{"id":"0593359c-72ea-4ba9-b55d-d72cb9e2b305","arxiv_id":"2506.23218","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every compact Kähler manifold with nef anti-canonical bundle admits a locally trivial fibration over a Calabi-Yau manifold with rationally connected fibers.","lead":"This paper proves that every compact Kähler manifold whose anti-canonical bundle is nonnegative splits into a rationally connected part over a Calabi-Yau base, a conjecture previously known only in the projective case. It also proves a new flatness criterion for pseudo-effective sheaves and derives several applications, including Hacon-McKernan type inequalities and generic nefness of tangent bundles.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.7's construction of G depends on an MMP over the graph Γ' that may not be available in arbitrary dimension, so the proof of Theorem 1.2 as stated is incomplete.","rationale":"The reader's verdict identified the reliance on the projectivity criterion [CH24] as the weakest assumption. While that is indeed a critical external dependency, our stress-test found a more concretely problematic step: the proof of Lemma 3.7 invokes a relative MMP to construct the auxiliary variety Γ. This MMP is used to obtain the relatively φ-big line bundle G with the specific properties (1)–(3) that are needed for the rest of the argument. The cited theorem [DHP24] has a dimensional restriction (four-dimensional Kähler varieties), and it is unclear whether [Fuj22] supplies an arbitrary-dimensional relative MMP over an analytic base. If this step fails, the construction of G fails, and with it Theorem 3.8, the flatness of E_m, and the splitting theorem. The concern is not that the main result is false; rather, the proof as written does not currently establish the claimed generality. A concrete check of the referenced theorems could settle the issue. Because this is a gap in an essential lemma rather than a mere need for additional verification of a plausible criterion, the verdict should move from CONDITIONAL to UNVERDICTED until the dimension question is resolved.","tokens_in":33394,"tokens_out":24932,"duration_ms":256176,"concrete_test":"Inspect the statements of [DHP24, Theorem 1.4] and [Fuj22, Theorem 1.7] and verify their hypotheses for the morphism α': M' → Γ' in Lemma 3.7. Specifically, determine whether the MMP for K_{M'}+(1−ε)Φ over Γ' is guaranteed when dim M' ≥ 5 and Γ' is only known to be a compact Kähler space. If neither theorem covers this case, test whether the contraction of the components of Φ meeting the good locus can be performed directly via [CH24, Lemma 2.11] or another all-dimensional criterion; if not, Theorem 1.2 is currently proved only in the dimensions covered by the cited MMP theorems.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The most load-bearing step is the construction of the relatively φ-big line bundle G in Lemma 3.7. There, the resolution M is factored through a variety Γ obtained by running the MMP for K_{M'}+(1−ε)Φ over the graph Γ', citing [DHP24, Theorem 1.4] and [Fuj22, Theorem 1.7]. The reference [DHP24] is a theorem for 4-dimensional Kähler varieties, and [Fuj22] may not provide a relative MMP for arbitrary-dimensional projective morphisms over non-algebraic bases. Since Theorem 1.2 is claimed for all dimensions, the existence of Γ—and hence of G satisfying properties (1)–(3)—is not established when dim X ≥ 5. All subsequent results (Theorem 3.8, numerical flatness of E_m, Theorem 4.1) depend on G. This is an internal gap in the proof, not a disagreement with the literature.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.2: for a klt pair (X, Δ) with X a compact Kähler manifold and −(K_X + Δ) nef, there exists a locally constant fibration f: X → Y whose general fiber is rationally connected and whose base has c1(Y) = 0. The strategy follows Cao–Höring and the projective work of the authors: one runs an MRC fibration, constructs a relatively big line bundle G, proves a flatness criterion for pseudo-effective sheaves, shows that certain direct image sheaves are numerically flat, and then deduces a splitting of the tangent bundle and applies Ehresmann's theorem. The paper also proves applications to the Beauville–Bogomolov decomposition, Hacon–McKernan-type inequalities, and generic nefness of tangent bundles.","tokens_in":33564,"tokens_out":5776,"duration_ms":62709,"significance":"If the proof can be completed, Theorem 1.2 