Pith. sign in

REVIEW 4 major objections 4 minor 50 references

On astheno-K\"ahler manifolds

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Astheno-Kähler metrics, defined by a weak Kähler identity, are the subject of a survey that compiles known rigidity theorems, cohomology bounds, examples, and PDE existence results.

desk verdict A survey of astheno-Kähler geometry with no new results, whose usefulness depends on transcription fidelity—and the fidelity breaks in its headline Theorem 5.2. read the letter →

arxiv 2506.06369 v1 pith:3NJ5FLVA submitted 2025-06-04 math.DG

Punam Gupta , Nidhi Yadav This is my paper
classification math.DG MSC 53C5532C35
keywords Complexmanifoldsastheno-KählermetricscohomologystrongKTmetricGauduchonFu-YauequationMonge-Ampèrenilmanifolds
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

Astheno-Kähler manifolds are Hermitian manifolds whose Kähler form $\Omega$ obeys $\partial\bar\partial\Omega^{m-2}=0$, a weak Kähler condition that is automatic in complex dimension 2 and reduces to the strong Kähler-with-torsion condition in dimension 3. The paper is a survey: it collects the classical rigidity theorems, cohomological bounds, bundle results, and PDE existence statements known for this class, and it assembles a catalogue of examples and counterexamples, including products of Sasakian manifolds, Calabi-Eckmann manifolds, nilmanifolds, and torus fibrations. The authors' claim is that this map is accurate and useful, and that the examples and PDE pointers are faithfully transcribed from the cited literature. A reader would care because astheno-Kähler metrics sit at the boundary between Kähler and non-Kähler geometry, and the survey makes that boundary visible in one place.

What carries the argument

The load-bearing object is the differential identity $\partial\bar\partial\Omega^{m-2}=0$, where $\Omega$ is the fundamental 2-form of a Hermitian metric on a complex manifold of complex dimension $m$. The exponent $m-2$ is what makes the condition weak: it is vacuous for $m=2$, it is the strong Kähler-with-torsion condition for $m=3$, and when paired with the strong Kähler-with-torsion condition for $m=4$ it forces the Gauduchon condition. Almost every result in the survey is reached by checking this identity or combining it with another structural condition such as balance, Gauduchon, conformal balance, or k-Gauduchon, so the identity acts as the switchboard connecting rigidity, cohomology, bundle theory, and PDE estimates.

What would settle it

Take any load-bearing quotation, such as Theorem 5.2, and compare it line-by-line with the cited original; the claim that the survey is a reliable map fails if the original has no role for the function $F'$ or if another core statement is materially altered.

Watch

Extended reading notes

Core claim

On the paper's own terms, the contribution is synthesis, not new mathematics. The authors set out to show that the single condition $\partial\bar\partial\Omega^{m-2}=0$ organizes a wide body of results: rigidity theorems for harmonic maps and homotopy-equivalent manifolds, the conclusion that compact balanced astheno-Kähler manifolds are Kähler, an inequality between Bott-Chern and Aeppli cohomology, finiteness and numerical flatness results for vector bundles, and existence theorems for the Fu-Yau and Monge-Ampère equations. The examples range across products of low-dimensional Sasakian and cosymplectic manifolds, Calabi-Eckmann manifolds, nilmanifolds with rational structure constants, and torus bundles over Kähler bases; the counterexamples include compact Vaisman manifolds, the complex-parallelizable Nakamura manifold, and Oeljeklaus-Toma manifolds. If the survey is right, it gives a trustworthy entry point and reference map for the subfield.

Load-bearing premise

The survey's usefulness as a reference map depends on every quoted theorem and example being transcribed faithfully from its cited source, and visible slips such as a smooth function $F'$ introduced in Theorem 5.2 but never used in its conclusion, two misspelled author names, and an absent Figure 1 make that fidelity the point to check.

