{"id":"284875a0-a346-46bf-af77-15c329ea0bd9","arxiv_id":"2505.05250","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Assuming the non-vanishing conjecture, the abundance conjecture holds for all smooth projective varieties of numerical dimension at most one; unconditionally, it holds in dimension at most five whenever κ≥0 and ν≤1.","lead":"On a projective variety, the abundance conjecture says the Kodaira dimension equals the numerical dimension; this paper shows that the non-vanishing conjecture alone settles abundance in numerical dimension one, and proves abundance unconditionally up to dimension five in that case. The result narrows the gap between two central conjectures of the minimal model program and gives the first unconditional abundance in some remaining fourfold and fivefold cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 asserts an integral linear equivalence aK_U∼bS with positive a,b but never derives it; in the coefficient-1 case the asserted positive b does not exist and the cover step is left unverified.","rationale":"The reader's verdict is CONDITIONAL, and I agree that Theorem 3.1 is the load-bearing spot, but my diagnosis differs from the reader's stated 'sign typo' reading. The asserted relation aK_U∼bS can actually be derived from (2) and (4): on a neighborhood avoiding B−S, K_U+S∼R a0S and integrality of principal divisors forces a0∈Z, giving K_U∼(a0−1)S. The real defect is that when a0=1 the coefficient b=a0−1 is zero, so 'positive integers' is false and the cover lemma is invoked without checking that b=0 is permitted. This is a precise, addressable point rather than a fatal flaw, so the conditional verdict should stand. The reader's separate concern about [Bir10, Lemmas 3.6,3.8] affects only the unconditional dimension-5 statement, not the central conditional Theorem 1.4, and the cited results are standard for this community; I do not make that the main attack. The proposed concrete test settles the only non-obvious point: whether the b=0 case of the cover construction goes through.","tokens_in":22673,"tokens_out":36755,"duration_ms":398210,"concrete_test":"Re-derive the cover step of Theorem 3.1 in the coefficient-1 case a0=1, where K_U∼0 in a neighborhood of S. Apply [Kol+92, 11.3.6 Lemma] with G=a1S and b'=0, and verify that the resulting finite cover \tilde U→U, étale in codimension 1, still satisfies (i)–(iv) of the proof, in particular \tilde S∼a'\\tilde G and \tilde K_U∼0, and that [Kol+92, 11.3.7 Lemma] then yields κ(S)≥1. If the lemma requires b>0, the proof must supply a separate argument for the a0=1 case or the theorem is incomplete; if it works, the one-line fix is to replace 'positive integers' with 'nonnegative integers' and to add the derivation of the integral relation from principal-divisor coefficient integrality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the key special case Theorem 3.1 depends on the unexplained sentence: 'there exist two positive integers a and b such that aK_U∼bS and b>−a.' This is more than a sign typo. From assumptions (2) and (4), on a neighborhood U of S avoiding B−S, we have D|_U=a0S for the coefficient a0>0 of S in D, hence K_U+S∼R a0S on U, so K_U∼R(a0−1)S. Because K_U and S are integral Weil divisors, the principal divisor (K_U+S)−a0S has integral coefficients, forcing a0 to be an integer. Thus the integral relation holds with a=1 and b=a0−1, which is nonnegative. When a0=1, no positive integer b exists, so the statement as written is false; the proof tacitly needs b=0. The subsequent application of [Kol+92, 11.3.6 Lemma] constructs a finite cover using the exact relation and produces \tilde K_U∼b'\\tilde G with b'=b/c. For b=0 this is \tilde K_U∼0, a case the text has not verified is covered by the lemma. Since Theorem 4.1 invokes Theorem 3.1 after Lemma 4.7, and Theorem 1.4 passes through Theorem 4.1, the whole chain is sensitive to this point. If the intended condition is 'nonnegative a,b with b>−a', the proof should state it and check the b=0 case of the cover lemma.