Complexity one varieties are cluster type

Jennifer Li; Jos\'e Ignacio Y\'a\~nez; Joshua Enwright

REVIEW 3 major objections 4 minor 3 cited by

Complexity one varieties are cluster type

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that log Calabi–Yau pairs of index one and fine complexity at most one are cluster type: crepant-birational to a toric log Calabi–Yau pair, extracting only log canonical places.

desk verdict Strong program paper with a real, load-bearing gap in the proof of Theorem 6: fine complexity one does not imply the ordinary complexity one that Lemma 2.38 requires. read the letter →

arxiv 2504.17369 v2 pith:SAAWPKAW submitted 2025-04-24 math.AG

Joshua Enwright , Jennifer Li , José Ignacio Yáñez This is my paper · ORCID

classification math.AG MSC 14E3014M25

keywords Calabi–YaupairscomplexitytoricgeometryclustertypevarietiesCoxringsFanoT-varietiesMorifibrespaces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Complexity is a number attached to a log Calabi–Yau pair that measures how far the variety is from being toric: complexity zero is toric, and the paper proves the next level of the hierarchy. Its main theorem says that a log Calabi–Yau pair of index one and fine complexity at most one is cluster type, meaning it admits a crepant birational map from a toric log Calabi–Yau pair that extracts only log canonical places. The paper also proves that varieties whose absolute complexity is exactly one are non-toric Fano type varieties with a cA-type singularity at the vertex of their Cox ring, and that each such variety has a finite surjective cover of degree at most two by a cluster type variety. Along the way, complexity below two forces Fano type, complexity below 3/2 forces a log Calabi–Yau structure of complexity one and index at most two, and Fano type T-varieties of T-complexity one admit a degree-60 cover by a variety carrying an open torus.

What carries the argument

The load-bearing object is the fine complexity $c(X,B)=\dim X+\operatorname{rank} W\operatorname{Div}^{alg}(X)-|B|$, together with its behaviour under birational contractions. For Theorem 6, the mechanism is a Mori fibre space sequence with at least $\dim X-1$ conic fibrations supplied by an external theorem; repeatedly applying the canonical bundle formula shows that both complexity and index-one status survive each fibration step, so a hypothetical complexity-one pair would push down to $\mathbb{P}^1$ and produce an impossible log Calabi–Yau pair. For Theorems 3 and 7, the mechanism is the Cox ring: a cA-type singularity at the vertex forces $\operatorname{Cox}(X)$ to be a hypersurface $K[x_1,\ldots,x_{d+1}]/f$ with either an $x_1x_2$ or an $x_1^2+x_2^2$ leading term; the first case gives an index-one cluster type pair directly, while the second case is resolved by a degree-two quasi-étale quotient that kills the 2-torsion difference between the two linear terms.

What would settle it

Run the minimal model program on the examples of Theorem 4 and on a complexity-one index-one threefold from the cited constructions: if any such pair has a Mori fibre space sequence with fewer than $\dim X-1$ conic fibrations, the quoted external input fails and the proof collapses. Independently, exhibiting an index-one fine-complexity-one log Calabi–Yau pair with no crepant birational map from a toric pair would refute Theorem 6 itself.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 6: if $(X,B)$ is a log Calabi–Yau pair of index one and fine complexity at most one, then $(X,B)$ is cluster type, i.e. there is a crepant birational map from a toric log Calabi–Yau pair to $(X,B)$ that extracts only log canonical places. The proof passes to a crepant birational model $X_0$ whose Mori fibre space sequence contains $\dim X-1$ conic fibrations; if the complexity were one, it would propagate down this sequence and land on $\mathbb{P}^1$ with a log Calabi–Yau structure of complexity one, which is impossible, so the complexity must be zero and the model toric. Theorem 7 extends this: when the absolute complexity equals one, a surjective finite morphism of degree at most two from a cluster type variety to $X$ exists. For Fano type $T$-varieties of $T$-complexity one, the same circle of ideas yields a finite morphism of degree at most 60 from a variety with a log Calabi–Yau sub-pair containing an open torus.

Load-bearing premise

The proof of the main cluster-type theorem rests on an unpublished result that every pair in question can be birationally simplified so that all but at most one step of its Mori fibre space sequence is a conic fibration; if that result fails or is misapplied, the conclusion of Theorem 6 is unsupported.