Editorial extensions

If this is right

  • Every compact balanced astheno-Kähler manifold is Kähler, so genuinely non-Kähler examples must violate balance; this is a sharp separation line.
  • Any compact Vaisman manifold of dimension at least three carries no astheno-Kähler metric, which rules out the standard Hopf-type candidates.
  • Products of two compact complex surfaces admit an astheno-Kähler metric exactly when at least one factor is Kähler, and blow-ups preserve the condition only under extra assumptions such as $\partial\bar\partial\Omega=0$ and $\partial\bar\partial\Omega^2=0$.
  • The cohomological inequality $h^{0,1}_{BC}(M)\le h^{0,1}_{A}(M)\le h^{0,1}_{BC}(M)+1$ holds on compact astheno-Kähler manifolds, and the equality cases detect obstructions, including the failure of invariance under modifications.
  • The Fu-Yau equation has a unique solution on compact astheno-Kähler manifolds for sufficiently small normalization constant $A$, giving a PDE foothold on these non-Kähler spaces.

Reading between the lines

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

  • A natural extension the paper leaves implicit is to treat the family of conditions $\partial\bar\partial\Omega^k=0$ as a ladder interpolating between strong Kähler with torsion and Gauduchon metrics; the survey's examples suggest the existence theory changes sharply near $k=m-2$.
  • Because the condition is automatic in complex dimension 2, the first genuinely constrained dimension is 3, and the survey suggests that the rigidity lemmas quoted for astheno-Kähler manifolds should apply verbatim to strong Kähler-with-torsion threefolds.
  • The survey's cohomological inequality can be used as a quick obstruction test: a compact complex manifold with $h^{0,1}_{A}(M)>h^{0,1}_{BC}(M)+1$ cannot carry an astheno-Kähler metric, even if other geometric candidates look plausible.
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

4 major / 4 minor

Summary. This paper is a survey of astheno-Kähler manifolds. Sections 2 and 3 collect definitions and cited results, including cohomological inequalities, criteria for astheno-Kähler metrics to be Kähler, nilmanifold constructions, and theorems on vector bundles. Section 4 presents examples drawn from Matsuo–Takahashi, Fino–Tomassini, Fino–Grantcharov–Vezzoni, Matsuo, and Chiose–Rasdeaconu. Section 5 reports PDE results on the Fu–Yau equation, the complex Monge–Ampère equation, and the Hermitian–Yang–Mills flow. The authors state no original theorems; the paper's value depends entirely on the fidelity of its transcriptions and references.

Significance. If the transcriptions are accurate, the paper is a useful reference map of a mature subfield, and the worked examples in Section 4 provide a convenient, mostly reproducible collection of computations. The paper has no original derivation, machine-checked proof, parameter-free construction, or falsifiable prediction, so its significance is that of a survey rather than a research contribution. Its strengths are the breadth of the bibliography and the explicit presentation of several known examples. However, because the manuscript is nothing but transcriptions, accuracy of quotation is load-bearing, and the defects noted below prevent a reader from using the text as a reliable reference at present.