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the abundance conjecture in the case of numerical dimension at most one. The main results are: (1) an unconditional proof that smooth projective varieties of dimension at most five with κ≥0 and ν≤1 have a good minimal model, hence κ=ν; and (2) a conditional theorem that the non-vanishing conjecture in dimension d implies the abundance conjecture for all smooth projective varieties of dimension ≤d with ν≤1. The technical core is a reduction, via a special termination conjecture (Conjecture 1.7), to a special case (Theorem 3.1) modeled on Kawamata's threefold approach, combined with the Du Bois property of slc singularities and recent MMP results. The paper also contains applications to fourfolds with nonzero Euler characteristic and to relative klt pairs.","tokens_in":22962,"tokens_out":20399,"duration_ms":196639,"significance":"If the proof is correct after repairing the issues below, this is a substantial advance: the conditional implication from non-vanishing to abundance for ν≤1 is new, and the unconditional fivefold result is the first higher-dimensional case beyond known threefold abundance in this numerical-dimension range. The paper is carefully organized, the reduction chain is explicit and does not appear to feed the abundance conclusion back into its hypotheses, and the authors are transparent about the role of Kawamata's withdrawn note. The main strengths are the clean separation of the special termination input and the use of recent MMP results; the proof is algebraic rather than analytic, which is a useful structural contribution. The paper is not accompanied by machine-checked code, but its statements are precise and the logical skeleton is testable at each lemma.","major_comments":[{"comment":"The assertion 'there exist two positive integers a and b such that aK_U∼bS and b>−a' is not derived and is false as stated. From assumptions (2) and (4), on a neighborhood U of S avoiding B−S, we have K_U+S∼_R D|_U=a0S, hence K_U∼_R(a0−1)S. Since K_U and S are integral Weil divisors, a0 must be an integer. If a0=1, then K_U∼_R0 on U and no positive b can satisfy aK_U∼bS; the proof tacitly needs b=0. The subsequent application of [Kol+92, 11.3.6] constructs a cover using the exact relation, and in the b=0 case it becomes K_{\\tilde U}∼0, a case that is not verified. This is load-bearing because Theorem 4.1 invokes Theorem 3.1 after Lemma 4.7. Please restate the relation as 'a>0, b≥0, and b>−a' and check the cover lemma in the b=0 case.","section":"Theorem 3.1"},{"comment":"The displayed consequence 'By Theorem 2.22, there exists E≥0 such that K_X+B−ε/2S≡E' does not follow from the hypotheses as written. With the natural data L=K_X+B, D=εS, F=K_X+B−εS, the theorem yields D+sF≡E for s∈(0,1], so for s=1/2 one obtains 2E−εS≡K_X+B, not K_X+B−ε/2S≡E. The uniqueness contradiction in the next lines relies on the stated numerical equivalence, so this is a genuine gap in the written proof. The argument appears repairable by using the explicit form of E from the proof of Theorem 2.22 and taking 2E−εS; please supply the correct derivation.","section":"Lemma 4.4, Case 1"},{"comment":"The unconditional dimension-five result depends on the statement 'Conjecture 1.7 holds in dimension≤5 by [Bir10, Lemmas 3.6, 3.8]', but the proof is not reproduced and it is not explained why those lemmas apply to Q-factorial dlt pairs with R-divisors and to the exact formulation of Conjecture 1.7. Since this citation is the only unconditional input for Theorem 1.2, please either reproduce the argument or state precisely how the cited lemmas cover the needed special termination statement.","section":"Theorem 5.1"}],"minor_comments":[{"comment":"In the proof, the displayed birational map should be φ:(X,B)99K(X′,B′), not φ:(X′,B′)99K(X′,B′).","section":"Lemma 4.7"},{"comment":"In the sentence 'It is clear that dimZ < dimZ', the second dimZ should be dimX.","section":"Lemma 4.5"},{"comment":"The phrase 'Lets≫0' should read 'Let s≫0'.","section":"Lemma 4.2"},{"comment":"The word 'irreducble componet' should be 'irreducible component'.","section":"Lemma 4.7"},{"comment":"In the sentence 'we done by [LM21, Corollary 1.2]', the intended phrase is 'we are done'.","section":"Corollary 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is promising and the overall reduction is coherent; the issues listed in the major comments are local and appear fixable without changing the architecture of the proof. The false positive-b assertion in Theorem 3.1 is the most serious point because it is the pivot of the special-case argument, but the intended statement with b=0 is clear and the cover lemma should still apply. I do not see circularity in the use of non-vanishing, and the unconditional fivefold result is an attractive contribution if the Birkar citation is confirmed. