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Special non-K\"ahler metrics -- old and new

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Pith's one-line read This survey reports that on compact nilmanifolds with left-invariant complex structure, balanced, pluriclosed, and LCK metrics cannot coexist unless the manifold is a complex torus.

desk verdict A careful survey that compiles known incompatibility results; its strongest claim leans on an unstated positivity theorem from OV25, so treat it as a map, not a proof. read the letter →

arxiv 2505.01795 v1 pith:ZJ6ENAUV submitted 2025-05-03 math.DG

Liviu Ornea , Miron Stanciu This is my paper · ORCID
classification math.DG MSC 53C5522E2532J18
keywords HermitianmetricbalancedlocallyconformallyKählerpluriclosedastheno-KählerGauduchonnilmanifoldcoexistenceofspecialmetrics
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 survey assembles the known results about special non-Kähler Hermitian metrics—balanced, pluriclosed, locally conformally Kähler (LCK), Gauduchon, astheno-Kähler, and Hermitian-symplectic—and focuses on the question of which pairs can coexist on the same compact complex manifold. Its central message is that incompatibility is the norm: on compact nilmanifolds with left-invariant complex structure, a balanced metric together with a pluriclosed metric forces the manifold to be a complex torus, and the same collapse happens if an LCK metric is paired with a balanced, left-invariant k-Gauduchon, or pluriclosed metric. The paper additionally records how these metric classes behave under blow-up and small deformations, and describes a general recipe for building explicit examples from structure equations on nilpotent Lie algebras. A sympathetic reader should care because these coexistence and stability results provide some of the few structural organizing principles in the otherwise vast non-Kähler landscape.

What carries the argument

The machinery is the left-invariant reduction: on a compact nilmanifold, if a special metric exists, an averaging argument produces a left-invariant one; then a classification of left-invariant complex structures on nilmanifolds turns the coexistence question into linear algebra on structure equations. For LCK metrics, the Lee 1-form and its strict positivity with respect to any Gauduchon metric carries the obstruction. The survey also presents a general recipe for constructing 2-step nilmanifold examples by prescribing rational structure constants, extending them to a compact quotient, and then writing down left-invariant fundamental forms.

What would settle it

A compact complex nilmanifold that is not a torus, with a left-invariant complex structure and both a balanced and a pluriclosed metric compatible with that structure, would disprove Theorem 4.7; the Section 3 recipe, prescribing rational structure constants for a 2-step nilpotent Lie algebra, offers a concrete search space for such an example.

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Extended reading notes

Core claim

On its own terms, the survey's central discovery is the collection of incompatibility theorems it reports. Theorem 4.7 states that a compact complex nilmanifold with a left-invariant complex structure carrying both a balanced and a pluriclosed metric compatible with that structure must be a complex torus. Theorem 4.8 extends the same conclusion to LCK metrics: under the same left-invariant setting, an LCK metric cannot coexist with a balanced, a left-invariant k-Gauduchon, or a pluriclosed metric unless the manifold is a torus. At a broader level, Theorem 4.5 says that no known non-Kähler LCK manifold admits a balanced metric, and Theorem 4.11 excludes k-Gauduchon, Hermitian-symplectic, and balanced metrics on compact LCK manifolds with parallel Lee form. The survey also presents stability results: pluriclosed metrics survive blow-up, Hermitian-symplectic metrics survive point blow-up and small deformations, while the full LCK class is not stable under deformation.

Load-bearing premise

The survey's conclusions rest on the cited theorems being correct and accurately transcribed; in particular, the claim that known LCK manifolds cannot carry balanced metrics depends on an unproved technical positivity property of a certain 1-form associated to the metric, not derived in this survey.

Editorial extensions

If this is right

  • On a compact complex nilmanifold with left-invariant complex structure, a non-torus manifold cannot admit both a balanced and a pluriclosed metric compatible with that structure.
  • The same obstruction applies to LCK metrics: together with a balanced, left-invariant k-Gauduchon, or pluriclosed metric, they force a complex torus.
  • None of the currently known classes of non-Kähler LCK manifolds admits a balanced metric.
  • Compact LCK manifolds with parallel Lee form of dimension at least three exclude all k-Gauduchon and Hermitian-symplectic metrics, and in any dimension exclude balanced metrics.
  • Pluriclosed and Hermitian-symplectic metrics survive blow-up in the stated situations, while the full LCK class does not survive small deformations.

