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K\"ahlerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures

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Pith's one-line read If the (2,0) part of the complexified real Strominger-Bismut-Ricci curvature vanishes and the second SB-Ricci curvature obeys a torsion-corrected non-positivity bound, then a compact Hermitian surface is Kähler.

desk verdict New Bismut-connection Kählerness criteria with a genuinely useful torsion identity, but the main theorems lean on an unproved imported identity and Section 6 has a real gap. read the letter →

arxiv 2510.06648 v4 pith:RLWHYH6P submitted 2025-10-08 math.DG math.CV

Liangdi Zhang This is my paper · ORCID
classification math.DGmath.CV MSC 53C55
keywords HermitiansurfacesStrominger-BismutconnectionBismut-RiccicurvatureKählersurfacetorsionChernnumberidentitiescompactcomplexnon-Kählergeometry
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 several Kähler criteria for compact Hermitian surfaces: under certain semi-definiteness conditions on the Ricci curvatures of the Strominger-Bismut connection, the surface must actually be Kähler, meaning its torsion vanishes. The central mechanism is a set of explicit identities that tie these Ricci curvatures to the anti-holomorphic piece of the torsion form ∂̄*ω. In the main theorem, vanishing (2,0) Ricci together with non-positivity of Ric^{SB(2)} + (7/2)√-1 ∂̄*ω∧∂*ω forces the squared norm of ∂̄∂̄*ω to be zero, which implies ∂ω = 0. These results reinterpret and extend the curvature–torsion identities previously discovered for the Levi-Civita connection, and if correct they rule out all non-Kähler compact complex surfaces under the stated hypotheses.

What carries the argument

The Strominger-Bismut connection is the unique Hermitian connection with totally skew-symmetric torsion; its real Ricci curvature, after complexification, splits into (2,0), (1,1), and (0,2) parts. The engine of the paper is a curvature–torsion identity (Lemma 3.4) expressing ∥∂̄∂̄*ω∥² + ∥Λ∂̄∂̄*ω∥² as a sum of pairings of the SB-Ricci curvatures with the nonnegative form √-1 ∂̄*ω∧∂*ω, together with squares of the (2,0)-Ricci part and torsion terms. Combining this identity with the assumed inequality converts the geometric curvature condition into an L² estimate that leaves no room for nonzero torsion. The Chern-number identities of Section 4 play the analogous role in the parallel-torsion an

What would settle it

Take a compact non-Kähler Hermitian surface with an explicitly given metric and numerically compute both sides of the quoted identity (3.32); any discrepancy disproves the main theorems. Alternatively, search for a compact Hermitian surface satisfying Ric^{SB,C}_{(2,0)} = 0 and Ric^{SB(2)} + (7/2)√-1 ∂̄*ω∧∂*ω ≤ 0 that is not Kähler; its existence would directly refute Theorem 1.1.

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Extended reading notes

Core claim

The paper's central claim is that on a compact Hermitian surface (M,ω), the combination Ric^{SB,C}_{(2,0)} = 0 and Ric^{SB(2)} + (7/2)√-1 ∂̄*ω∧∂*ω ≤ 0 forces the identity ∥∂̄∂̄*ω∥² + ∥Λ∂̄∂̄*ω − 3|∂̄*ω|²∥² ≤ 0, so ∂̄∂̄*ω = 0 and hence ∂ω = 0; therefore (M,ω) is Kähler. Variants with the third and fourth Strominger-Bismut-Ricci curvatures, and with Gauduchon metrics, relax the constant 7/2 to 3/2 or 5/2. A further theorem shows that if the Strominger-Bismut connection has parallel torsion, semi-definiteness of any of the natural SB-Ricci curvatures implies the surface is projective or Calabi-Yau.

