Pith. sign in

REVIEW 1 major objections 4 minor 10 cited by

A characterization of uniruled compact K\"ahler manifolds

T0 review · 1 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.

desk verdict Important new Kähler uniruled criterion; the main theorem is likely correct, but Lemma 8.1 is false as stated and needs a routine repair. read the letter →

arxiv 2501.18088 v1 pith:OOEFXC7X submitted 2025-01-30 math.AG math.CV

Wenhao Ou This is my paper · ORCID
classification math.AGmath.CV MSC 14E3032Q1514M20
keywords uniruledmanifoldscompactKählerpseudoeffectivecanonicalbundlefoliationsalgebraicintegrabilityLelongnumbersmeromorphicmapsformalextensions
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 proves a complete positivity characterization of uniruled compact Kähler manifolds: $X$ is covered by rational curves if and only if its canonical line bundle $\omega_X$ is not pseudoeffective, meaning no positive closed current can represent its class. The projective version of this statement was already known, so the new content is the converse direction in the full Kähler category, where algebraic techniques such as Chow's theorem are unavailable. The proof develops a Kähler analogue of an algebraicity criterion for germs, then uses it to show that foliations whose cotangent bundle is not pseudoeffective are induced by meromorphic maps, and finally runs a dimension-by-dimension contradiction built on holomorphic 2-forms, relative Albanese reductions, and positivity of relative canonical bundles. If correct, the theorem turns uniruledness into a cohomological positivity condition that can be checked without knowing the rational curves themselves.

What carries the argument

The engine is Theorem 1.2, a Kähler counterpart of the projective algebraicity criterion. It says that if a locally closed submanifold $S_0$ contains a Zariski open piece $C_0$ of a submanifold $C$ and extends formally along $C$ — meaning the pair survives an infinite sequence of blowups along the successive intersections — then for any $\lambda>0$ there is an $\omega$-plurisubharmonic function $\varphi$ with analytic singularities whose restriction Lelong numbers along $C_0$ exceed $\lambda$. The mass-concentration and regularization theorems for plurisubharmonic functions produce such functions, and Lemma 2.5 extracts their vanishing order by an explicit blowup cascade. Theorem 1.3 then turns largeness of restriction Lelong numbers into an algebraicity conclusion: $S_0$ has the same dimension as its Zariski closure whenever the extended conormal bundle is non-pseudoeffective. For a foliation, $S_0$ is the analytic graph, and Lemma 5.6 shows it extends formally along the diagonal, so Theorem 1.4 follows: a foliation with non-pseudoeffective cotangent, or with positive minimal slope with respect to a movable class, is induced by a meromorphic map. The final section imports Bergman-kernel positivity of relative canonical bundles to force pseudoeffectivity of $\omega_{X/W}$ and close the induction.

What would settle it

A compact Kähler manifold $X$ with $\omega_X$ not pseudoeffective that is not dominated by rational curves would refute Theorem 1.1, and the paper's final contradiction argument is precisely an attempt to rule out such an object.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for every compact Kähler manifold $X$, $X$ is uniruled if and only if $\omega_X$ is not pseudoeffective. One direction is standard: a free rational curve intersects $\omega_X$ negatively, so a positive current representing $\omega_X$ would have to be nonnegative on that curve, forcing the canonical class to be pseudoeffective only when no such curve exists. The reverse direction is the paper's achievement: assuming $\omega_X$ is not pseudoeffective, it constructs the rational curves that dominate $X$. The bridge is a new algebraicity criterion for locally closed submanifolds of a compact Kähler manifold, proved through formal extensions and restriction Lelong numbers, and applied to the analytic graph of a foliation so that non-pseudoeffective cotangent bundles force the foliation to come from a meromorphic map. With that in hand, the proof rules out the non-uniruled alternative by an induction on dimension that uses holomorphic 2-forms, relative Albanese fibrations, and Bergman-kernel positivity.

Load-bearing premise

The load-bearing premise is that the germ $S_0$ extends formally along $C$, equivalently that it survives infinitely many blowups, and that for the analytic graph of a foliation this formal extension is guaranteed by Lemma 5.6; if that property failed in any needed case, the blowup cascade and the Lelong-number contradiction that prove Theorems 1.2 and 1.3 would collapse.