Reading between the lines

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

  • If the nilmanifold incompatibility theorems reflect a deeper cohomological principle, a testable extension is to check the same pairings on solvmanifolds, where left-invariant structures are harder to classify.
  • The structure-equation recipe in Section 3 could be used to systematically search for the first known counterexample or for new coexisting pairs among 2-step nilmanifolds, since existence is reduced to linear algebra on the constants.
  • The strict positivity of the Lee-form degree behind Theorem 4.5 suggests that any future LCK manifold with non-positive Lee degree would be a candidate to break the balanced obstruction, so computing this degree for new LCK examples would directly test the scope of the result.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This survey reviews special Hermitian metrics on compact complex manifolds, focusing on locally conformally Kähler (LCK) metrics (including Vaisman, LCK with potential, Kato, and Oeljeklaus–Toma manifolds), Gauduchon and k-Gauduchon metrics, pluriclosed, astheno-Kähler, balanced, locally conformally balanced, Hermitian-symplectic, and locally conformally symplectic structures. It presents a Lie-algebraic recipe for constructing nilmanifold examples, collects results on the incompatibility of different metric classes on the same manifold, and surveys stability under blow-up and deformation. The central message is that, apart from special cases such as complex tori or explicit coexistence examples, these metric classes are largely mutually incompatible.

Significance. If the cited results are accurate, this survey is a useful and well-organized reference that collects the main known incompatibility statements (Theorems 4.1, 4.5, 4.7, 4.8, 4.11) and the blow-up/deformation results (Theorems 5.1–5.4) in one place. It includes valuable caveats, such as Remark 4.10, which explicitly notes that Theorem 4.8 fails without left-invariance for the k-Gauduchon metric. The exposition of the nilmanifold construction recipe in Section 3 is a helpful didactic contribution. The paper contains no original proofs, so its value as a reliable reference depends entirely on the faithful transcription of the cited theorems; this is where the main risk lies, particularly for statements imported from the authors' own recent papers.

major comments (2)
  1. [Section 4, Theorem 4.5 and Remark 4.6] Theorem 4.5 is the survey's most general LCK/balanced incompatibility statement, covering all known LCK manifolds up to bimeromorphism. The supporting fact described in Remark 4.6, the strict positivity of the degree of the Lee form with respect to any Gauduchon metric, is not stated precisely, and the 'degree of a 1-form' is never defined in the paper. Moreover, the theorem as written extends the cited [OV25, Theorem 4.17] to manifolds bimeromorphic to the known LCK classes, yet the bimeromorphic invariance of the obstruction is not discussed. Since no proofs are included, the reader must rely on the transcription being exactly correct. Please add a formal definition of the degree of a 1-form, state the positivity theorem as a clearly labeled assertion (with a reference), and either justify the bimeromorphic invariance or quote the original theorem verbatim without broadening it.
  2. [Section 4.1, Theorem 4.1] The statement that LCK and k-Gauduchon conditions are 'mutually incompatible in a given conformal class' is ambiguous, because the k-Gauduchon condition is not conformally invariant. The theorem should be formulated for a single Hermitian metric rather than for a conformal class. As written, the reader could incorrectly infer that no metric in any conformal class can satisfy both conditions, which is also contradicted by Kähler metrics. Please rephrase, for example: 'A Hermitian metric on a compact complex manifold cannot be both LCK and k-Gauduchon unless it is Kähler', and similarly for the balanced/k-Gauduchon claim.
minor comments (4)
  1. [Remark 4.12] The name 'Calaby–Eckmann' should be 'Calabi–Eckmann'.
  2. [Remark 4.6] The phrase 'on the m' is incomplete and should read 'on the manifold'.
  3. [Theorem 5.4(iii)] The cohomology group 'H^{n−1,n−21}_{BC}' appears to be a typo; the intended bidegree should be stated correctly, likely H^{n−1,n−2}_{BC} or H^{n−2,n−1}_{BC} depending on the convention.
  4. [References] The reference [AI03] is labeled with the year 2003 but cites Differ. Geom. Appl. 14 (2001); please harmonize the year in the citation key and the bibliographic entry.

