{"id":"3513e1bb-07af-4d60-b982-bcc200c8c5e3","arxiv_id":"2502.02183","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New pseudo-effectivity theorems for relative canonical bundles and new algebraicity and uniruledness criteria for foliations on compact Kähler manifolds are established.","lead":"This paper proves a new partial result on the pseudo-effectivity of twisted relative canonical bundles for Kähler families, and gives an analytic proof of Ou's algebraicity criterion for foliations, plus new Kähler extensions of uniruledness criteria. These are contributions at the interface of complex and algebraic geometry that strengthen the dictionary between curvature positivity and algebraicity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1(2) relies on an unproved extension of Theorem 1.2 to a singular relative Kähler current; the 'goes through' assertion is the weakest load-bearing step.","rationale":"The reader's verdict is CONDITIONAL, and I agree with that overall. However, I would locate the main load-bearing risk differently. The reader's weakest_assumption points to the existence of the pair (ω,σ) and to the weaker analytic definition of pseudo-effectivity. The pair (ω,σ) is a standard, cited device (Campana [12]) and its use in Theorem 5.1 is plausible; the weak psef definition is explicitly mitigated by Proposition 6.8, which proves the needed slope consequence. The most concrete vulnerability is the unproved passage in the proof of Theorem 5.1(2), where Theorem 1.2 is applied with a singular current in place of a Kähler metric. The manuscript itself flags this by saying 'the proof of Theorem 1.2 goes through', without supplying the required L2/k extension argument. This step is load-bearing for the uniruledness criterion and for Corollary 5.4, and it is exactly the kind of compressed step that an independent expert should verify. My proposed test directly checks whether the Bergman-kernel construction survives with a singular background current. Since the rest of the core argument, especially Theorem 1.3, appears carefully structured, I do not think the verdict should be changed from CONDITIONAL; but the conditional should explicitly include this missing verification.","tokens_in":21905,"tokens_out":27491,"duration_ms":286085,"concrete_test":"Prove the promised generalization of Theorem 1.2 for a holomorphic fibration q : Z → Y equipped with a closed positive (1,1) current α such that α ≥ cω_Z for a Kähler metric ω_Z and α is smooth off q^{-1}(Σ) for a divisor Σ. Verify the local L2/k Ohsawa–Takegoshi estimate used in Step four of Theorem 3.1 with the singular weight α, including a uniform constant independent of k and m0 when m0 is large. If the estimate requires extra hypotheses on the singularities of α, the 'goes through' step in Theorem 5.1 is not justified as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 5.1(2), after constructing the relative MRC fibration r : X → Z and s : Z → Y, the authors need to prove that det(H)^* is psef, where H = ker(ds). They do this by applying Theorem 1.2 to Z → Y with a 'Kähler metric' replaced by the closed (1,1) current r_*ω^{k+1}. The text says this current is 'greater than a Kähler metric on Z' and smooth away from a divisor, and then states: 'The local L2/k-extension in Step four still works when m0 is large enough. Then the proof of Theorem 1.2 goes through.' This is exactly the step that yields det(H)^* psef, which in turn forces H = 0 and gives rational connectedness of the fibers. The concern is not that the conclusion is false, but that the invoked generalization of Theorem 1.2 is neither stated nor proved. Theorem 1.2 and Theorem 3.1 use a genuine smooth Kähler metric to set up the Bergman-kernel L2/k norms and the Ohsawa–Takegoshi extension estimate in Step four. Here the background form is only a closed positive current with possible singularities over the discriminant locus. The local weights and the L2/k integrability of sections can change when the background has singular support; uniform constants in the extension theorem are not automatic. If this extension fails, the proof of Theorem 5.1(2) and Corollary 5.4 collapses. The authors explicitly acknowledge the setup is 'a bit more general' and do not supply the missing argument, so this is a genuine gap in the central chain of the paper, not an ungenerous reading.