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arxiv: 2605.03852 · v1 · submitted 2026-05-05 · 🧮 math.DG

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Holomorphically parallelizable solvmanifolds with special metrics and their deformations

Ettore Lo Giudice , Lapo Rubini , Adriano Tomassini
Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:11 UTC · model grok-4.3

classification 🧮 math.DG
keywords holomorphically parallelizable solvmanifoldsdeformationsstrong Kähler with torsion metricsbalanced metricsKuranishi spaceIwasawa manifoldNakamura manifoldnilmanifolds
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The pith

Deformations of the holomorphically parallelizable Nakamura manifold with non-left-invariant complex structures admit balanced metrics but no strong Kähler with torsion metrics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the persistence of special Hermitian metrics under deformations of complex structures on holomorphically parallelizable solvmanifolds, focusing on the Iwasawa manifold and the Nakamura manifold. It establishes that deformations of the Nakamura manifold producing non-left-invariant complex structures always allow balanced metrics yet prohibit strong Kähler with torsion metrics. The work further constructs the Kuranishi space for a four-dimensional holomorphically parallelizable solvmanifold to classify small deformations and their metric properties, while deriving existence results for metrics satisfying the conditions partial bar partial omega equals zero and partial bar partial omega squared equals zero on certain two-step nilpotent nilmanifolds. These findings clarify how metric existence can depend on whether a deformation preserves left-invariance or the underlying parallelizability.

Core claim

The paper shows that the class of deformations of the holomorphically parallelizable Nakamura manifold yielding a non-left-invariant complex structure admits a balanced metric but does not admit any strong Kähler with torsion metric. It investigates the existence of strong Kähler with torsion metrics along deformations of the Iwasawa manifold and of the holomorphically parallelizable Nakamura manifold, constructs the Kuranishi space of a 4-dimensional holomorphically parallelizable solvmanifold to study whether small deformations admit SKT metrics, and provides results on the existence of metrics satisfying partial bar partial omega equals zero and partial bar partial omega squared equals 0.

What carries the argument

The Kuranishi space of infinitesimal deformations of the complex structure on holomorphically parallelizable solvmanifolds, combined with explicit checks of the torsion condition partial bar partial omega equals zero that defines strong Kähler with torsion metrics.

If this is right

  • Deformations of the Iwasawa manifold admit SKT metrics only in certain directions within the deformation space.
  • Non-left-invariant deformations of the Nakamura manifold consistently support balanced metrics.
  • Small deformations inside the Kuranishi space of the four-dimensional solvmanifold admit SKT metrics only for some parameters.
  • On the considered two-step nilpotent nilmanifolds, metrics exist that simultaneously satisfy partial bar partial omega equals zero and partial bar partial omega squared equals zero.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The loss of left-invariance under deformation appears to be the mechanism that eliminates SKT metrics while preserving balanced ones.
  • The same deformation analysis could be applied to other classes of solvmanifolds to test whether the balanced-but-not-SKT pattern is general.
  • Explicit coordinate computations on low-dimensional examples would provide concrete test cases for the metric conditions.

Load-bearing premise

The solvmanifolds are holomorphically parallelizable so that their deformations and the resulting Hermitian metrics can be described explicitly using standard deformation theory.

What would settle it

An explicit construction of a non-left-invariant deformation of the Nakamura manifold together with a Hermitian metric omega satisfying partial bar partial omega equals zero would show that the non-existence claim for SKT metrics does not hold in that class.

read the original abstract

We investigate the existence of strong K\"ahler with torsion metrics along deformations of the Iwasawa manifold and of the holomorphically parallelizable Nakamura manifold. We also show that the class of deformations of the holomorphically parallelizable Nakamura manifold yielding a non-left-invariant complex structure admits a balanced metric but does not admit any strong K\"ahler with torsion metric. We then construct the Kuranishi space of a $4$-dimensional holomorphically parallelizable solvmanifold and study whether small deformations of such a manifold admit SKT metrics. Finally, we provide some results concerning the existence of metrics satisfying $\partial \bar{\partial} \omega = 0$, $\partial \bar{\partial} \omega^2 = 0$ on a particular class of $2$-step nilpotent nilmanifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates the existence of strong Kähler with torsion (SKT) metrics along deformations of the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold. It shows that deformations of the Nakamura manifold yielding non-left-invariant complex structures admit balanced metrics but no SKT metrics. The Kuranishi space of a 4-dimensional holomorphically parallelizable solvmanifold is constructed and small deformations are studied for the existence of SKT metrics. Results are also given on the existence of metrics satisfying ∂∂̄ω = 0 and ∂∂̄ω² = 0 for a class of 2-step nilpotent nilmanifolds.

