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Hodge decomposition and Hard Lefschetz Condition on almost K\"{a}hler manifolds

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Pith's one-line read If the Nijenhuis tensor is spectrally small, almost Kähler manifolds regain Hodge decomposition and the Hard Lefschetz Condition, and in real dimension four the structure is forced to be Kähler.

desk verdict The spectral-condition proofs check out, but the paper's advertised non-integrable regime may be empty, since no example reaches the c>20 threshold. read the letter →

arxiv 2506.06402 v1 pith:Z5NSTAEK submitted 2025-06-06 math.DG

Teng Huang , Weiwei Wang This is my paper
classification math.DG MSC 53C5553D0558A1432Q60
keywords HodgedecompositionalmostKählermanifoldsharmonicformsNijenhuistensorHardLefschetzConditioncomplexC∞-pure-and-fullspectralgapfour-manifoldrigidity
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 paper asks when the Hodge-theoretic structure of Kähler manifolds survives on almost Kähler manifolds — symplectic manifolds with a compatible almost complex structure that need not be integrable. It establishes that if the non-integrability, measured through the Nijenhuis tensor, is small in a precise spectral sense — namely $X\in M(k,c)$ for a universal constant with $c>20$ explicit — then $H^k_d(X) = \oplus_{p+q=k} H^{p,q}_d(X)$, the harmonic $k$-forms split by type. From this decomposition the paper derives that $J$ is complex $C^\infty$-pure-and-full in degree $k$, and that when $X\in\cap_{k=1}^{n-1}fM(k,\tilde c)$, the Hard Lefschetz Condition holds on $H^\bullet_d$. In real dimension four, the same spectral smallness with $b^+_2\ge 2$ forces $J$ to be integrable, so the ambient manifold is actually Kähler.

What carries the argument

The central object is the spectral class $M(k,c)$: closed almost Kähler manifolds for which the ratio inequality $\langle(\Delta_{\bar\partial}+\Delta_\mu)\alpha,\alpha\rangle \ge c\langle(\Delta_\mu+\Delta_{\bar\mu})\alpha,\alpha\rangle$ holds on the $L^2$-orthogonal complement of $\oplus_{p+q=k}H^{p,q}_d$. The operator $\Delta_\mu+\Delta_{\bar\mu}$ measures the Nijenhuis tensor and vanishes exactly when $J$ is integrable, so the ratio quantifies how non-integrable $J$ is. The proof engine is Proposition 3.4, an AM-GM estimate on the commutator terms supplied by the almost Kähler identities, giving $\langle\Delta_d\alpha,\alpha\rangle \ge \tfrac43(c-20)\langle(\Delta_\mu+\Delta_{\bar\mu})\beta,\beta\rangle$; once $c>20$, every harmonic form's component orthogonal to the pure spaces has zero Nijenhuis energy and vanishes by Lemma 3.6. For the Hard Lefschetz half, the companion class $fM(k,\tilde c)$ uses $\Delta_{\bar\partial+\mu}=\tfrac14(\Delta_d+\Delta_{d\Lambda})$, and Lemma 5.1 bounds $\|d\Lambda\alpha\|^2+\|(d\Lambda)^*\alpha\|^2$ in terms of Nijenhuis terms, yielding the explicit thresholds $\tilde c>2$ for $k=1$ and $\tilde c>4$ for $k\ge 2$. Theorem 1.7 completes the picture by combining the Hodge decomposition with the four-dimensional vanishing of $H^{2,0}_d$ and $H^{0,2}_d$ for non-integrable almost complex structures.

What would settle it

A concrete computation to attempt: on the Thurston-type compact 4-manifold of Example 5.4, evaluate the ratio $\langle(\Delta_{\bar\partial}+\Delta_\mu)\alpha,\alpha\rangle / \langle(\Delta_\mu+\Delta_{\bar\mu})\alpha,\alpha\rangle$ and search for a degree-$k$ form orthogonal to $\oplus_{p+q=k}H^{p,q}_d$ with ratio exceeding $20$ while $H^k_d$ still fails to equal $\oplus_{p+q=k}H^{p,q}_d$; such a form would refute Theorem 1.1. Checking the same manifold for a non-integrable $J$ with $b^+_2\ge 2$ satisfying the $M(2,c)$ inequality would test the rigidity of Theorem 1.7.

