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Discreteness of volumes of divisors on Calabi-Yau type varieties

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For fixed dimension and singularity bound, every integral divisor on a Calabi–Yau pair has volume in a discrete set depending only on those two numbers.

desk verdict Important conjecture, plausible strategy, but the written proof has a false lemma and a load-bearing induction gap—worth refereeing, not accepting as is. read the letter →

arxiv 2508.05082 v1 pith:LMLRYXPX submitted 2025-08-07 math.AG

Junpeng Jiao This is my paper · ORCID
classification math.AG MSC 14J3214E3014C20
keywords Calabi–YaupairsdivisorvolumeslogcanonicalsingularitiesboundednessMorifiberspacesbundleformulaCartierindexBirkarconjecture
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

The paper claims that divisor volumes on Calabi–Yau pairs are rigid: once the dimension $d$ and a lower bound $\epsilon$ on the log canonical threshold are fixed, every integral divisor on every $d$-dimensional $\epsilon$-lc Calabi–Yau pair has volume in a discrete set $\mathcal{C}$ depending only on $d$ and $\epsilon$. If true, this means the possible volumes cannot accumulate and are determined without knowing the coefficients of the boundary divisor. The proof establishes a stronger geometric statement: the family of such pairs equipped with an ample divisor of bounded volume is log bounded, first up to codimension-one isomorphism and then outright. This boundedness also proves the conjecture, attributed in the paper to Birkar, that polarized log Calabi–Yau pairs of fixed dimension, singularity, and degree bound form a bounded family.

What carries the argument

The carrying object is the family $\mathcal{C}(d,v,\epsilon)$: $d$-dimensional $\epsilon$-lc Calabi–Yau type couples $(X,A)$ with $A$ ample integral and $\mathrm{vol}(A)\le v$. The engine is a chain of boundedness results. Theorem 3.2 fixes $\delta>0$ such that $K_X+\delta A$ is never pseudo-effective. Theorem 3.3 proves $\mathcal{C}(d,v,\epsilon)$ is log bounded in codimension one by induction on dimension: it runs a $K_X$-MMP with scaling of $A$ to reach a Mori fiber space $W\to Z$, applies the canonical bundle formula to produce a generalized pair $(Z,C+R)$, and invokes boundedness of Fano-type fibrations to control $W'$ and the strict transform of $A$. Theorem 3.5 upgrades this to full l

What would settle it

Check the induction step in Theorem 3.3 on the base $(Z,C')$ of a Mori fiber space: if $K_Z+C+R$ is ample and not $\mathbb{Q}$-linearly trivial, and the dimension-$(d-1)$ log boundedness statement applied to $(Z,A_Z)$ fails for some family with fixed $d,\epsilon,v$, the induction collapses. Concretely, look for a family where $A_Z=p(K_Z+C+R)$ has bounded volume but unbounded Cartier index, since the proof needs that index bounded.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for fixed $d\in\mathbb{N}$ and $\epsilon\in(0,1)$, there is a discrete set $\mathcal{C}=\mathcal{C}(d,\epsilon)$ such that for every $d$-dimensional $\epsilon$-lc Calabi–Yau pair $(X,B)$—meaning $K_X+B\sim_{\mathbb{Q}}0$ and the pair has log discrepancy at least $\epsilon$—and every integral divisor $A$ on $X$, one has $\mathrm{vol}(A)\in\mathcal{C}$. Theorem 1.2 packages the same result as boundedness: for fixed $d,\epsilon,v$, the varieties admitting an $\epsilon$-lc Calabi–Yau boundary $B$ and an ample integral divisor $A$ with $A^d\le v$ form a bounded family. The proof introduces the auxiliary set $\mathcal{C}(d,v,\epsilon)$ of couples $(X,A)$ with $X$

Load-bearing premise

The proof's induction step assumes the lower-dimensional statement also applies to the base of a Mori fiber space, but that base has ample log canonical class rather than being Calabi–Yau; the paper states the induction only for Calabi–Yau type pairs, so a stronger unstated induction hypothesis is needed.

Editorial extensions

If this is right

  • For fixed $d$ and $\epsilon$, all integral divisor volumes on $d$-dimensional $\epsilon$-lc Calabi–Yau pairs belong to one discrete set, so no infinite accumulation of volumes can occur.
  • Within any bounded range $0\le\mathrm{vol}(A)\le v$, only finitely many volumes are possible, with the finite list depending only on $d,\epsilon,v$.
  • Polarized $\epsilon$-lc Calabi–Yau pairs of fixed dimension and degree bound form a bounded family, confirming the paper's named conjecture.
  • This boundedness holds without assuming the coefficients of the boundary $B$ lie in a finite set.
  • A divisor of bounded volume on such a pair has uniformly bounded Cartier index and becomes very ample after a bounded multiple, so the volume is actually the degree of a fixed embedding in a bounded family.