Circularity Check

0 steps flagged · score 0.0 of 10

Survey of quoted theorems; no derivation reduces to its inputs.

full rationale

I find no circularity. This is a survey, not a derivation: Section 2 gives definitions, Section 3 describes a construction mechanism, and Sections 4 and 5 transcribe published theorems. The only proof given, Proposition 2.1, constructs the Lee form from the defining equation and is not circular. The most load-bearing results, Theorem 4.5 and Theorem 4.8, are quoted from the authors' own papers [OV25] and [OOS23]; these are self-citations, but they report published, parameter-free theorems with proofs in those papers, and nothing in the present survey defines those theorems' conclusions into their hypotheses. Under the review rules, such independent external evidence does not raise the circularity score. There is a completeness caveat: Remark 4.6 attributes Theorem 4.5 to strict positivity of the degree of the Lee form, while the term 'degree' is not defined in the survey and the bimeromorphic hypothesis of Theorem 4.5 is not unpacked. That is an omitted-support or verification issue, not an equivalence-by-construction. No fitted input is called a prediction and no known result is renamed as a new derivation, so no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The survey introduces no new free parameters or entities. It relies on standard Lie theory and on the correctness of the many cited theorems; the most structurally important background inputs are listed.

assumptions (4)
  • standard math Lie's third theorem (existence of a Lie group for a given Lie algebra)
    Used in Section 3 to produce a simply connected Lie group from the structure equations via Serre [Se65, Page 152].
  • standard math Malcev's theorem (rational structure equations yield a cocompact lattice)
    Used in Section 3 to produce the compact nilmanifold Gamma-G from the rational structure constants.
  • domain assumption Sawai's structure theorem for nilmanifolds with left-invariant complex structures
    Used in Remark 4.9 to classify non-Kähler nilmanifolds as quotients of the real Heisenberg group.
  • domain assumption Accuracy of the cited theorems from IP13, FV16, OOS23, OV25, etc.
    The survey's claims rest on the correctness of these prior results, which are taken as black boxes without proof.

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Cite this review

Pith. "Pith review of Special non-K\"ahler metrics -- old and new." pith.science (2026). https://pith.science/paper/ZJ6ENAUV

@misc{pith2026250501795,
  author       = {Pith},
  title        = {Pith review of: Special non-K\"ahler metrics -- old and new},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJ6ENAUV}},
  note         = {Machine review of arXiv:2505.01795}
}
read the original abstract

We give an account of old and new results concerning many types of non-K\"ahler metrics, with focus on the problem of their coexistence on compact complex manifolds, and their behaviour at deformations and blow-up. We also describe a mechanism that several authors have used to construct examples of nilmanifolds admitting metrics with certain properties.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

56 extracted references · 54 canonical work pages

  1. [1]

    Alessandrini, G

    L. Alessandrini, G. Bassanelli, Metric properties of manifolds bimeromorphic to compact K\"ahler spaces , J. Differ. Geom. 37 (1993), 95–121

    work page 1993

  2. [2]

    Alexandrov, S

    B. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds , Differ. Geom. Appl. 14 (2001), 251–265

    work page 2001

  3. [3]

    Andrada, A

    A. Andrada, A. Tolcachier, Harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds , arXiv:2301.09706v2

    arXiv

  4. [4]

    Angella, A

    D. Angella, A. Otiman, A note on compatibility of special Hermitian structures , arXiv:2306.02981v2

    arXiv

  5. [5]

    Apostolov, G

    V. Apostolov, G. Dloussky, Twisted differentials and Lee classes of locally conformally symplectic complex surfaces , Math. Z. 303 (2023), Art. 76

    work page 2023

  6. [6]

    F. A. Belgun, On the metric structure of non-K\"ahler complex surfaces , Math. Ann. 317 (2000), 1–40

    work page 2000

  7. [7]

    Bismut, A local index theorem for non-K\"ahler manifolds , Math

    J.-M. Bismut, A local index theorem for non-K\"ahler manifolds , Math. Ann. 284 (1989), 681–699

    work page 1989

  8. [8]

    Brunella, Locally conformally K\"ahler metrics on Kato surfaces , Nagoya Math

    M. Brunella, Locally conformally K\"ahler metrics on Kato surfaces , Nagoya Math. J. 202 (2011), 77–81

    work page 2011

Show all 56 references
  1. [9]

    Cavalcanti, Hodge theory of SKT manifolds , Adv

    G. Cavalcanti, Hodge theory of SKT manifolds , Adv. Math. 374 (2020), Art. 107270

    2020

  2. [10]

    Ciulic a , A

    C. Ciulic a , A. Otiman, M. Stanciu, Special non-K\"ahler metrics on Endo-Pajitnov manifolds , Ann. Mat. Pura Appl. (2024), https://doi.org/10.1007/s10231-024-01533-0

    2024 doi

  3. [11]

    Chiose, R

    I. Chiose, R. R a sdeaconu, I. S uvaina, Balanced manifolds and SKT metrics , Ann. Mat. Pura Appl. 201 (2022), 2505–2517

    2022

  4. [12]

    Ciulic a , Special Hermitian metrics , arXiv:2411.02567 (2024)