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Extended reading notes

Core claim

The paper's central claim is that a spectral smallness condition on the non-integrability of $J$ recovers the two signature properties of Kähler Hodge theory on the harmonic spaces. Precisely, there is a universal constant $c$ — with $c>20$ as the explicit sufficient value from inequality (3.4) — such that if a closed $2n$-dimensional almost Kähler manifold $(X,J,\omega)$ lies in $M(k,c)$, meaning that $\langle(\Delta_{\bar\partial}+\Delta_\mu)\alpha,\alpha\rangle \ge c\langle(\Delta_\mu+\Delta_{\bar\mu})\alpha,\alpha\rangle$ for every $k$-form $\alpha$ orthogonal to $\oplus_{p+q=k}H^{p,q}_d$, then $H^k_d(X)=\oplus_{p+q=k}H^{p,q}_d(X)$. Under the same hypothesis $J$ is complex $C^\infty$-pure-and-full at stage $k$. For the Hard Lefschetz half, the paper introduces the companion class $fM(k,\tilde c)$ built from $\Delta_{\bar\partial+\mu}=\tfrac14(\Delta_d+\Delta_{d\Lambda})$; membership in $\cap_{k=1}^{n-1}fM(k,\tilde c)$ gives $H^k_d=H^k_{d\Lambda}$, and hence the maps $L^{n-k}:H^k_d\to H^{2n-k}_d$ are isomorphisms. Finally, in dimension four with $b^+_2\ge 2$, membership in $M(2,c)$ forces $J$ to be integrable: for a non-integrable 4-dimensional almost complex structure $H^{2,0}_d=H^{0,2}_d=\{0\}$, which together with the Hodge decomposition would force $b^+_2=1$, contradicting $b^+_2\ge 2$.

Load-bearing premise

The load-bearing premise is the spectral hypothesis $X\in M(k,c)$ (or $X\in fM(k,\tilde c)$): the $\bar\partial$-plus-$\mu$ Laplacian must dominate the Nijenhuis-tensor Laplacian by a fixed constant on forms orthogonal to the very pure harmonic spaces whose decomposition the theorem is trying to prove, so the condition is only checkable once a Hodge decomposition is essentially known, and the paper does not exhibit a non-integrable example meeting the $c>20$ threshold or tie the ratio to a concrete pointwise bound on the Nijenhuis tensor.

Editorial extensions

If this is right

  • If $X\in M(k,c)$ with the universal constant, then $H^k_d(X)=\oplus_{p+q=k}H^{p,q}_d(X)$, so the Betti number $b_k$ equals $\sum_{p+q=k}h^{p,q}$.
  • Under the same hypothesis, $J$ is complex $C^\infty$-pure-and-full in $k$-stage: every class in $H^k_{dR}(X)$ has a representative of pure bi-degree, and the pure subspaces intersect trivially.
  • If $X\in\cap_{k=1}^{n-1}fM(k,\tilde c)$, then $H^k_d(X)=H^k_{d\Lambda}(X)$ and the Hard Lefschetz maps $L^{n-k}:H^k_d(X)\to H^{2n-k}_d(X)$ are isomorphisms; the ordinary Hard Lefschetz Condition on de Rham cohomology follows.
  • In dimension four, a closed almost Kähler manifold with $b^+_2(X)\ge 2$ and $X\in M(2,c)$ must have $J$ integrable, so a Kähler metric exists.
  • The sufficient constants are explicit: $c>20$ for the Hodge decomposition, and $\tilde c>2$ for $k=1$ or $\tilde c>4$ for $k\ge 2$ for the Lefschetz statement; Corollary 3.7 also bounds the spectral gap $\lambda_{2k+1}$ by a curvature constant when $b_{2k+1}$ is odd.

Reading between the lines

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

  • A testable extension is to replace the abstract spectral classes by geometric hypotheses: uniform $L^2$ or pointwise bounds on the Nijenhuis tensor (or on $\nabla J$) might imply the ratio inequality, making the theorems applicable without first knowing the pure harmonic spaces.
  • The explicit thresholds invite sharpness analysis: the Thurston nilmanifold of Example 5.4 sits exactly at the HLC threshold $\tilde c=2$ in degree one, so it would be informative to build families that approach $c=20$ in higher degrees and test whether the Hodge decomposition is lost exactly at the threshold.
  • The four-dimensional rigidity suggests a broader principle: for closed symplectic 4-manifolds with $b^+_2\ge 2$, several notions of 'small Nijenhuis tensor' should force integrability; the paper leaves open whether the spectral membership $M(2,c)$ is equivalent to a more geometric smallness condition, such as smallness in $L^1$ of the type studied for the Calabi-Yau equation.
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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

4 major / 6 minor

Summary. The paper studies harmonic forms on closed almost Kähler manifolds. It introduces two families M(k,c) and fM(k,c̃) of almost Kähler manifolds defined by spectral ratio inequalities involving the Laplacians Δ∂̄+Δµ and Δµ+Δ¯µ on the L2-orthogonal complement of certain harmonic subspaces. The main results are: (1) Theorem 1.1, if X belongs to M(k,c) with a uniform c>20, then the space of d-harmonic k-forms splits as a direct sum of (p,q)-harmonic spaces; (2) Theorem 1.4, an analogous condition in fM(k,c̃) forces H^k_d = H^k_{dΛ}, which yields the Hard Lefschetz condition on harmonic forms; (3) Theorem 1.7, in dimension 4 with b+_2 ≥ 2, membership in M(2,c) forces J to be integrable. The paper also establishes complex-C∞-pure-and-full properties and gives an example of a non-integrable Thurston manifold satisfying fM(1,2).