Reading between the lines

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

  • The induction step in Theorem 3.3 is applied to the base $(Z,C')$ of a Mori fiber space, but $K_Z+C+R$ is ample rather than $\mathbb{Q}$-linearly trivial. The paper's stated induction hypothesis covers only Calabi–Yau type pairs, so as written the proof needs a stronger, unstated induction statement for log-canonical models of general type.
  • The discrete set $\mathcal{C}$ is shown to exist but is not produced explicitly; effective versions would require explicit constants from the Cartier-index and base-point-freeness inputs, which the paper does not compute.
  • The same route should work for generalized pairs, since the canonical bundle formula already outputs a generalized pair $(Z,C+R)$; a generalized-pair induction hypothesis would make the argument formally uniform.
  • A bounded family of polarized Calabi–Yau pairs is a natural input for moduli and stability questions, although the paper itself stops at boundedness.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper claims two theorems about epsilon-lc Calabi-Yau pairs of fixed dimension d. Theorem 1.1 asserts that the volumes of all integral divisors on such a pair form a discrete set depending only on d and epsilon. Theorem 1.2 asserts Birkar's boundedness conjecture for polarized log Calabi-Yau pairs: for fixed d, epsilon, v, the varieties X admitting an epsilon-lc Calabi-Yau pair (X,B) and an ample integral divisor A with A^d <= v form a bounded family. The strategy is to introduce the set C(d,v,epsilon) of couples (X,A), prove it is bounded in codimension one via the recent result [JJZ25], then prove log boundedness in codimension one (Theorem 3.3) using a pseudo-effectivity threshold statement (Theorem 3.2) and an induction on dimension over a Mori fiber space. A final Cartier-index argument (Theorem 3.5) upgrades this to log boundedness, from which both main theorems follow.

Significance. If the proof is completed, Theorem 1.1 gives a strong, purely numerical discreteness result for divisor volumes on Calabi-Yau type varieties, and Theorem 1.2 proves a conjecture of Birkar. The manuscript's overall architecture is attractive: it reduces the discreteness of volumes to a uniform boundedness statement in codimension one and then to a uniform bound on Cartier indices. It also makes explicit use of many recent advances, including Birkar's boundedness results and [JJZ25]. As written, however, the proof has two load-bearing gaps: Theorem 3.2 is false as stated for pairs with B=0, and the induction in Theorem 3.3 applies a hypothesis to a base that is not of Calabi-Yau type. These gaps are not merely cosmetic; they occur exactly at the steps that establish log boundedness in codimension one.

major comments (2)
  1. [Section 3, Theorem 3.2 (and final paragraph of its proof)] Theorem 3.2 is false as stated. The set C(d,v,epsilon) includes varieties X of Calabi-Yau type with B=0, for instance K3 surfaces or abelian varieties, where K_X is numerically trivial. For such X and any ample integral divisor A, K_X + delta A is big for every delta>0, so no positive delta can make K_X + delta A non-pseudo-effective. In the proof, the contradiction 'K_{X_i} ~_Q -B_i is not pseudo-effective' uses implicitly that B_i is nonzero, but this is not part of the hypothesis. The argument can likely be repaired by stating Theorem 3.2 under the additional assumption that the chosen complement B is nonzero (equivalently, K_X is not pseudo-effective). Since Theorem 3.3 explicitly handles B=0 before invoking Theorem 3.2, the main proof may survive this repair, but the theorem as written must be corrected.
  2. [Section 3, Theorem 3.3, paragraph beginning 'Note we assume the result in dimension d-1'] The induction hypothesis is not strong enough for the application to the base of the Mori fiber space. The theorem being proved by induction concerns C(d-1,v',epsilon'), whose elements are epsilon'-lc Calabi-Yau type pairs, i.e. pairs satisfying K + B ~_Q 0. In the proof, however, after the canonical bundle formula the base satisfies K_Z + C' ~_Q K_Z + C + R, which is ample, not numerically trivial. Thus (Z,C') is not a Calabi-Yau type pair and (Z,A_Z) is not an element of the set to which the induction hypothesis applies. To make this step valid, one would need to formulate and prove a stronger induction statement covering pairs with K+B either numerically trivial or ample and with bounded volume of an ample integral divisor, or to obtain the desired log boundedness of the base by a separate argument. This is load-bearing because it is the step from which log boundedness in codimension
minor comments (5)
  1. [Lemma 2.4] The statement says 'there exists r in N depending only on d, t, vsuch that', but the data in the lemma are d, t, and alpha; the symbol v is not introduced and should be alpha. The proof correctly uses alpha.
  2. [Proof of Theorem 1.1] In the sentence 'then vol( v) is in a finite set', the argument of vol should be A, not v.
  3. [Section 2.2 heading] Typo: 'We need to following definition' should read 'We need the following definition'.
  4. [Section 1 and Definition 2.1] The paper uses 'integral divisor' for divisors that may not be Q-Cartier, while volume is normally defined only for Q-Cartier divisors. The authors should clarify whether all integral divisors considered are assumed Q-Cartier, or should state the convention used for vol(A) when A is only a Weil divisor.
  5. [Theorem 3.2, paragraph after equation h^*_i(mA_i)=g^*_i H_i+F_i] The transition 'After passing to a stratification of T, we may assume W -> T has a fiberwise log resolution' would benefit from a brief justification that the stratification can be chosen so that the construction of H' and the effectivity condition on E_t are preserved; this is standard but not immediate.