resolves a long-standing conjecture for compact Kähler manifolds with nef anti-canonical bundle, generalizing the projective results of Cao–Höring and the authors' earlier work. The flatness criterion (Theorem 2.1), stating that a pseudo-effective torsion-free sheaf with c1 = 0 has locally free numerically flat reflexive hull, is a substantial independent contribution that goes beyond the projective setting. The paper is carefully organized and contains detailed arguments in Sections 2–4, and the proof does not assume the desired fibration. However, several load-bearing steps are delegated to prior papers and recent preprints, and at least one such step appears incomplete as written; therefore the central claim is not yet fully established.","major_comments":[{"comment":"The construction of the intermediate model Γ, and hence of the relatively φ-big line bundle G, is not justified in arbitrary dimension. The text applies the MMP for K_{M'} + (1−ε)Φ over Γ′ citing [DHP24, Theorem 1.4] and, with 'cf.', [Fuj22, Theorem 1.7]. The former is a four-dimensional Kähler MMP statement, and the latter is not stated precisely or shown to cover projective morphisms of arbitrary relative dimension over a non-projective base such as Γ′. Since properties (1)–(3) of G are used in every subsequent step (Theorem 3.8, Propositions 3.13 and 3.16, and Theorem 4.1), Theorem 1.2 as stated for arbitrary dimension is not proven. The authors should either supply a valid MMP statement that applies here or explain how [Fuj22, Theorem 1.7] gives the required relative MMP in this generality.","section":"Lemma 3.7"},{"comment":"Theorem 2.5, which is the key input for the flatness criterion, is not proved in the paper: the proof says that it is 'proved by suitably adapting the arguments of [Wu22]' and gives a sketch of the modifications. The subsequent proof of Proposition 2.6 and Theorem 2.1 depends on the exact Lelong-number estimate (2.3) and on the extension of the current across Z, neither of which is verified in the text. Since the flatness criterion is a central new ingredient of the paper, the full proof or an explicit reduction to a stated result in [Wu22] should be included.","section":"Theorem 2.5"},{"comment":"Corollary 3.15 is used in Propositions 3.16 and 3.17 to conclude weak positivity of U_{c,m} and V_{c,m} and to prove Theorem 3.8, but its proof is omitted with the remark that it is 'completely the same' as [Wan22, Corollary 3.1]. This transfer from the projective to the Kähler setting is not automatic, because the Kähler proof relies on localized positivity of direct images. The proof should be written out, or the precise statement of [Wan22, Corollary 3.1] should be quoted and the reduction explained.","section":"Corollary 3.15"},{"comment":"The argument depends essentially on the projectivity criterion of Claudon–Höring [CH24, Theorem 1.1 and Corollary 4.2], a recent preprint. In Proposition 3.1 the morphism f is asserted to be projective by [CH24, Theorem 1.1], and Lemma 3.7 uses [CH24, Corollary 4.2] to produce a relatively ample line bundle. The manuscript does not state the exact hypotheses of the criterion or verify them in the present setting. Because this is the first step that replaces the missing ample line bundle on X, Theorem 1.2 is conditional on the validity of [CH24]. The authors should state the precise result they use and confirm that all hypotheses are satisfied, or provide a proof in the Kähler setting.","section":"Proposition 3.1 and Lemma 3.7"}],"minor_comments":[{"comment":"In the proof of Theorem 5.3, the sentence 'The implication from (1) to (2) is obvious' should read '(2) implies (1)', since the subsequent argument proves (1) ⇒ (2).","section":"Theorem 5.3"},{"comment":"In the proof of Theorem 1.4, the symbol m is used both for a sufficiently large divisible integer and for dim Y (in the line 'n := dim X and m := dim Y'); this is confusing and should be fixed.","section":"Proof of Theorem 1.4"},{"comment":"The statement of Theorem 2.5 contains the typo 'Käher' for 'Kähler'; also, the notation 'π' is used for the projection in Theorem 2.5 while π is used for the desingularization map in Setting 2.2, which may cause ambiguity.","section":"Theorem 2.5"},{"comment":"Theorem 5.1 is stated as a theorem but is essentially a variant of [CP25, Theorem 1.2] proved by a short gluing argument. The applications in Theorems 5.2 and 5.3 depend on this result, so the statement should be