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies positivity of relative canonical bundles and algebraicity criteria for foliations on compact Kähler manifolds. The main results are: (i) Theorem 3.1, a conditional version of the Cao–Höring conjecture on the pseudo-effectivity of K_{X/Y}+L-D(p) under the existence of a rational class ω+σ+σ̄ with σ vanishing on the generic fibers; (ii) Theorem 1.3, an analytic proof of Ou's algebraicity criterion for conormal bundles; and (iii) Theorem 1.4 / Corollary 1.5, extensions of uniruledness criteria. The paper also contains an appendix on stability of coherent sheaves with respect to Gauduchon metrics.","tokens_in":22168,"tokens_out":30236,"duration_ms":256725,"significance":"If the gaps identified below are repaired, the paper would constitute a meaningful step toward the Cao–Höring conjecture and provide a self-contained analytic proof of Ou's criterion, with a weaker analytic notion of pseudo-effectivity for sheaves. The appendix's treatment of slopes with Gauduchon metrics and Proposition 6.8 are useful contributions. However, two load-bearing points need work: an explicit error in Proposition 4.1 and an unproved generalization of Theorem 1.2 in Section 5.","major_comments":[{"comment":"The function f is defined as f = (1/m)(n-1)/n log(|z1|^2 + |z2|^{2m} + ... + |zn|^{2m}). For a submanifold S0 contained in {z1 = 0}, the restriction f|S0 has Lelong number (n-1)/n at x, independent of m, because the coefficient and the exponent m cancel. The text asserts ν(f|S0, x) = 2 m^{1/n}, which is incompatible with this definition. Moreover, the multiplier ideal computation that follows uses a threshold proportional to kC m^{(n-1)/n}, which would correspond to a different coefficient. As Proposition 4.1 is the starting point for making the Lelong numbers arbitrarily large in the proof of Theorem 1.3, this discrepancy is load-bearing. The formula for f should be corrected and the subsequent estimates rechecked.","section":"§4, Proposition 4.1"},{"comment":"The proof applies Theorem 1.2 to the fibration s: Z → Y with the closed positive current r_*ω^{k+1} in place of a smooth Kähler metric. The text states that “the local L^{2/k}-extension in Step four still works when m0 is large enough” and that “the proof of Theorem 1.2 goes through.” No argument is supplied to justify the Ohsawa–Takegoshi L^{2/k} extension with uniform constants when the background form has singularities over a divisor, nor is it shown that the class remains rational and that the (2,0)-part vanishes on the generic fibers of s after the push-forward. This step is essential to conclude that det(H)^* is pseudo-effective and hence that the fibers are rationally connected. The same unproved application appears in the proof of Corollary 5.4.","section":"§5, proof of Theorem 5.1(2)"},{"comment":"The formula r_*(ω+σ+σ̄)^{k+1} = r_*ω^{k+1} + k·r_*(ω^k∧σ + ω^k∧σ̄) omits the term involving ω^{k-1}∧σ∧σ̄, which is of bidegree (2,2) and contributes a (1,1)-current after push-forward over k-dimensional fibers. This omission affects the identification of the Kähler current and the rationality argument in the reduction to Theorem 1.2. The authors should either justify that the omitted term vanishes or include it in the verification of the hypotheses.","section":"§5, expansion of r_*((ω+σ+σ̄)^{k+1})"}],"minor_comments":[{"comment":"The notation “Σ (m_i - 1)[W_i ∩ X]” is ambiguous; it should be “Σ (m_i - 1)[W_i]” (or specify the intersection with the local chart) so that the current's divisor is unambiguous.","section":"§3, Step three"},{"comment":"The phrase “r_*ω^{k+1} is closed, greater than a Kähler metric on Z” should read “greater than a Kähler form” or “a Kähler current,” since a current is not a metric.","section":"§5, proof of Theorem 5.1"},{"comment":"There is a typo in the final line: “the ananlytic graphe” should be “the analytic graph.”","section":"§4, proof of Theorem 1.3"},{"comment":"The constants K̃1 and K̃2 appear without definition; their provenance from Proposition 4.2 and Lemma 4.5 should be stated explicitly.","section":"§4, Proposition 4.4"}],"recommendation":"major_revision","confidential_remarks":"The Proposition 4.1 issue appears to be a simple typographical error in the definition of f; the intended scaling can likely be recovered from the multiplier ideal computation. The Section 5 gap is more substantive and may require a genuine argument or a reformulation of the reduction to Theorem 1.2. I recommend major revision rather than rejection, because the central ideas are promising and the flaws seem addressable within the manuscript's framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2502.02183. First, the