Hermitian manifolds with nonpositive holomorphic sectional curvature

Kai Tang

REVIEW 4 major objections 4 minor 30 references

Hermitian manifolds with nonpositive holomorphic sectional curvature

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read On any compact Kähler manifold, a Hermitian metric with nonpositive holomorphic sectional curvature forces the canonical bundle to be nef; if the curvature vanishes, the first Chern class vanishes and a Ricci-flat Kähler metric exists.

desk verdict A clean, honest proof of nefness for Hermitian holomorphic sectional curvature, conditional on a recent birational-geometry preprint that the author openly relies on. read the letter →

arxiv 2607.23246 v1 pith:RO7K3LXV submitted 2026-07-25 math.DG

Kai Tang This is my paper · ORCID

classification math.DG MSC 53C5532Q1532Q4514E30

keywords HermitianmanifoldsholomorphicsectionalcurvaturecanonicalbundlenefnessrationalcurvesChern–LuformulaGauduchonmetricBott–Chernclass

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 proves that a single curvature sign controls the positivity of the canonical bundle even when the metric is an arbitrary Hermitian metric on a Kähler manifold, with no Kähler or pluriclosed requirement. The main theorem states that nonpositive holomorphic sectional curvature forces the canonical bundle to be nef, and identically vanishing curvature forces the first Chern class to vanish, so the manifold admits a Ricci-flat Kähler metric. In complex dimension two, negative holomorphic sectional curvature upgrades nefness to ampleness, making the surface projective of general type. A companion rigidity theorem says that on any compact complex manifold with vanishing first Bott–Chern class, a Hermitian metric with nonnegative holomorphic sectional curvature must actually have vanishing holomorphic sectional curvature.

What carries the argument

Three mechanisms carry the argument. (1) A Chern–Lu formula for maps from P^1 to X: for h with H_h ≤ 0, the energy density u of a nonconstant map satisfies Δ u ≥ τ u, so integrating on P^1 forces u≡0; this rules out all rational curves. (2) A birational-geometry bridge: a recent characterization states that a compact Kähler manifold is uniruled exactly when its canonical bundle fails to be pseudoeffective; combined with a rational-curve existence theorem for pseudoeffective-but-not-nef canonical bundles, the absence of rational curves implies K_X is nef. (3) For the vanishing and rigidity theorems, the Berger averaging formula (which expresses the average of H_h in terms of two scalar curvat

What would settle it

Construct a Hermitian metric with nonpositive holomorphic sectional curvature on P^2 (or any uniruled compact Kähler manifold). The paper's Chern–Lu lemma forbids nonconstant maps from P^1 under this curvature condition, so such a manifold cannot carry any rational curve; a concrete metric of this kind on P^2 would directly contradict Theorem 1.1(1), since P^2 has non-nef canonical bundle.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the holomorphic sectional curvature of a Hermitian metric — no Kähler condition imposed — determines birational positivity of a compact Kähler manifold: if H_h ≤ 0 then K_X is nef, and if H_h ≡ 0 then c_1(X)=0. The nefness theorem drops the pluriclosed hypothesis used in earlier Schwarz-lemma arguments and weakens the real-bisectional-curvature condition used by prior authors, achieving the conclusion through a Chern–Lu estimate that forbids rational curves, followed by a birational-geometric bridge that turns the absence of rational curves into pseudoeffectivity of K_X and then cites a theorem that non-nef pseudoeffective K_X would pro

Load-bearing premise

The proof of the main nefness theorem rests on a recent birational-geometry characterization: a compact Kähler manifold is uniruled exactly when its canonical bundle is not pseudoeffective. If that characterization fails in any dimension used by the induction, the contradiction step that forces K_X to be nef collapses — the curvature estimate itself only rules out rational curves and does not directly imply nefness.