Significance. If the results are correct, the paper supplies concrete examples distinguishing the existence of balanced metrics from the non-existence of SKT metrics on solvmanifolds with non-left-invariant complex structures. The explicit deformation analysis and Kuranishi space construction add to the deformation theory of special Hermitian metrics on homogeneous non-Kähler manifolds.

major comments (2)
  1. [Deformations of the holomorphically parallelizable Nakamura manifold] The central non-existence claim for SKT metrics on the non-left-invariant deformations of the Nakamura manifold (stated in the abstract and developed in the corresponding section) requires a global obstruction applying to arbitrary Hermitian metrics. The argument must be checked to ensure it does not rely on averaging or Nomizu-type reductions that presuppose left-invariance of the complex structure J, as such reductions fail once J is deformed to be non-left-invariant.
  2. [Kuranishi space construction] In the section constructing the Kuranishi space of the 4-dimensional solvmanifold, the criteria used to determine which small deformations admit SKT metrics should be stated explicitly (e.g., via the relevant cohomology classes or the ∂∂̄-lemma failure), together with the dimension of the space and the parameter range considered.
minor comments (2)
  1. [Abstract] The abstract refers to 'a particular class of 2-step nilpotent nilmanifolds' without further identification; adding a brief characterizing property or reference would aid readability.
  2. [Notation and preliminaries] Notation for the Hermitian form ω and the complex structure should be introduced uniformly before the deformation sections to avoid ambiguity when moving between invariant and non-invariant cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below. Where clarification is needed, we will revise the text accordingly in the next version.

read point-by-point responses

  1. Referee: The central non-existence claim for SKT metrics on the non-left-invariant deformations of the Nakamura manifold (stated in the abstract and developed in the corresponding section) requires a global obstruction applying to arbitrary Hermitian metrics. The argument must be checked to ensure it does not rely on averaging or Nomizu-type reductions that presuppose left-invariance of the complex structure J, as such reductions fail once J is deformed to be non-left-invariant.

    Authors: The non-existence of SKT metrics on these deformations is established via a global cohomological obstruction that holds for any Hermitian metric and does not invoke averaging over the group or any Nomizu-type theorem. The argument relies instead on the holomorphic parallelizability of the deformed manifold together with the explicit failure of the ∂∂̄-lemma on the underlying solvmanifold, which persists under the deformation to non-left-invariant complex structures. We will insert a short clarifying paragraph immediately after the statement of the theorem to emphasize that the obstruction is intrinsic and independent of left-invariance. revision: yes

  2. Referee: In the section constructing the Kuranishi space of the 4-dimensional solvmanifold, the criteria used to determine which small deformations admit SKT metrics should be stated explicitly (e.g., via the relevant cohomology classes or the ∂∂̄-lemma failure), together with the dimension of the space and the parameter range considered.

    Authors: We agree that the criteria and the geometric data of the Kuranishi space should be stated more explicitly. In the revised manuscript we will add a dedicated paragraph listing the precise conditions (vanishing of a certain class in H^{1,1} and the explicit failure of the ∂∂̄-lemma) that characterize the SKT locus. The Kuranishi space is two-dimensional, and we consider deformations inside a sufficiently small ball in the parameter space; these details will be recorded together with the explicit equations defining the SKT subset. revision: yes

Circularity Check

0 steps flagged

No circularity: standard deformation theory and explicit constructions without self-referential reductions

full rationale

The paper's claims rest on direct application of Kuranishi deformation theory, explicit metric constructions for balanced metrics, and standard obstructions (such as failure of the ∂∂̄-lemma or non-vanishing classes) for non-existence of SKT metrics. These are applied to deformations of holomorphically parallelizable solvmanifolds, including non-left-invariant cases on the Nakamura manifold. No equations or arguments reduce by construction to fitted parameters renamed as predictions, self-definitions, or load-bearing self-citations whose validity depends on the current paper. The central non-existence result for SKT metrics is presented as following from global obstructions independent of left-invariance, and balanced-metric existence is shown constructively. This is self-contained against external benchmarks in complex geometry, consistent with the reader's assessment of no circularity indicators in the abstract or setup.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract; the work relies on standard definitions and theorems from complex geometry and deformation theory.

pith-pipeline@v0.9.0 · 5443 in / 1156 out tokens · 115123 ms · 2026-05-07T13:11:55.519905+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

60 extracted references

  1. [1]

    Abbena and A

    E. Abbena and A. Grassi,Hermitian left invariant metrics on complex Lie groups and cosymplectic Hermitian mani- folds, Boll. Un. Mat. Ital. A (6)5(1986), no. 3, 371–379.↑2