Load-bearing premise

The whole argument leans on an L² identity that is quoted from another preprint without proof; a sign or coefficient error in that identity would invalidate the main theorems.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, any compact Hermitian surface with vanishing (2,0) part of the complexified real SB-Ricci curvature and with Ric^{SB(2)} + (7/2)√-1 ∂̄*ω∧∂*ω ≤ 0 is Kähler, so its underlying complex surface lies in the Kähler class.
  • Under the same vanishing (2,0) assumption, the alternative bounds of Theorem 1.2 on Ric^{SB(3)} + Ric^{SB(4)} or on Ric^{SB,C}_{(1,1)} plus its conjugate also force Kählerity.
  • When the metric is Gauduchon, the torsion-correction constant can be lowered to 3/2 or 5/2 and the conclusion still holds, giving stronger statements for the standard conformal class on compact complex surfaces.
  • If the Strominger-Bismut connection has parallel torsion, semi-definiteness of any of the four natural SB-Ricci curvatures forces the surface to be either projective or Calabi-Yau (a torus or a K3 surface).

Reading between the lines

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

  • The proof is essentially an L² gap argument, so the same mechanism might yield Kähler criteria in higher dimensions once an analogue of the curvature–torsion identity is established for Hermitian manifolds of arbitrary dimension.
  • The constant 7/2 in Theorem 1.1 is likely not optimal; parameterizing the torsion correction in the identity could reveal the sharp threshold beyond which non-Kähler metrics are possible.
  • A direct numerical check of the imported identity on an explicit compact non-Kähler Hermitian surface (one with an explicit metric) would settle the proof's reliance on the companion preprint independently of geometric intuition.
  • If these criteria hold, they provide a curvature-only obstruction to non-Kählerity, which could be useful for designing geometric flows that preserve the inequality and converge to a Kähler metric.
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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 / 5 minor

Summary. The paper establishes Kählerness criteria for compact Hermitian surfaces under semidefiniteness assumptions on natural Ricci curvatures of the Strominger–Bismut connection. The main theorems (1.1–1.4) assume that the (2,0)-component of the complexified real Bismut–Ricci curvature vanishes and that a suitable corrected (1,1)-Bismut–Ricci form is non-positive, e.g. Ric^{SB(2)} + (7/2)√−1 ∂̄*ω∧∂*ω ≤ 0. The proofs reduce the hypotheses via integral identities in §3 to the conclusion that a sum of squares involving ∂̄∂̄*ω is ≤ 0; hence ∂̄∂̄*ω=0 and, by (5.3), ∂ω=0. Section 4 derives Chern-number identities, used together with a parallel-torsion hypothesis in Theorem 1.5 to remove the (2,0)-vanishing condition and to classify the Kähler limit as projective or Calabi–Yau. Section 6 states boundedness variants with a constant a.

Significance. If the main identities are correct, the results are strong and geometrically natural: they give Bismut–Ricci analogues of Yang's Riemannian criteria and, conditional on the (2,0)-vanishing hypothesis, rule out non-Kähler compact complex surfaces under explicit semidefinite curvature inequalities. The paper's strengths are the explicit nature of the torsion/curvature identities (Lemma 3.1), the careful bookkeeping of coefficients, and the Bochner-type final argument, which makes the theorems falsifiable and the constants precise. The main correctness risk is the reliance of Lemma 3.4 on the imported identity (3.32) from the concurrent preprint [36]; the local checks I made of the subsequent algebra (e.g., (5.2)) are consistent, but the foundation must be independently verifiable in the present manuscript.