Editorial extensions

If this is right

  • Uniruledness and canonical pseudoeffectivity now split the class of compact Kähler manifolds into two disjoint families: every non-uniruled example has a pseudoeffective canonical bundle.
  • Foliations with non-pseudoeffective cotangent bundle on a compact Kähler manifold are induced by meromorphic maps, so their leaves are algebraic; the same holds if the foliation has positive minimal slope with respect to some movable class.
  • For a foliation restricted to a subvariety, non-pseudoeffectivity of the pulled-back tangent distribution forces the local leaves through that subvariety to have Zariski closures of the same dimension.
  • The known threefold case is recovered, and the uniruledness criterion is now uniform in every dimension across projective and Kähler manifolds.

Reading between the lines

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

  • The paper leaves implicit that the classical conjecture 'uniruled iff Kodaira dimension is negative' is not formally settled by this theorem: the theorem uses the stronger hypothesis 'canonical bundle not pseudoeffective', so any remaining gap would be a non-uniruled manifold whose canonical class is pseudoeffective yet admits no pluricanonical sections.
  • The formal-extension and Lelong-number machinery is a transferable tool: the same two conditions — formal extendability plus non-pseudoeffectivity of the conormal bundle — could be tested for other algebraicity questions, such as which germs or leaves are algebraic under weaker positivity hypotheses.
  • Because the proof separates the obstruction into formal extension, conormal non-pseudoeffectivity, and relative canonical positivity, a natural next step is to check whether the same trichotomy survives for mildly singular Kähler spaces or for foliations with only weak positivity of their cotangent bundles.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper proves Theorem 1.1, which gives a complete characterization of uniruled compact Kähler manifolds by the non-pseudoeffectivity of their canonical bundle: X is uniruled if and only if ω_X is not pseudoeffective. The proof develops a Kähler analogue of Bost's algebraicity criterion for germs, introduces a notion of formal extension of subvarieties along a submanifold, establishes criteria for algebraic integrability of foliations (Theorems 1.3–1.5), and then applies these to a fibration induced by a holomorphic 2-form. The final step uses relative Bergman kernel metrics to force the relative canonical bundle to be pseudoeffective and derives a contradiction by descent and induction on dimension. The argument is long and carefully structured, with many auxiliary lemmas, some of which are delegated to external references.

Significance. If correct, Theorem 1.1 resolves a central conjecture in the classification of compact Kähler manifolds, extending the projective theorem of Boucksom–Demailly–Păun–Peternell to the non-projective Kähler setting. The paper introduces several potentially influential tools: the formal extension criterion for subvarieties, quasi-psh functions with large restricted Lelong numbers, and a positivity concept for torsion-free sheaves on Kähler manifolds. The proof is largely self-contained and the structural organization is clear. However, as discussed below, a key lemma (Lemma 8.1) is stated in false generality, and although the applications appear to satisfy the missing hypothesis, the manuscript as written contains a genuine gap that needs to be addressed.

major comments (1)
  1. [8.A, Lemma 8.1] Lemma 8.1 is false as stated. The lemma claims that for any smooth Hermitian metric h on a line bundle L, the Bergman kernel metric g on ω_{X/Y}^{⊗m} satisfies Θ_g ≥ -Θ_h. This is equivalent to positivity of the Bergman kernel metric H on ω_{X/Y}^{⊗m}⊗L, which requires the curvature of h to be semipositive; the cited source [BP10, Theorem 0.1] indeed assumes semipositivity of the metric on L. Without that hypothesis the statement is contradicted, for example, by X = C × E, Y = C, L = O_X, and h = e^{-|y|^2 v(w)} where v is a smooth function on the elliptic curve E with average 1 and minimum < 1/2: at points where v(w) < 1/2, Θ_g = i(v(w)-1)dy∧d\bar{y} does not dominate -Θ_h = -i v(w)dy∧d\bar{y}. In the only application of the lemma, Proposition 8.8, the metrics h_i have Θ_{h_i} = ω|_{V_i}, a Kähler form, so the semipositivity hypothesis is satisfied there and the proof of Theorem 1.1 can be repaired by adding that hypothesis to Lemma 8.1. Nevertheless, the paper currently invokes Lemma 8.1 in its false full generality, so the written proof of Theorem 1.1 contains a genuine gap that must be corrected.
minor comments (4)
  1. [1. Introduction, Theorem 1.3] The statement of Theorem 1.3 contains a grammatical error: "Then S0 is has the same dimension" should read "Then S0 has the same dimension".
  2. [8.D, Proposition 8.8 proof] The phrase "We endow ω_{X/Y}^{⊗ab}|_{V_i} the metric g_i" is awkward; it should say "we endow ω_{X/Y}^{⊗ab}|_{V_i} with the metric g_i".
  3. [8.A, Lemma 8.1] The proof of the first paragraph of Lemma 8.1 is a bare citation to [BP10, Theorem 0.1] and [Cao17, Theorem 3.5]; the statement should explicitly record the semipositivity hypothesis so that the reader can verify the cited result applies.
  4. [4., Lemma 4.4] The proof of Lemma 4.4 refers to external results (GKP16, Toma16) without giving the precise statements; a brief indication of how these results combine would improve readability, though the argument itself appears sound.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the uniruledness criterion is proved by an independent induction, with only a non-load-bearing self-citation in a local blowup computation.