Editorial extensions

If this is right

The known result for Kähler metrics — nonpositive holomorphic sectional curvature implies nef canonical bundle — now extends to arbitrary Hermitian metrics on Kähler manifolds, with no Kähler or pluriclosed assumptions.
In complex dimension two, a Hermitian metric of negative holomorphic sectional curvature forces the surface to be projective of general type with ample canonical bundle, so it is Kobayashi hyperbolic.
A compact Kähler manifold admitting a Hermitian metric with identically zero holomorphic sectional curvature must have vanishing first Chern class and therefore carries a Ricci-flat Kähler metric; the given metric itself need not be Kähler or flat.
The rigidity statement: on a compact complex manifold with vanishing first Bott–Chern class, nonnegative holomorphic sectional curvature is only possible if it vanishes identically; the Fermat quartic K3 shows that c_1=0 alone does not guarantee a metric with vanishing curvature, so the nonnegativity assumption is essential.
The negative-curvature case in dimension two is one step toward the conjecture, stated in the introduction, that negative or quasi-negative holomorphic sectional curvature for Hermitian metrics should imply ampleness of the canonical bundle.

Reading between the lines

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

The structure of the proof suggests a transfer principle: on compact Kähler manifolds, any Hermitian curvature condition that (i) satisfies a Chern–Lu-type estimate ruling out rational curves and (ii) can be paired with the birational-geometric characterization of uniruledness will force K_X to be nef. Other curvature notions (e.g., k-Ricci curvature) may be amenable to the same two-step argument.
Because the nefness step leans on the birational-geometric characterization of uniruled Kähler manifolds (a recent arXiv preprint cited in the paper), the theorem's proof is conditional on that characterization in every dimension; if that characterization were to fail, the argument would break even though the curvature-to-nefness conclusion might still be true by other means.
The vanishing-case proof goes through a Gauduchon conformal change and integral identities; a natural testable extension is whether the conclusion can be strengthened to the given Hermitian metric being Chern-flat (or the manifold being Chern–Kähler-flat) under an additional integrability or torsion condition, rather than merely the existence of a Ricci-flat Kähler metric.
Theorem 1.2's rigidity has a possible analogue for noncompact manifolds or for one-parameter families of Hermitian metrics deforming a flat one: nonnegative holomorphic sectional curvature with vanishing first Bott–Chern class may force the family to remain in the zero-curvature locus, not merely the vanishing of H_h at each member.

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.

Desk Editor's Note

read the letter

Kai Tang's paper does exactly what it advertises: it proves that a compact Kähler manifold carrying any Hermitian metric with nonpositive holomorphic sectional curvature has nef canonical bundle, removing the pluriclosed assumption from Broder-Stanfield and replacing real bisectional curvature with the weaker holomorphic sectional condition in the Yang-Zheng program. The proof route is smart: exclude rational curves with a Chern-Lu calculation, then invoke recent birational geometry (Ou, Cao-Höring) to get nefness. The curvature part is clean and correctly done.

What is genuinely new: Theorem 1.1(1) is a real step forward, and Theorem 1.1(2)—vanishing HSC forces c1=0, hence a Ricci-flat Kähler metric—is a nice consequence. The surface corollary (negative HSC ⇒ K_X ample) follows naturally from nefness plus classification, and the Hodge index step is correct. Theorem 1.2 is a tidy partial converse using Bott-Chern class zero; the argument is short and sound. The K3 example showing that c1=0 alone doesn't guarantee a vanishing-HSC Hermitian metric is a good addition.

The soft spot is the one the author openly flags: the proof of Theorem 1.1(1) leans on Ou's characterization of uniruled compact Kähler manifolds ([17]) and on Cao-Höring's rational-curve theorem, whose hypothesis requires that same characterization in all lower dimensions. If Ou's preprint is incomplete or only projective, the nefness argument collapses. The paper does not attempt to verify or even sketch that input; it treats it as a black box. That's a legitimate structural dependency, not a hidden flaw, but it means the theorem's validity is tied to an external result that is itself recent and not yet widely vetted. A referee should be explicitly asked to check that chain. The hyperbolicity step in Corollary 1.1 also relies on Yau's Schwarz lemma, which is standard but cited tersely.

My take: the internal geometry is sound, the writing is honest and clear, and the contribution is significant if the birational inputs hold. I would send this to a serious referee, with special attention to [17] and [6, Thm. 1.3]. The likely outcome is a solid paper, maybe with a request to expand the discussion of the external inputs.

