{"id":"55a7a410-1594-4ab7-961f-9a08e660cf5a","arxiv_id":"1908.06421","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Projective manifolds with pseudo-effective tangent bundle admit a smooth fibration to a flat projective manifold with rationally connected general fiber, and all minimal such surfaces are classified.","lead":"This paper proves that any projective manifold whose tangent bundle is pseudo-effective, a weak positivity condition, must fiber smoothly over a flat projective base with rationally connected general fibers. It then classifies which algebraic surfaces have pseudo-effective tangent bundles, including blow-ups of Hirzebruch surfaces along up to three general points.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 relies at several key steps on the asserted equivalence between the metric Definition 2.1 and the algebraic Definition 2.2(5), cited from [Iwa18] rather than proved; if that equivalence fails for non-locally-free sheaves, the theorem and its applications may concern…","rationale":"The reader's weakest assumption identifies exactly the same external equivalence [Iwa18, Theorem 1.3] as the most load-bearing point. I agree: the proof of Theorem 1.1 uses algebraic characterizations and descent properties of pseudo-effectivity that are not derived from Definition 2.1, and the only bridge is a cited equivalence that is not reproved and is applied to possibly non-locally-free sheaves. The Lelong-number inequality in Proposition 4.5(2) is also a genuine error, but it affects the surface classification rather than the central structure theorem, so it is not the most load-bearing concern about Theorem 1.1. Since the equivalence is an external citation rather than an internal contradiction, the appropriate disposition remains CONDITIONAL; my stress-test does not change the reader's verdict.","tokens_in":24247,"tokens_out":25351,"duration_ms":271671,"concrete_test":"Verify the hypotheses of [Iwa18, Theorem 1.3] for the two sheaves that occur in the proof of Theorem 3.10: the reflexive sheaf Q = (π_*φbar^*Ω_Y)^∨ in (3.3), and the restriction TX|F to a general fiber F. Concretely, inspect the proof of [Iwa18, Theorem 1.3] for any essential use of local freeness of E or of the minimal extension property. If either hypothesis is needed, the equivalence does not cover the sheaves used here and the algebraic steps in Section 3.3 require separate justification; if neither is needed, the concern is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is stated for \"pseudo-effective tangent bundle\" without fixing a definition. In Section 2, Definition 2.1 is the metric notion, and the text asserts equivalence with Definition 2.2(5) via [Iwa18, Theorem 1.3]. The proof of Theorem 3.10 uses algebraic consequences of pseudo-effectivity that are immediate for Definition 2.2(5) but are not proved for Definition 2.1: the non-nef locus of O_{P(TX)}(1) is a proper subset of X (used for conclusion (4)); quotients of pseudo-effective sheaves are pseudo-effective (used for Q in (3.3) and for the surjection Λ^{m+1}TX|F → TF); and pseudo-effectivity restricts to a general fiber F. In addition, the sheaf Q in (3.3) is defined as the reflexive dual of π_*φbar^*Ω_Y, and it is only generically a quotient of TX; the step \"Q is pseudo-effective\" depends on the equivalence in a situation where Q may not be locally free. The paper does not reprove [Iwa18, Theorem 1.3], and that theorem is not checked for reflexive non-locally-free sheaves. If the equivalence has a hidden hypothesis, Theorem 1.1 could hold for the metric notion while the classification and examples in Section 4 use a different class, and conclusion (4) would not transfer.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24562,"tokens_out":25751,"duration_ms":271292,"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":[{"comment":"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.","section":"§3.3, proof of Theorem 3.10"},{"comment":"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.","section":"§3.1, Lemma 3.2"},{"comment":"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.","section":"§4.2, Proposition 4.5(2), around Eq. (4.8)"},{"comment":"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.","section":"§3.2, proof of Theorem 3.9(2)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"§2.1, after Definition 2.2"},{"comment":"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.","section":"§4.1, Proposition 4.2"},{"comment":"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.","section":"§4.2, proof of Proposition 4.8"}],"recommendation":"major_revision","confidential_remarks":"The main structure theorem is plausible and the surface examples are valuable, but the proof currently rests on two fragile pillars: the equivalence between metric and algebraic pseudo-effectivity for non-locally-free sheaves, cited from the unpublished preprint [Iwa18], and a Lelong-number semicontinuity assertion in Proposition 4.5(2). The first issue is not merely cosmetic, because the sheaf Q in (3.3) may not be locally free. The second issue affects the classification, though Theorem 1.1 might survive if Proposition 4.5 is modified. I recommend major revision rather than rejection, since these gaps appear fixable in principle, but they require real work and should be addressed explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the structure theorem is the real thing and deserves a serious referee; the non-minimal surface section has a genuine flaw, and the paper leans on a cited equivalence you should check before relying on it.\n\nWhat's new: Theorem 1.1 extends the nef tangent bundle structure theory of Campana–Peternell and Demailly–Peternell–Schneider to pseudo-effective tangent bundles. That is a genuine step, and removing the minimal extension property in Theorems 1.3 and 1.4 compared to HPS18 is a real improvement. The proof route is sensible: take the MRC fibration, build the reflexive sheaf Q, show Q is numerically flat via Theorem 1.2, then apply Lemma 3.1 and the Beauville–Bogomolov decomposition. Theorem 1.5, the classification of minimal surfaces, is also solid and uses explicit ruled-surface constructions.