full rationale

The central derivation is self-contained and not circular. Theorem 1.1 is proved by induction on dimension, with the projective case imported from BDPP13 as an external benchmark; the induction step uses the theorem only in strictly smaller dimensions and never assumes the target equivalence for X itself. Theorems 1.2, 1.3, and 1.4 form a forward chain: formal extension (Lemma 5.6) plus Demailly mass concentration/regularization produces large restriction Lelong numbers, which contradict non-pseudoeffectivity via divisorial Zariski decompositions. No step renames its input as a prediction. The positivity of relative canonical bundles in Section 8 is imported from Berndtsson–Păun and Cao, and in the actual applications the metrics have Kähler curvature; even though Lemma 8.1 as stated omits the semipositivity hypothesis and is therefore false in that generality, the applications in Proposition 8.8 satisfy the missing hypothesis, making this a repairable correctness gap rather than circular reasoning. The only self-citation is [Ou24] in Lemma 7.2 for a local blowup computation; the computation is carried out in the text and the citation is illustrative, not load-bearing. No prediction reduces by construction to a fitted parameter, and no uniqueness claim is imported from the author's own prior work.

Assumptions & free parameters 0 free parameters · 11 assumptions · 0 invented entities

The central claim is not supported by fitting or by invented objects. It rests on a large stack of imported structure theorems: divisorial Zariski decompositions, Demailly regularization and mass concentration, Bergman kernel positivity, Ohsawa-Takegoshi extension, BDPP's projective criterion, GHS sections, relative Albanese reduction, and Barlet cycle theory. These are treated as axioms from the literature; the paper's contribution is the architecture that combines them for Kähler germs and foliations.

assumptions (11)
  • domain assumption Boucksom divisorial Zariski decomposition and modified nef cones on compact Kähler manifolds.
    Invoked throughout Section 3 and in Lemma 6.1 as [Bou04, Theorem 3.12, Proposition 3.9].
  • domain assumption Demailly regularization of closed positive currents with control of Lelong numbers.
    Theorem 2.4 is stated and used in Lemma 6.1 and in the proof of Theorem 1.2, based on [Dem92] and [Dem12].
  • domain assumption Demailly mass concentration lemma extends from projective manifolds to compact Kähler manifolds.
    Lemma 6.2 generalizes [Dem93, Section 6] to Kähler; the paper checks the Monge-Ampère mass inequality needed for the method.
  • domain assumption Positivity of direct images of adjoint relative canonical sheaves in the form of HPS18, PT18, BP10, and Cao17.
    Used in Lemma 4.6 and Lemma 8.1. Lemma 8.1 explicitly says its first paragraph is a consequence of [BP10, Theorem 0.1].
  • standard math Ohsawa-Takegoshi L2 extension theorem, including multiplier ideal sheaf restriction to fibers.
    Used inside Lemma 4.6 to pass from nonvanishing on a general fiber to nonvanishing of a global direct image sheaf.
  • domain assumption BDPP projective uniruled criterion: a projective manifold is uniruled iff its canonical bundle is not pseudoeffective.
    Used in the first case of the proof of Theorem 1.1 when H2,0(X,C) = 0 and X is projective by Kodaira's embedding theorem.
  • domain assumption Graber-Harris-Starr theorem: fibrations to curves with rationally connected general fibers admit sections.
    Used in Lemma 8.10 to show the rational quotient of a uniruled Kähler manifold is not uniruled.
  • standard math Ueno theorem: a compact Kähler manifold of maximal Albanese dimension has nonnegative Kodaira dimension.
    Used in Lemma 8.7 to produce nonzero pluricanonical sections on general fibers.
  • domain assumption Relative Albanese reduction and Stein factorization for compact Kähler fibrations.
    Used in Proposition 8.6 and cited to [Cam85] and [Fuj83].
  • domain assumption Barlet space theory: compact irreducible components and analytic families of cycles.
    Used in Lemma 7.5 to turn the analytic graph of a foliation into a meromorphic map.
  • domain assumption Local freeness of higher direct images and relative duality for R1 f_* O_X.
    Used in Lemma 8.9 and cited to [Tak95], [RRV71], and [Kol86].

