{"id":"e55f206a-2339-47c7-94f9-a6ab719465f7","arxiv_id":"2607.24986","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Big gklt Kähler pairs admit a full MMP with scaling, and a nef canonical class with modified-big boundary is semiample (Tosatti’s conjecture).","lead":"The paper proves the minimal model program for big gklt Kähler pairs and establishes Tosatti’s transcendental base-point-free conjecture. This extends the BCHM theorem from projective varieties into the analytic Kähler setting for a large natural class of pairs.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The non-big half of the induction rests on Theorem 2.52 — cited only as \"M. Paun, personal communication, to appear\" — for nefness/modified-bigness of the moduli part; with no public proof, Theorem 2.53 and everything built on it is presently unverifiable.","rationale":"The reader's weakest_assumption — the modified-big hypothesis — is a scope restriction the authors themselves flag openly (Conjecture 1.14), and I agree it is not a correctness concern; hence only partial agreement. The reader's rationale does gesture at the real issue (\"rests on concurrent preprints by the same circle of authors\"), but it understates it: exactly one ingredient, Theorem 2.52, is not a preprint at all but a personal communication, and it sits on the critical path of the non-big induction (Theorem 2.52 → Theorem 2.53 → Theorem 4.1 Case 2 / Proposition 5.13 / Proposition 6.1 Case 2), which is where the paper's central novelty lives. Everything else in the inductive machine is either proved in the text, classical ([CT15], [Bou04], [Ar70], [Fuj75]), or at least publicly available ([Ou25], [HLX26], [HP24], [DH24]). Credit where due: the logical graph is explicit and well-founded, the big/non-big separation is clean, and where a result is attributed to an in-preparation manuscript (Lemma 5.2 ↔ [DH26]) the proof is included. My recommendation is CONDITIONAL — accept contingent on the public appearance and independent verification of [Pau26] (and secondarily [Ou25]), with the re-derivation of Theorem 2.53's modified-big conclusion from published sources as the decisive check. If one reads the reader's \"moderate confidence ACCEPT\" as already pricing in this dependency, UNCHANGED is defensible; but the gate is specific, single-point, and cheap to state, so making it an explicit condition is the more honest verdict.","tokens_in":63253,"tokens_out":5741,"duration_ms":175341,"concrete_test":"When [Pau26] appears (or from the authors now), check that its hypotheses are met in the exact configuration used in Theorem 2.53's proof — a projective morphism f′: (X′, B′_δ) → Z′ between compact Kähler manifolds satisfying Definition 2.50 (Kawamata's condition) with β′_δ Kähler — and that its conclusion gives γ_δ nef for all 0 < δ ≪ 1, so that lim_{δ→0} γ_δ = δ_{Z′} − μ*ω_Z is nef. Independently, attempt to re-derive \"δ_Z is modified big\" in Theorem 2.53 using only published sources ([FF14], [Fuj22 §21], [HP24]); if this fails, the main theorems are gated on [Pau26]. As a sanity check, run the Theorem 4.1 Case 2 reduction for dim X = 2 (where semiample-ness is classically known) and confirm it recovers the known answer without circular use of Theorem 2.52.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The genuinely new content of the paper is the non-big case: Theorem 4.1 Case 2 (semiample-ness of nef non-big α) and Proposition 5.13 (log terminal models when K_X+B+β_X is psef but not big). Both arguments reduce, via Ou's uniruledness theorem and the MRC fibration, to a lower-dimensional gklt pair (Z′, B_{Z′}+γ) whose moduli part γ must be modified big so that the inductive hypotheses (Theorems 1.4_{n−1}, 1.5_{n−1}, and the standing modified-big assumption) apply on the base. That pair structure comes entirely from the canonical bundle formula, Theorem 2.53, whose proof applies Theorem 2.52 to the perturbed pairs (X′, B′_δ), β′_δ to obtain a positive current with zero Lelong numbers in the moduli class, concluding γ_δ nef and hence δ_{Z′} − μ*ω_Z nef, i.e. δ_Z modified big. The same modified-big moduli part is used again in Proposition 6.1 Case 2. The problem: Theorem 2.52 is [Pau26], listed in the references as \"personal communication, July 2026, to appear.