    C. Ciulic a , Special Hermitian metrics , arXiv:2411.02567 (2024)

    2024 arXiv

  5. [13]

    Dinew, D

    S. Dinew, D. Popovici, A generalised volume invariant for Aeppli cohomology classes of Hermitian-symplectic metrics , Adv. Math. 393 (2021), 108056

    2021

  6. [14]

    Dinew, D

    S. Dinew, D. Popovici, A variational approach to SKT and balanced metrics , J. Math. Pures Appl. 175 (2023), 237–268

    2023

  7. [15]

    Dloussky, Structure des surfaces de Kato , M\'em

    G. Dloussky, Structure des surfaces de Kato , M\'em. Soc. Math. France (N.S.) 14 (1984)

    1984

  8. [16]

    H. Endo, A. Pajitnov, On generalized Inoue manifolds , Proc. Int. Geom. Center 13 (2020), 24–39

    2020

  9. [17]

    A. Fino, G. Grantcharov, L. Vezzoni, Astheno-K\"ahler and balanced structures on fibrations , Int. Math. Res. Not. 2019 (2019), 7093–7117

    2019

  10. [18]

    A. Fino, A. Tomassini, Blow-ups and resolutions of strong K\"ahler with torsion metrics , Adv. Math. 221 (2009), 914–935

    2009

  11. [19]

    A. Fino, A. Tomassini, On astheno-K\"ahler metrics , J. Lond. Math. Soc. 83 (2011), 290–308

    2011

  12. [20]

    A. Fino, L. Vezzoni, On the existence of balanced and SKT metrics on nilmanifolds , Proc. Amer. Math. Soc. 144 (2016), 2455–2459

    2016

  13. [21]

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

    2013

  14. [22]

    Gauduchon, Le th\'eor\`eme de l’excentricité nulle , C

    P. Gauduchon, Le th\'eor\`eme de l’excentricité nulle , C. R. Acad. Sci. Paris S\'er. A–B 285 (1977), A387–A390

    1977

  15. [23]

    Gauduchon, Structures de Weyl et th\'eorèmes d’annulation sur une vari\'et\'e conforme autoduale , Ann

    P. Gauduchon, Structures de Weyl et th\'eorèmes d’annulation sur une vari\'et\'e conforme autoduale , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1991), 563–629

    1991

  16. [24]

    A. Gray, L. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants , Ann. Mat. Pura Appl. 123 (1980), 35–58

    1980

  17. [25]

    Inoue, On surfaces of Class VII_0 , Invent

    M. Inoue, On surfaces of Class VII_0 , Invent. Math. 24 (1974), 269–310

    1974

  18. [26]

    Istrati, A

    N. Istrati, A. Otiman, M. Pontecorvo, On a class of Kato manifolds , Int. Math. Res. Not. 2021 (2021), 5366–5412

    2021

  19. [27]

    Ivanov, G

    S. Ivanov, G. Papadopoulos, Vanishing theorems on (l|k) -strong K\"ahler manifolds with torsion , Adv. Math. 237 (2013), 147–164

    2013

  20. [28]

    Jost, S.-T

    J. Jost, 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

  21. [29]

    Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds , Bull

    H. Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds , Bull. Lond. Math. Soc. 45 (2013), 15–26

    2013

  22. [30]

    Kato, Compact complex manifolds containing global spherical shells , Proc

    Ma. Kato, Compact complex manifolds containing global spherical shells , Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), 15–16

    1977

  23. [31]

    ahler compact complex manifolds with balanced and astheno-K\

    A. Latorre, L. Ugarte, On non-K\"ahler compact complex manifolds with balanced and astheno-K\"ahler metrics , C. R. Math. Acad. Sci. Paris 336 (2017), 90–93

    2017

  24. [32]

    A. I. Mal’cev, On a class of homogeneous spaces , Trans. Amer. Math. Soc. 9 (1962), 276–307

    1962

  25. [33]

    Marchidanu, Complex geometry and Hermitian metrics on the product of two Sasakian manifolds , J

    V. Marchidanu, Complex geometry and Hermitian metrics on the product of two Sasakian manifolds , J. Geom. Phys. 199 (2024), Art. 105134

    2024

  26. [34]

    M. L. Michelsohn, On the existence of special metrics in complex geometry , Acta Math. 149 (1982), 261–295

    1982

  27. [35]

    Oeljeklaus, M

    K. Oeljeklaus, M. Toma, Non-K\"ahler compact complex manifolds associated to number fields , Ann. Inst. Fourier 55 (2005), 1291–1300

    2005

  28. [36]