how reviews work

0 comments
Cite this review

Pith. "Pith review of A characterization of uniruled compact K\"ahler manifolds." pith.science (2026). https://pith.science/paper/OOEFXC7X

@misc{pith2026250118088,
  author       = {Pith},
  title        = {Pith review of: A characterization of uniruled compact K\"ahler manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OOEFXC7X}},
  note         = {Machine review of arXiv:2501.18088}
}
read the original abstract

We adapt Bost's algebraicity characterization to the situation of a germ in a compact K\"ahler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-P\u{a}un and of Druel to foliations on compact K\"ahler manifolds. As an application, we prove that a compact K\"ahler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 10 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the K\"ahler MMP and the transcendental base-point-free theorem

    math.AG 2026-07 accept novelty 7.0 of 10

    Big gklt Kähler pairs admit a full MMP with scaling, and a nef canonical class with modified-big boundary is semiample (Tosatti’s conjecture).

  2. Hermitian manifolds with nonpositive holomorphic sectional curvature

    math.DG 2026-07 accept novelty 7.0 of 10

    Nonpositive Hermitian holomorphic sectional curvature on a compact Kähler manifold implies the canonical bundle is nef; vanishing curvature implies vanishing first Chern class.

  3. Universal affine bundles for compact complex manifolds

    math.DS 2026-07 accept novelty 7.0 of 10

    A universal affine bundle over compact complex manifolds with invariant measures provides canonical potentials for dd^c-closed (1,1)-forms and reduces torsor-lifting obstructions to compact subtori.

  4. Classification of Smooth Minimal K\"ahler Fourfolds Without Effective Divisors and Surfaces

    math.AG 2026-07 conditional novelty 7.0 of 10

    Compact Kähler fourfolds with pseudo-effective K_X and no codimension-1 or -2 subvarieties have torsion K_X, hence are torus quotients or IHS manifolds.

  5. The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors

    math.AG 2025-07 conditional novelty 7.0 of 10

    For projective klt varieties with big canonical or anticanonical divisor, the Miyaoka-Yau Chern class inequality holds when intersections are taken with the non-pluripolar product.

  6. Compact K\"ahler manifolds with nef anti-canonical bundle

    math.AG 2025-06 conditional novelty 7.0 of 10

    Every compact Kähler manifold with nef anti-canonical bundle admits a locally trivial fibration over a Calabi-Yau manifold with rationally connected fibers.

  7. On compact K\"ahler manifolds with pseudo-effective tangent bundle

    math.AG 2025-02 conditional novelty 7.0 of 10

    Compact Kähler manifolds with pseudo-effective tangent bundle admit a smooth fibration whose base is an étale quotient of a torus and whose fibers are rationally connected.

  8. Semipositivity of the orbifold second Chern class in Fujiki's class

    math.AG 2026-07 conditional novelty 6.0 of 10

    For compact normal analytic varieties in Fujiki's class, Miyaoka's inequality holds when the canonical divisor is nef, and the orbifold second Chern class is semipositive when the anti-canonical divisor is nef, under ...