major comments (2)
  1. [Section 3, Lemma 3.4, Eq. (3.32)] This identity is the engine for Theorems 1.1–1.4, and it is quoted from the concurrent preprint [36] without proof. Equations (3.29)–(3.31), and therefore the decisive estimate (5.2), depend on its exact coefficients and signs. A missing or erroneous term would change the constants 7/2, 3/2, 6, 5 in Theorems 1.1–1.4. Please include a complete proof of (3.32), or a detailed derivation in the notation of §3, rather than sending the reader to [36].
  2. [Section 5, proof of Theorem 1.5, after (5.9)] The line 'It follows from (2.21) and (3.2) that R^{SB,C}_{ij} = T_iT_j = 0' is not immediate. From (3.2), parallel torsion gives ∇_j T_i = 0 and hence R^{SB,C}_{ij} = T_iT_j; the conclusion T_iT_j = 0 needs an additional argument, presumably from the Kähler-like symmetry in [41,42]. Without that, the reduction to (5.10) and the classification in Theorem 1.5 rest on an unstated fact. Please supply the missing argument or give the precise statement in [41,42].
minor comments (5)
  1. [Section 6, Eq. (6.1)] I do not see the claimed gap concerning vanishing of the denominator. The right-hand side of (6.1) is the global integral (|∂̄*ω|^4,1), not a pointwise denominator. If the integral is positive, compactness gives a finite constant a; if it is zero, then ∂̄*ω=0, M is Kähler, and the left side is also zero. The phrase 'throughout M' is confusing, but the argument is valid.
  2. [Global] There are several typos: the Section 3 heading reads 'Stromonger-Bismut'; the abstract has 'for achieve these results' instead of 'for achieving'; and (1.3) has 'of of'. These should be corrected.
  3. [Lemma 4.3] The sentence '(4.13) follows by (2.2)' appears to refer to the wrong equation; it should cite Lemma 2.2 or equations (2.19)–(2.20).
  4. [Theorems 1.2 and 1.4] The notation Ric^{SB,C}_{(1,1)} + Ric^{SB,C}_{(1,1)} in (1.6), (1.9), and (5.4) is hard to read in the typeset version; if the second term is the conjugate, please make the overline visible. If the two terms are indeed identical, the displayed redundancy should be explained.
  5. [Proof of Proposition 3.2, Eq. (3.17)] The equality ∥T_iT_j∥² = (|∂̄*ω|⁴,1) is used several times later; stating it explicitly after (3.17) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems follow from a Bochner-type identity argument; the load-bearing identity is imported from an external preprint, not from the paper's own assumptions.

full rationale

The central derivation chain is not circular. The hypotheses of Theorems 1.1–1.4 are curvature inequalities, and the conclusion ∂ω = 0 is obtained by showing that a sum of non-negative L2 norms is forced to be ≤ 0. For example, in Theorem 1.1, equation (5.1) follows algebraically from the hypothesis Ric^{SB,C}_{(2,0)}=0 and the identity (3.2), and equation (5.2) is then an algebraic manipulation of the quoted identity (3.29). The right-hand side is non-positive by (1.4), while the left-hand side is a sum of squares, so both vanish and (5.3) gives ∂ω=0. No parameter in the proof is fitted to the target conclusion, and the Chern number identities in Section 4 are derived from topological invariants rather than assumed to match the theorem. The key external input is Yang's identity (3.32), quoted in Lemma 3.4 and used to obtain (3.29)–(3.31). This is a load-bearing import from a concurrent preprint, and the paper does not prove (3.32) or give the reader a way to verify its signs and coefficients. However, this is a completeness and provenance concern, not a circularity: the cited identity is an identity about Hermitian surfaces, not the Kählerness conclusion, and the cited author is not an author of the present paper. Similarly, the Section 6 compactness assertion (6.1) with the constant a is an unproved existence claim when |∂̄*ω|^4 vanishes, but it is a hypothesis of the boundedness theorems, not a fitted prediction. No self-definitional reduction, no fitted-input-called-prediction, and no author-overlapping self-citation was found. The paper is therefore assigned a circularity score of 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's contribution is a clean algebraic reformulation of Yang's Levi-Civita program, built on four external inputs from concurrent preprints (Yang [35,36], Zhao–Zheng [41,42]) plus one ad hoc constant a in §6 whose existence is not established. No fitted parameters and no new physical or mathematical entities are introduced.

