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On projective manifolds with pseudo-effective tangent bundle

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Pith's one-line read A projective manifold with pseudo-effective tangent bundle admits a smooth fibration to a base finitely covered by an abelian variety, with rationally connected general fibers; under a positively curved singular hermitian metric the…

desk verdict The main structure theorem is a genuine extension of the nef tangent bundle theory and deserves peer review, but the blow-up section has a false Lelong number inequality and the whole paper rests on an equivalence cited from elsewhere. read the letter →

arxiv 1908.06421 v1 pith:ZTIMZ3QA submitted 2019-08-18 math.AG math.CVmath.DG

Genki Hosono , Masataka Iwai , Shin-ichi Matsumura This is my paper · ORCID
classification math.AGmath.CVmath.DG MSC 32J2514J2658A30
keywords SingularhermitianmetricsPseudo-effectivevectorbundlesTangentRationallyconnectedvarietiesAbelianMRCfibrationsNumericallyflatClassificationofsurfaces
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

The paper establishes a structure theorem for projective manifolds whose tangent bundle is pseudo-effective: such a manifold is smoothly fibered over a base that is finitely covered by an abelian variety, and the general fiber is rationally connected. This mirrors older structure results for nef tangent bundles and semi-positive curvature, but in the weaker pseudo-effective setting, where the tangent bundle need not be nef. It also classifies minimal surfaces with pseudo-effective tangent bundle: ruled surfaces over the projective line or over an elliptic curve, with the elliptic cases necessarily minimal. The machinery is the theory of singular hermitian metrics on vector bundles, which the paper develops to prove splitting and numerical-flatness results that carry the main argument.

What carries the argument

The central object is the tangent bundle $T_X$ equipped with a singular hermitian metric, defined on the open set where the sheaf is locally free and required to satisfy that $\log|u|_{g^\vee}$ is plurisubharmonic for local sections of the dual. Pseudo-effectivity is the existence, for every $m>0$, of such metrics on $\operatorname{Sym}^m E$ whose curvature currents are bounded below by a fixed hermitian form. The argument is carried by three consequences of this notion: Theorem 1.2, that pseudo-effectivity plus $c_1=0$ forces local freeness and numerical flatness; Theorem 1.3, that positively curved exact sequences with $c_1(Q)=0$ split; and Theorem 1.4, the reflexive-sheaf version. These results, applied to the relative tangent sequence of an MRC fibration, force the fibration to be smooth and its base to have a numerically flat tangent bundle.

What would settle it

Construct a rank-two reflexive sheaf on a smooth projective threefold that is not locally free, is weakly positive in the sense of Definition 2.2(5), but admits no sequence of singular hermitian metrics on its symmetric powers satisfying the curvature bound of Definition 2.1; such a sheaf would disprove the equivalence on which the paper's main theorem and its surface classification rest.

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Extended reading notes

Core claim

The paper's central claim is that pseudo-effectivity of the tangent bundle imposes a rigid structure on a projective manifold $X$: there exists a smooth surjective morphism $\varphi:X\to Y$ with connected fibers, where $Y$ is smooth and admits a finite étale cover by an abelian variety, and a general fiber $F$ is rationally connected and itself has pseudo-effective tangent bundle. This is Theorem 1.1. The proof goes through three structural results for singular hermitian metrics: a pseudo-effective reflexive sheaf with $c_1=0$ is locally free and numerically flat (Theorem 1.2); a positively curved exact sequence of vector bundles with $c_1(Q)=0$ splits (Theorem 1.3); and the same splitting holds for reflexive sheaves on compact Kähler manifolds (Theorem 1.4). When $T_X$ carries a positively curved singular hermitian metric, the tangent sequence splits and $\varphi$ is locally trivial. As an application, the paper classifies minimal ruled surfaces with pseudo-effective tangent bundle (base $\mathbb{P}^1$ or an elliptic curve) and studies blow-ups of Hirzebruch surfaces, showing that pseudo-effectivity persists for blow-ups along up to three general points.

Load-bearing premise

The paper relies on an external theorem asserting that its metric definition of pseudo-effectivity agrees with the algebraic weak-positivity definition for projective manifolds; if that equivalence fails for reflexive or singular sheaves, the structure theorem and the surface classification may be speaking about different classes of bundles.

Editorial extensions

If this is right

  • Every projective manifold with pseudo-effective tangent bundle is smoothly fibered over a flat projective base, so the MRC fibration can be chosen holomorphic and without singular fibers.
  • The base of this fibration is finitely covered by an abelian variety, so its tangent bundle is numerically flat and its canonical bundle is torsion.
  • A general fiber $F$ again has pseudo-effective tangent bundle, so the class is closed under taking general fibers of the MRC fibration.
  • If $T_X$ admits a positively curved singular hermitian metric, the tangent sequence splits and $\varphi$ is locally trivial, giving a genuine fiber bundle structure over the flat base.
  • The surface results determine the ruled case completely: pseudo-effectivity forces the base to be $\mathbb{P}^1$ or an elliptic curve, forces smoothness of the ruling over an elliptic curve, and holds for all minimal ruled surfaces over those bases.