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

    math.CV 2025-02 conditional novelty 6.0 of 10

    New pseudo-effectivity theorems for relative canonical bundles and new algebraicity and uniruledness criteria for foliations on compact Kähler manifolds are established.

  10. Fundamental groups of compact K\"ahler manifolds with semi-positive holomorphic sectional curvature

    math.DG 2025-02 conditional novelty 5.0 of 10

    A compact Kähler manifold with semi-positive holomorphic sectional curvature is a locally trivial fibration over a finite étale quotient of a torus with rationally connected projective fibers.

Reference graph

Works this paper leans on

48 extracted references · 39 canonical work pages · cited by 10 Pith papers

  1. [1]

    The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension

    S \'e bastien Boucksom, Jean-Pierre Demailly, Mihai P a un, and Thomas Peternell. The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension. J. Algebraic Geom. , 22(2):201--248, 2013

    work page 2013

  2. [2]

    Curvature of vector bundles associated to holomorphic fibrations

    Bo Berndtsson. Curvature of vector bundles associated to holomorphic fibrations. Ann. of Math. (2) , 169(2):531--560, 2009

    work page 2009

  3. [3]

    urgen Bingener. On deformations of K \

    J\"urgen Bingener. On deformations of K \"ahler spaces. II . Arch. Math. (Basel) , 41(6):517--530, 1983

    work page 1983

  4. [4]

    On the O hsawa- T akegoshi extension theorem

    Zbigniew B o cki. On the O hsawa- T akegoshi extension theorem. Univ. Iagel. Acta Math. , (50):53--61, 2013

    work page 2013

  5. [5]

    Rational curves on foliated varieties

    Fedor Bogomolov and Michael McQuillan. Rational curves on foliated varieties. In Foliation theory in algebraic geometry , Simons Symp., pages 21--51. Springer, Cham, 2016

    work page 2016

  6. [6]

    Complex analytic cycles

    Daniel Barlet and J \'o n Magn \'u sson. Complex analytic cycles. I ---basic results on complex geometry and foundations for the study of cycles , volume 356 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Cham; Soci\'et\'e Math\'ematique de France, Paris, [2019] 2019. Translated from the 2014...

    work page 2019

  7. [7]

    Algebraic leaves of algebraic foliations over number fields

    Jean-Beno\^it Bost. Algebraic leaves of algebraic foliations over number fields. Publ. Math. Inst. Hautes \'Etudes Sci. , (93):161--221, 2001

    work page 2001

  8. [8]

    Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems

    Jean-Beno\^it Bost. Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems. In Geometric aspects of D work theory. V ol. I , II , pages 371--418. Walter de Gruyter, Berlin, 2004

    work page 2004

Show all 48 references
  1. [9]

    Divisorial Z ariski decompositions on compact complex manifolds

    S\'ebastien Boucksom. Divisorial Z ariski decompositions on compact complex manifolds. Ann. Sci. \'Ecole Norm. Sup. (4) , 37(1):45--76, 2004

    2004

  2. [10]

    Bergman kernels and the pseudoeffectivity of relative canonical bundles

    Bo Berndtsson and Mihai P a un. Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. , 145(2):341--378, 2008

    2008

  3. [11]

    Bergman kernels and subadjunction

    Bo Berndtsson and Mihai P a un. Bergman kernels and subadjunction. arXiv preprint arXiv:1002.4145 , 2010

    2010 arXiv

  4. [12]

    A positivity property for foliations on compact K \"ahler manifolds

    Marco Brunella. A positivity property for foliations on compact K \"ahler manifolds. Internat. J. Math. , 17(1):35--43, 2006

    2006

  5. [13]

    F. Campana. R\'eduction d' A lban\`ese d'un morphisme propre et faiblement k\"ahl\'erien. I . Compositio Math. , 54(3):373--398, 1985

    1985

  6. [14]

    F. Campana. Connexit \'e rationnelle des vari \'e t \'e s de F ano. Ann. Sci. \'Ecole Norm. Sup. (4) , 25(5):539--545, 1992

    1992

  7. [15]

    Orbifolds, special varieties and classification theory: an appendix

    Fr\'ed\'eric Campana. Orbifolds, special varieties and classification theory: an appendix. Ann. Inst. Fourier (Grenoble) , 54(3):631--665, 2004