\n\nSoft spots, in order.\n\nFirst, the paper uses two definitions of pseudo-effectivity. Definition 2.1 is metric; Definition 2.2(5) is algebraic weak positivity. The text asserts equivalence via [Iwa18] but does not prove it. The proof of Theorem 1.1 switches freely between them, for example when it calls Q pseudo-effective and when it uses the non-nef locus of O_P(TX)(1). If [Iwa18] has hidden hypotheses for reflexive sheaves, the structure theorem and the surface classification could be about different classes. This is a dependency, not a demonstrated error, but a referee should confirm the equivalence covers the reflexive sheaves used here.\n\nSecond, Proposition 4.5(2) has a genuine mistake. The proof says Lelong numbers increase after taking a weak limit. That is backwards: for positive closed currents, the Lelong number of the weak limit is at most the liminf of the Lelong numbers, not at least the limsup. Without the inequality, the bound ν(S,p0) ≥ 1/2 does not follow, and the claim ♯Σ ≤ 4 is unproven. This does not touch Theorem 1.1 or the minimal surface classification, but it affects the blow-up examples and the abstract claim about non-minimal surfaces.\n\nThird, several computations in section 4.2 are asserted rather than written, for example 'an involved but straightforward computation' and explicit θ sections without verification. That is minor and normal for this kind of paper, but combined with the Lelong issue, the non-minimal surface part is the weakest part.\n\nThe central theorem looks plausible and the citation pattern is honest. No fitted parameters or normalization tricks. The main theorems are proved in the text. I would send this to a serious referee, with instructions to fix or weaken Proposition 4.5(2) and to verify the [Iwa18] equivalence in the reflexive setting. The paper is for people working on positivity of tangent bundles and singular hermitian metrics; I would cite the main theorem with a caveat about the definitional equivalence.","headline":"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.","tokens_in":25135,"tokens_out":6294,"would_cite":true,"duration_ms":57308,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32J25","14J26","58A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Singular hermitian metrics","Pseudo-effective vector bundles","Tangent bundles","Rationally connected varieties","Abelian varieties","MRC fibrations","Numerically flat vector bundles","Classification of surfaces"],"falsifier":"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.","tokens_in":24027,"feed_emoji":"","tokens_out":9585,"duration_ms":83970,"temperature":0.7,"pith_summary":"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.","feed_headline":"Pseudo-effective tangent bundle forces smooth rational fibration","feed_subtitle":"Proves the manifold fibers over a flat base with rationally connected fibers, and classifies the minimal surfaces.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the equivalence between the paper's metric definition of pseudo-effectivity and the algebraic weak-positivity definition used in the surface classification.","marker":"[Iwa18]"},{"why":"Provides the underlying MRC-fibration argument and the splitting technique that Theorem 1.1 builds on.","marker":"[Mat18b]"},{"why":"Admissible Hermitian-Einstein metrics are used in Theorem 1.2 to conclude that a pseudo-effective reflexive sheaf with $c_1=0$ is locally free and numerically flat.","marker":"[BS94]"},{"why":"Establishes the framework of singular hermitian metrics on vector bundles and proved the splitting results under a stronger minimal-extension assumption that the paper removes.","marker":"[HPS18]"},{"why":"Supplies the nef tangent bundle structure theory, numerical flatness facts, and the surface classification that the new pseudo-effective results extend.","marker":"[DPS94]"},{"why":"Gives the pseudo-effective cone criterion and non-nef locus results used to show the base has pseudo-effective canonical bundle and general fibers inherit pseudo-effectivity.","marker":"[BDPP13]"},{"why":"Corollary 2.11 turns the constructed foliation with rationally connected leaves into a holomorphic smooth MRC fibration.","marker":"[Hör07]"},{"why":"Theorem 1.6 defines the curvature current of a positively curved singular hermitian metric when the determinant is smooth, a key step in Lemma 3.5.","marker":"[Rau15]"},{"why":"Classifies surfaces with nef tangent bundle, identifying the elliptic ruled surfaces $S_0, A_0, A_{-1}$ as nef and providing the baseline for the surface classification.","marker":"[CP91]"}],"fun_headline_variants":["Pseudo-effective tangent bundle forces rational fibration","Tangent bundle positivity implies smooth fibration","Pseudo-effectivity yields flat base and rational fibers","Minimal surfaces classified by tangent bundle condition","Structure theorem: pseudo-effective tangent bundle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Pseudo-effective tangent bundle forces rational fibration","Tangent bundle positivity implies smooth fibration","Pseudo-effectivity yields flat base and rational fibers","Minimal surfaces classified by tangent bundle condition","Structure theorem: pseudo-effective tangent bundle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1347,"prompt_tokens":876,"completion_tokens":471,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":404}},"tokens_in":492,"tokens_out":471,"duration_ms":4834,"temperature":1.0,"reasoning_tokens":404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:47:43.356489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}