Reading between the lines

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

  • If the metric/algebraic equivalence used in the paper extends from projective manifolds to compact Kähler manifolds, the same MRC argument should give a Kähler version of Theorem 1.1 with a torus base; the paper already proves the Albanese version, Theorem 3.11.
  • The splitting theorem suggests a practical test for positive curvature on ruled surfaces: the dimension count $h^0(X,T_X)=h^0(X,T_{X/Y})+h^0(X,\varphi^*T_C)$ that rules out positively curved metrics on $S_n$ could be applied to other projectivized flat bundles to detect when the tangent sequence must split.
  • The surface examples suggest a threshold phenomenon: blowing up a Hirzebruch surface at up to three general points preserves pseudo-effectivity of the tangent bundle, while no more than two points can preserve generic global generation; understanding the fourth point likely requires controlling the position of the blown-up points rather than just their number.
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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

4 major / 4 minor

Summary. The paper develops singular hermitian metrics on torsion-free sheaves and applies them to study projective manifolds with pseudo-effective tangent bundle. The main result (Theorem 1.1) asserts that such a manifold X admits a smooth MRC fibration X -> Y over a smooth base Y with a finite etale abelian cover, whose general fiber is rationally connected and itself has pseudo-effective tangent bundle; under a positively curved singular metric assumption, the tangent sequence splits and the fibration is locally trivial. The paper also proves splitting theorems for positively curved vector bundles (Theorems 1.3 and 1.4), characterizes numerically flat sheaves (Theorem 1.2), and gives a partial classification of surfaces with pseudo-effective tangent bundle: minimal ruled surfaces over P^1 or an elliptic curve, plus a study of blow-ups of Hirzebruch surfaces.

Significance. If the main theorem is correct, it is a substantial extension of the nef tangent bundle theory of Campana-Peternell and Demailly-Peternell-Schneider to the pseudo-effective regime, with an elegant statement: the base of the MRC fibration is abelian up to etale cover, and the fibers are rationally connected. The splitting theorems for singular positively curved metrics and the explicit examples of ruled surfaces with pseudo-effective but not nef tangent bundle are useful and interesting in their own right. The paper also provides concrete computations distinguishing pseudo-effectivity, generic global generation, and nefness. However, several load-bearing steps rely on an external equivalence between metric and algebraic pseudo-effectivity for non-locally-free sheaves, and on a semicontinuity assertion for Lelong numbers that is not proved. These gaps affect both the proof of the structure theorem and the surface classification.

major comments (4)
  1. [§3.3, proof of Theorem 3.10] The main theorem is stated for the metric Definition 2.1, but the proof uses algebraic consequences of pseudo-effectivity that are immediate for Definition 2.2(5) and not proved for Definition 2.1. In particular, the sentence 'The morphism r is generically surjective, and thus the reflexive sheaf Q is also pseudo-effective' is used for the sheaf Q in (3.3), which is only generically a quotient of TX and may not be locally free before Theorem 1.2 is applied. Similarly, the proof of conclusion (4) uses the fact that the non-nef locus of O_{P(TX)}(1) has proper image in X, which is an algebraic statement. The paper cites [Iwa18, Theorem 1.3] for the equivalence of Definitions 2.1 and 2.2(5), but does not verify that this equivalence holds for reflexive non-locally-free sheaves. Since the classification and the structure theorem could refer to different classes if that equivalence fails, this point is load-bearing and must be either proved in the text or stated with complete hypotheses.
  2. [§3.1, Lemma 3.2] In the proof of Lemma 3.2(1), after defining f_m = (1/m) log |τ^m|_{h_m^∨} and observing √-1∂∂ f_m ≥ -ω/m, the text asserts that 'its weak limit (after we take a subsequence) should be zero'. This does not follow from the displayed curvature bound: a sequence of quasi-psh functions with second derivatives bounded below by -ω/m can converge weakly to a non-zero psh function. The subsequent contradiction using Lelong numbers depends on this unproved convergence to zero. Lemma 3.2 is used in the proof of Theorem 3.11 for the smoothness of the Albanese map, so this is not a purely cosmetic gap.
  3. [§4.2, Proposition 4.5(2), around Eq. (4.8)] The proof of the bound ν(S,p_0) ≥ 1/2 relies on the assertion 'Lelong numbers will also increase after taking a weak limit of currents', yielding ν(T,p') ≥ lim sup ν(√-1Θ_{g_m},p'). This is not a general property of weakly convergent closed positive currents; the inequality needs a proof or a precise reference valid for the specific sequence constructed here. If this inequality fails, the bound ν(S,p_0) ≥ 1/2 and hence the conclusion #Σ ≤ 4 are unsupported. Since Proposition 4.5(2) is used in the surface classification, this is a load-bearing gap in the classification part of the paper.
  4. [§3.2, proof of Theorem 3.9(2)] The proof of conclusion (2) applies Lemma 3.1 to the injection Q^∨ → Ω_X induced by (3.3). Lemma 3.1 requires the relevant sheaf to be almost nef. In the setting of Theorem 3.10, Ω_X is not known to be almost nef from the metric pseudo-effectivity of TX without invoking the same equivalence with Definition 2.2(5) discussed above, and the paper does not supply a direct metric argument. Please clarify the exact logical chain or provide a self-contained proof of this step.
minor comments (4)
  1. [Throughout] There are repeated typographical errors: 'Chen class' should be 'Chern class' in Theorems 1.2, 1.4, and Lemma 3.5, and 'Ehrensmann' should be 'Ehresmann' in the proof of Theorem 3.10.
  2. [§2.1, after Definition 2.2] The sentence 'The above definition is equivalent to the definition (5) below' should state precisely which class of sheaves (locally free, reflexive, or all torsion-free) is covered by [Iwa18, Theorem 1.3], and whether that reference has appeared in final form.
  3. [§4.1, Proposition 4.2] The construction of sections of Sym^m(T_X) ⊗ φ^*O(2p) via condition (4.7) is very hard to follow; a more conceptual explanation or a worked-out example for small m would improve readability.
  4. [§4.2, proof of Proposition 4.8] The explicit formulas for θ_1, θ_2, θ_3 in the cases n=1 and n≥2 are extremely long. Consider moving them to an appendix or providing a computer-algebra verification file, as the current presentation is difficult to check by hand.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are proved in the text, the cited definitional equivalence is external rather than an input-to-prediction reduction, and no fitted parameter or self-citation chain forces the conclusions.

