On the normalized local volume of a non-closed point

Abstract: In this paper, we show that the normalized local volume of a non-closed point can be expressed in terms of the normalized local volumes of closed points. This confirms a folklore conjecture stating that the normalized local volume carries no additional information at a non-closed point.

• The paper establishes a scaling law demonstrating that the normalized local volume at a non-closed point can be precisely reconstructed from volumes at general closed points.
• It employs étale and henselian techniques to verify a long-standing conjecture, confirming that non-closed points carry no extra singularity data.
• The result simplifies calculations in birational geometry, enhancing methods in studying K-stability, Kähler–Einstein metrics, and the minimal model program.

Normalized Local Volume at Non-Closed Points: A Reduction to the Closed Case

Introduction

The paper "On the normalized local volume of a non-closed point" (2604.27484) establishes a precise relationship between the normalized local volume at a non-closed point in an algebraic variety and the corresponding volumes at general closed points of its closure. The normalized local volume, originally defined for closed points, is a key invariant in birational geometry and the study of singularities, playing essential roles in recent advances on K-stability, Kähler–Einstein metrics, and the minimal model program. By rigorously confirming a conjecture that the invariant at non-closed points carries no new information beyond that of closed points, the work completes an important aspect of the theory and justifies the practice of restricting attention to closed points when investigating local volumes.

Background and Problem Statement

Given a klt pair $(X, \Delta)$ over an uncountable field, the normalized local volume at a closed point $x$ is defined by

$$\widehat{\mathrm{vol}}(x, X, \Delta) = \inf_{\nu \in \mathrm{Val}^*_{X,x}} A_{X, \Delta}(\nu)^{n} \cdot \mathrm{vol}(\nu),$$

where $A_{X, \Delta}(\nu)$ denotes the log discrepancy and $\mathrm{vol}(\nu)$ the valuation volume. This concept, originally motivated by the study of metric tangent cones to Kähler–Einstein singularities and by volume minimization problems [Li18, Li17], proved central in the proof of existence and uniqueness of minimizers [Blu18, XZ21], ACC properties [HLQ23], and deformation-invariance results [BL21, XZ25].

While the definition is standard at closed points, attempts to generalize $\widehat{\mathrm{vol}}$ to non-closed points led to a natural question (cf. [LLX20, LX20]): does this generalization encode any new data or can the value at a non-closed point be reconstructed from those at nearby closed points?

Main Theorem and Its Consequences

The core result proves that the normalized local volume at a non-closed point is entirely determined by the values at general closed points of its closure. Specifically, if $x \in X$ is a (possibly non-closed) point with closure $Z = \overline{\{x\}}$, $d = \dim Z$, $n = \dim X$, and $x$0 is a general closed point, then

$x$1

This exactly confirms [LLX20, Conj. 6.7] and establishes a scaling law for local volumes under passage from the non-closed to closed points of the closure.

A crucial aspect of the proof is the globalization and specialization techniques, utilizing étale neighborhoods, henselization, and the machinery of $x$2-Gorenstein families of klt singularities. The argument incorporates the lower semicontinuity of normalized volumes [BL21], the flatness of Kollár components, and the invariance of singularity invariants in families [Xu20, HLQ23]. The perspective is deeply rooted in the reduction to "very general" points, which is necessary for technical reasons and relies on the uncountability of the ground field.

Technical Highlights

The study invokes several deep results and tools, most notably:

• Graded sequences of primary ideals and Samuel multiplicities: The construction of normalized local volume fundamentally involves passing to multiplicities along primary ideals, and multiplicity limits as in [Cut14, ELS03].
• Valuative criterion and approximation: The infimum in the volume definition can be taken over divisorial valuations, and log canonical thresholds can be computed by such valuations [LX20].
• Étale and henselian methods: Following [Kim26q] and standard étale descent (cf. Stacks Project), the authors build étale neighborhoods and sections to pass from non-closed to closed points, ensuring normality and the $x$3-Gorenstein property persists in families.
• Kollár component globally in families: Flatness and the spreadability of Kollár components are used to control normalized volumes in families and ensure the crucial scaling property.

A detailed calculation in the proof shows that the Samuel multiplicity of specially constructed families of ideals agrees with the expected scaled volume, using results on flatness and homological algebra.

Implications and Outlook

The main result definitively establishes that, from the standpoint of normalized local volume, no extra data is contained at a non-closed point compared to sufficiently general closed points in its closure. This not only streamlines future research—allowing restriction to closed points—but also clarifies the topological robustness of the normalized local volume invariant under specialization and generalization.

Practically, this scaling property facilitates computations and constructions in birational geometry and moduli problems, where invariants are required to be compatible in families and under degenerations. The result harmonizes with the deformation and boundedness properties of singularities established in recent years [BL21, XZ25, HLQ23].

From a theoretical standpoint, this theorem suggests that any subtle extensions of the local volume theory must come from entirely new invariants, rather than from extensions to non-closed loci. It further strengthens the connection between singularity theory, valuation theory, and algebraic families, with implications for the study of moduli of Fano varieties, K-stability, and higher-dimensional minimal model program.

Future directions may include generalized volume invariants in mixed characteristic, singularities beyond klt, and the behavior in non-reduced or stack-theoretic contexts.

Conclusion

This paper settles a fundamental conjecture regarding the normalized local volume at non-closed points by showing it reduces, via an explicit scaling law, to the volumes at general closed points of the closure. It attests to the sufficiency of studying closed points for normalized volumes in birational geometry and further consolidates the invariant's deformation and specialization properties. This clarification strengthens the conceptual and technical toolkit available for the investigation of singularities and their moduli in algebraic geometry.