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine step forward. It reduces the abundance conjecture for ν(X)≤1 to the non-vanishing conjecture, and it unconditionally proves abundance for dim≤5 with κ≥0 and ν≤1. That is new, and the reduction chain from non-vanishing to special termination to abundance is coherent. The use of recent MMP results (Hashizume, Liu–Tsakanikas, Meng–Zhuang) is appropriate, and the paper is careful about replacing analytic inputs with algebraic ones. Credit where due: the proof of Theorem 4.1 is a substantial argument, and the unconditional dimension-5 result, if it survives referee scrutiny, is a real advance.\n\nThe soft spots are real but narrow. First, the assertion in Theorem 3.1 that there exist positive integers a,b with aK_U∼bS is not justified as written. From K_U+S∼_R a0S on the neighborhood avoiding B−S, you get K_U∼_R(a0−1)S, and integrality forces a0 integer. If a0=1, then b=0, not positive. The stress-test note is right that the b=0 case needs to be checked in the cover lemma [Kol+92, 11.3.6]; the paper does not explicitly verify it. This looks like a fixable oversight—probably the intended statement is nonnegative with b>−a, which is automatic—but it is load-bearing for Theorem 3.1 and therefore for the whole chain. Second, the unconditional dimension-5 result (Theorem 5.1) relies on Conjecture 1.7 in dimension≤5, asserted via [Bir10, Lemmas 3.6, 3.8] without proof. The citation is standard and likely correct, but the paper does not reproduce the argument or state exactly why those lemmas cover the needed Q-factorial dlt pairs with R-divisors. That is a clarity issue, not evidence of a hole.\n\nCitation pattern is fine; self-citations like [LX23] and [HLS24] are used as inputs for standard lemmas, not to hide a circularity. No step appears to assume abundance. The paper is honest about where it is conditional and about Kawamata's withdrawn note.\n\nBottom line: this deserves a serious referee. I would send it to review with a request that the authors fix the integer b issue in Theorem 3.1 and either prove or precisely cite the Bir10 lemmas. If the b=0 case fails, the main theorem may still hold by a small perturbation, but right now the write-up has an error in a key step. I would not desk-reject.","headline":"A serious and mostly sound step on the ν≤1 case of abundance; the sign/integrality glitch in Theorem 3.1 needs a fix, but the main reduction chain is coherent and the paper deserves peer review.","tokens_in":23509,"tokens_out":1985,"would_cite":true,"duration_ms":21397,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","32J27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the non-vanishing conjecture implies the abundance conjecture whenever the numerical dimension is at most one, and settles abundance for smooth projective fivefolds with κ ≥ 0 and ν ≤ 1.","keywords":["Abundance conjecture","Non-vanishing conjecture","Numerical dimension","Kodaira dimension","Minimal model program","Good minimal model","Log canonical pairs","Fivefold"],"falsifier":"A concrete check is dimension five: a smooth projective fivefold $X$ with $\\kappa(X)=0$, $\\nu(X)=1$, and no good minimal model would refute Theorem 1.2 directly. A second, more targeted check is to exhibit a $\\mathbb{Q}$-factorial effective dlt pair of dimension five for which every $(K_X+B)$-MMP fails to terminate near the divisorial part of the image of $\\lfloor B\\rfloor$; that would break the proof of the unconditional statement at its cited termination input. A third check is on the proof itself: verify whether the relation $aK_U\\sim bS$ used inside Theorem 3.1 holds with the stated signs on a concrete retracting tubular neighborhood, since the entire cover argument rests on it.","tokens_in":22446,"feed_emoji":"📐","tokens_out":10006,"duration_ms":92529,"temperature":0.7,"pith_summary":"The