Valuative independence for Calabi--Yau varieties

Abstract: We construct valuatively independent bases for the space of sections of an ample line bundle on a log Calabi--Yau pair over a discretely valued field and the space of regular functions on an affine CY pair with maximal boundary. While the bases are not in general unique, they induce canonical functions on the respective skeletons and are expected to agree with tropicalizations of theta functions when they exist. The proof uses techniques from the study of higher rank degenerations in K-stability.

• The paper presents that for any ample line bundle on a Calabi–Yau variety, there exists a basis whose valuation structure mimics the toric monomial–valuation pairing.
• It introduces multi-degeneration techniques by constructing flat multigraded Rees algebras to achieve simultaneous diagonalization of valuations.
• The work bridges algebraic, non-Archimedean, and metric geometry, providing key insights for mirror symmetry, K-stability, and moduli compactifications.

Valuative Independence for Calabi–Yau Varieties: An Expert Synthesis

Introduction and Motivations

The paper "Valuative independence for Calabi–Yau varieties" (2604.27890) by Blum and Liu establishes fundamental existence and structural results concerning canonical bases in the function theory of Calabi–Yau (CY) varieties and their degenerations. Central to the investigation is the notion of valuative independence: the property that a basis of sections of (powers of) an ample line bundle, or a basis of regular functions on a log CY pair, interacts with all valuations in the essential skeleton in a manner analogous to the monomial–valuation pairing in toric geometry.

Canonical bases, exemplified by theta functions in the Gross–Siebert program [GHK15, GS26], play a critical role in the construction and understanding of mirror symmetry, particularly in the explicit description of mirrors via graded section rings. Valuative independence articulates a refinement: not only do distinguished bases exist, but their interaction with non-Archimedean and birational geometric invariants (the essential skeleton) admits a strong combinatorial description. Previously, such bases existed only in formal toric, low-dimensional, or cluster-theoretic settings [Man16, CMMM25, KY24].

Main Results

The authors prove two principal existence results:

1. Existence of Valuatively Independent Bases in Pandegeneration and Affine Settings.

Given a projective log canonical (lc) CY pair $(X_K, B_K)$ over the fraction field $K$ of a (possibly complete) DVR $R$ (essentially of finite type over an alg. closed field $k$ of char $0$), and an ample line bundle $L_K$, there exists, for each $m > 0$, a $K$-basis $\{\theta_i\}$ of $H^0(X_K, mL_K)$ such that for every $K$0 in the essential skeleton $K$1,

$K$2

for coefficients $K$3 [Theorem 1].

In affine situations—specifically, for open loci $K$4 in a (possibly non-compact) CY pair $K$5, where $K$6 is a maximal boundary or contains a sufficiently ample effective divisor—the space of regular functions $K$7 possesses a similarly valuatively independent basis [Theorem 2].

Notably, these bases are not unique; however, each basis determines a well-defined, piecewise-affine tropicalization map on the essential skeleton, invariant under changes of basis up to permutation and translation.

2. Uniform Simultaneous Diagonalization via K-Stability and Degeneration Techniques

Valuative independence is interpreted as a simultaneous diagonalization property for the filtrations of the section space induced by each divisorial valuation in the skeleton. The technical heart of the paper constructs higher-rank degenerations—families over multi-parameter bases $K$8—to simultaneously encode the data of multiple degenerations associated to different valuations. Flatness of the corresponding multigraded Rees algebras is established via the minimal model program for lc pairs over multidimensional bases [HX13, Xu21], resulting in simultaneous diagonalizability (valuative independence) of the corresponding filtrations.

Technical Contributions

Construction of Multi-Degenerations and Multigraded Flatness

A key innovation lies in the explicit construction of families over

$K$9

gluing together a collection of degenerate models so that combinatorial data of all relevant valuations is respected (Section 4). This enables realization of the Rees algebra (and associated graded) of multiple filtrations as the relative section ring of a globally flat family, whose flatness properties directly yield simultaneous diagonalization.

By systematically transferring the problem to Rees algebras, criteria from commutative algebra and equivariant geometry ensure that a basis diagonalizing all filtrations exists. This approach extends to the setting where $R$0 is a complete DVR via Artin approximation and the theory of algebraic stacks, ensuring the applicability of the results to general degenerations.

Uniqueness of Induced Tropical Functions

An important structural property, thoroughly proved, is that while valuatively independent bases are generally not unique, the tropicalizations they induce on the essential skeleton are canonical—invariant up to natural changes. The independence from choices of models and metrics is made precise, and in the case of maximal degenerations, the tropical differences of basis elements are shown to be non-constant, reflecting the expected combinatorial richness (cf. maximal dimension of skeletons in the context of the SYZ conjecture).

Functoriality and Relation to Mirror Symmetry and the SYZ Conjecture

The existence of valuatively independent bases is linked to tropical and canonical bases arising in the Gross–Siebert approach and beyond, and is shown to be consistent with expectations from mirror symmetry, notably the existence and behavior of theta functions [GHK15, GS26, KY24]. The algebraic existence results here are demonstrated to be crucial inputs for recent advances on the metric SYZ conjecture [Li25, Li26], where they underlie the comparison property for non-Archimedean Monge–Ampère equations, critical for constructing semi-flat metrics and realizing the non-Archimedean SYZ fibration [Li23, HJMM24].

Implications and Future Directions

This paper reveals that valuative independence is a robust and general phenomenon for CY degenerations, that is deeply rooted in birational and non-Archimedean geometry, and is structurally controlled by properties that enjoy stability under suitable moduli. It demonstrates that the hidden toric combinatorics anticipated in mirror symmetry not only arise in the function theory of toric and cluster varieties but are native to all sufficiently degenerate Calabi–Yau spaces.

Implications include:

• Mirror Symmetry: The existence of canonical, tropical bases provides a strong algebraic foundation, expected to synchronize with enumerative constructions and theta functions in explicit mirror constructions.
• Metric Geometry: These valuations and the basis properties translate, via non-Archimedean geometry, to data about the limiting behavior of Ricci-flat metrics, thus feeding directly into the geometric SYZ program.
• K-stability and Moduli: The methods used—multi-degenerations, flat Rees modules, and MMP over higher-dimensional bases—are anticipated to extend to finer questions about K-stability, moduli compactifications, and singularity invariants in the minimal model program.

Key numerical and structural conclusions: The bases exist for all powers and, when the boundary is maximal, the induced functions exhibit full combinatorial variability rather than degeneracy; in all cases, the function theory recovers maximal toric-type behavior in a non-toric setting.

Conclusion

Blum and Liu (2604.27890) systematically establish the existence and core properties of valuatively independent bases on CY degenerations, using K-stability-degeneration techniques, flatness of multigraded modules, and deep geometric insights. This resolves several speculative points in both mirror symmetry and the minimal model program, and lays a foundation for further developments in the algebraic, non-Archimedean, and metric theory of CY varieties and their mirrors. The results reinforce the paradigm that combinatorial and toric structures underlie a wide class of degenerations and should be regarded as intrinsic to their function theory.