\" There is no preprint. In the analytic category this semipositivity is precisely the delicate point — the classical proofs ([FF14], [Fuj22 §21], [Kol07]) are Hodge-theoretic/algebraic, and the Kähler analogue is recent, subtle work. If the zero-Lelong-number conclusion requires extra hypotheses (e.g. fails outside Kawamata's condition, or only gives δ psef rather than nef), then δ need not be modified big, the induction hypothesis cannot be invoked on Z, and the non-big direction of the transcendental base-point-free theorem — the headline result — has an actual gap rather than a stylistic one. This is not an internal inconsistency; the deduction from Theorem 2.52 to Theorem 2.53 as written looks correct. It is a verification gap at the one point where the argument leaves publicly available mathematics. (Secondary dependencies [Ou25], [HLX26], [HP24] are at least public preprints/papers; Lemma 5.2, though attributed to the in-preparation [DH26], has its proof included in full.)","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper proves the minimal model program for compact Kähler generalized klt pairs (X, B+β) with B+β_X modified big: if K_X+B+β_X is not pseudo-effective there is a Mori fiber space, and if it is pseudo-effective (with B+β_X big, or itself big) there is a good log terminal model (Theorem 1.1). The central new input is the transcendental base-point-free theorem (Theorem 1.4): nef α = K_X+B+β_X is semiample, realized by a Moishezon contraction with α ≡ f*γ for a Kähler class γ on the base. The proof is an explicit dimension induction linking the contraction theorem (1.6), base-point-freeness (1.4), and the MMP with scaling (1.5). The big case of the contraction theorem (Section 3) proceeds via jumping numbers of multiplier ideals, adjunction onto successive non-klt strata, and gluing of contractions via push-out diagrams, extending the resulting morphism off the null locus using Artin–Fujiki blowing-down. The non-big case uses Ou's uniruledness criterion, the MRC fibration, and a canonical bundle formula (Theorem 2.53) whose key analytic input is a semipositivity statement (Theorem 2.52) attributed to a personal communication of Paun.","tokens_in":63703,"tokens_out":5631,"duration_ms":163863,"significance":"If the arguments hold, the paper establishes the minimal model program for big gklt Kähler pairs (Theorem 1.1/1.5) and proves Tosatti's transcendental base-point-free conjecture for nef K_X+B+β_X with modified-big boundary (Theorem 1.4), together with the Kähler criterion (Theorem 7.1), geography of log canonical models (Theorem 5.15), and several consequences in the Kähler–Ricci flow literature (Theorems 1.12, 1.13). This is the natural extension of [BCHM10] to the transcendental setting and would be a landmark result. Particular strengths: the proof is a genuine induction (1.6_{n−1} ⇒ 1.6_{n,big} ⇒ 1.4_n ⇒ 1.5_n ⇒ 1.6_n) with the big and non-big cases cleanly separated; Section 3 (the big-case contraction theorem via jumping numbers, adjunction on non-reduced strata, and Artin–Fujiki extension) is substantial new technical work; and the dependence on prior results (flips [DH23], cone theorem [HP24, HLX26], Ou's uniruledness criterion) is as black-box lemmas with independent proofs, not circular restatements. The paper also states its boundary precisely: the modified-big hypothesis is flagged as essential, with the general case left as Conjecture 1.14.","major_comments":[{"comment":"Theorem 2.52 is cited only as 'M. Paun, personal communication, July 2026, to appear' [Pau26]; there is no public preprint. This is not a peripheral input. Theorem 2.53 (the canonical bundle formula producing a gklt pair (Z, ∆+δ) with δ modified big) is proved by applying Theorem 2.52 to the perturbed pairs (X′, B′_δ), β′_δ to obtain a positive current with zero Lelong numbers in the moduli class. Theorem 2.53 is in turn the load-bearing input for the entire non-big half of the induction: Theorem 4.1 Case 2 (semiample-ness of nef non-big α, p. 31), Proposition 5.13 (existence of log terminal models when K_X+B+β_X is psef but not big, p. 41), and Proposition 6.1 Case 2 (p. 47) all reduce, via Ou's theorem and the MRC fibration, to a lower-dimensional gklt pair on the base whose moduli part must be modified big for the inductive hypotheses 1.4_{n−1} and 1.5_{n−1} to apply — and those hypot","section":"§2.12, Theorem 2.52 / Theorem 2.53; used in §4 (Thm 4.1 Case 2), §5.2 (Prop 5.13), §6 (Prop 6.1)"},{"comment":"Theorem 1.7 (the Kähler criterion: for a gklt pair with α = K_X+B+β_X nef and big, α·C > 0 for all rational curves implies α Kähler) is stated in the introduction with no proof and no citation, but it is invoked at a critical point in the proof of Theorem 4.1, Case 1 (p. 30: 'By Theorem 1.7, [γ] is a Kähler class'). The in-paper variant, Theorem 7.1, cannot substitute: its proof uses Theorem 1.5, which depends on Theorem 1.4_n, so invoking it inside Theorem 4.1 would be circular. Presumably Theorem 1.7 is imported from [HLX26], but this needs to be stated explicitly, with the hypotheses matched (note that Theorem 1.7 as stated requires no NQC assumption, unlike Theorem 7.1), and the dependency made clear in the inductive scheme of §1.1.","section":"Theorem 1.7; used in §4, proof of Theorem 4.1, Case 1"}],"minor_comments":[{"comment":"Reference [Pau26]: 'peronal communication' → 'personal communication'.","section":"References"},{"comment":"Typo: 'satisfy the hypotesis of Theorem 2.52' → 'hypothesis'.","section":"Proof of Theorem 2.53"},{"comment":"Typo: 'and and D a Cartier divisor'.","section":"Theorem 2.3 (statement)"},{"comment":"Typo: 'log