cross-checked against the precise hypotheses of [CP25], and a reference to the published version should be added once available.","section":"Section 5.2 and Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and the overall strategy is plausible, but the number of delegations to unpublished preprints and to 'suitable adaptations' of previous arguments makes it hard for a referee to certify correctness. The editor may wish to ask the authors to include full proofs for Theorem 2.5 and Corollary 3.15, to state and verify the exact projectivity and MMP results used from [CH24], [DHP24], and [Fuj22], and to confirm that the MMP step in Lemma 3.7 is valid in arbitrary dimension. If these points are addressed, the paper would be a major contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper claims the full Kähler case of the Cao-Höring conjecture for klt pairs: a compact Kähler manifold with nef anti-canonical bundle fibers locally trivially over a Calabi-Yau base with rationally connected fiber. If correct, that is a major result, and the proof has real substance.\n\nThe genuinely new piece is Theorem 2.1, a flatness criterion for pseudo-effective torsion-free sheaves with c1=0 on compact Kähler manifolds. The Segre current argument from Wu22 is extended substantially, and the localization of the extension theorem via relative Bergman kernels is a genuine technical innovation. The applications to Hacon-McKernan type inequalities and generic nefness of the tangent bundle are nontrivial.\n\nThat said, I share the reader's conditional verdict. The proof leans on very recent preprints: CH24 for projectivity of fibrations with rationally connected fibers, Ou25 for pseudo-effectiveness of KY, and CP25 for the positivity theorem in Section 5. Those are not yet publicly verified. More concretely, the stress-test note about Lemma 3.7 is a real concern. The construction of the relatively big line bundle G requires running an MMP for K_M' + (1−ε)Φ over the graph Γ', citing DHP24 Thm 1.4 and Fuj22 Thm 1.7. DHP24 is explicitly a 4-dimensional theorem. If Fuj22 does not supply the necessary arbitrary-dimensional relative MMP over a non-algebraic base, then Step 1 of the proof is incomplete for dim X ≥ 5. The paper does not address this dimension issue. That is a load-bearing gap, not a stylistic quibble.\n\nAlso, Theorem 2.5 is proved by \"suitably adapting\" Wu22, and Corollary 3.15 is omitted as identical to Wan22. These may be fine, but they make independent checking harder. To the paper's credit, it openly says the singular klt case is only a first step and that technical issues remain, so it is not overselling.\n\nWho is this for: anyone working on the structure of Kähler manifolds with nef anti-canonical bundle or on positivity of direct images. It deserves a serious referee, but the referee will need to check the MMP step and the three recent preprints. My recommendation: send to a strong journal, ask for a report that verifies Lemma 3.7 in all dimensions or restricts the main theorem to dim ≤ 4, and require the authors to supply the omitted proofs or point to precise statements in the cited preprints. This is conditional acceptance, not desk reject.","headline":"Kähler case of the Cao-Höring conjecture is plausibly resolved, but the proof rests on an under-verified MMP step and several recent preprints.","tokens_in":34147,"tokens_out":3111,"would_cite":true,"duration_ms":30098,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32Q30","14C30","14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A compact Kähler manifold with nef anti-canonical bundle admits a locally constant fibration with rationally connected fibers over a Calabi–Yau base, and the paper proves this for klt pairs.","keywords":["nef anti-canonical bundle","compact Kähler manifold","klt pair","locally constant fibration","rationally connected","pseudo-effective sheaf","numerically flat sheaf","Segre currents"],"falsifier":"A concrete counterexample would decide the matter: a compact Kähler manifold X carrying a klt pair (X, Δ) with -(K_X + Δ) nef whose MRC fibration base Y has c1(Y) ≠ 0, or whose general fiber is not rationally connected, would refute Theorem 1.2. Alternatively, a pseudo-effective torsion-free sheaf E on a compact Kähler manifold with c1(E) = 0 but whose reflexive hull E** is not locally free, or not numerically flat, would refute the flatness criterion Theorem 2.1 that powers the proof. Since the projective case