    1986

  2. [2]

    Alessandrini,Proper modifications of generalizedp-K¨ ahler manifolds, J

    L. Alessandrini,Proper modifications of generalizedp-K¨ ahler manifolds, J. Geom. Anal.27(2017), no. 2, 947–967.↑4

    2017

  3. [3]

    Alessandrini and G

    L. Alessandrini and G. Bassanelli,Small deformations of a class of compact non-K¨ ahler manifolds, Proc. Amer. Math. Soc.109(1990), no. 4, 1059–1062.↑3 HOLOMORPHICALLY PARALLELIZABLE SOLVMANIFOLDS, SPECIAL METRICS AND DEFORMATIONS 21

    1990

  4. [4]

    ,Smooth proper modifications of compact K¨ ahler manifolds, Complex analysis (Wuppertal, 1991), 1991, pp. 1– 7.↑2

    1991

  5. [5]

    Alexandrov and S

    B. Alexandrov and S. Ivanov,Vanishing theorems on Hermitian manifolds, Differential Geom. Appl.14(2001), no. 3, 251–265.↑2

    2001

  6. [6]

    Angella,The cohomologies of the Iwasawa manifold and of its small deformations, The Journal of Geometric Analysis23(December 2013), 1355–1378.↑3

    D. Angella,The cohomologies of the Iwasawa manifold and of its small deformations, The Journal of Geometric Analysis23(December 2013), 1355–1378.↑3

    2013

  7. [7]

    Angella, M

    D. Angella, M. G. Franzini, and F. A. Rossi,Degree of non-K¨ ahlerianity for 6-dimensional nilmanifolds, Manuscripta Math.148(2015), no. 1-2, 177–211.↑3

    2015

  8. [8]

    Angella, V

    D. Angella, V. Guedj, and C. H. Lu,Plurisigned Hermitian metrics, Trans. Amer. Math. Soc.376(2023), no. 7, 4631–4659.↑2

    2023

  9. [9]

    Angella and A

    D. Angella and A. Tomassini,On cohomological decomposition of almost-complex manifolds and deformations, Journal of Symplectic Geometry9(2011), 403–428.↑3, 11

    2011

  10. [10]

    Math.192(2013), no

    ,On the∂ ∂-lemma and Bott-Chern cohomology, Invent. Math.192(2013), no. 1, 71–81.↑3

    2013

  11. [11]

    Angella and L

    D. Angella and L. Ugarte,On small deformations of balanced manifolds, Differential Geom. Appl.54(2017), 464–474. ↑3

    2017

  12. [12]

    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), no. 4, 681–699.↑1

    1989

  13. [13]

    Ciulic˘ a,A new class of non-K¨ ahler metrics, Complex Manifolds12(2025), no

    C. Ciulic˘ a,A new class of non-K¨ ahler metrics, Complex Manifolds12(2025), no. 1, Paper No. 20250014, 14.↑3, 8

    2025

  14. [14]

    Ciulic˘ a, A

    C. Ciulic˘ a, A. Otiman, and M. Stanciu,Special non-K¨ ahler metrics on Endo-Pajitnov manifolds, Ann. Mat. Pura Appl. (4)204(2025), no. 4, 1425–1441.↑3

    2025

  15. [15]

    Cordero, M

    L. Cordero, M. Fern´ andez, A. Gray, and L. Ugarte,Nilpotent complex structures on compact nilmanifolds, Rend. Circ. Mat. Palermo (2) Suppl.49(1997), 83–100.↑6

    1997

  16. [16]

    L. A. Cordero, M. Fern´ andez, A. Gray, and L. Ugarte,Compact nilmanifolds with nilpotent complex structures: Dol- beault cohomology, Trans. Amer. Math. Soc.352(2000), no. 12, 5405–5433.↑6

    2000

  17. [17]

    Ehresmann,Sur les espaces fibr´ es diff´ erentiables, C

    C. Ehresmann,Sur les espaces fibr´ es diff´ erentiables, C. R. Acad. Sci. Paris224(1947), 1611–1612.↑5

    1947

  18. [18]

    A. Fino, G. Grantcharov, and L. Vezzoni,Astheno-K¨ ahler and balanced structures on fibrations, Int. Math. Res. Not. IMRN22(2019).↑2

    2019

  19. [19]

    A. Fino, H. Kasuya, and L. Vezzoni,SKT and tamed symplectic structures on solvmanifolds, Tohoku Math. J. (2)67 (2015), no. 1, 19–37.↑2

    2015

  20. [20]