major comments (4)
  1. [§5.2, Theorem 5.2] The statement introduces a smooth function F′ and a unique constant b′, but neither appears in the displayed conclusion. The displayed equation Ω_u^{m−1} = Ω_0^{m−1} + i∂∂̄u ∧ Ω^{m−2} is a linear equation in u, not a Monge–Ampère equation; it does not determine b′, does not use F′, and does not state the normalization condition needed for uniqueness. As printed, the theorem cannot be used, and it does not faithfully report the result of [47] described in the preceding paragraph.
  2. [§4, Example 1] The opening sentence, 'Any product manifold of curves and surfaces is astheno-Kähler', is false under Definition 2.2 if the surface factor is an arbitrary Hermitian surface: for M = C × S with Ω = ω_C + ω_S and m = 3, we have ∂∂̄Ω = ∂∂̄ω_S, which need not vanish on a compact complex surface. If the intended statement assumes Kähler factors, that hypothesis is missing. The almost-contact construction that follows is unrelated to the initial claim, so the example is internally inconsistent as written.
  3. [§4, Example 7] The exclusion 'with s ≥ 2 and s = 1' is impossible for any integer s, so the class of manifolds being discussed is empty and the sentence has no determinate content. Since the survey's purpose is to report [9, Corollary 5.3] accurately, this is not merely a cosmetic typo but a fidelity failure that makes the cited result unusable.
  4. [§2.1, Theorem 2.13] The hypothesis 'compact almost-Calabi-Yau manifold with torsion form ≥ 3' is not meaningful in the notation of Definitions 2.10 and 2.11; no object called a torsion form is defined, and the inequality '≥ 3' is attached to no quantity. The disjunction 'either almost-pluriclosed (Gauduchon) or almost-astheno-Kähler' is followed by a conclusion that does not specify which of the alternative conditions is being used. The theorem cannot be checked as printed and should be compared line-by-line with [24, Theorem 1.1, 1.2].
minor comments (4)
  1. [§2, Figure 1] The text 'Figure 1: flow chart' appears after Definition 2.2, but no figure is included in the manuscript, and there is no caption explaining the chart's content.
  2. [§2.1 and §5.2] The names 'Grantcherov' and 'Weincove' are misspelled; they should be 'Grantcharov' (as in [12]) and 'Weinkove' (as in [47]).
  3. [§5.1, Eq. (5.1)] The Fu–Yau equation is attributed to '[37, 17]', but [37] is Phong–Picard–Zhang and [17] is Garcia-Fernandez; the original references [14,15] should be cited for the introduction of the equation.
  4. [§2, Definition 2.11] The word 'Hessain' should be 'Hessian', and the phrase 'trace of the Hessian ∇²f' should be punctuated so that it reads as a definition of the tension field rather than of the Laplacian.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is a survey that attributes every theorem, example, and equation to external cited sources, with no original derivation whose output could reduce to its input.

full rationale

The paper makes no original mathematical claim and performs no derivation chain of its own. Every substantive result is introduced with an external citation: Jost–Yau [22, 23], Fino–Tomassini [13], Matsuo–Takahashi [32], Chiose–Rasdeaconu [9], Biswas [3], Podestà [38], Chen [8], Li–Nie–Zhang [28], Tosatti–Weinkove [46, 47], Nie–Zhang [35], and others. The survey's role is transcription and organization, not prediction or fitting, so there is no parameter that is fitted to one subset of data and then 'predicted' elsewhere, no quantity defined in terms of the claimed output, and no uniqueness theorem imported from the authors' own prior work to force a choice. The authors of the survey, Gupta and Yadav, do not appear as the sources of any cited result; the cited 'the authors in [46]' refers to Tosatti and Weinkove, not to the survey authors. The apparent defects flagged elsewhere, such as the garbled statement of Theorem 5.2 in Section 5.2 (where F' and b' are introduced but do not occur in the displayed conclusion), the misspelled surnames in references [12] and [47], and the missing flow chart in Figure 1, are transcription-fidelity or correctness concerns, not circularity. They do not establish that any result in the paper is equivalent to its own input by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The survey rests on standard definitions of Hermitian geometry (astheno-Kähler, SKT, Gauduchon, balanced, Bott-Chern and Aeppli cohomology) and on the correctness of the cited results it reproduces. It introduces no free parameters and no invented entities. The only load-bearing nonstandard assumption is that the transcriptions of quoted results are faithful, which the manuscript itself undermines in places (Theorem 5.2, Section 5.2).

assumptions (3)
  • standard math An almost complex structure is integrable if and only if its Nijenhuis tensor vanishes (Newlander-Nirenberg theorem).
    Invoked in Section 2 (Preliminaries) to define complex manifolds; standard result, cited as [34] and [49].
  • domain assumption On a compact astheno-Kähler manifold, every holomorphic 1-form is closed (Jost-Yau Lemma 2.1).
    Repeatedly used in the existence obstructions presented in Section 2 (for example, Biswas's result on G/Γ); the survey takes it as established from [22, Lemma 6].
  • domain assumption The quoted results of the cited papers are correct and are transcribed without material error.
    The survey's content is a chain of transcriptions; visible defects such as the incomplete statement of Theorem 5.2 (Section 5.2) make this assumption the fragile one.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On astheno-K\"ahler manifolds." pith.science (2026). https://pith.science/paper/3NJ5FLVA