paper tries to establish that the abundance conjecture—the prediction that the Kodaira dimension $\\kappa(X)$ and the numerical dimension $\\nu(X)$ of any smooth projective variety coincide—is true in the case $\\nu(X)\\le 1$ provided a weaker statement, the non-vanishing conjecture, holds. Its main theorem shows that if non-vanishing is assumed in dimension $d$, then every smooth projective variety of dimension at most $d$ with $\\nu(X)\\le 1$ has a good minimal model or a Mori fiber space, so in particular $\\kappa(X)=\\nu(X)$. The same conclusion is proved unconditionally in dimensions up to five: every such fivefold with $\\kappa(X)\\ge 0$ and $\\nu(X)\\le 1$ has a good minimal model. The significance, if the argument is correct, is that the numerical-dimension-one case of abundance is reduced to a single, widely believed conjecture, without analytic extension theorems.","feed_headline":"Non-vanishing proves abundance in numerical dimension one","feed_subtitle":"Five-dimensional case is unconditional: κ ≥ 0 and ν ≤ 1 now guarantee a good minimal model.","key_machinery":"The load-bearing object is Conjecture 1.7, a special-termination statement for the minimal model program: for a projective $\\mathbb{Q}$-factorial effective dlt pair one can run a $(K_X+B)$-MMP that terminates near the divisorial part of the image of $\\lfloor B\\rfloor$. This is the external input needed to make the chain of reductions converge. The paper's own engine is Theorem 3.1, a special-case lemma that converts the numerical hypothesis into motion of a divisor: if $K_X+B$ is nef, numerically one-dimensional, and $\\mathbb{R}$-linearly equivalent to an effective divisor supported on a reduced boundary $B$ with an isolated component $S$, then adjunction yields $K_S+B_S\\equiv 0$, a finite cover étale in codimension one makes $S$ a Cartier divisor with trivial dualizing sheaf, and the Du Bois property of slc singularities—a controlled class of possibly nonnormal singularities—gives the cohomological surjectivity that makes $S$ move infinitesimally. The conclusion $\\kappa(S)\\ge 1$ turns the equality $\\kappa=\\nu$ from a global statement into a local infinitesimal-movement check.","core_discovery":"The central claim, stated on the paper's own terms, is a reduction theorem: for any projective lc pair $(X,B)$ with $\\kappa_\\iota(K_X+B)\\ge 0$ and $\\kappa_\\sigma(K_X+B)\\le 1$, assuming a weak special-termination conjecture in the same dimension, the pair has a good minimal model. The proof funnels the problem through a chain of MMP reductions to the special case in which $K_X+B\\sim_{\\mathbb{R}}D\\ge 0$ is nef, $D$ is supported on a reduced boundary $B$, and one irreducible component $S$ of $B$ is disjoint from $B-S$. In that special case adjunction gives $K_S+B_S\\equiv 0$, hence $K_S+B_S\\sim_{\\mathbb{Q}}0$; after a finite cover that is étale in codimension one, $S$ becomes a Cartier divisor with trivial dualizing sheaf, and the Du Bois property of the resulting slc pair forces $S$ to move infinitesimally, giving $\\kappa(S)\\ge 1$ and therefore $\\kappa(K_X+B)=1$. Theorem 1.2 is the special case of this reduction in dimension at most five, where the termination input is available; Theorem 1.4 replaces that input by the non-vanishing conjecture.","pith_inferences":["A reader might infer that special termination is the real bottleneck: the paper's Theorem 1.6 would make Conjecture 1.7 in dimension $d$ the only missing ingredient for abundance with $\\nu\\le 1$ in dimension $d$, even if the full non-vanishing conjecture is never proved.","One testable extension is to run the same special-case mechanism for generalized pairs or for pairs with $\\nu-\\kappa\\le 1$ outside the projective setting; the paper already sketches a relative klt version, and the Du Bois infinitesimal-movement step is the part that would need to survive.","Because the proof is algebraic rather than analytic, it suggests that the numerical-dimension-one abundance pattern is governed by termination and positivity of the boundary rather than by metric methods; if so, one might expect a similar reduction in any characteristic where a replacement for the Du Bois cohomological input is available."],"forward_implications":["For