genuinely new result is Theorem 3.1: under the existence of a Kähler metric ω and a holomorphic 2-form σ with ω+σ+conjugate σ rational and σ vanishing on the generic fiber, the twisted relative canonical bundle K_{X/Y}+L−D(p) is pseudoeffective. This is a conditional advance on the Cao–Höring conjecture, and the proof is mostly self-contained. The gluing lemma (Lemma 3.3) is a solid technical contribution. Second, the proof of Theorem 5.1(2) has a load-bearing gap: the authors apply Theorem 1.2 to a family where the background form is the closed positive current r_*ω^{k+1}, which is singular over the discriminant locus, and assert that the L2/k extension step \"goes through.\" That assertion is not proved. The singular background can affect the Bergman-kernel norms and the uniformity of the Ohsawa–Takegoshi constants, so this is a genuine missing argument, not a stylistic complaint. It does not invalidate Theorem 3.1 or the analytic proof of Theorem 1.3, but Theorem 5.1(2) and Corollary 5.4 remain conditional unless the step is filled.\n\nWhat the paper does well: the analytic proof of Ou's algebraicity criterion is a real alternative to Ou's algebraic method, using Lelong numbers and avoiding heavy desingularization. The authors are transparent that their Definition 2.4 of psef for sheaves is a priori weaker than the algebraic one, and Proposition 6.8 provides the slope consequence they need for the applications. The handling of the divisor D(p) in Theorem 3.1 is careful.\n\nThe softer spots: the reliance on a network of advanced cited results is normal for this area, but it does mean the paper will be hard to referee quickly. The definition of psef for sheaves deserves scrutiny: Proposition 6.8 shows maximal slope is nonnegative for mobile classes, which suffices here, but the gap between analytic and algebraic psef is not fully clarified. Minor: the proof of Theorem 5.1 also uses an approximation argument to get σ; that is standard and harmless.\n\nWho is this for: complex geometers working on positivity, relative canonical bundles, and foliations. It deserves a serious referee. I would send it to peer review, with the request that the authors prove the singular-current generalization of Theorem 1.2 or adjust the statement of Theorem 5.1(2). The paper is worth engaging with.","headline":"A worthwhile conditional result with a real gap in one proof—deserves refereeing, not desk rejection.","tokens_in":22788,"tokens_out":4129,"would_cite":true,"duration_ms":38598,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32Q15","37F75"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a conditional positivity theorem for twisted relative canonical bundles and derives algebraicity and rational connectedness for positive slope foliations on compact Kähler manifolds.","keywords":["relative canonical bundle","pseudo-effective","Kähler manifolds","holomorphic foliations","algebraicity criterion","conormal bundle","uniruledness","Bergman kernel metrics"],"falsifier":"To test Theorem 3.1, take a compact Kähler fibration satisfying (i)–(iii) for which the class $c_1(K_{X/Y}+L-D(p))$ can be computed explicitly—for example a Lagrangian fibration on a hyperkähler manifold—and check whether that class lies in the pseudo-effective cone; the theorem asserts it always does, so a negative computation would disprove it. To test the boundary, the same computation on a fibration with no such form $\\sigma$ would indicate whether the extra hypothesis is removable.","tokens_in":21612,"feed_emoji":"🌿","tokens_out":16201,"duration_ms":139672,"temperature":0.7,"pith_summary":"This paper addresses the question of when a fibration of compact Kähler manifolds has a positive twisted relative canonical bundle. The central claim is conditional: if the fibration admits a Kähler metric $\\omega$ and a holomorphic 2-form $\\sigma$ such that $\\omega+\\sigma+\\overline{\\sigma}$ is a rational class and $\\sigma$ vanishes on the generic fiber, and if the adjoint bundle restricted to the generic fiber is pseudo-effective, then $K_{X/Y}+L-D(p)$ is pseudo-effective. This is a step toward the conjecture that the same conclusion holds without the auxiliary form. The same circle of ideas yields an analytic proof of an algebraicity criterion: a submanifold whose Zariski closure has larger dimension forces its conormal bundle to be pseudo-effective, so any holomorphic foliation with non-pseudo-effective dual is induced by a meromorphic map. A final layer shows that