@misc{pith2026250606369,
  author       = {Pith},
  title        = {Pith review of: On astheno-K\"ahler manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NJ5FLVA}},
  note         = {Machine review of arXiv:2506.06369}
}
read the original abstract

This survey explores a range of classical findings and recent developments related to our understanding of astheno-K\"ahler manifolds. Furthermore, we provide various examples of astheno-K\"ahler manifolds and analyze the challenges associated with their existence.

Figures

Figures reproduced from arXiv: 2506.06369 by the authors.

Figure 1
Figure 1. flow chart Definition 2.3 [32] Let ∇ be a connection on a smooth manifold M. The torsion tensor T is a (1,2)-tensor (alternatively, a vector-valued 2-form) defined as: T (X,Y ) = ∇X Y − ∇Y X −[X,Y ] 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

50 extracted references · 49 canonical work pages

  1. [47]

    Tosatti and B

    V . Tosatti and B. Weinkove, Hermitian metrics, (n−1,n−1) form and Monge-Ampère Equations, Journal für die reine und angewandte Mathematik (Crelles Journal), 2019 (2019), no. 755, 67- 101

    work page 2019

  2. [1]

    Ballmann, Lectures on Kähler manifolds, ESI Lectures in mathematics and physics, (2006)

    W . Ballmann, Lectures on Kähler manifolds, ESI Lectures in mathematics and physics, (2006)

    work page 2006

  3. [2]

    Biswas and J

    I. Biswas and J. Loftin, Hermitian-Einstein connections on principal bundles over flat affine manifolds, Int. Jour. Math., 23 (2012), no. 4, 1250039, 23 pp

    work page 2012

  4. [3]

    Biswas, On Hermitian structure on G/Γ, Bull

    I. Biswas, On Hermitian structure on G/Γ, Bull. Sci. Math., 137 (2013), 716–717

    work page 2013

  5. [4]

    I.Biswas and V . P . Pingali, A characterization of finite vector bundles on Gauduchon astheno- Kähler manifolds, Épijournal de Géométrie Algébrique, 2 (2018), Art. 6, 13pp

    work page 2018

  6. [5]

    Bott and S.S

    R. Bott and S.S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math., 114 (1965), no. 1, 71-112

    work page 1965

  7. [6]

    Calamai, Positive projectively flat manifolds are locally conformally flat-Kähler Hopf mani- folds, Pure Appl

    S. Calamai, Positive projectively flat manifolds are locally conformally flat-Kähler Hopf mani- folds, Pure Appl. Math. Q., 17 (2021), no.3, 1139-1154

    work page 2021

  8. [7]

    H. Chen, L. Chen and X. Nie, Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics, Sci. China Math., 64 (2021), no.4, 763-780

    work page 2021

Show all 50 references
  1. [8]

    Chen, A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifold, C

    Y. Chen, A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifold, C. R. Math. Acad. Sci. Paris, 359 (2021), 523-531

    2021

  2. [9]

    Chiose and R

    I. Chiose and R. Rasdeaconu, Remarks on astheno-Kähler manifolds, bott-chern and aeppli cohomology groups, Ann. Glob. Anal. Geom., 63 (2023), no. 24

    2023

  3. [10]

    J. Chu , L. Huang and X. Zhu, The Fu–Yau equation on compact astheno-Kähler manifolds, Adv. Math., 346 (2019), 908-945

    2019

  4. [11]

    A. Fino, G. Grantcharov and M. Verbitsky, Algebraic dimension of complex nilmanifolds, J. Math. Pures Appl., 118 (2018), 204–218

    2018

  5. [12]

    A. Fino, G. Grantcharov and L. Vezzoni, Astheno–Kähler and Balanced Structures on Fibrations, Int. Math. Research Notices, 2019 (2019), no. 22, 7093–7117

