$ is constructible and lower-semicontinuous (Xu, Blum–Liu).
• Lipschitz-type estimates: The local volume varies Lipschitz continuously under small changes to boundary coefficients within a fixed analytic germ and coefficients in $[\varepsilon,1]$ (Theorem 5.1).
• Truncation: If the local volume is bounded below and coefficients are $\geq \varepsilon > 0$, one can truncate divisors by analytic jets (for large $k$) while preserving local volume and existence of $\delta$-plt blow-ups (Theorem 6.2). This enables reduction from families with DCC boundary to ones with finite coefficients.
• Volume/lct bounds: Sub-additivity and Izumi-type inequalities provide

$$\operatorname{vol}(x,(X,\Delta)) \leq c \cdot \operatorname{lct}(X,\Delta;\Delta)$$

with uniform $c$ for a bounded family (Theorem 4.1), allowing passage from volume bounds to log-canonical threshold (lct) bounds.

• Cartier index bounding: Existence of uniform $\delta$-plt blow-ups and boundedness of complements (Birkar) give a uniform bound on the Cartier index within a log-bounded family (Theorem 1.10).

6. Examples and Sharp Constants

Illustrative examples include:

• Surface case sharp bounds: For $(\mathbb{A}^2,\Delta)$,

$$8\,\operatorname{lct}(0,\mathbb{A}^2,\Delta) \geq \operatorname{vol}(0,\mathbb{A}^2,\Delta)$$

so the sharp constant $c_{\min}(2)=8$.

• Quotient surface singularities: Type $\frac{1}{m+1}(1,m)$ surface singularities display volume accumulation at $0$ but nowhere else.
• Threefold case: Three-dimensional Gorenstein terminals reduce to the 2-dimensional cone over $(\mathbb{P}^1, \frac{1}{2}D)$ via index-one covers and involution, linking threefold terminal volumes to the surface ACC.

7. Significance and Concluding Remarks

The NRVE-Acc property formalizes the rigidity of local volumes within classes of klt germs, showing that, up to accumulation at zero, only finitely many values arise in analytically bounded families or for surfaces and threefold terminals. Existence of minimizers, their link to K-semistability (Kollár components), volume-lct relations, and explicit reduction via truncation are essential features in the argument. These results underlie further advances in the theory of singularities and birational geometry, provide uniformity statements for families of singularities, and connect to the study of K-stability and moduli of Fano varieties (Han et al., 2020).