Editorial extensions

If this is right

If a log canonical pair has complexity $<2$, its underlying variety is Fano type and the vertex of $\operatorname{Spec}\operatorname{Cox}(X)$ is a compound Du Val singularity; for log Calabi–Yau pairs the boundary components generate $\operatorname{Cl}(X)_{\mathbb{Q}}$.
A normal projective variety has absolute complexity exactly one precisely when it is non-toric Fano type with a cA-type Cox-ring singularity, and in that case it carries a reduced log Calabi–Yau structure of complexity one and index at most two.
For every $n\ge 3$ there are Fano $n$-folds with cyclic divisor class group whose absolute complexity is exactly $3/2$, so $3/2$ is the next value after one.
If $(X,B)$ has index one and fine complexity at most one, it is cluster type; in the $\mathbb{Q}$-factorial case, $X\setminus B$ is covered up to codimension two by two algebraic tori.
A variety of absolute complexity one admits a surjective finite morphism of degree at most two from a cluster type variety, and a Fano type $T$-variety of $T$-complexity one admits a degree $\le 60$ cover by a normal projective variety with a log Calabi–Yau sub-pair and an open $n$-dimensional torus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the degree-two cover in Theorem 7 behaves like an étale resolution of the index obstruction; one testable refinement is that a crepant degree-two cover can always be chosen, making the 2-torsion in the divisor class group the only obstruction to $X$ itself being cluster type.
Since cluster type varieties are rational, Theorem 6 implies that every index-one, fine-complexity-one log Calabi–Yau pair is rational; the Cox-ring hypersurface description in Lemma 2.52 gives a concrete rationality test for such pairs.
The degree-60 bound in Theorem 8 is tied to the icosahedral orbifold fundamental group of $\mathbb{P}^1$ with weights $1/2,2/3,4/5$; one could search for examples showing whether that bound is sharp or whether smaller uniform bounds hold in special cases.
The hierarchy established here suggests a general principle beyond the proved statements: bounded complexity should force bounded covers by cluster type varieties, with the cover degree controlled by torsion in the divisor class group.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the good news: most of this paper is real progress. Theorem 2 extends the Brown–McKernan–Svaldi–Zong framework to complexity < 2, Theorem 3 gives a clean Cox-ring characterization of absolute complexity one, Theorem 4 supplies explicit examples in every dimension, and Theorem 8 is a clever T-variety application. Lemma 2.52 is a genuinely reusable statement, and the Cox-ring computations in Section 3 are detailed and convincing. If you work on the complexity program, this is a paper you want to read.

The problem is Theorem 6, the flagship result. The theorem assumes fine complexity ≤ 1. The proof invokes [29, Thm 5.6] to get a crepant model (X0,B0) with fine complexity ≤ 1 and a sequence of Mori fiber spaces with many conic fibrations. Then it says: assume the complexity equals 1, and "It follows from Lemma 2.38 that (X,B) has two components horizontal over X1." Lemma 2.38 is a statement about ordinary complexity: it uses |B| = dim X + ρ(X) − 1 and counts actual components of the boundary. Fine complexity one only gives the existence of a decomposition with norm dim X + ρ(X) − 1; the actual boundary can have larger norm. The paper itself notes α(X,B) ≤ c(X,B), so α = 1 implies c ≥ 1, not equality. There is no argument anywhere bridging fine complexity to ordinary complexity on the model (X0,B0).

So either [29] actually guarantees ordinary complexity one, in which case Theorem 6 is stated with the wrong hypothesis, or [29] only gives fine complexity one, in which case Lemma 2.38 is simply inapplicable. Either way, the contradiction step and the induction "continuing in this way" are unsupported as written. There is also a likely typo: the proof says "(X,B) has two components" where it almost certainly means "(X0,B0)", which adds to the confusion.

The dependence on recent preprints ([29], [31]) is not in itself a reason for suspicion, but it means the referee cannot just trust the cited theorems; they have to verify what notion of complexity appears in the hypotheses of [29, Thm 5.6]. That is exactly the kind of thing that can be a one-line fix or a genuine obstruction. I do not see in the paper any evidence of the bridge claim, so at the moment the proof of Theorem 6 does not go through.

Bottom line: this paper deserves a serious referee. Theorems 2, 3, 4, 5, and 8 are strong enough on their own to justify refereeing, and the shortcoming in Theorem 6 is probably fixable—but it needs to be fixed before the paper is accepted. A referee should focus on Section 4 and ask the authors to either (a) prove that index-one log CY pairs with fine complexity one admit a crepant model with ordinary complexity one, or (b) restate Theorem 6 with the stronger hypothesis. As written, the flagship theorem is not proved.