free parameters (1)
  • a (boundedness constant, §6) = unspecified / asserted finite
    Introduced in (6.1) to bound ∥R_{ij}+R_{ji}−3T_iT_j∥² by a·(|∂̄*ω|⁴,1); claimed to exist 'by compactness', but the ratio can blow up where the torsion vanishes, so existence and value are unproven. Theorems 6.1–6.3 carry a explicitly in their hypotheses (e.g., 'a + 3/2').
assumptions (5)
  • domain assumption Yang's identity (3.32): ∥∂̄∂̄*ω∥² + ∥Λ∂̄∂̄*ω∥² = 2(Ric^{(1,1)}, √-1∂̄*ω∧∂*ω) + 2∥Ric^{(2,0)}∥² + ½(|∂̄*ω|⁴,1), plus the Chern-number identity [36, Thm 7.5]
    Imported as a black box from arXiv:2508.11171 (Aug 2025, unrefereed). Lemmas 3.4 and 4.1–4.3, and hence Theorems 1.1–1.4, are algebraic reformulations of it; a coefficient or sign error there propagates to every main theorem.
  • domain assumption Lemma 2.2 (from [35]): the (1,1)-component of the complexified real Bismut-Ricci satisfies Ric^{SB,C}_{(1,1)} = Ric^{SB(3)}, with the conjugate equal to Ric^{SB(4)}
    Imported from Yang's preprint arXiv:2507.15002; it converts conditions on Ric^{SB,C}_{(1,1)} in the abstract into the (1,1)-form inequalities of Theorems 1.2 and 1.4 (via (5.4)).
  • ad hoc to paper Existence of a constant a satisfying (6.1)
    POSTULATED in §6 under the assertion 'by compactness'. Since R_{ij} = −∇_jT_i + T_iT_j is generically O(1) where T_i = 0, the pointwise ratio ∥R+...∥²/(|∂̄*ω|⁴) can diverge, so (6.1) is an extra unproven assumption. Theorems 6.1–6.3 are conditional on it.
  • domain assumption Parallel torsion ⟺ Kähler-like (Zhao–Zheng [41,42]); Kähler-like implies all four SB-Ricci forms coincide and the (2,0)-Ricci vanishes
    Imported from two preprints (2023, 2024). Used in Theorem 1.5 at (5.9)–(5.10). The paper never states 'Kähler-like ⟹ R^{2,0} = 0', yet (5.10) silently uses it, and (5.7) is used without the Gauduchon hypothesis (salvageable via parallel torsion, but unstated).
  • standard math Enriques–Kodaira classification of compact complex surfaces and the Kodaira embedding theorem
    Background for the conclusions of Theorem 1.5 ('projective or Calabi-Yau'). Standard, well-established results.

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Cite this review

Pith. "Pith review of K\"ahlerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures." pith.science (2026). https://pith.science/paper/RLWHYH6P

@misc{pith2026251006648,
  author       = {Pith},
  title        = {Pith review of: K\"ahlerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLWHYH6P}},
  note         = {Machine review of arXiv:2510.06648}
}
read the original abstract

We prove several K\"ahlerness criteria for compact Hermitian surfaces under semi-definiteness assumptions on natural Ricci curvatures of the Strominger-Bismut connection. The key tools for proving these results are explicit identities relating these Ricci curvatures to the torsion of the Strominger-Bismut connection, together with corresponding Chern number identities on compact Hermitian surfaces. The results may be viewed as Strominger-Bismut analogues and reformulations of Yang's K\"ahlerness criteria for compact complex surfaces.

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Forward citations

Cited by 1 Pith paper

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

  1. First eigenvalue estimates on complete balanced Hermitian manifolds

    math.DG 2025-11 conditional novelty 6.5 of 10

    On complete balanced Hermitian manifolds, curvature lower bounds for the Strominger–Bismut connection imply eigenvalue lower bounds of Lichnerowicz–Obata, Li–Yau, and Zhong–Yang type.

Reference graph

Works this paper leans on

43 extracted references · 3 linked inside Pith · cited by 1 Pith paper

  1. [36]

    Yang, Chern number identities on compact complex surface and applications, arXiv:2508.11171v1 [math.DG]

    X. Yang, Chern number identities on compact complex surface and applications, arXiv:2508.11171v1 [math.DG]

    arXiv

  2. [1]

    Andrada, R

    A. Andrada, R. Villacampa, Bismut connection on Vaisman manifolds, Math. Z. 302 (2022), 1091-1126

    2022

  3. [2]

    Angella, A

    D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with K¨ ahler-like curvatures, Comm. Anal. Geom. 30(5) (2022), 961-1006