is already settled, the search space is the genuinely non-projective Kähler territory: nowhere-ampleness, non-projective MRC fibrations, or sheaves whose Segre currents misbehave.","tokens_in":33173,"feed_emoji":"🌀","tokens_out":8330,"duration_ms":76639,"temperature":0.7,"pith_summary":"Every compact Kähler manifold whose anti-canonical bundle is nef (numerically effective) admits a locally trivial fibration over a Calabi–Yau manifold, with rationally connected fibers, and the paper proves this for klt pairs (X, Δ), not just smooth projective varieties. This resolves the Kähler case of a conjecture that, outside a few low-dimensional partial results, was previously known only in the projective setting. A new flatness criterion is the engine: any pseudo-effective torsion-free sheaf with vanishing first Chern class on a compact Kähler manifold has a locally free, numerically flat reflexive hull. If true, the result reduces the classification of nef-anticanonical Kähler manifolds to two well-understood building blocks — rationally connected varieties and Calabi–Yau manifolds.","feed_headline":"Kähler manifolds with nef anticanonical bundle fiber over Calabi–Yau","feed_subtitle":"A flatness criterion for pseudo-effective sheaves carries the proof from projective to Kähler manifolds.","key_machinery":"The engine is Theorem 2.1, a flatness criterion: if E is a pseudo-effective torsion-free sheaf on a compact Kähler manifold with c1(E) = 0, its reflexive hull E** is locally free and numerically flat. The proof generalizes the Segre current construction to sheaves: via singular Hermitian metrics and standard regularization theorems, it shows the second Chern class of E is represented by a semi-negative (2,2)-current when c1(E) = 0; together with sheaf stability and the classical Hermitian–Einstein local freeness criterion, this forces E** to be locally free and numerically flat. On the way, the paper builds a relatively big line bundle G on a resolution of the MRC fibration, using the projectivity criterion of [CH24], so that the direct image sheaf E_m = π_* φ_* V_m has vanishing first Chern class and is weakly positively curved.","core_discovery":"The paper establishes a structure theorem: for a klt pair (X, Δ) with X a compact Kähler manifold and -(K_X + Δ) nef, there exists a locally constant fibration f: X → Y such that the general fiber F is a rationally connected manifold and Y is a compact Kähler manifold with c1(Y) = 0. The proof proceeds by constructing, along a resolution of the MRC fibration, a direct image sheaf that is pseudo-effective with vanishing first Chern class, then applying the new flatness criterion to conclude it is locally free and numerically flat. The flatness of this sheaf yields a flat connection that descends to a splitting of the tangent bundle of X into the fibration direction and a numerically trivial part; Ehresmann's theorem then upgrades the fibration to a locally constant one. The same circle of ideas yields the Beauville–Bogomolov–Yau decomposition for klt Kähler pairs, the Hacon–McKernan inequalities in the Kähler setting, and the characterization of equality c2 = 0 in terms of tori and P1-bundles over tori.","pith_inferences":["If the flatness criterion is robust, it should extend to compact Kähler spaces with klt singularities, since the paper's obstacles are analytic rather than cohomological; the authors themselves flag this as forthcoming, and a natural test is to run the same Segre current argument on normal compact Kähler spaces with quotient singularities.","The structure theorem suggests viewing nef-anticanonical Kähler manifolds as 'almost Fano' in a fibration sense: the rationally connected fibers carry the Fano-like positivity and the Ricci-flat base carries the calabi-Yau part; this picture could be probed on examples such as projectivized vector bundles over tori with nef anti-canonical class, which should appear exactly in the c2-equality case.","The flatness criterion, phrased purely in terms of pseudo-effectivity and c1 = 0, may provide a new tool for abundance-type questions on compact Kähler manifolds: any pseudo-effective direct image sheaf with numerically trivial determinant is a flat bundle, which is exactly the input needed for the ramified covering trick in the abundance literature."],"forward_implications":["Conjecture 1.1 is resolved in full for klt pairs on compact Kähler manifolds: the nef anticanonical class forces a locally