foliations with positive slope on compact Kähler manifolds are algebraic and have rationally connected leaves, which implies uniruledness.","feed_headline":"Positive foliations on Kähler manifolds are algebraic","feed_subtitle":"Extends an algebraicity criterion from projective to Kähler manifolds and yields uniruledness.","key_machinery":"The load-bearing construction is the fiberwise Bergman kernel metric with minimal singularities. For a projective fiber $Z$ and an ample line bundle $A$, Lemma 3.3 shows that the extremal metric defined by $L^{2/k}$-normalized sections of $k(K_Z+F+(1/m_0)A)+\\rho$ is independent of the flat twist $\\rho\\in\\operatorname{Pic}^0(Z)$; this independence is what lets the metric glue across coordinate charts of the base, and Theorem 3.4 turns it into a positively curved metric on $K_{X/U}+L+(1/m_0)H_U$. The ramification divisor $D(p)$ is subtracted because the curvature of the fiberwise $L^{2/k}$ metric gains a current $(m_i-1)[W_i]$ along the components where $p$ has multiplicity $m_i$. For the algebraicity criterion, the mechanism is the comparison of Lelong numbers under blow-up and push-forward (Lemmas 2.2 and 2.3): it converts a very large Lelong number at one point of $S_0$ into a large generic Lelong number along $C_0$, and that in turn produces positive currents in the class of $\\mathcal{O}_{\\mathbb{P}(N^*_{C_0/S_0})}(1)$.","core_discovery":"The paper's main objective is to establish three linked statements. Theorem 3.1 says that under hypotheses (i)–(iii) of the introduction—a surjective map $p:X\\to Y$ of compact Kähler manifolds, a Kähler metric $\\omega$ and holomorphic 2-form $\\sigma$ with $[\\omega+\\sigma+\\overline{\\sigma}]$ rational and $\\sigma|_{X_y}=0$ for generic $y$, and a line bundle $L$ whose adjoint restriction $K_{X_y}+L|_{X_y}$ is pseudo-effective for generic $y$—the twisted relative canonical bundle $K_{X/Y}+L-D(p)$ is pseudo-effective, where $D(p)=\\sum (m_k-1)W_k$ records the ramification of $p$. Theorem 1.3 is the algebraicity criterion: if a compact submanifold $C$ of a compact Kähler manifold $X$ is contained, over a dense open $C_0$, in a locally closed submanifold $S_0$ whose Zariski closure $M$ has $\\dim M>\\dim S_0$, then the conormal bundle $N^*_{C_0/S_0}$ is pseudo-effective in the sense of Definition 2.4; the corollary is that a holomorphic foliation whose conormal sheaf is not pseudo-effective is induced by a meromorphic map. The final theorem uses these two to show that a foliation with positive slope has an algebraic maximal destabilizing subsheaf whose generic leaves are rationally connected, so $X$ is uniruled and $K_X$ is not pseudo-effective.","pith_inferences":["The proof suggests the rationality hypothesis is removable: it is used only to obtain local projectivity, so any independent glueing mechanism for the fiberwise Bergman metrics would prove the full conjecture.","The analytic notion of pseudo-effectivity for sheaves is explicitly weaker than the algebraic one; if the two coincide, the algebraicity criterion recovers the exact algebraic statement, and Proposition 6.8 is the first bridge.","The constants in Proposition 4.2 are not made explicit; a quantitative version would bound the degree of the Zariski closure of $S_0$ in terms of the ambient curvature, giving an effective algebraicity criterion.","The movable-slope characterization of uniruledness suggests a cohomological test: finding a movable class on which $T_X$ has positive slope should certify the existence of rational curves through every point."],"forward_implications":["Under the rationality condition, the positivity conjecture for twisted relative canonical bundles holds: $K_{X/Y}+L-D(p)$ is pseudo-effective whenever the adjoint bundles on the generic fibers are pseudo-effective.","If $K_X$ is pseudo-effective on a compact Kähler manifold, then every torsion-free quotient of $\\otimes^m T_X^*$ has pseudo-effective determinant, and $\\nu(X,L)\\le \\nu(X,K_X)$ for any line bundle $L$ injecting into $\\otimes^m T_X^*$.","A holomorphic foliation with $\\mu_{\\alpha,\\min}(F)>0$ for some movable class $\\alpha$ is algebraic, its generic leaves are rationally connected, and $X$ is uniruled.","Uniruledness of a compact Kähler manifold is equivalent to $\\mu_{\\alpha,\\max}(T_X)>0$ for some movable class $\\alpha$, and also to $\\mu_{\\alpha}(T_X)>0$.","In a proper holomorphic submersion over a disk, pseudo-effectivity of the canonical bundle of the central fiber propagates to every fiber, giving a special case