{"id":"36cf0f67-d411-4be9-8818-5bbf1de889e5","arxiv_id":"2508.05082","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Volumes of integral divisors on epsilon-lc Calabi-Yau pairs lie in a fixed discrete set, settling Birkar's boundedness conjecture for polarized log Calabi-Yau pairs.","lead":"This paper proves that volumes of divisors on Calabi-Yau varieties with mild singularities can take only finitely many possible values, depending only on dimension and singularity strength. It also proves a boundedness conjecture of Birkar for such varieties with a fixed polarization degree.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Induction in Theorem 3.3 applies to a base with K_Z+C' ample, not Calabi-Yau type; the stated induction hypothesis does not cover it","rationale":"The reader's weakest assumption identifies exactly the same obstacle: the induction hypothesis in Theorem 3.3 is applied to a base whose canonical class is ample rather than numerically trivial. This is load-bearing because the entire proof of log boundedness in Theorem 3.3 rests on this inductive step. The paper does not state or prove a stronger induction hypothesis covering log Fano type bases, so as written the argument is incomplete. The false statement of Theorem 3.2 for B=0 is also real but appears less critical because Theorem 3.3 handles the B=0 case separately before using Theorem 3.2; nonetheless it shows the theorem statements need tightening. Given the importance of the claimed results and the possibility that a stronger induction statement could be established, CONDITIONAL is the appropriate verdict: the central argument is plausible but the current text does not support the theorems as stated. My confidence is moderate; the gap may be fixable if the stronger induction can be formulated and verified.","tokens_in":9890,"tokens_out":7213,"duration_ms":91823,"concrete_test":"Formulate the stronger induction statement S(d): for every ε∈(0,1), v>0, every d-dimensional ε-lc pair (Y,B_Y) with K_Y+B_Y semiample (either ∼_Q 0 or ample) and every ample integral divisor A_Y with vol(A_Y)≤v is log bounded in codimension one. Then check whether the Mori fiber space step of Theorem 3.3 produces a base (Z,C') satisfying S(d−1): explicitly verify that (Z,C') is ε'/2-lc, K_Z+C' is semiample (here ample), and vol(A_Z)≤v' with v' depending only on d,ε,v. If all verifications go through, the induction closes; if the construction of C' from the moduli part does not preserve the required hypotheses, the proof has a genuine gap that no minor rewriting can fix.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main inductive step of Theorem 3.3 is used to prove that C(d,v,ε) is log bounded in codimension one. After running a K_X-MMP and a Mori fiber space h:W→Z, the paper writes K_W+tA_W ∼_Q h^*(K_Z+C+R) with K_Z+C+R ample. It then chooses C'∼_Q C+R with (Z,C') ε'/2-lc and says: “Note we assume the result in dimension d−1. Because (Z,C') is ε'/2-lc and A_Z is ample and integral and vol(A_Z)≤v', then (Z,A_Z) is log bounded in codimension one.” But the induction hypothesis, as stated, applies only to ε-lc Calabi–Yau pairs, i.e. pairs with K+boundary ∼_Q 0. Here K_Z+C' is ample, not numerically trivial, so (Z,C') is not Calabi–Yau type and the induction does not formally apply. To make the argument work one must prove a strictly stronger induction statement covering pairs with K+B either numerically trivial or ample, with bounded volume of an ample divisor; this is not stated or proved. The nearby false statement of Theorem 3.2 (for B=0, K_X+δA is always big, so K_X+δA is pseudo-effective) is a separate defect, but it is less central because Theorem 3.3 explicitly excludes B=0 before invoking Theorem 3.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10172,"tokens_out":14978,"duration_ms":177754,"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":[{"comment":"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.","section":"Section 3, Theorem 3.2 (and final paragraph of its proof)"},{"comment":"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","section":"Section 3, Theorem 3.3, paragraph beginning 'Note we assume the result in dimension d-1'"}],"minor_comments":[{"comment":"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.","section":"Lemma 2.4"},{"comment":"In the sentence 'then vol( v) is in a finite set', the argument of vol should be A, not v.","section":"Proof of Theorem 1.1"},{"comment":"Typo: 'We need to following definition' should read 'We need the following definition'.","section":"Section 2.2 heading"},{"comment":"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.","section":"Section 1 and Definition 2.1"},{"comment":"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.","section":"Theorem 3.2, paragraph after equation h^*_i(mA_i)=g^*_i H_i+F_i"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on several very recent preprints, including [Bir23b], [Bir24], and the author's own [JJZ25], as well as [HLQZ25]. Reliance on recent work is not itself a defect, but the editor may wish to ensure that the cited results are independently verified before publication. The two main gaps identified above are internal to the argument and can be addressed in a revision, but one of them requires a genuine strengthening of the induction statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper proves a real conjecture: Birkar's boundedness of polarized log Calabi-Yau pairs, and the discreteness of divisor volumes (Thm 1.1). Second, the written proof is not sound as is: Theorem 3.2 is false as stated, and the induction step in Theorem 3.3 jumps to a stronger statement than the one being proved.