Referee Report

4 major / 4 minor

Summary. The paper proves several results on compact Kähler manifolds admitting Hermitian metrics with nonpositive holomorphic sectional curvature. Theorem 1.1(1) states that the canonical bundle K_X is nef; the proof uses a Chern–Lu computation to exclude rational curves, then invokes Ou's characterization of uniruled compact Kähler manifolds to conclude K_X is pseudoeffective, and finally uses Cao–Höring's theorem to rule out the non-nef case by producing a K_X-negative rational curve. Theorem 1.1(2) shows that vanishing holomorphic sectional curvature forces c1(X)=0, giving a Ricci-flat Kähler metric; the proof uses a conformal Gauduchon metric and a limiting argument based on nefness. Theorem 1.2 gives a partial converse for compact complex manifolds with vanishing first Bott–Chern class and nonnegative holomorphic sectional curvature. Corollary 1.1 shows that negative holomorphic sectional curvature implies ampleness of K_X for compact Kähler surfaces. The paper is clearly written; the main caveat is that the nefness theorem depends on an external preprint [17].

Significance. If correct, the paper makes a genuine advance in the Wu–Yau program for Hermitian metrics: it removes the pluriclosed assumption from Broder–Stanfield's nefness theorem and replaces the nonpositive real bisectional curvature assumption in Yang–Zheng's theorem by the weaker and more natural nonpositive holomorphic sectional curvature. The vanishing result and its partial converse are also new and clearly proved. The Chern–Lu and conformal-Gauduchon computations are standard; the proofs are transparent and honestly flag the reliance on recent birational geometry (Ou, Cao–Höring). The principal uncertainty is external, not internal: Theorem 1.1(1) collapses if Ou's characterization fails. The paper also benefits from explicit, checkable curvature computations and a well-organized structure.

major comments (4)

[§2.1, definition of H_g] The displayed definition H_g(ξ) = R_{i\bar j k\bar \ell} ξ^i ξ^j ξ^k ξ^\ell / |ξ|^4 is not the holomorphic sectional curvature for the curvature tensor defined two lines above; with that tensor H_g is not real-valued and cannot be used in the Berger averaging formula. It should read H_g(ξ)=R_{i\bar j k\bar \ell} ξ^i \bar ξ^j ξ^k \bar ξ^\ell / |ξ|^4 (or the equivalent index ordering). This is the central curvature hypothesis of the paper and must be corrected.
[§3, Eq. (3.1)] The curvature contraction in (3.1) is written as R^h_{α\bar β γ\bar δ} f^α_1 f^β_1 \bar f^γ_1 \bar f^δ_1. With the curvature tensor convention in §2.1, the Chern–Lu holomorphic sectional curvature term should be R^h_{α\bar β γ\bar δ} f^α_1 \bar f^β_1 f^γ_1 \bar f^δ_1. The subsequent identification with H_h(ξ)u^2 is only valid in this second form. Please correct the indices in the display.
[§3, Eqs. (3.2)/(3.5)] The conformal Gauduchon identity is quoted with a factor f in the integrand, but the Gauduchon metric was defined as ω_G = f^{1/(n-1)}ω. Unless Yang's [25, Eq. (3.8)] uses a different convention, the conformal factor in the first integral should be f^{1/(n-1)}. Please verify the formula against [25] and make the notation consistent. The later arguments use only positivity of the factor, so the conclusions are unaffected, but the displayed identity is notational and dimensionally suspect.
[§1 and §3, proof of Thm 1.1(1)] The nefness proof depends essentially on Ou's theorem [17] in two places: to get K_X pseudoeffective from non-uniruledness, and to satisfy the induction hypothesis of Cao–Höring [6, Thm 1.3] in all lower dimensions. Since [17] is an arXiv preprint, the paper should state the exact theorem used, confirm it applies to all compact Kähler manifolds in every dimension, and flag the preprint status. This is an acknowledged structural dependency, but it is load-bearing: if [17] is invalid or circular, Theorem 1.1(1) has no proof.

minor comments (4)

[Title/Abstract] The title contains 'CUR V A TURE' with extra spaces; this should be corrected to 'CURVATURE'.
[Example 3.1] There is a typo: 'compact. and the adjunction formula' should be 'compact, and the adjunction formula'.
[Throughout] The author's name in citations appears as both 'Broder-Stanfield' and 'Broder-Stanfield' (e.g., Abstract vs. reference [3]); please ensure consistent spelling.
[§2.3] The Brody criterion is stated as 'X is Kobayashi hyperbolic if and only if it is Brody hyperbolic'; for compact complex manifolds this is correct, but it may be helpful to note that the equivalence holds for compact complex spaces.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central derivation uses acknowledged external birational-geometry inputs; the only self-citations are contextual and not load-bearing.