    2022

  4. [3]

    Apostolov, G

    V. Apostolov, G. Barbaro, K. H. Lee, J. Streets, Rigidity results for non-K¨ ahler Calabi-Yau geometries on threefolds, arXiv:2408.09648v4 [math.DG]

    arXiv

  5. [4]

    Barth, K

    W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, 2nd ed., Springer, Berlin, (2004)

    2004

  6. [5]

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

    2000

  7. [6]

    J. M. Bismut, A local index theorem for non-K¨ ahler manifolds, Math. Ann. 284(4) (1989), 681-699

    1989

  8. [7]

    Boling, Homogeneous solutions of pluriclosed flow on closed complex surfaces, J

    J. Boling, Homogeneous solutions of pluriclosed flow on closed complex surfaces, J. Geom. Anal. 26 (2016) 2130-2154

    2016

Show all 43 references
  1. [8]

    Edwards, The Chern-Ricci flow on primary Hopf surfaces, Math

    G. Edwards, The Chern-Ricci flow on primary Hopf surfaces, Math. Z. 299 (2021), 1689-1702

    2021

  2. [9]

    Enriques, Federigo Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di generep 1 = 1, Atti

    F. Enriques, Federigo Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di generep 1 = 1, Atti. Acc. Lincei V Ser. 23 (1914)

    1914

  3. [10]

    Enriques, Le Superficie Algebriche, Nicola Zanichelli, Bologna, (1949)

    F. Enriques, Le Superficie Algebriche, Nicola Zanichelli, Bologna, (1949)

    1949

  4. [11]

    A. Fino, A. Otal, L. Ugarte, Six-dimensional solvmanifolds with holomorphically trivial canonical bundle, Int. Math. Res. Not. IMRN 2015(24) (2015), 13757-13799

    2015

  5. [12]

    A. Fino, N. Tardini, L. Vezzoni, Pluriclosed and Strominger K¨ ahler-like metrics com- patible with abelian complex structures, Bull. London Math. Soc. 54(5) (2022), 1862- 1872

    2022

  6. [13]

    A. Fino, L. Ugarte, On generalized Gauduchon metrics, Proc. Edinb. Math. Soc. (2) 56(3) (2013), 733-753

    2013

  7. [14]

    Garcia-Fernandez, J

    M. Garcia-Fernandez, J. Jordan, J. Streets, Non-K¨ ahler Calabi-Yau geometry and pluriclosed flow, J. Math. Pures Appl. 177 (2023), 329-367. 24

    2023

  8. [15]

    Gauduchon, S

    P. Gauduchon, S. Ivanov, Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension 4, Math. Z. 226(2) (1997) 317-326

    1997

  9. [16]

    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-B 285 (1977), A387-A390

    1977

  10. [17]

    Kodaira, On the structure of compact complex analytic surfaces, I, Amer

    K. Kodaira, On the structure of compact complex analytic surfaces, I, Amer. J. Math. 86(4) (1964), 751-798

    1964

  11. [18]

    Kodaira, On the structure of compact complex analytic surfaces, II, Amer

    K. Kodaira, On the structure of compact complex analytic surfaces, II, Amer. J. Math. 88(3) (1966), 682-721

    1966

  12. [19]

    Kodaira, On the structure of compact complex analytic surfaces, III, Amer

    K. Kodaira, On the structure of compact complex analytic surfaces, III, Amer. J. Math. 90(1) (1968), 55-83

    1968

  13. [20]

    Kodaira, On the structure of compact complex analytic surfaces, IV, Amer

    K. Kodaira, On the structure of compact complex analytic surfaces, IV, Amer. J. Math. 90(4) (1968), 1048-1066

    1968

  14. [21]

    T. J. Li, W. Zhang, Comparing tamed and compatible symplectic cones and coho- mological properties of almost complex manifolds, Comm. Anal. Geom. 17(4) (2009), 651-683

    2009

  15. [22]

    K. Liu, X. Yang, Geometry of Hermitian manifolds, Internat. J. Math. 23(6) (2012), 1250055

    2012

  16. [23]