constant fibration with rationally connected fibers over a Calabi–Yau base.","The Beauville–Bogomolov–Yau decomposition extends to klt Kähler pairs with numerically trivial anti-log canonical class: a finite étale cover splits as a rationally connected factor times a torus times strict Calabi–Yau and holomorphic symplectic factors.","The Hacon–McKernan inequalities hold in the Kähler setting: the Kodaira dimension and numerical dimension of -(K_X + Δ) are bounded above by those of the restriction to a general MRC fiber.","The tangent bundle of a compact Kähler manifold with nef -K_X is generically nef, and the second Chern class inequality c2(T_X) · ω_1 ··· ω_{n-2} ≥ 0 holds for all Kähler classes.","Equality in the c2 inequality characterizes tori and P1-bundles over tori up to finite étale cover."],"supporting_citations":[{"why":"Supplies the projectivity criterion for Kähler morphisms with rationally connected general fibers, which produces the relatively ample line bundle that starts Step 1.","marker":"[CH24]"},{"why":"Establishes the decomposition for smooth projective varieties, the blueprint adapted here to the Kähler setting.","marker":"[CH19]"},{"why":"Provides the projective klt-pair version and the definition and technology of locally constant fibrations with respect to a pair.","marker":"[MW21]"},{"why":"Introduces the Segre current machinery for pseudo-effective sheaves and proves numerical flatness for locally free sheaves, here generalized to torsion-free sheaves.","marker":"[Wu22]"},{"why":"Supplies the projective klt-pair strategy that Step 3 follows and extends to the Kähler case.","marker":"[CCM21]"},{"why":"Gives the positivity of direct image sheaves via Bergman kernel metrics, used in Proposition 3.1.","marker":"[PT18]"},{"why":"Provides the theorem on weakly positively curved direct images applied over Kähler bases.","marker":"[HPS18]"},{"why":"Gives the stability-to-local-freeness criterion via Hermitian–Einstein metrics that converts c2 semi-negativity into local freeness.","marker":"[BS94]"},{"why":"Shows rationally connected families are dominated by rational curves, yielding the non-uniruledness of the MRC base used in Proposition 3.11.","marker":"[GHS03]"},{"why":"Supplies the foliation-to-fibration and Ehresmann arguments that turn the tangent splitting into a locally constant fibration.","marker":"[Hör07]"}],"fun_headline_variants":["Nef anticanonical bundle forces Calabi–Yau fibration","Kähler manifolds with nef anticanonical bundle admit Calabi–Yau fibrations","Nef anticanonical Kähler manifolds fiber over Calabi–Yau","Compact Kähler with nef anticanonical fibers over Calabi–Yau"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the projectivity criterion that a fibration between compact Kähler manifolds whose general fibers are rationally connected must be a projective morphism and carry a relatively ample line bundle; if that criterion fails, the construction of the relatively big line bundle and the whole chain of direct image sheaves collapses.","fun_headline_variants_meta":{"raw":{"variants":["Nef anticanonical bundle forces Calabi–Yau fibration","Kähler manifolds with nef anticanonical bundle admit Calabi–Yau fibrations","Nef anticanonical Kähler manifolds fiber over Calabi–Yau","Compact Kähler with nef anticanonical fibers over Calabi–Yau"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000808,"raw_usage":{"total_tokens":3518,"prompt_tokens":886,"completion_tokens":2632,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":2541}},"tokens_in":502,"tokens_out":2632,"duration_ms":21339,"temperature":1.0,"reasoning_tokens":2541,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:47:20.405540+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would decide the matter: a compact Kähler manifold X carrying a klt pair (X, Δ) with -(K_X + Δ) nef whose MRC fibration base Y has c1(Y) ≠ 0, or whose general fiber is not rationally connected, would refute Theorem 1.2. Alternatively, a pseudo-effective torsion-free sheaf E on a compact Kähler manifold with c1(E) = 0 but whose reflexive hull E** is not locally free, or not numerically flat, would refute the flatness criterion Theorem 2.1 that powers the proof. Since the projective case is already settled, the search space is the genuinely non-projective Kähler territory: nowhere-ampleness, non-projective MRC fibrations, or sheaves whose Segre currents misbehave.","supporting_citations":[],"review_version":1}