\n\nWhat's good. The main results are new and important. The reduction of discreteness to log boundedness plus effective base point freeness is clean: Theorem 3.5, with Lemma 3.4, is a nice argument. The paper uses the latest machinery (Han-Liu-Qi-Zhuang, the author's joint preprint [JJZ25], Birkar's boundedness of Fano fibrations) in a way that suggests the author knows the area and the tools well. For a specialist, the outline of the strategy is convincing.\n\nWhere it falls short. Theorem 3.2 says K_X+δA is not pseudo-effective for (X,A) in C(d,v,ε). That is false when B=0: a K3 or abelian variety has K_X numerically trivial, so K_X+δA is big (hence pseudo-effective) for every positive δ. The proof relies on K_X not being pseudo-effective, which only holds when B≠0. The author later treats B=0 separately in Theorem 3.3, so this is fixable by restating Theorem 3.2 with B≠0, but as printed it's simply wrong.\n\nMore serious: the induction in Theorem 3.3. After the MMP and the canonical bundle formula, the base (Z,C') satisfies K_Z+C' ample, not K_Z+C' ~Q 0. The stated induction hypothesis only covers ε-lc Calabi-Yau pairs (K+B ~Q 0). The line \"Because (Z,C') is ε'/2-lc and A_Z is ample and integral and vol(A_Z)≤v', then (Z,A_Z) is log bounded in codimension one\" does not follow from the induction as stated. You'd need a strictly stronger induction statement covering pairs with K+B ample, or a separate boundedness theorem for such pairs. That's load-bearing: the whole proof depends on the base being bounded. It may be repairable by proving the stronger statement, but it's not in the paper.\n\nThere's also a hand-wavy use of invariance of plurigenera near the end of Theorem 3.2; that is minor compared to the above but should be checked.\n\nBottom line: the paper is worth engaging with. The main theorems are plausible and important, and the flaws look like they might be repairable—but they are real and load-bearing. I'd send it to a serious referee, and I'd expect the referee to bounce it back for a revision that fixes Theorem 3.2 and makes the induction explicit.","headline":"Important conjecture, plausible strategy, but the written proof has a false lemma and a load-bearing induction gap—worth refereeing, not accepting as is.","tokens_in":10683,"tokens_out":4396,"would_cite":false,"duration_ms":43560,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J32","14E30","14C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Calabi–Yau pairs","divisor volumes","log canonical singularities","boundedness","Mori fiber spaces","canonical bundle formula","Cartier index","Birkar conjecture"],"falsifier":"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.","tokens_in":9729,"feed_emoji":"📐","tokens_out":16826,"duration_ms":177206,"temperature":0.7,"pith_summary":"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.","feed_headline":"Divisor volumes on Calabi–Yau pairs are discrete","feed_subtitle":"Dimension and singularity strength alone control divisor volumes, proving the boundedness conjecture for polarized pairs.","key_machinery":"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","core_discovery":"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 $X$","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the input that $\\mathcal{C}(d,v,\\epsilon)$ is bounded in codimension one, the starting point for the log boundedness proof.","marker":"[JJZ25]"},{"why":"Provides the boundedness theorems for polarized varieties, the fixed $m$ making $|mA|$ birational, and the finite-set result for the threshold $t$ used in Theorems 3.2, 3.3, and 3.5.","marker":"[Bir23a]"},{"why":"Gives the structural template for Theorem 3.3 and Corollary 1.3 on bounded multiplicities of a Fano fibration over codimension-one points.","marker":"[Bir23b]"},{"why":"Boundedness of Fano-type fibrations, applied to conclude that $W'$ is bounded and that $(X',h'^*A_{X'})$ is log bounded in Theorem 3.3.","marker":"[Bir24]"},{"why":"Theorem 1.8 is used to produce small lc perturbations of the boundary in Theorem 3.3 and Lemma 3.4.","marker":"[Bir21]"},{"why":"Provides the bound on the Cartier index of integral divisors via graded linear series (Lemmas 2.3 and 2.4), used in Theorem 3.5 to make $A$ very ample.","marker":"[HLQZ25]"},{"why":"Lemma 3.2 and invariance of plurigenera in Theorem 3.2 control the exceptional and pushforward terms that force $K_X+\\delta A$ not to be pseudo-effective.","marker":"[HMX13]"},{"why":"The moduli b-divisor result used in Theorem 3.3 to convert the generalized pair $(Z,C+R)$ into a genuine $\\mathbb{Q}$-divisor $C'$ on $Z$.","marker":"[Amb05]"},{"why":"Effective base point freeness used in Theorem 3.5 to pass from a bounded Cartier index to a fixed very ample multiple $r'A$.","marker":"[Kol93]"},{"why":"Proposition 3.7 transports the pair $(W,B_W)$ through the codimension-one isomorphism to $W'\\to Z'$ in Theorem 3.3.","marker":"[BDCS20]"}],"fun_headline_variants":["Volume discreteness for divisors on Calabi-Yau pairs","Calabi-Yau divisor volumes land in discrete sets","Fixed singularity strength forces divisor volumes discrete","Birkar's boundedness conjecture proved for log Calabi-Yau pairs","Discrete divisor volumes on Calabi-Yau pairs resolve conjecture"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Volume discreteness for divisors on Calabi-Yau pairs","Calabi-Yau divisor volumes land in discrete sets","Fixed singularity strength forces divisor volumes discrete","Birkar's boundedness conjecture proved for log Calabi-Yau pairs","Discrete divisor volumes on Calabi-Yau pairs resolve conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000527,"raw_usage":{"total_tokens":2332,"prompt_tokens":648,"completion_tokens":1684,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":392,"completion_tokens_details":{"reasoning_tokens":1603}},"tokens_in":392,"tokens_out":1684,"duration_ms":13900,"temperature":1.0,"reasoning_tokens":1603,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T23:35:11.913943+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}