    A. Fino, M. Parton, and S. Salamon,Families of strong KT structures in six dimensions, Comment. Math. Helv.79 (2004), no. 2, 317–340.↑3, 7

    2004

  21. [21]

    Fino and A

    A. Fino and A. Tomassini,Blow-ups and resolutions of strong K¨ ahler with torsion metrics, Adv. Math.221(2009), no. 3, 914–935.↑2, 3, 11

    2009

  22. [22]

    ,On astheno-K¨ ahler metrics, J. Lond. Math. Soc. (2)83(2011), no. 2, 290–308.↑2, 3, 4, 8

    2011

  23. [23]

    Fino and L

    A. Fino and L. Vezzoni,Special Hermitian metrics on compact solvmanifolds, J. Geom. Phys.91(2015), 40–53.↑2

    2015

  24. [24]

    ,On the existence of balanced and SKT metrics on nilmanifolds, Proc. Amer. Math. Soc.144(2016), no. 6, 2455–2459.↑2, 12

    2016

  25. [25]

    M. G. Franzini,Deformazioni di variet` a bilanciate e loro propriet` a coomologiche, 2011. Tesi di Laurea Magistrale, Universit` a di Parma.↑13

    2011

  26. [26]

    Gauduchon,Fibr´ es hermitiens ` a endomorphisme de Ricci non n´ egatif, Bull

    P. Gauduchon,Fibr´ es hermitiens ` a endomorphisme de Ricci non n´ egatif, Bull. Soc. Math. France105(1977), no. 2, 113–140.↑1

    1977

  27. [27]

    Ann.267(1984), no

    ,La1-forme de torsion d’une vari´ et´ e hermitienne compacte, Math. Ann.267(1984), no. 4, 495–518.↑1

    1984

  28. [28]

    ,Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B (7)11(1997), no. 2, 257–288.↑2

    1997

  29. [29]

    Goto,Deformations of generalized complex and generalized K¨ ahler structures, J

    R. Goto,Deformations of generalized complex and generalized K¨ ahler structures, J. Differential Geom.84(2010), no. 3, 525–560.↑3

    2010

  30. [30]

    Gray and L

    A. Gray and L. M. Hervella,The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4)123(1980), 35–58.↑1

    1980

  31. [31]

    Guan and Q

    B. Guan and Q. Li,Complex Monge-Amp` ere equations and totally real submanifolds, Adv. Math.225(2010), no. 3, 1185–1223.↑2

    2010

  32. [32]

    Harvey and A

    R. Harvey and A. W. Knapp,Positive(p, p)forms, Wirtinger’ inequality, and currents, Value distribution theory (Proc. Tulane Univ. Program, Tulane Univ., New Orleans, La., 1972-1973), Part A, 1974, pp. 43–62.↑4

    1972

  33. [33]

    Hasegawa,Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds, Differential Geom

    K. Hasegawa,Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds, Differential Geom. Appl.28(2010), no. 2, 220–227.↑14

    2010

  34. [34]

    R. Hind, C. Medori, and A. Tomassini,Families of almost complex structures and transverse(p, p)-forms, J. Geom. Anal.33(2023), no. 10, Paper No. 334, 23.↑12

    2023

  35. [35]

    Huybrechts,Complex Geometry:An Introduction, Universitext, Springer Berlin Heidelberg, 2005.↑5

    D. Huybrechts,Complex Geometry:An Introduction, Universitext, Springer Berlin Heidelberg, 2005.↑5

    2005

  36. [36]

    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), no. 2, 221–254.↑1

    1993

  37. [37]

    Kasuya,de Rham and Dolbeault cohomology of solvmanifolds with local systems, Math

    H. Kasuya,de Rham and Dolbeault cohomology of solvmanifolds with local systems, Math. Res. Lett.21(2014), no. 4, 781–805.↑15

    2014

  38. [38]

    Global Anal

    ,Generalized deformations and holomorphic Poisson cohomology of solvmanifolds, Ann. Global Anal. Geom. 51(2017), no. 2, 155–177.↑15 22 ETTORE LO GIUDICE, LAPO RUBINI AND ADRIANO TOMASSINI

    2017

  39. [39]

    Kodaira,Complex Manifolds and Deformations of Complex Structures, Classics in Mathematics, Springer Berlin, Heidelberg, 2005.↑3, 5

    K. Kodaira,Complex Manifolds and Deformations of Complex Structures, Classics in Mathematics, Springer Berlin, Heidelberg, 2005.↑3, 5

    2005

  40. [40]

    Kodaira and J

    K. Kodaira and J. Morrow,Complex Manifolds, AMS Chelsea Publishing, 2006.↑5

    2006

  41. [41]