Referee Report

3 major / 4 minor

Summary. The paper studies the complexity invariants of log Calabi-Yau pairs introduced by Brown-McKernan-Svaldi-Zong. It proves Theorem 2, asserting that a log canonical pair with nef negative log canonical class and complexity less than 2 is of Fano type and has a Cox ring with a compound Du Val singularity at the vertex; Theorem 3, characterizing absolute complexity in [1, 3/2) through non-toric Fano type varieties with cA-type Cox singularities; Theorem 4, giving hypersurface examples with absolute complexity 3/2 in every dimension at least 3; and Theorem 5, showing that for Fano type varieties the absolute complexity is attained. The central result is Theorem 6: an index-one log Calabi-Yau pair of fine complexity at most one is cluster type. The paper then derives Theorem 7, giving a finite cover of degree at most two by a cluster type variety when the absolute complexity is one, and Theorem 8, a related statement for T-varieties of T-complexity one.

Significance. If the central arguments can be made sound, this is a substantial contribution to the geometric complexity program. Theorems 2-5 give new structural results and explicit examples, and Theorem 6 would extend the toric characterization of low-complexity pairs to the cluster type setting. The paper is ambitious and draws on sophisticated machinery: MMP, canonical bundle formula, Cox ring techniques, and recent work of Moraga on conic fibrations. The main theorem is genuinely novel and the finite-cover statements are appealing consequences. The proofs are detailed and the exposition is mostly careful. The paper would be significantly strengthened if the version of the key result from [29] used here were stated explicitly, since the proof currently depends on a preprint with no local statement of its hypotheses.

major comments (3)

[Section 4, proof of Theorem 6] The proof applies Lemma 2.38 to a pair (X0,B0) for which only the fine complexity is known to be 1. Lemma 2.38 requires ordinary complexity c(X0,B0)=1, and its proof uses the identity |B| = dim X + ρ(X) - 1, which is a statement about the actual coefficient sum of B, not about the existence of a decomposition with total norm and rank summing to dim X + ρ(X) - 1. Since fine complexity is bounded above by ordinary complexity, the assumption that fine complexity is 1 only gives c(X0,B0) >= 1, not equality. If [29, Theorem 5.6] also guarantees only fine complexity one, then the invocation of Lemma 2.38 is unjustified; if it guarantees ordinary complexity one, the paper must first prove that the fine complexity hypothesis implies ordinary complexity one on the chosen model. No such bridge is provided, so the contradiction argument and the induction that follows are unsupported.
[Section 2.5, Lemma 2.38] The proof assumes, after choosing the unique horizontal component S, that S is a log canonical center of (X,B), and later counts 'dim X + ρ(X) - 2 components of B' vertical. Both steps require coeff_S(B)=1 and, more generally, require all components of B to have coefficient one. These facts are not established by the hypotheses. The condition c(X,B)=1 bounds the weighted sum |B|, not the number of prime components, and a horizontal multi-section of degree d>2 would have coefficient 2/d from the relative canonical bundle formula and would not be a log canonical center. Since Lemma 2.38 is used in the proof of Theorem 6, this gap is load-bearing and should be repaired or the lemma reproved under an explicit reducedness assumption.
[Section 4, proof of Theorem 6] The reduction 'we may assume r = dim X - 1 and dim X_r = 1' is terse. If the Mori fiber space sequence produced by [29, Theorem 5.6] ends in a point rather than a curve, the proof should explain why the final step can be discarded while preserving the property that at least dim X - 1 of the remaining steps are conic fibrations. This is likely true by a dimension count, but without the explanation the later assertion that X_r is a smooth curve has no basis.

minor comments (4)

[Section 4, proof of Theorem 6] The sentence 'It follows from Lemma 2.38 that (X,B) has two components that are horizontal over X1' should refer to the crepant model (X0,B0), not to the original pair (X,B), unless the notation has been changed without comment.
[Throughout] The distinction between fine complexity and ordinary complexity is not always visually clear in the text, since both are typeset as c(X,B). A consistent notation such as an underlined symbol for fine complexity should be used in the published version to make arguments such as the one in Theorem 6 auditable.
[Section 4 and Section 5] The paper relies on [29, Theorem 5.6] as a black box, but does not state the precise hypothesis of that theorem or indicate whether it concerns ordinary complexity, fine complexity, or some birational variant. Since [29] is a recent preprint and is load-bearing for Theorem 6, a self-contained statement or at least a precise quotation should be included.
[Section 2.5, Lemma 2.38] In the proof of Lemma 2.38, the phrase 'There are dim X + ρ(X) - 2 components of B which are vertical' should say 'the total coefficient of the vertical part is ...' unless reducedness has been established; this is related to the major comment above.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; the cluster type theorem rests on external structural input [29] and the toric classification [8], with self-citations confined to terminology and a technical lemma.

