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Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in K\"ahler context

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A worthwhile conditional result with a real gap in one proof—deserves refereeing, not desk rejection. read the letter →

arxiv 2502.02183 v2 pith:XY6IVIKR submitted 2025-02-04 math.CV math.AG

Junyan Cao , Mihai Păun This is my paper · ORCID
classification math.CVmath.AG MSC 32Q1537F75
keywords relativecanonicalbundlepseudo-effectiveKählermanifoldsholomorphicfoliationsalgebraicitycriterionconormalunirulednessBergmankernelmetrics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\mathcal{O}_{\mathbb{P}(N^*_{C_0/S_0})}(1)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§4, Proposition 4.1] 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.
  2. [§5, proof of Theorem 5.1(2)] 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.
  3. [§5, expansion of r_*((ω+σ+σ̄)^{k+1})] 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.
minor comments (4)
  1. [§3, Step three] 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.
  2. [§5, proof of Theorem 5.1] 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.
  3. [§4, proof of Theorem 1.3] There is a typo in the final line: “the ananlytic graphe” should be “the analytic graph.”
  4. [§4, Proposition 4.4] The constants K̃1 and K̃2 appear without definition; their provenance from Proposition 4.2 and Lemma 4.5 should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central positivity and algebraicity claims are derived from independent prior theorems, with no step reducing to its own input; one unproved extension in Theorem 5.1(2) is a gap, not circularity.

full rationale

I find no circular step that would make a claimed prediction equivalent to its input. Theorem 3.1's proof applies the Bergman-kernel theorem of Berndtsson-Păun [1] and the technical lemmas of [39] and [30] as external results; their assumptions do not include the conclusion K_{X/Y}+L-D(p) psef, and the gluing step (Lemma 3.3) is argued directly. Theorem 1.3 and Proposition 4.4 construct currents with large Lelong numbers via Demailly mass concentration and Collins-Tosatti extension, and then read off pseudo-effectivity of N*_{C0/S0} from the inequality defining the current; this is the standard equivalence, not a self-definitional reduction. The algebraicity consequences are explicit applications of Theorem 1.3. The one genuinely fragile load-bearing sentence is in the proof of Theorem 5.1(2): 'the (1,1) current r_*omega^{k+1} is closed, greater than a Kähler metric on Z and smooth ... 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 asserts an unproved generalization of Theorem 1.2 to a singular background current; if it failed, the proof of det(H)^* psef would collapse. But that is a correctness/completeness gap, not circularity: the conclusion is not assumed, and the cited Theorem 1.2 remains an independent input. Self-citations such as [1], [6], [15], [39] are load-bearing in places, but they are established theorems with independent proofs and assumptions that do not include the target results, so under the stated rules they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper is a theory paper. No numerical parameters are fitted to data, and no new objects (particles, forces, dimensions) are postulated. The proofs depend on a large network of cited theorems in pluripotential theory, Bergman kernel theory, and stability theory, several authored or co-authored by the present authors (Păun in [1], [30], [39], and both in [15]). These are established results with independent proofs, so they are listed as axioms rather than as circular inputs. The only new conceptual machinery introduced is the analytic notion of pseudo-effectivity for sheaves (Definition 2.4), which is explicitly noted to be weaker than the algebraic version; this is captured under domain assumptions.

assumptions (9)
  • standard math El Mir extension theorem and support theorem for closed positive currents (Demailly, Complex analytic and differential geometry, Ch. III, Thm 2.3 and 2.10; cited in Proposition 2.1)
    Used to extend positive currents across subsets of codimension at least 2 and to identify currents that agree outside such subsets (e.g., in Lemma 2.2 and Proposition 4.2).
  • standard math Demailly's mass concentration theorem (Demailly, J. Differential Geom. 37 (1993), Cor 6.8; cited in Proposition 4.1)
    Provides ωX-psh functions with arbitrarily large Lelong number at a prescribed point, used in Proposition 4.1 and the proof of Theorem 1.3.
  • standard math Collins-Tosatti extension theorem for Kähler currents with analytic singularities (Collins and Tosatti, Ann. Fac. Sci. Toulouse 23 (2014), Thm 1.1; invoked in proof of Theorem 1.3)
    Extends the ωX-psh function constructed on the Zariski closure M to the ambient Kähler manifold X; without it Proposition 4.1 could not be applied globally.
  • standard math Berndtsson-Păun theorem on Bergman kernels and pseudo-effectivity of relative canonical bundles (Berndtsson and Păun, Duke Math. J. 145 (2008); quoted as Theorem 3.4)
    The core input in Step two of the proof of Theorem 3.1; it makes the fibrewise minimal-singularity metric hmin psh along the fibers.
  • standard math Hacon-Păun canonical bundle formula for generalized Kähler pairs (Hacon and Păun, arXiv:2404.12007 [30]; the proof of Theorem 3.1 is said to be contained in the ideas of [30] and [39])
    The paper explicitly says the main ideas of Theorem 3.1 are contained in [30] and [39], so these results are relied upon as background rather than reproved.
  • standard math Nadel vanishing theorem and Shokurov's trick (cited in Lemma 3.3)
    Used in Lemma 3.3 to show that sections of k0(KZ+F+1/m0 A)+ρ can be lifted to sections with ρ0 through multiplication by powers of s, guaranteeing the constructed limiting metric exists.
  • standard math Bruasse's Harder-Narasimhan filtration for coherent torsion-free sheaves on compact complex manifolds with Gauduchon metric (Bruasse, Internat. J. Math. 12 (2001) and Ann. Inst. Fourier 53 (2003); quoted as Theorem 6.5)
    Underlies the definition of the maximal α-destabilizing subsheaf used in all slope arguments in Section 5 and the appendix.
  • standard math Campana-Peternell additivity of maximal and minimal slopes for reflexive tensor powers (Campana and Peternell, Bull. Soc. Math. France 139 (2011) with appendix by Toma; quoted as Theorem 6.6)
    Used in Proposition 6.8 and Corollary 6.7 to pass from slope positivity of tensor powers to slope positivity of F, and to control μ_{α,min} of Grassmannian tensor spaces.
  • domain assumption Existence of relative MRC fibration in the Kähler context via compactness of Chow-Barlet spaces (stated in a footnote in Section 5)
    The footnote asserts the relative rational quotient exists on compact Kähler manifolds because the Chow-Barlet space is compact; this is not proved in the paper, and Corollary 5.4 depends on it.

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Pith. "Pith review of Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in K\"ahler context." pith.science (2026). https://pith.science/paper/XY6IVIKR

@misc{pith2026250202183,
  author       = {Pith},
  title        = {Pith review of: Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in K\"ahler context},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XY6IVIKR}},
  note         = {Machine review of arXiv:2502.02183}
}
read the original abstract

In this note, motivated by the recent preprint of W. Ou, we pursue three main objectives. The first is to make progress towards the positivity of the relative canonical bundle in the K\"ahler setting. In the second part, we provide a proof of Ou's algebraicity criterion. Finally, based on the two previous parts, we slightly extend his uniruledness criterion.

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