    Kodaira and D

    K. Kodaira and D. C. Spencer,On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2)71(1960), 43–76.↑3

    1960

  42. [42]

    Latorre and L

    A. Latorre and L. Ugarte,On non-K¨ ahler compact complex manifolds with balanced and astheno-K¨ ahler metrics, C. R. Math. Acad. Sci. Paris355(2017), no. 1, 90–93.↑2

    2017

  43. [43]

    A. I. Malcev,On a class of homogeneous spaces, Amer. Math. Soc. Translation Ser. 19(1962), 276–307.↑9

    1962

  44. [44]

    Matsuo and T

    K. Matsuo and T. Takahashi,On compact astheno-K¨ ahler manifolds, Colloq. Math.89(2001), no. 2, 213–221.↑2

    2001

  45. [45]

    M. L. Michelsohn,On the existence of special metrics in complex geometry, Acta Math.149(1982), no. 3-4, 261–295. ↑1, 2

    1982

  46. [46]

    Nakamura,Complex parallelisable manifolds and their small deformations, J

    I. Nakamura,Complex parallelisable manifolds and their small deformations, J. Differential Geometry10(1975), 85– 112.↑3, 4, 10, 12, 13, 14, 15

    1975

  47. [47]

    Piovani and T

    R. Piovani and T. Sferruzza,Deformations of Strong K¨ ahler with torsion metrics, Complex Manifolds8(2021), no. 1, 286–301.↑3, 8, 9

    2021

  48. [48]

    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), no. 3, 515–534.↑3

    2013

  49. [49]

    S. Rao, X. Wan, and Q. Zhao,On local stabilities ofp-K¨ ahler structures, Compos. Math.155(2019), no. 3, 455–483. ↑3

    2019

  50. [50]

    J.246(2022), 305–354.↑3

    ,Power series proofs for local stabilities of K¨ ahler and balanced structures with mild∂ ∂-lemma, Nagoya Math. J.246(2022), 305–354.↑3

    2022

  51. [51]

    Rao and Q

    S. Rao and Q. Zhao,Extension formulas and deformation invariance of Hodge numbers, C. R. Math. Acad. Sci. Paris 353(2015), no. 11, 979–984.↑4, 5, 6

    2015

  52. [52]

    ,Several special complex structures and their deformation properties, J. Geom. Anal.28(2018), no. 4, 2984– 3047.↑4, 5, 6

    2018

  53. [53]

    Sferruzza,Deformations of balanced metrics, Bull

    T. Sferruzza,Deformations of balanced metrics, Bull. Sci. Math.178(2022), Paper No. 103143, 23.↑3

    2022

  54. [54]

    1, Paper No

    ,Deformations of astheno-K¨ ahler metrics, Complex Manifolds10(2023), no. 1, Paper No. 20230102, 19.↑3, 4, 8

    2023

  55. [55]

    Sferruzza and A

    T. Sferruzza and A. Tomassini,On cohomological and formal properties of strong K¨ ahler with torsion and astheno- K¨ ahler metrics, Math. Z.304(2023), no. 4, Paper No. 55, 27.↑2, 4, 7

    2023

  56. [56]

    ,Bott-Chern formality and Massey products on strong K¨ ahler with torsion and K¨ ahler solvmanifolds, J. Geom. Anal.34(2024), no. 11.↑3, 4, 15, 19

    2024

  57. [57]

    Wang,Complex parallisable manifolds, Proceedings of the American Mathematical Society5(1954), 771–776

    H.-C. Wang,Complex parallisable manifolds, Proceedings of the American Mathematical Society5(1954), 771–776. ↑2

    1954

  58. [58]

    Wu,On the geometry of superstrings with torsion, ProQuest LLC, Ann Arbor, MI, 2006

    C.-C. Wu,On the geometry of superstrings with torsion, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Harvard University.↑3

    2006

  59. [59]

    Xiao,On strongly Gauduchon metrics of compact complex manifolds, J

    J. Xiao,On strongly Gauduchon metrics of compact complex manifolds, J. Geom. Anal.25(2015), no. 3, 2011–2027. ↑3, 12

    2015

  60. [60]

    Ulisse Dini

    T. Yamada,A construction of compact pseudo-K¨ ahler solvmanifolds with no K¨ ahler structures, Tsukuba J. Math.29 (2005), no. 1, 79–109.↑4 (Ettore Lo Giudice)Dipartimento di Scienze Matematiche, Fisiche e Informatiche Unit`a di Matematica e Infor- matica, Universit`a degli Studi di Parma, Parco Area delle Scienze 53/A, 43124, Parma, Italy &, Dipartimento ...

    2005