full rationale

The proof of Theorem 1.1(1) is not circular. The Chern–Lu calculation (Eq. 3.1) is a direct curvature estimate that excludes rational curves under H_h ≤ 0. The subsequent birational-geometry steps are explicitly external: the paper invokes Ou's uniruledness criterion [17] and Cao–Höring's theorem [6], neither authored by Tang. The Cao–Höring result is peer-reviewed, and Ou's preprint is an external input whose correctness is a legitimate dependency but not a circular one. The paper openly flags this in the introduction: 'The proof of Theorem 1.1(1) is based on recent results of Ou [17] and Cao-Höring [6].' No parameter is fitted and then renamed as a prediction; no conclusion is built into a definition; no self-citation is used as the load-bearing justification. The only self-citations (Broder–Tang [4], Tang [19], Tang [20]) are historical/contextual — e.g., 'The vanishing case has also been studied by Broder-Tang [4]' — and none is used as a premise in the proofs. Thus the derivation is self-contained apart from acknowledged, independent external results.

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

No free parameters or fitted constants appear anywhere in the paper. The central proofs are deductive; the main assumptions are external theorems in birational geometry and Hermitian geometry plus standard background. No new geometric objects are invented.

assumptions (7)

standard math Chern–Lu formula for holomorphic maps from Riemann surfaces into Hermitian manifolds
Used in (3.1) to compute Δu and derive the contradiction excluding rational curves; standard in Hermitian geometry.
standard math Berger averaging formula for Hermitian metrics: average H = (s + b_s)/(n(n+1))
Invoked in Theorem 1.1(2) and Theorem 1.2 to pass from H≥0 or H=0 to sign conditions on the scalar curvatures s and b_s.
domain assumption Gauduchon's existence theorem and conformal identity [25, Eq. (3.8)]
The paper chooses ω_G = f^{1/(n-1)}ω in the conformal class and uses equation (3.2)/(3.5) to relate scalar curvatures; the identity is cited from Yang [25] rather than proved.
domain assumption Ou's theorem: a compact Kähler manifold is uniruled iff its canonical bundle is not pseudoeffective [17]
Load-bearing in Theorem 1.1(1): after excluding rational curves, the paper concludes K_X is pseudoeffective; this relies on [17], a recent arXiv preprint.
domain assumption Cao–Höring Theorem 1.3: pseudoeffective but non-nef K_X gives a K_X-negative rational curve in compact Kähler manifolds, assuming the lower-dimensional uniruled–pseudoeffective equivalence [6]
Used to produce the contradiction in Theorem 1.1(1); the paper checks the hypothesis via Ou's theorem.
domain assumption Yau's Schwarz lemma [27] and Brody criterion implying Kobayashi hyperbolicity from negative holomorphic sectional curvature
Used in Corollary 1.1 to rule out κ=0 and κ=1 surfaces; the paper cites [27] without proving the Hermitian-target version.
standard math Calabi–Yau theorem, Enriques–Kodaira classification, Hodge index theorem, Nakai–Moishezon criterion
Standard tools invoked in Theorem 1.1(2) and Corollary 1.1; not proved in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Hermitian manifolds with nonpositive holomorphic sectional curvature." pith.science (2026). https://pith.science/paper/RO7K3LXV

@misc{pith2026260723246,
  author       = {Pith},
  title        = {Pith review of: Hermitian manifolds with nonpositive holomorphic sectional curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RO7K3LXV}},
  note         = {Machine review of arXiv:2607.23246}
}

read the original abstract

We study compact K\"ahler manifolds admitting Hermitian metrics with nonpositive holomorphic sectional curvature. We prove that the canonical bundle of such a manifold is nef, removing the pluriclosed assumption from the corresponding nefness result of Broder-Stanfield \cite{BroderStanfield}. In complex dimension two, we further show that negative holomorphic sectional curvature implies the ampleness of the canonical bundle. We also prove that vanishing holomorphic sectional curvature forces the first Chern class to vanish. As a partial converse, we show that on a compact complex manifold with vanishing first Bott-Chern class, every Hermitian metric with nonnegative holomorphic sectional curvature must have vanishing holomorphic sectional curvature.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 2 linked inside Pith

[17]