terminimal model' → 'log terminal model'.","section":"Proof of Lemma 2.20"},{"comment":"Grammar: 'Conjectures 1.1, 1.2, and 1.3 of [TZ18] holds' → 'hold'.","section":"Theorem 1.13 (statement)"},{"comment":"The first sentence ('α is big if α ≥ ω where ω is a Kähler current') is garbled and only the second formulation (existence of a Kähler current T with [T] ∈ α) is meaningful; please rewrite.","section":"Definition 2.8"},{"comment":"The notation E ∧ Z = 0 (no common components) is used without having been defined; a one-line definition would help.","section":"Lemma 2.4 (proof)"},{"comment":"The section preamble lists Theorems 1.4_{n−1}, 1.5_{n−1}, and 1.6_{n−1} as assumptions, while Theorem 3.1 states that 1.6_{n−1} alone implies 1.6_{n,big}. Please align the two, and check whether the results cited inside Proposition 3.2 (e.g. the relative vanishing inputs) require the additional hypotheses.","section":"§3, opening paragraph vs. Theorem 3.1"},{"comment":"'Next we recall our famous Abundance conjecture' — the phrasing is odd for a standard conjecture; suggest neutral wording.","section":"§1.3, Conjecture 1.16"}],"recommendation":"major_revision","confidential_remarks":"The paper's main new content depends on [Pau26], which is listed as a personal communication dated July 2026 — the same month as submission — with no preprint. I could not verify Theorem 2.52, and I do not see how any referee could. Given that the non-big direction is the paper's headline advance, I would recommend not accepting until a proof of Theorem 2.52 is part of the public record (ideally included here as an appendix). The big-case half of the paper appears independent of this input and is, in my reading, sound. The citation pattern is otherwise appropriate: the heavy use of the first author's joint work (DH23, DH24, HP24, HLX26) reflects the fact that this paper caps that program, and each cited item is used as a proved lemma rather than restated."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finishes the MMP for big gklt Kähler pairs and proves Tosatti’s transcendental base-point-free theorem: if α = K + B + β is nef and B + β is modified big, then α is semiample via a Moishezon contraction to a Kähler space. That is the real advance. Cone and flips were already available; the new pieces are the contraction theorem (big and non-big), the BPF itself, and termination with scaling via a geography result for weak log canonical models.\n\nThe inductive skeleton is clean and explicit (1.6_{n-1} ⇒ big contraction ⇒ BPF_n ⇒ MMP_n ⇒ full contraction_n). Big and non-big cases are separated properly. Analytic tools (Ou uniruledness, Null = E_nK, Nadel/KV vanishing for g-pairs, Artin–Fujiki extension, multiplier ideals) are used in the expected places. Self-citations function as ordinary lemmas, not circularity. The modified-big hypothesis is load-bearing and honestly flagged as open beyond that range (Conjecture 1.14).\n\nThe soft spot is real and concentrated. The non-big half (Theorem 4.1 Case 2 and Proposition 5.13) reduces via MRC and the canonical bundle formula to a lower-dimensional pair whose moduli part must be modified big so induction applies. That modified-bigness comes from Theorem 2.53, which rests on Theorem 2.52 — Paun, personal communication, July 2026, to appear, no preprint. In the Kähler setting this semipositivity (positive current with zero Lelong numbers) is exactly the delicate analytic step; classical algebraic proofs do not automatically transfer. If that statement needs extra hypotheses or only yields pseudo-effectiveness, the non-big direction has a verification gap rather than a stylistic one. Everything else is public or proved in-line (including the Boucksom–Zariski lemmas attributed to the in-prep DH26).\n\nThis is for specialists already working on analytic MMP or Kähler abundance. It deserves a serious referee who can either confirm the Paun input or force it into the open. I would cite the big-case results and the overall framework immediately; I would wait on the non-big claims until the moduli positivity is public. Send it to peer review.","headline":"Major completion of the Kähler MMP for big gklt pairs and Tosatti’s BPF, but the non-big half hangs on an unpublished Paun personal communication for the moduli part.","tokens_in":62211,"tokens_out":568,"would_cite":true,"duration_ms":11671,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","32J27","14J40"],"pacs":[],"model":"grok-4.5","headline":"The minimal model program works for big gklt Kähler pairs, and a nef class of that form is semiample via a Moishezon contraction.","keywords":["Kähler MMP","transcendental base-point-free","generalized klt pairs","Moishezon contraction","modified big","multiplier ideals","canonical bundle formula"],"falsifier":"Exhibit a compact Kähler gklt pair with B+β_X modified big and α = K_X+B+β_X nef such that no Moishezon contraction realizes α as the pullback of a Kähler class, or produce an