Circularity Check

1 steps flagged · score 2.0 of 10

No circular reduction found; main proof has an induction gap and a non-load-bearing self-citation.

  1. other [Section 3, proof of Theorem 3.3, paragraph beginning 'Note we assume the result in dimension d−1']
    "Note we assume the result in dimension d−1. Because (Z,C′) is ε′/2-lc and AZ is ample and integral and vol(AZ)≤v′, then (Z,AZ) is log bounded in codimension one."

    The induction hypothesis stated in Theorem 3.3 applies only to C(d−1,v′,ε′/2), whose members are (d−1)-dimensional ε′-lc Calabi–Yau type pairs, i.e. pairs with K+B∼Q0. But earlier in the same proof, K_Z+C+R is shown to be ample, and C′∼Q C+R, so K_Z+C′ is ample, not numerically trivial. Hence (Z,C′) is not Calabi–Yau type and the stated induction does not apply. The step silently requires a stronger induction statement covering pairs with K+B ample, which is neither stated nor proved. This is a formal gap in the proof, not a circular reduction of the theorem to itself.

full rationale

The paper's central claim (Theorem 1.1) is obtained by reducing to log boundedness (Theorem 3.5), which is proved from Theorem 3.3 and external boundedness results (Birkar, Hacon–McKernan–Xu, etc.). I found no step in which a predicted quantity is defined as the fitted input or in which a conclusion is assumed as a hypothesis. The citation to the author's own preprint [JJZ25, Thm 1.3] asserts boundedness in codimension one for C(d,v,ε); however, this assertion is not used in the proof of Theorem 3.3, which proves the stronger log boundedness by induction and external results, so the self-citation is not load-bearing. The one substantive defect is an induction gap in Theorem 3.3: the induction hypothesis covers Calabi–Yau type pairs, but it is applied to (Z,C′) with K_Z+C′ ample. This is an omitted proof of a stronger induction statement, not a circularity. The false statement of Theorem 3.2 for B=0 is also a correctness issue but is avoided in Theorem 3.3 by the B≠0 reduction. Overall, no significant circularity; score 2 reflects minor self-citation only.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters or invented entities. The proof rests on a network of deep external theorems, several of them very recent or unpublished. The axioms are standard for the MMP program but are strong tools that the paper does not prove. The most fragile is the application of invariance of plurigenera and the induction hypothesis in Theorem 3.3.

assumptions (8)
  • domain assumption Birkar's theorem [Bir23a, Thm 1.1] on birationality of |mA| for bounded polarized epsilon-lc pairs
    Invoked in Theorem 3.2 and Theorem 3.3 to obtain a fixed m such that |mA| defines a birational map. This is a deep external result.
  • domain assumption Birkar-BAB theorem for epsilon-lc Fano varieties
    Used in Theorem 3.3 to conclude that general fibers (W_g, tA_W_g) form a log bounded family.
  • domain assumption Canonical bundle formula for lc-trivial fibrations [Amb05]
    Used in Theorem 3.3 to define the generalized pair (Z,C+R) and to construct the Q-divisor C'.
  • domain assumption Boundedness of Cartier index [HLQZ25, Lemmas 3.13 and 3.14]
    Lemma 2.3 rests entirely on these two lemmas, which bound the Cartier index of integral divisors via the log canonical volume. These are from a very recent preprint.
  • domain assumption Birkar's boundedness of Fano type fibrations [Bir24, Theorem 1.2]
    Used in Theorem 3.3 to conclude that W' is bounded and that (X', ...) is log bounded.
  • domain assumption Birkar's theorem [Bir21, Theorem 1.8] on singularities of linear systems
    Used in Lemma 3.4 to find delta such that (X,B+delta D) remains lc.
  • domain assumption Invariance of plurigenera (log version, [HMX13, Theorem 1.8])
    Invoked in the proof of Theorem 3.2 to pass from big-ness of K+R+delta_i B to pseudo-effectivity of K+R as delta_i -> 0. The precise framework (possibly singular family) is not stated.
  • domain assumption [Bir23b, Theorem 1.1 and Corollary 1.3] on generalized pairs and multiplicities
    Used to establish that (Z,C+R) is generalized epsilon'-lc and to bound multiplicities of the fibration.

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Pith. "Pith review of Discreteness of volumes of divisors on Calabi-Yau type varieties." pith.science (2026). https://pith.science/paper/LMLRYXPX

@misc{pith2026250805082,
  author       = {Pith},
  title        = {Pith review of: Discreteness of volumes of divisors on Calabi-Yau type varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMLRYXPX}},
  note         = {Machine review of arXiv:2508.05082}
}
abstract

We study the volumes of divisors in Calabi--Yau type varieties. We show that given a klt Calabi--Yau pair $(X,B)$ and an integral divisor $A$ on $X$, the volume of $A$ is in a fixed discrete set depending only on the dimension and singularities of $(X,B)$. As an application, we prove a boundedness result of polarized log Calabi--Yau pairs which was conjectured by Birkar.

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