Ou,A characterization of uniruled compact K¨ ahler manifolds, arXiv:2501.18088

W. Ou,A characterization of uniruled compact K¨ ahler manifolds, arXiv:2501.18088

arXiv
[25]

Yang,Hermitian manifolds with semi-positive holomorphic sectional curvature, Math

X. Yang,Hermitian manifolds with semi-positive holomorphic sectional curvature, Math. Res. Lett.23(2016), no. 3, 939–952

2016
[1]

Barth, K

W. Barth, K. Hulek, C. Peters and A. Van de Ven,Compact Complex Surfaces, second enlarged edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 4, Springer-Verlag, Berlin, 2004

2004
[2]

M. Berger,Sur les vari´ et´ es d’Einstein compactes, in: Comptes rendus de la IIIe R´ eunion du Groupement des Math´ ematiciens d’Expression Latine (Namur, 1965), Librairie Universitaire, Louvain, 1966, 35–55

1965
[3]

Broder and J

K. Broder and J. Stanfield,A general Schwarz lemma for Hermitian manifolds, Bull. Lond. Math. Soc.58 (2026), no. 1, Paper No. e70269, 13 pp

2026
[4]

Broder and K

K. Broder and K. Tang,On Hermitian manifolds with vanishing curvature, Math. Z.309(2025), no. 1, Paper No. 12

2025
[5]

Brody,Compact manifolds and hyperbolicity, Trans

R. Brody,Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc.235(1978), 213–219

1978
[6]

Cao and A

J. Cao and A. H¨ oring,Rational curves on compact K¨ ahler manifolds, J. Differential Geom.114(2020), no. 1, 1–39

2020

Show all 30 references

[7]

J. Chu, M. Lee, L. Tam,K¨ ahler manifolds and mixed curvature, Trans. Amer. Math. Soc.375(2022), no. 11, 7925-7944

2022
[8]

Diverio and S

S. Diverio and S. Trapani,Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle, J. Differential Geom.111(2019), no. 2, 303–314

2019
[9]

Gauduchon,Le th´ eor` eme de l’excentricit´ e nulle, C

P. Gauduchon,Le th´ eor` eme de l’excentricit´ e nulle, C. R. Acad. Sci. Paris S´ er. A-B285(1977), no. 5, A387– A390

1977
[10]

Heier, S

G. Heier, S. S. Y. Lu and B. Wong,On the canonical line bundle and negative holomorphic sectional curvature, Math. Res. Lett.17(2010), no. 6, 1101–1110

2010
[11]

Kamenova, S

L. Kamenova, S. Lu and M. Verbitsky,Kobayashi pseudometric on hyperk¨ ahler manifolds, J. Lond. Math. Soc. (2)90(2014), no. 2, 436–450. HERMITIAN MANIFOLDS WITH NONPOSITIVE HOLOMORPHIC SECTIONAL CUR V ATURE 11

2014
[12]

Kobayashi,Hyperbolic Complex Spaces, Grundlehren der Mathematischen Wissenschaften, vol

S. Kobayashi,Hyperbolic Complex Spaces, Grundlehren der Mathematischen Wissenschaften, vol. 318, Springer-Verlag, Berlin, 1998

1998
[13]

Lazarsfeld,Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series, Ergeb- nisse der Mathematik und ihrer Grenzgebiete, 3

R. Lazarsfeld,Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series, Ergeb- nisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 48, Springer-Verlag, Berlin, 2004

2004
[14]

Lee and J

M.-C. Lee and J. Streets,Complex manifolds with negative curvature operator, Int. Math. Res. Not. IMRN (2021), no. 24, 18520–18528

2021
[15]

C. Li, L. Ni and X. Zhu,An application of aC 2-estimate for a complex Monge-Amp` ere equation, Internat. J. Math. 32 (2021), no. 12, Paper No. 2140007, 13 pp

2021
[16]

Nomura,K¨ ahler manifolds with negative holomorphic sectional curvature, K¨ ahler-Ricci flow approach, Int

R. Nomura,K¨ ahler manifolds with negative holomorphic sectional curvature, K¨ ahler-Ricci flow approach, Int. Math. Res. Not. IMRN (2018), no. 21, 6611–6616

2018
[18]

H. L. Royden,The Ahlfors–Schwarz lemma in several complex variables, Comment. Math. Helv.55(1980), no. 4, 547–558

1980
[19]