    K. Liu, X. Yang, Ricci curvatures on Hermitian manifolds, T. Am. Math. Soc. 369(7) (2017), 5157-5196

    2017

  17. [24]

    Streets, Pluriclosed flow and the geometrization of complex surfaces, in: Geometric Analysis, Progress in Mathematics, 333

    J. Streets, Pluriclosed flow and the geometrization of complex surfaces, in: Geometric Analysis, Progress in Mathematics, 333. Birkh¨ auser, Cham, (2020)

    2020

  18. [25]

    Streets, G

    J. Streets, G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Notices 16 (2010), 3101-3133

    2010

  19. [26]

    Streets, G

    J. Streets, G. Tian, Regularity results for pluriclosed flow, Geom. Topol. 17(4) (2013), 2389-2429

    2013

  20. [27]

    Strominger, Superstrings with torsion, Nuclear Phys

    A. Strominger, Superstrings with torsion, Nuclear Phys. B 274(2) (1986), 253-284

    1986

  21. [28]

    Tosatti, B

    V. Tosatti, B. Weinkove, The Chern-Ricci flow on complex surfaces, Compos. Math. 149(12) (2013), 2101-2138

    2013

  22. [29]

    Tosatti, B

    V. Tosatti, B. Weinkove, On the evolution of a Hermitian metric by its Chern-Ricci form, J. Differential Geom. 99(1) (2015), 125-163

    2015

  23. [30]

    Tosatti, B

    V. Tosatti, B. Weinkove, X. Yang, Collapsing of the Chern-Ricci flow on elliptic surfaces, Math. Ann. 362(3-4) (2015), 1223-1271

    2015

  24. [31]

    J. Wang, X. Yang, Curvatures of real connections on Hermitian manifolds, Pac. J. Math. 337(2) (2025), 365-391

    2025

  25. [32]

    Q. Wang, B. Yang, F. Zheng, On Bismut flat manifolds, Trans. Amer. Math. Soc. 373 (2020), 5747-5772

    2020

  26. [33]

    B. Yang, F. Zheng, On curvature tensors of Hermitian manifolds, Comm. Anal. Geom. 26(5) (2018), 1195-1222

    2018

  27. [34]

    B. Yang, F. Zheng, On compact Hermitian manifolds with flat Gauduchon connec- tions, Acta Math. Sin. (Engl. Ser.) 34(8) (2018), 1259-1268

    2018

  28. [35]

    Yang, Comparison theorems in Hermitian geometry I, arXiv:2507.15002v1 [math.DG]

    X. Yang, Comparison theorems in Hermitian geometry I, arXiv:2507.15002v1 [math.DG]

    arXiv

  29. [37]

    S. T. Yau, Q. Zhao, F. Zheng, On Strominger K¨ ahler-like manifolds with degenerate torsion, Trans. Amer. Math. Soc. 376(5) (2023), 3063-3085

    2023

  30. [38]

    Ye, Pluriclosed flow and Hermitian-symplectic structures, J

    Y. Ye, Pluriclosed flow and Hermitian-symplectic structures, J. Geom. Anal. 34 (2024), 110

    2024

  31. [39]

    Ye, Bismut Einstein metrics on compact complex manifolds, J

    Y. Ye, Bismut Einstein metrics on compact complex manifolds, J. Fuct. Anal. 288 (2025), 110805

    2025

  32. [40]

    Q. Zhao, F. Zheng, Complex nilmanifolds and K¨ ahler-like connections, J. Geom. Anal. 146 (2019), 103512. 25

    2019

  33. [41]

    Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics, J. reine angew. Math. 796 (2023), 245-267

    2023

  34. [42]

    Q. Zhao, F. Zheng, Curvature characterization of Hermitian manifolds with Bismut parallel torsion, arXiv:2407.10497v1 [math.DG]

    arXiv

  35. [43]

    Q. Zhao, F. Zheng, On balanced Hermitian threefolds with parallel Bismut torsion, arXiv:2506.15141v1 [math.DG]. (Liangdi Zhang) Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, P. R. China Email address:ldzhang91@163.com

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