every $d$, the non-vanishing conjecture in dimension $d$ implies $\\kappa(X)=\\nu(X)$ and the existence of a good minimal model or a Mori fiber space for every smooth projective $d$-fold with $\\nu(X)\\le 1$.","Unconditionally, every smooth projective variety of dimension at most five with $\\kappa(X)\\ge 0$ and $\\nu(X)\\le 1$ has a good minimal model; in particular $\\kappa(X)=\\nu(X)$.","The same conclusions hold for log canonical pairs with $\\kappa_\\iota(K_X+B)\\ge 0$ and $\\kappa_\\sigma(K_X+B)\\le 1$ in dimension at most five, and for klt pairs over a base when $\\nu-\\kappa\\le 1$ and one of the dimension bounds in Theorem 6.6 holds.","Combined with existing results on $\\chi(\\mathcal{O}_X)\\neq 0$, a fourfold with nonvanishing Euler characteristic and $\\nu\\le 1$ has a good minimal model.","The dlt extension conjecture is no longer needed for abundance when $\\nu\\le 1$; only non-vanishing, or special termination, is required."],"supporting_citations":[{"why":"Supplies the two lemmas used to conclude that special termination holds in dimension at most five, the key input of the unconditional theorem.","marker":"[Bir10]"},{"why":"Provides the reduction that abundance for a variety follows from the existence of good minimal models, used throughout the introduction and proofs.","marker":"[GL13]"},{"why":"Supplies the threefold abundance approach and the special-case strategy adapted to the $\\nu=1$ setting.","marker":"[Kaw92]"},{"why":"Proves that slc singularities are Du Bois, the cohomological input that makes the test divisor move infinitesimally.","marker":"[KK10]"},{"why":"Provides the Du Bois property and slc pair foundations used in the cover argument of the special case.","marker":"[Kol13]"},{"why":"States that numerically trivial log canonical divisors of slc pairs are $\\mathbb{Q}$-linearly trivial, used to pass from $K_S+B_S\\equiv 0$ to linear equivalence.","marker":"[Gon13]"},{"why":"Gives the minimal-model existence criteria for lc pairs and the equivalence used in Lemma 2.9.","marker":"[HH20]"},{"why":"Used in the proof that non-vanishing in lower dimensions implies the existence of minimal models, hence special termination.","marker":"[LT22]"},{"why":"Used to transfer good minimal models between pairs with the same boundary support.","marker":"[MZ23]"}],"fun_headline_variants":["Non-vanishing conjecture forces abundance when ν≤1","Abundance proved in dimension five for κ≥0, ν≤1","Good minimal models exist in dim 5 for κ≥0, ν≤1","Numerical dimension one: non-vanishing suffices for abundance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The unconditional five-dimensional theorem depends on an unproved assertion that the special-termination conjecture holds in dimension at most five; the paper cites two lemmas from an earlier work for this and does not reproduce the proof.","fun_headline_variants_meta":{"raw":{"variants":["Non-vanishing conjecture forces abundance when ν≤1","Abundance proved in dimension five for κ≥0, ν≤1","Good minimal models exist in dim 5 for κ≥0, ν≤1","Numerical dimension one: non-vanishing suffices for abundance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001238,"raw_usage":{"total_tokens":5026,"prompt_tokens":835,"completion_tokens":4191,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":4114}},"tokens_in":451,"tokens_out":4191,"duration_ms":30325,"temperature":1.0,"reasoning_tokens":4114,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:10:13.229364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is dimension five: a smooth projective fivefold $X$ with $\\kappa(X)=0$, $\\nu(X)=1$, and no good minimal model would refute Theorem 1.2 directly. A second, more targeted check is to exhibit a $\\mathbb{Q}$-factorial effective dlt pair of dimension five for which every $(K_X+B)$-MMP fails to terminate near the divisorial part of the image of $\\lfloor B\\rfloor$; that would break the proof of the unconditional statement at its cited termination input. A third check is on the proof itself: verify whether the relation $aK_U\\sim bS$ used inside Theorem 3.1 holds with the stated signs on a concrete retracting tubular neighborhood, since the entire cover argument rests on it.","supporting_citations":[],"review_version":1}