    2019

  6. [13]

    Fino and A

    A. Fino and A. Tomassini, On astheno-Kähler metrics, Jour. Lond. Math. Soc., 83 (2011), no. 2, 290-308

    2011

  7. [14]

    Fu and S.-T

    J.-X. Fu and S.-T . Yau, A Monge-Ampère-type equation motivated by string theory, Comm. Anal. Geom., 15 (2007), no. 1, 29–75

    2007

  8. [15]

    Fu and S.-T

    J.-X. Fu and S.-T . Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation, Jour. Diff. Geom., 78 (2008), no. 3, 369–428. 15

    2008

  9. [16]

    J. Fu, Z. Wang and D. Wu, Semilinear equations, the function, and generalized Gauduchon metrics, Jour. Eur. Math. Soc., 15 (2013), 659-680

    2013

  10. [17]

    Garcia-Fernandez, Lectures on the Strominger system, Travaux mathématiques, 24 (2016), 7-61

    M. Garcia-Fernandez, Lectures on the Strominger system, Travaux mathématiques, 24 (2016), 7-61

    2016

  11. [18]

    Gauduchon, La 1-forme de torsion d’une variété hermitienne compacte, Math

    P . Gauduchon, La 1-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), no. 8, 495–518

    1984

  12. [19]

    Goldstein and S

    E. Goldstein and S. Prokushkin, Geometric model for complex non-Kähler manifolds with SU (3) structure, Comm. Math. Phys., 251 (2004), no. 1, 65–78

    2004

  13. [20]

    Hatcher, Vector Bundles and K-theory, Version 2.1, (May 2009)

    A. Hatcher, Vector Bundles and K-theory, Version 2.1, (May 2009)

    2009

  14. [21]

    Hironaka, An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures, Ann

    H. Hironaka, An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures, Ann. Math., 75 (1962), no. 2, 190–208

    1962

  15. [22]

    Jost and S.-T

    J. Jost and S.-T . Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math., 170 (1993), 221–254

    1993

  16. [23]

    Jost and S.T

    J. Jost and S.T . Yau, Correction to A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math., 173 (1994), 307

    1994

  17. [24]

    Kawamura, On the conformally balanced condition on almost Hermitian manifolds and the quasi-Kählerity, J

    M. Kawamura, On the conformally balanced condition on almost Hermitian manifolds and the quasi-Kählerity, J. Geom., 112 (2021), no.2, paper no. 20, 13 pp

    2021

  18. [25]

    Kawamura, An a priori C 0-estimate for the Fu-Yau equation on compact almost astheno- Kähler manifolds, Complex Manifolds, 9 (2022), no

    M. Kawamura, An a priori C 0-estimate for the Fu-Yau equation on compact almost astheno- Kähler manifolds, Complex Manifolds, 9 (2022), no. 1, 223-237

    2022

  19. [26]

    Kobayashi, Kanô Memorial Lectures 5, Differential geometry of Complex vector bundles, Iwanami Shoten, Publishers and Princeton University Press (1987)

    S. Kobayashi, Kanô Memorial Lectures 5, Differential geometry of Complex vector bundles, Iwanami Shoten, Publishers and Princeton University Press (1987)

    1987

  20. [27]

    Latorre and L

    A. Latorre and L. Ugarte, On non-Kähler compact complex manifolds with balanced and astheno-Kähler metrics, C. R. Math. Acad. Sci. Paris, 355 (2017), no. 1, 90–93

    2017

  21. [28]

    C. Li, Y. Nie and X. Zhang, Numerically flat holomorphic bundles over non-Kähler manifolds, Jour. Reine Angew. Math., 790 (2022), 267–285

    2022

  22. [29]

    Nijenhuis, Xn−1 forming sets of eigenvectors, Nederl

    A. Nijenhuis, Xn−1 forming sets of eigenvectors, Nederl. Akad. Wet., Proc., Ser. A, 54 (1951), 200-212; Indagationes Math. 13 (1951), 200-212