of the invariance of plurigenera."],"supporting_citations":[{"why":"It supplies the $L^{2/k}$ Bergman kernel method and the psh variation of minimal-singularity metrics used to build the relative metric.","marker":"[1]"},{"why":"It provides the comparison $h_{\\min}=h_{\\min,\\rho}$ and the boundedness arguments that make the metric glue on overlapping coordinate charts.","marker":"[39]"},{"why":"It states the algebraicity and uniruledness criteria that the paper reproves and extends to the Kähler setting.","marker":"[37]"},{"why":"It provides the mass concentration theorem used to construct $\\omega_X$-psh functions with arbitrarily large Lelong numbers at a point of $S_0$.","marker":"[20]"},{"why":"It supplies the extension theorem for Kähler currents with analytic singularities used to pass from $S_0$ to $X$.","marker":"[18]"},{"why":"It gives the projective proof that positive-slope foliations are algebraic with rationally connected leaves, adapted here to Kähler manifolds.","marker":"[11]"},{"why":"It establishes local projectivity for the relevant fibrations, justifying the rationality condition involving the holomorphic 2-form.","marker":"[12]"},{"why":"It shows that a non-pseudo-effective canonical bundle implies uniruledness, used to identify generic fibers as rationally connected.","marker":"[6]"}],"fun_headline_variants":["Ou's algebraicity criterion proven for Kähler manifolds","From projective to Kähler: algebraicity for foliations","Kähler foliations: Ou's algebraicity criterion proved","Kähler positivity yields algebraic foliations","Uniruledness and algebraicity for Kähler foliations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the existence of the auxiliary 2-form $\\sigma$ making $[\\omega+\\sigma+\\overline{\\sigma}]$ a rational class with $\\sigma$ vanishing on the generic fiber; the paper expects this to be unnecessary, but without it the relative Bergman metric construction cannot be globalized.","fun_headline_variants_meta":{"raw":{"variants":["Ou's algebraicity criterion proven for Kähler manifolds","From projective to Kähler: algebraicity for foliations","Kähler foliations: Ou's algebraicity criterion proved","Kähler positivity yields algebraic foliations","Uniruledness and algebraicity for Kähler foliations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000991,"raw_usage":{"total_tokens":4198,"prompt_tokens":943,"completion_tokens":3255,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":3171}},"tokens_in":559,"tokens_out":3255,"duration_ms":22611,"temperature":1.0,"reasoning_tokens":3171,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:02:58.879650+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test Theorem 3.1, take a compact Kähler fibration satisfying (i)–(iii) for which the class $c_1(K_{X/Y}+L-D(p))$ can be computed explicitly—for example a Lagrangian fibration on a hyperkähler manifold—and check whether that class lies in the pseudo-effective cone; the theorem asserts it always does, so a negative computation would disprove it. To test the boundary, the same computation on a fibration with no such form $\\sigma$ would indicate whether the extra hypothesis is removable.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the $L^{2/k}$ Bergman kernel method and the psh variation of minimal-singularity metrics used to build the relative metric."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the comparison $h_{\\min}=h_{\\min,\\rho}$ and the boundedness arguments that make the metric glue on overlapping coordinate charts."},{"cited_title":"Differential Geom., 37(2):323–374, 1993","cited_arxiv_id":null,"evidence_quote":"It provides the mass concentration theorem used to construct $\\omega_X$-psh functions with arbitrarily large Lelong numbers at a point of $S_0$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the extension theorem for Kähler currents with analytic singularities used to pass from $S_0$ to $X$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the projective proof that positive-slope foliations are algebraic with rationally connected leaves, adapted here to Kähler manifolds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes local projectivity for the relevant fibrations, justifying the rationality condition involving the holomorphic 2-form."},{"cited_title":"Algebraic Geom","cited_arxiv_id":null,"evidence_quote":"It shows that a non-pseudo-effective canonical bundle implies uniruledness, used to identify generic fibers as rationally connected."}],"review_version":1}