full rationale

Walking the claimed derivation chain, I find no step in which a stated prediction is equivalent to its own input by construction. Theorem 6 is proved by first invoking [29, Theorem 5.6] to obtain a crepant model (X0,B0) with a Mori fiber space sequence and fine complexity at most one, then ruling out fine complexity one and concluding fine complexity zero, and finally applying [8, Theorem 1.2] to identify the zero-complexity model with a toric log Calabi--Yau pair. That is an application of an external structural theorem followed by an external toric classification, and it is exactly the defining property of a cluster type pair from Definition 2.23. The self-citations in the paper are not load-bearing for the central claim: [12] supplies the definition and terminology of cluster type, while [11] supplies Lemma 2.18 on behavior of complexity under birational contractions, which is used in secondary results such as Theorems 2, 3, and 5 rather than in the proof of Theorem 6. Neither of these citations is the conclusion of Theorem 6, and the proof does not reduce the cluster type assertion to a previously asserted equivalent formulation by the same authors. The principal external input [29] is a preprint by Moraga and is not authored by the present paper's authors, so it is not part of a self-citation chain. The referee-style concern that Lemma 2.38 is stated for ordinary complexity one while [29] guarantees only fine complexity one is a possible correctness or robustness gap in the proof, but it is an unproved bridge between input and conclusion, not a circular identification of input with output. Overall circularity is minimal: a low score reflecting minor, non-load-bearing self-citation, with no exhibited equation reducing to itself by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new geometric or physical entities. Its main external assumptions are a set of standard MMP results and several recent preprints in the same research program. The free-parameter count is zero; the axioms listed capture the key unproved inputs the reader must accept.

assumptions (6)

standard math Standard MMP results (existence of dlt modifications, Mori fiber spaces, basepoint-free theorem)
Used throughout; quoted from [22], [23], [7].
standard math Canonical bundle formula for log pairs as in [14]
Used in Lemmas 2.31, 2.32 and in proofs of Theorems 6 and 8.
standard math Theorem 1 of Brown-McKernan-Svaldi-Zong [8, Theorem 1.2, Corollary 1.3]
Used to identify toric pairs when complexity < 1 (e.g., end of Theorem 6).
domain assumption Lemma 2.18 from [11] (Enwright-Figueroa)
Self-cited result on complexity under birational contraction, used in Lemma 2.22 and Theorem 5.
domain assumption [29, Theorem 5.6] (Moraga, arXiv:2403.17251)
Existence of a crepant model with a Mori fiber space sequence with many conic fibrations; central input for Theorem 6. Recent preprint, not flagged as such.
domain assumption [31, Theorem 3.25] (Moraga)
Used in the proof of Theorem 8.

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Pith. "Pith review of Complexity one varieties are cluster type." pith.science (2026). https://pith.science/paper/SAAWPKAW

@misc{pith2026250417369,
  author       = {Pith},
  title        = {Pith review of: Complexity one varieties are cluster type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAAWPKAW}},
  note         = {Machine review of arXiv:2504.17369}
}

abstract

The complexity of a Calabi-Yau pair $(X,B)$ is an invariant that relates the dimension of $X$, the rank of the group of divisors, and the coefficients of $B$. If the complexity is less than one, then $X$ is a toric variety. We prove that if the complexity is less than two, then $X$ is a Fano type variety. Furthermore, if the complexity is less than 3/2, then $X$ admits a Calabi-Yau structure of complexity one and index at most two, and it admits a finite cover $Y \to X$ of degree at most 2, where $Y$ is a cluster type variety. In particular, if the complexity is one and the index is one, $(X,B)$ is cluster type. Finally, we establish a connection with the theory of $T$-varieties. We prove that a variety of $T$-complexity one admits a similar finite cover from a cluster type variety.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [29]

J. Moraga. Birational complexity and conic ﬁbrations, 2024, arXiv:2403.17251

arXiv 2024

[2] [31]

J. Moraga. Coregularity of Fano varieties. Geom. Dedicata, 218(2):Paper No. 40, 55, 2024. doi:10.1007/s10711-023-00882-z. 26 J. ENWRIGHT, J. LI, AND J. I. Y ´A ˜NEZ

work page doi:10.1007/s10711-023-00882-z 2024

[3] [1]

Abramovich and K

D. Abramovich and K. Karu. W eak semistable reduction in c haracteristic 0. Invent. Math. , 139(2):241–273, 2000. doi:10.1007/s002229900024

doi:10.1007/s002229900024 2000

[4] [2]

Altmann and J

K. Altmann and J. Hausen. Polyhedral divisors and algebr aic torus actions. Math. Ann. , 334(3):557–607, 2006. doi:10.1007/s00208-005-0705-8

doi:10.1007/s00208-005-0705-8 2006

[5] [3]

Altmann, J

K. Altmann, J. Hausen, and H. S¨ uss. Gluing aﬃne torus act ions via divisorial fans. Transform. Groups, 13(2):215–242,

work page

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