    1951

  23. [30]

    J. Li, S.T . Yau and F . Zheng, On projectively flat Hermitian manifolds, Comm. Anal. Geom., 2 (1994), no. 1, 103–109

    1994

  24. [31]

    Loustau, Harmonic maps from Kähler manifolds, https://arxiv.org/abs/2010.03545v1, (2020)

    B. Loustau, Harmonic maps from Kähler manifolds, https://arxiv.org/abs/2010.03545v1, (2020)

    2020 arXiv

  25. [32]

    Matsuo and T

    K. Matsuo and T . Takahashi, On compact astheno-Kähler manifolds, Colloq. Math., 89 (2001), 213– 221

    2001

  26. [33]

    Matsuo, Astheno-Kähler structures on Calabi–Eckmann manifolds, Colloq

    K. Matsuo, Astheno-Kähler structures on Calabi–Eckmann manifolds, Colloq. Math., 115 (2009), 33–39

    2009

  27. [34]

    Newlander and L

    A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. Math., 65 (1957), 391-404. 16

    1957

  28. [35]

    Nie and X

    Y. Nie and X. Zhang, The limiting behaviour of the Hermitian-Yang-Mills flow over compact non-Kähler manifolds, Sci. China Math., 63 (2020), no. 7, 1369-1390

    2020

  29. [36]

    M. V . Nori, On the representations of the fundamental group, Compositio Math., 33 (1976), no. 1, 29–41

    1976

  30. [37]

    Phong, S

    D.H. Phong, S. Picard and X. Zhang, The Fu-Yau equation with negative slope parameter, Invent. Math., 209 (2017), no. 2, 541–576

    2017

  31. [38]

    Podestà, Homogeneous Hermitian manifolds and special metrics, Transformation Groups, 23 (2018), no

    F . Podestà, Homogeneous Hermitian manifolds and special metrics, Transformation Groups, 23 (2018), no. 4, 1129–1147

    2018

  32. [39]

    Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull

    D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull. Soc. Math. Fr., 143 (2015), no. 4, 763-800

    2015

  33. [40]

    Z.Shen, Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds, Bull. Sci. Math., 187 (2023), paper No. 103288, 21 pp

    2023

  34. [41]

    Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Annals Math., 112 (1980), 73-111

    Y.T . Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Annals Math., 112 (1980), 73-111

    1980

  35. [42]

    Streets, and G

    J. Streets, and G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN, 2010 (2010), no.16, 3101–3133

    2010

  36. [43]

    Swann, Twisting Hermitian and hypercomplex geometries, Duke Math

    A. Swann, Twisting Hermitian and hypercomplex geometries, Duke Math. Jour., 155 (2010), no. 2, 403-431

    2010

  37. [44]

    Sferruzza, Deformations of astheno-Kähler metrics, Complex Manifolds, 10 (2023), no

    T . Sferruzza, Deformations of astheno-Kähler metrics, Complex Manifolds, 10 (2023), no. 1, 19 pp

    2023

  38. [45]

    Sferruzza and A

    T . Sferruzza and A. Tomassini, On cohomological and formal properties of strong Kähler with torsion and astheno-Kähler metrics, Math. Z., 304 (2023), paper no. 55, 27pp

    2023

  39. [46]

    Tosatti and B

    V . Tosatti and B. Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, Jour. Amer. Math. Soc., 23 (2010), no.4, 1187–1195

    2010

  40. [48]

    Vaisman, On locally conformal almost Kaehler manifolds, Isr

    I. Vaisman, On locally conformal almost Kaehler manifolds, Isr. J. Math., 24 (1976), 338-351

    1976

  41. [49]

    Yano and M

    K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, World Scientific, Singapore, Distr. by John Wiley and Sons Ltd., Chichester. IX, 3 (1984), 508 pp

    1984

  42. [50]

    Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm

    S.T . Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math., 31 (1978), no.3, 339–411. 17

    1978

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.