full rationale

The paper's central derivation chain is self-contained. Theorem 1.2 is proved directly in Section 3.1 by induction using Bando-Siu admissible Hermitian-Einstein metrics; Theorems 1.3 and 1.4 are proved in Section 3.2; Theorem 3.10 (= Theorem 1.1) is then proved by running the MRC fibration argument and applying these proved results together with standard external facts from [Hör07], [BDPP13], [GHS03], [Bea83] and [DPS94]. No equation is normalized to force a conclusion, and no fitted constant is renamed as a prediction. The only definitional bridge is the assertion after Definition 2.2 that Definition 2.1 is equivalent to Definition 2.2(5) for projective manifolds, cited to [Iwa18, Theorem 1.3]. Although [Iwa18] is a preprint by a co-author, it is a parameter-free theorem with stated hypotheses that do not include the target structure theorem, so under the stated rules it counts as independent support rather than circularity. A possible mathematical concern is that this equivalence is not reproved and is used for reflexive sheaves that may not be locally free, but that is a correctness or verification risk, not a circularity of the derivation. The surface classifications in Section 4 are obtained by explicit section constructions and applications of the proved splitting theorem, again without reducing the conclusion to its own input.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper is proof-based; it introduces no fitted constants and no new entities. It relies on standard deep results in complex algebraic geometry, listed below, and on one external equivalence result from [Iwa18].

assumptions (7)
  • domain assumption Equivalence between metric pseudo-effectivity, Definition 2.1, and weak positivity, Definition 2.2(5)
    Invoked via [Iwa18, Theorem 1.3] immediately after Definition 2.2; the paper does not prove the equivalence.
  • standard math Beauville-Bogomolov decomposition applies to compact Kahler manifolds with numerically flat tangent bundle
    Used in Theorem 3.10(2) to rule out hyperkahler and Calabi-Yau components and conclude Y is covered by an abelian variety.
  • standard math Bando-Siu admissible Hermitian-Einstein metrics exist for stable reflexive sheaves and imply local freeness and hermitian flatness when c1 and c2 vanish
    Used in Theorem 3.4 in the rank S equals rank E case and in Lemma 3.5(2).
  • standard math Rau fi's theorem: a positively curved singular metric with smooth determinant has a well-defined curvature current
    Used in Lemma 3.5(1) to compute the curvature of the metric on Q.
  • standard math Höring's criterion: an integrable subbundle of TX with rationally connected general leaves yields a smooth MRC fibration
    Used in Theorem 3.10(1), cited as [Hör07, Corollary 2.11].
  • standard math The MRC base has pseudo-effective canonical bundle by [BDPP13], and rationally connected fibrations over curves have sections by [GHS03]
    Used in Theorem 3.10(1) and (2) to control c1(Q) and to justify rational connectivity of general fibers.
  • standard math Classification results for ruled surfaces over P1 and elliptic curves from [CP91], [Ati55], [Ati57], and [Suw69]
    Used in section 4.1 to enumerate minimal ruled surfaces and to determine which have nef tangent bundle.

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Pith. "Pith review of On projective manifolds with pseudo-effective tangent bundle." pith.science (2026). https://pith.science/paper/ZTIMZ3QA

@misc{pith2026190806421,
  author       = {Pith},
  title        = {Pith review of: On projective manifolds with pseudo-effective tangent bundle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTIMZ3QA}},
  note         = {Machine review of arXiv:1908.06421}
}
abstract

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general non-minimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

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