Tang,On real bisectional curvature and K¨ ahler-Ricci flow, Proceedings of the American Mathematical Society.147(2019), 793–798

K. Tang,On real bisectional curvature and K¨ ahler-Ricci flow, Proceedings of the American Mathematical Society.147(2019), 793–798

2019
[20]

Tang,On almost nonpositive k-Ricci curvature, The Journal of Geometric Analysis

K. Tang,On almost nonpositive k-Ricci curvature, The Journal of Geometric Analysis. (2022) 32:306

2022
[21]

Tosatti and X

V. Tosatti and X. Yang,An extension of a theorem of Wu–Yau, J. Differential Geom.107(2017), no. 3, 573–579

2017
[22]

Wong,The uniformization of compact K¨ ahler surfaces of negative curvature, J

B. Wong,The uniformization of compact K¨ ahler surfaces of negative curvature, J. Differential Geom.16 (1981), no. 3, 407–420

1981
[23]

Wu and S.-T

D. Wu and S.-T. Yau,Negative holomorphic curvature and positive canonical bundle, Invent. Math.204 (2016), no. 2, 595–604

2016
[24]

Negative holomorphic curvature and positive canonical bundle

D. Wu and S.-T. Yau,A remark on our paper “Negative holomorphic curvature and positive canonical bundle”, Comm. Anal. Geom.24(2016), no. 4, 901–912

2016
[26]

Yang and F

X. Yang and F. Zheng,On real bisectional curvature for Hermitian manifolds, Trans. Amer. Math. Soc.371 (2019), no. 4, 2703–2718

2019
[27]

Yau,A general Schwarz lemma for K¨ ahler manifolds, Amer

S.-T. Yau,A general Schwarz lemma for K¨ ahler manifolds, Amer. J. Math.100(1978), no. 1, 197–203

1978
[28]

Yau,On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge–Amp` ere equation, I, Comm

S.-T. Yau,On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge–Amp` ere equation, I, Comm. Pure Appl. Math.31(1978), no. 3, 339–411

1978
[29]

Zhang,Remark on semi-positive holomorphic sectional curvature and quasi-negativek-Ricci curvature, arXiv:2508.17237

S. Zhang,Remark on semi-positive holomorphic sectional curvature and quasi-negativek-Ricci curvature, arXiv:2508.17237

arXiv
[30]

Zhang,Holomorphic sectional curvature, nefness and Miyaoka-Yau type inequality, Math

Y. Zhang,Holomorphic sectional curvature, nefness and Miyaoka-Yau type inequality, Math. Z.298(2021), no. 3-4, 953-974. Kai Tang. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China Email address:kaitang001@zjnu.edu.cn

2021

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [17]

Ou,A characterization of uniruled compact K¨ ahler manifolds, arXiv:2501.18088

W. Ou,A characterization of uniruled compact K¨ ahler manifolds, arXiv:2501.18088

arXiv

[2] [25]

Yang,Hermitian manifolds with semi-positive holomorphic sectional curvature, Math

X. Yang,Hermitian manifolds with semi-positive holomorphic sectional curvature, Math. Res. Lett.23(2016), no. 3, 939–952

2016

[3] [1]

Barth, K

W. Barth, K. Hulek, C. Peters and A. Van de Ven,Compact Complex Surfaces, second enlarged edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 4, Springer-Verlag, Berlin, 2004

2004

[4] [2]

M. Berger,Sur les vari´ et´ es d’Einstein compactes, in: Comptes rendus de la IIIe R´ eunion du Groupement des Math´ ematiciens d’Expression Latine (Namur, 1965), Librairie Universitaire, Louvain, 1966, 35–55

1965

[5] [3]

Broder and J

K. Broder and J. Stanfield,A general Schwarz lemma for Hermitian manifolds, Bull. Lond. Math. Soc.58 (2026), no. 1, Paper No. e70269, 13 pp

2026

[6] [4]

Broder and K

K. Broder and K. Tang,On Hermitian manifolds with vanishing curvature, Math. Z.309(2025), no. 1, Paper No. 12

2025

[7] [5]

Brody,Compact manifolds and hyperbolicity, Trans

R. Brody,Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc.235(1978), 213–219

1978

[8] [6]

Cao and A

J. Cao and A. H¨ oring,Rational curves on compact K¨ ahler manifolds, J. Differential Geom.114(2020), no. 1, 1–39

2020