    Ornea, A

    L. Ornea, A. Otiman, M. Stanciu, Compatibility between non-K\"ahler structures on non-complex (nil)manifolds , Transform. Groups 28 (2023), 1669–1686

    2023

  29. [37]

    Ornea, M

    L. Ornea, M. Verbitsky, Locally conformally K\"ahler manifolds with potential , Math. Ann. 348 (2010), 25–33

    2010

  30. [38]

    ahler geometry , Progress in Math. 354 , Birkh\

    L. Ornea, M. Verbitsky, Principles of locally conformally K\"ahler geometry , Progress in Math. 354 , Birkh\"auser (2024)

    2024

  31. [39]

    Ornea, M

    L. Ornea, M. Verbitsky, The Lee–Gauduchon cone on complex manifolds , arXiv:2411.05595

  32. [40]

    Ornea, M

    L. Ornea, M. Verbitsky, Balanced metrics and Gauduchon cone of locally conformally K\"ahler manifolds , Int. Math. Res. Not. rnaf014 (2025), https://doi.org/10.1093/imrn/rnaf014

    2025 doi

  33. [41]

    Ornea, M

    L. Ornea, M. Verbitsky, V. Vuletescu, Blow-ups of locally conformally K\"ahler manifolds , Int. Math. Res. Not. 2013 (2013), 2809–2821

    2013

  34. [42]

    Piovani, T

    R. Piovani, T. Sferruzza, Deformations of strong K\"ahler with torsion metrics , Complex Manifolds 8 (2021), 286–301

    2021

  35. [43]

    Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics , Invent

    D. Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics , Invent. Math. 194 (2013), 515–534

    2013

  36. [44]

    Popovici, Stability of strongly Gauduchon manifolds under modifications , J

    D. Popovici, Stability of strongly Gauduchon manifolds under modifications , J. Geom. Anal. 23 (2013), 653–659

    2013

  37. [45]

    Preda, M

    O. Preda, M. Stanciu, Vaisman’s theorem for lcK spaces , Ann. Sc. Norm. Super. Pisa Cl. Sci. 24 (2023), 2311–2321

    2023

  38. [46]

    Preda, M

    O. Preda, M. Stanciu, Vaisman’s theorem and local reducibility , arXiv:2405.04162 (2024). To appear in T\^ohoku Math. J

    2024 arXiv

  39. [47]

    Sawai, Locally conformal K\"ahler structures on compact nilmanifolds with left-invariant complex structures , Geom

    H. Sawai, Locally conformal K\"ahler structures on compact nilmanifolds with left-invariant complex structures , Geom. Dedicata 125 (2007), 93–101

    2007

  40. [48]

    Serre, Lie algebras and Lie groups: 1964 lectures given at Harvard University , Springer (1965)

    J.-P. Serre, Lie algebras and Lie groups: 1964 lectures given at Harvard University , Springer (1965)

    1965

  41. [49]

    Sferruzza, Deformations of balanced metrics , Bull

    T. Sferruzza, Deformations of balanced metrics , Bull. Sci. Math. 178 (2022)

    2022

  42. [50]

    Sferruzza, Deformations of astheno-K\"ahler metrics , Complex Manifolds 10 (2023)

    T. Sferruzza, Deformations of astheno-K\"ahler metrics , Complex Manifolds 10 (2023)

    2023

  43. [51]

    Shimobe, Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds , Geom

    H. Shimobe, Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds , Geom. Dedicata 197 (2018), 49–60

    2018

  44. [52]

    Streets, G

    J. Streets, G. Tian, A parabolic flow of pluriclosed metrics , Int. Math. Res. Not. 2010 (2010), 3101–3133

    2010

  45. [53]

    Vaisman, On locally and globally conformal Kähler manifolds , Trans

    I. Vaisman, On locally and globally conformal Kähler manifolds , Trans. Amer. Math. Soc. 262 (1980), 533–542

    1980

  46. [54]

    Verbitsky, Rational curves and special metrics on twistor spaces , Geom

    M. Verbitsky, Rational curves and special metrics on twistor spaces , Geom. Topol. 18 (2014), 897–909

    2014

  47. [55]

    Yang, On blow-ups and resolutions of Hermitian-symplectic and strongly Gauduchon metrics , Arch

    S. Yang, On blow-ups and resolutions of Hermitian-symplectic and strongly Gauduchon metrics , Arch. Math. 104 (2015), 441–450

    2015

  48. [56]

    S.-T. Yau, Q. Zhao, F. Zheng, On Strominger Kähler-like manifolds with degenerate torsion , arXiv:1908.05322v3

    1908 arXiv

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