    2004

  8. [16]

    Ohsawa- T akegoshi extension theorem for compact K \"ahler manifolds and applications

    Junyan Cao. Ohsawa- T akegoshi extension theorem for compact K \"ahler manifolds and applications. In Complex and symplectic geometry , volume 21 of Springer INdAM Ser. , pages 19--38. Springer, Cham, 2017

    2017

  9. [17]

    o ring. Rational curves on compact K \

    Junyan Cao and Andreas H \"o ring. Rational curves on compact K \"ahler manifolds. J. Differential Geom. , 114(1):1--39, 2020

    2020

  10. [18]

    Geometric stability of the cotangent bundle and the universal cover of a projective manifold

    Fr\'ed\'eric Campana and Thomas Peternell. Geometric stability of the cotangent bundle and the universal cover of a projective manifold. Bull. Soc. Math. France , 139(1):41--74, 2011. With an appendix by Matei Toma

    2011

  11. [19]

    Foliations with positive slopes and birational stability of orbifold cotangent bundles

    Fr\'ed\'eric Campana and Mihai P a un. Foliations with positive slopes and birational stability of orbifold cotangent bundles. Publ. Math. Inst. Hautes \'Etudes Sci. , 129:1--49, 2019

    2019

  12. [20]

    Regularization of closed positive currents and intersection theory

    Jean-Pierre Demailly. Regularization of closed positive currents and intersection theory. J. Algebraic Geom. , 1(3):361--409, 1992

    1992

  13. [21]

    A numerical criterion for very ample line bundles

    Jean-Pierre Demailly. A numerical criterion for very ample line bundles. J. Differential Geom. , 37(2):323--374, 1993

    1993

  14. [22]

    Analytic methods in algebraic geometry , volume 1 of Surveys of Modern Mathematics

    Jean-Pierre Demailly. Analytic methods in algebraic geometry , volume 1 of Surveys of Modern Mathematics . International Press, Somerville, MA; Higher Education Press, Beijing, 2012

    2012

  15. [23]

    A decomposition theorem for singular spaces with trivial canonical class of dimension at most five

    St\' e phane Druel. A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. Invent. Math. , 211(1):245--296, 2018

    2018

  16. [24]

    Relative algebraic reduction and relative A lbanese map for a fiber space in C

    Akira Fujiki. Relative algebraic reduction and relative A lbanese map for a fiber space in C . Publ. Res. Inst. Math. Sci. , 19(1):207--236, 1983

    1983

  17. [25]

    Families of rationally connected varieties

    Tom Graber, Joe Harris, and Jason Starr. Families of rationally connected varieties. J. Amer. Math. Soc. , 16(1):57--67, 2003

    2003

  18. [26]

    Movable curves and semistable sheaves

    Daniel Greb, Stefan Kebekus, and Thomas Peternell. Movable curves and semistable sheaves. Int. Math. Res. Not. IMRN , (2):536--570, 2016

    2016

  19. [27]

    Plurisubharmonische F unktionen in komplexen R \"aumen

    Hans Grauert and Reinhold Remmert. Plurisubharmonische F unktionen in komplexen R \"aumen. Math. Z. , 65:175--194, 1956

    1956

  20. [28]

    Variation of G ieseker moduli spaces via quiver GIT

    Daniel Greb, Julius Ross, and Matei Toma. Variation of G ieseker moduli spaces via quiver GIT . Geom. Topol. , 20(3):1539--1610, 2016

    2016

  21. [29]

    A solution of an L^2 extension problem with an optimal estimate and applications

    Qi'an Guan and Xiangyu Zhou. A solution of an L^2 extension problem with an optimal estimate and applications. Ann. of Math. (2) , 181(3):1139--1208, 2015

    2015

  22. [30]

    Algebraic integrability of foliations with numerically trivial canonical bundle

    Andreas H\" o ring and Thomas Peternell. Algebraic integrability of foliations with numerically trivial canonical bundle. Invent. Math. , 216(2):395--419, 2019

    2019

  23. [31]

    On the canonical bundle formula and adjunction for generalized kaehler pairs

    Christopher Hacon and Mihai P a un . On the canonical bundle formula and adjunction for generalized kaehler pairs. arXiv preprint arXiv:2404.12007 , 2024

    2024 arXiv

  24. [32]

    Algebraic fiber spaces over abelian varieties: around a recent theorem by C ao and P aun

    Christopher Hacon, Mihnea Popa, and Christian Schnell. Algebraic fiber spaces over abelian varieties: around a recent theorem by C ao and P aun. In Local and global methods in algebraic geometry , volume 712 of Contemp. Math. , pages 143--195. Amer. Math. Soc., [Providence], R...

