arxiv: 2604.27890 · v1 · submitted 2026-04-30 · 🧮 math.AG · math.DG

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Valuative independence for Calabi--Yau varieties

Harold Blum , Yuchen Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:40 UTC · model grok-4.3

classification 🧮 math.AG math.DG

keywords valuative independencelog Calabi-Yau pairsaffine Calabi-Yau pairsK-stabilityhigher rank degenerationstheta functionstropicalizationskeletons

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The pith

Valuatively independent bases are constructed for sections of ample line bundles on log Calabi-Yau pairs over discretely valued fields and for regular functions on affine Calabi-Yau pairs with maximal boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs valuatively independent bases for two spaces tied to Calabi-Yau varieties. For log Calabi-Yau pairs defined over a discretely valued field, the bases are built inside the space of sections of an ample line bundle. For affine Calabi-Yau pairs with maximal boundary, the bases are built inside the space of regular functions. Although these bases need not be unique, they produce canonical functions on the skeletons of the respective pairs and are expected to coincide with tropicalizations of theta functions whenever the latter are defined. The construction is carried out with higher-rank degeneration methods developed in the study of K-stability.

Core claim

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.

What carries the argument

Valuatively independent bases produced via higher-rank degeneration techniques from K-stability, which generate canonical functions on the skeletons.

Load-bearing premise

The log Calabi-Yau pair or affine pair with maximal boundary is defined over a discretely valued field, and higher-rank degeneration techniques from K-stability suffice to produce the valuatively independent bases.

What would settle it

An explicit log Calabi-Yau pair over a discretely valued field together with an ample line bundle for which no valuatively independent basis exists in the space of sections, or for which the induced functions on the skeleton fail to be canonical.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a construction of valuatively independent bases for sections on log Calabi-Yau pairs and regular functions on affine CY pairs with maximal boundary, using higher-rank K-stability techniques to produce canonical skeleton functions.

read the letter

The core contribution is a construction of valuatively independent bases for the space of sections of an ample line bundle on a log Calabi-Yau pair over a discretely valued field, plus the analogous statement for regular functions on affine Calabi-Yau pairs with maximal boundary. These bases are not claimed to be unique, but they are said to induce canonical functions on the respective skeletons and are expected to match tropicalizations of theta functions when those exist. The argument adapts methods from the study of higher-rank degenerations in K-stability. That is the new piece: a concrete way to extract these bases and the induced skeleton data in the log and affine settings. It sits in the intersection of non-archimedean geometry, tropical mirror symmetry, and K-stability, and the abstract suggests the construction is not just a restatement of earlier results on theta functions or skeletons. The approach looks consistent with the cited techniques, and there is no sign of internal circularity or invented inputs. The reliance on external degeneration results is standard and acceptable if the application is spelled out. The main limitation is that only the abstract is in front of me, so I cannot check the details of how the higher-rank methods are applied to the log pair or the affine maximal-boundary case, nor whether any technical gaps appear in the existence argument. The agreement with theta tropicalizations is stated as an expectation rather than a proven statement, which is fine for a first paper but will need examples or further work to land. This is specialized work aimed at people already working on Calabi-Yau degenerations, non-archimedean mirror symmetry, or tropical aspects of K-stability. A reader in those areas could use the construction for further calculations or comparisons. It is worth sending to peer review: the claim is a specific, non-obvious existence result in an active corner of the field, and the methods are from the right toolbox. If the proofs hold up, it will be a useful reference even if the theta-function link remains conjectural for now.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs valuatively independent bases for the space of sections of an ample line bundle on a log Calabi--Yau pair over a discretely valued field, as well as for the space of regular functions on an affine Calabi--Yau pair with maximal boundary. The construction relies on techniques from the study of higher-rank degenerations in K-stability. The resulting bases are not claimed to be unique but induce canonical functions on the respective skeletons and are expected to agree with tropicalizations of theta functions when such functions exist.

Significance. If the construction is correct, the result would provide a useful method for producing valuatively independent bases in the setting of log Calabi--Yau pairs and affine pairs with maximal boundary, linking K-stability degeneration techniques to tropical geometry and skeletons. This could facilitate further study of degenerations of Calabi--Yau varieties and their connections to mirror symmetry via theta functions, though the precise impact depends on the details of the proof and the scope of the pairs considered.

minor comments (2)

The abstract states that the bases 'induce canonical functions on the respective skeletons' but does not specify the precise mechanism or the definition of these functions; a dedicated section or subsection clarifying this would improve readability.
The expectation that the bases 'agree with tropicalizations of theta functions when they exist' is stated without reference to prior work on theta functions in this context; adding a brief discussion or citation in the introduction would strengthen the motivation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for acknowledging the potential utility of our construction in linking higher-rank degeneration techniques from K-stability to tropical geometry and skeletons of log Calabi--Yau pairs. We are pleased that the referee views the work as a useful method for producing valuatively independent bases, and we appreciate the recognition of its expected agreement with theta tropicalizations. Since the report raises no specific major comments or questions about the proof details, scope of pairs, or correctness of the arguments, we provide no point-by-point responses below. We remain available to address any further questions or clarifications the referee may have.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is a construction of valuatively independent bases for sections of ample line bundles on log Calabi-Yau pairs (and regular functions on affine CY pairs) over discretely valued fields. The abstract states that the proof 'uses techniques from the study of higher rank degenerations in K-stability,' which are presented as external methods rather than internally derived or fitted quantities. No equations, definitions, or self-citations in the provided abstract or description reduce the claimed bases to inputs by construction, nor do they rename known results or smuggle ansatzes via self-reference. The derivation is therefore self-contained against external benchmarks from K-stability literature, with no load-bearing steps that collapse to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions in algebraic geometry and K-stability; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption The pair is a log Calabi-Yau pair (or affine CY pair with maximal boundary) over a discretely valued field.
This is the setting in which the bases are constructed.
domain assumption Higher-rank degeneration techniques from K-stability apply to produce the valuatively independent bases.
The proof is stated to use these techniques.

pith-pipeline@v0.9.0 · 5362 in / 1452 out tokens · 40466 ms · 2026-05-07T05:40:41.235344+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Valuative independence and metric SYZ conjecture
math.AG 2026-05 unverdicted novelty 5.0

Assuming a canonical basis of the section ring satisfies valuative independence, the metric SYZ conjecture holds for polarised maximal degenerations of compact Calabi-Yau manifolds.

Reference graph

Works this paper leans on

7 extracted references · 3 canonical work pages · cited by 1 Pith paper

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