infinite sequence of flips with scaling that does not terminate.","tokens_in":61757,"feed_emoji":"📐","tokens_out":922,"duration_ms":15482,"temperature":0.7,"pith_summary":"The classical minimal model program (MMP) tells you how to simplify a projective variety by contracting curves and flipping until the canonical class becomes nef, then (under abundance) semiample. This paper extends that program to compact Kähler varieties, which need not be projective, for generalized klt pairs whose boundary-plus-b-divisor is modified big. The main theorems give a Mori fiber space when the canonical class is not pseudo-effective, and a good log terminal model when it is. As a direct consequence, any nef class of the form K_X + B + β_X is semiample: it is the pullback of a Kähler class under a Moishezon contraction to a compact Kähler space. That settles Tosatti’s transcendental base-point-free conjecture in this setting and supplies the missing contraction and termination steps needed to run the Kähler MMP with scaling.","feed_headline":"Kähler MMP works for big gklt pairs","feed_subtitle":"Nef canonical classes become semiample via Moishezon contractions, proving Tosatti’s conjecture","key_machinery":"An inductive package linking a non-gklt contraction theorem (Theorem 1.6), the transcendental base-point-free theorem (Theorem 1.4), and MMP-with-scaling termination (Theorem 1.5). The engine is successive jumping numbers of multiplier ideals along a Kähler current with singularities along Null(α), together with the canonical-bundle formula that reduces dimension when the pair is not big.","core_discovery":"For a compact Kähler gklt pair (X, B+β) with B+β_X modified big, the MMP holds: if K_X+B+β_X is not pseudo-effective there is a Mori fiber space; if it is pseudo-effective (or big) there is a good log terminal model. In particular, when α = K_X+B+β_X is nef it is semiample, realized by a Moishezon contraction f : X → Y onto a compact Kähler space with α ≡ f*γ for a Kähler class γ on Y (and f projective when X is strongly Q-factorial).","pith_inferences":["The modified-bigness hypothesis is the precise barrier to a full Kähler abundance statement; removing it would finish the program the authors leave as open conjectures.","The same inductive skeleton (jumping numbers + canonical bundle formula + NQC triviality) is likely to adapt to glc pairs once a suitable non-klt contraction theorem is in place.","Existence of the Moishezon contraction gives a concrete analytic substitute for the usual projective linear system, which should feed directly into analytic constructions of Kähler–Einstein metrics on the resulting models."],"forward_implications":["Tosatti’s transcendental base-point-free conjecture holds for big gklt Kähler pairs.","Flips and divisorial contractions of strongly Q-factorial big gklt Kähler pairs preserve the Kähler condition.","Good log terminal models and Mori fiber spaces exist, so the Kähler MMP with scaling can be run to completion in the big case.","Several earlier conjectures on the Kähler–Ricci flow and collapsing (Tosatti–Zhang, Tosatti) follow immediately.","Relative versions over a base and geography of weak log canonical models are available when the pair is modified big over the base."],"fun_headline_variants":["Kähler MMP holds for big gklt pairs","Transcendental base-point-free theorem proved for gklt Kähler pairs","Nef canonical classes on big gklt pairs are semiample","MMP and Moishezon contractions for compact Kähler gklt pairs","Tosatti conjecture settled via Kähler MMP for big gklt pairs"],"cache_read_input_tokens":49280,"weakest_assumption_plain":"Everything depends on the boundary-plus-b-divisor being modified big; without that hypothesis the vanishing, canonical-bundle formula, and termination arguments do not go through.","fun_headline_variants_meta":{"raw":{"variants":["Kähler MMP holds for big gklt pairs","Transcendental base-point-free theorem proved for gklt Kähler pairs","Nef canonical classes on big gklt pairs are semiample","MMP and Moishezon contractions for compact Kähler gklt pairs","Tosatti conjecture settled via Kähler MMP for big gklt pairs"]},"model":"grok-4.5","effort":"low","cost_usd":0.003215,"raw_usage":{"total_tokens":1021,"prompt_tokens":619,"num_sources_used":0,"completion_tokens":92,"cost_in_usd_ticks":32148000,"prompt_tokens_details":{"text_tokens":619,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":310,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":619,"tokens_out":92,"duration_ms":5869,"temperature":1.0,"reasoning_tokens":310,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T04:17:56.595427+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a compact Kähler gklt pair with B+β_X modified big and α = K_X+B+β_X nef such that no Moishezon contraction realizes α as the pullback of a Kähler class, or produce an infinite sequence of flips with scaling that does not terminate.","supporting_citations":[],"review_version":1}