    2018

  25. [33]

    Rationally connected varieties

    J \'a nos Koll \'a r, Yoichi Miyaoka, and Shigefumi Mori. Rationally connected varieties. J. Algebraic Geom. , 1(3):429--448, 1992

    1992

  26. [34]

    Higher direct images of dualizing sheaves

    J \'a nos Koll \'a r. Higher direct images of dualizing sheaves. II . Ann. of Math. (2) , 124(1):171--202, 1986

    1986

  27. [35]

    Rationally connected foliations after B ogomolov and M c Q uillan

    Stefan Kebekus, Luis Sol\'a Conde, and Matei Toma. Rationally connected foliations after B ogomolov and M c Q uillan. J. Algebraic Geom. , 16(1):65--81, 2007

    2007

  28. [36]

    Deformations of a morphism along a foliation and applications

    Yoichi Miyaoka. Deformations of a morphism along a foliation and applications. In Algebraic geometry, B owdoin, 1985 ( B runswick, M aine, 1985) , volume 46, Part 1 of Proc. Sympos. Pure Math. , pages 245--268. Amer. Math. Soc., Providence, RI, 1987

    1985

  29. [37]

    A numerical criterion for uniruledness

    Yoichi Miyaoka and Shigefumi Mori. A numerical criterion for uniruledness. Ann. of Math. (2) , 124(1):65--69, 1986

    1986

  30. [38]

    Orbifold modifications of complex analytic varieties

    Wenhao Ou. Orbifold modifications of complex analytic varieties. arXiv preprint arXiv:2401.07273 , 2024

    2024

  31. [39]

    Positivity of twisted relative pluricanonical divisors and their direct images

    Mihai P a un and Shigeharu Takayama. Positivity of twisted relative pluricanonical divisors and their direct images. J. Algebraic Geom. , 27:211--272, 2018

    2018

  32. [40]

    Singular hermitian metrics on holomorphic vector bundles

    Hossein Raufi. Singular hermitian metrics on holomorphic vector bundles. Ark. Mat. , 53(2):359--382, 2015

    2015

  33. [41]

    J. P. Ramis, G. Ruget, and J. L. Verdier. Dualit \'e relative en g \'e om \'e trie analytique complexe. Invent. Math. , 13:261--283, 1971

    1971

  34. [42]

    Analyticity of sets associated to L elong numbers and the extension of closed positive currents

    Yum Tong Siu. Analyticity of sets associated to L elong numbers and the extension of closed positive currents. Invent. Math. , 27:53--156, 1974

    1974

  35. [43]

    Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type

    Yum-Tong Siu. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In Complex geometry ( G \"ottingen, 2000) , pages 223--277. Springer, Berlin, 2002

    2000

  36. [44]

    Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper K \"ahler morphisms

    Kensho Takegoshi. Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper K \"ahler morphisms. Math. Ann. , 303(3):389--416, 1995

    1995

  37. [45]

    Bounded sets of sheaves on K \"ahler manifolds

    Matei Toma. Bounded sets of sheaves on K \"ahler manifolds. J. Reine Angew. Math. , 710:77--93, 2016

    2016

  38. [46]

    Classification theory of algebraic varieties and compact complex spaces , volume Vol

    Kenji Ueno. Classification theory of algebraic varieties and compact complex spaces , volume Vol. 439 of Lecture Notes in Mathematics . Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack

    1975

  39. [47]

    K\"ahler spaces and proper open morphisms

    Jean Varouchas. K\"ahler spaces and proper open morphisms. Math. Ann. , 283(1):13--52, 1989

    1989

  40. [48]

    On the curvature of compact H ermitian manifolds

    Shing Tung Yau. On the curvature of compact H ermitian manifolds. Invent. Math. , 25:213--239, 1974

    1974

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.