Significance. The conditional results are new and potentially significant: they give the first explicit spectral criteria that guarantee Hodge decomposition and HLC on the harmonic spaces of an almost Kähler manifold, with a quantitative threshold (c>20). Proposition 3.4's estimate is correct and the constant is explicit. The rigidity Theorem 1.7 is a novel conditional statement linking a spectral ratio to integrability. The Thurston example is a useful calibration for the constants in the fM family. However, the hypotheses are formulated in terms of the very harmonic spaces that the theorems aim to characterize, and no non-integrable example is provided for the large constants required by the main theorems; this limits the present significance.

major comments (4)
  1. [Section 5.2, Propositions 5.6 and 5.9] The proof shows only that no nonzero α in the complement satisfies equality in the ratio with c=1/2. On the infinite-dimensional complement, the infimum of the Rayleigh quotient need not be attained; therefore the conclusion that there exists a uniform c>1/2 does not follow. This invalidates the stated propositions and removes the only quantitative evidence in the paper that the M(k,c) condition is satisfiable for any c>1/2.
  2. [Section 1 (Theorems 1.1, 1.4, 1.7) and Section 5] The paper gives no example of a non-integrable almost Kähler manifold satisfying the spectral inequalities with the constants required by the theorems (c>20 for M(k,c), c̃>4 for fM(k,c̃)). Example 5.4 reaches only fM(1,2). Since the M(k,c) and fM(k,c̃) conditions are defined using the spaces ⊕H^{p,q}_d and H^k_{∂̄+µ} that are the targets of the conclusions, the hypotheses are not independently checkable and could be vacuous in the non-integrable regime. The authors should either construct such an example (e.g., by showing that sufficiently small Nijenhuis tensor implies membership with arbitrarily large c), or state explicitly that non-emptiness is open, and adjust the advertised claims accordingly.
  3. [Theorem 5.2, proof] The inequality "4c̃⟨(Δµ+Δ¯µ)α_J, α_J⟩ ≤ ∥dα_J∥²+∥d*α_J∥²" is obtained by applying the fM(k,c̃) condition, but this requires two unstated facts: (i) ∥dJα∥²+∥d*Jα∥² = ∥dΛα∥²+∥dΛ*α∥² for α∈H^k_d, and (ii) ⟨(Δµ+Δ¯µ)Jα,Jα⟩ = ⟨(Δµ+Δ¯µ)α,α⟩. The second is a pointwise statement about the isometry J and the type-shifting operators µ, ¯µ; the first is a non-obvious identity that should be proved. Without these, the proof is incomplete.
  4. [Theorem 3.13, proof] The step "there is a ∆d-harmonic (p,q)-form α^{p,q}_h such that α^{p,q} = α^{p,q}_h + d(exact)" does not follow from the preceding decomposition, because the harmonic projection of a (p,q)-form under Proposition 3.2 need not be of pure bi-degree (p,q) unless one uses the assumption H^k_d = ⊕H^{p,q}_d together with an orthogonality argument that is not written. The proof of surjectivity of H^{p,q}_d → H^{p,q} should be made explicit.
minor comments (6)
  1. [Section 1, definition of M(k,c)] The inequality is missing an opening angle bracket: it should read "⟨(Δ∂̄+Δµ)α,α⟩ ≥ c⟨(Δµ+Δ¯µ)α,α⟩".
  2. [Throughout] Typographical errors: "Nigenhuis" should be "Nijenhuis", "papproach" should be "approach", "symplectic symplectic" is duplicated, and "Demaill" in the references should be "Demailly".
  3. [Example 5.4] The statement "∥µ∥ = 1/4" is ambiguous; it should specify whether this is the operator norm or the pointwise norm of the Nijenhuis tensor.
  4. [Proposition 5.9] In the final sentence, the third inclusion should be "α2,0 ∈ H2,0_d" rather than repeating "α0,2 ∈ H0,2_d".
  5. [Theorems 1.1 and 1.4] The statements do not provide the explicit constants; the reader must infer c>20 and c̃>4 from Sections 3 and 5. It would improve readability to state these thresholds in the theorems.
  6. [Lemma 5.1] The notation "4Re(J^{-1}⟨µ+¯µ)Jα, dΛ*α⟩" is confusing; it should be written as "4Re⟨J^{-1}(µ+¯µ)Jα, dΛ*α⟩".

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorems are genuine conditional implications from explicit spectral-gap hypotheses that do not assert their own conclusions.

full rationale

The paper's central results are conditionals of the form: if a spectral ratio exceeds a uniform constant, then a Hodge-theoretic equality follows. The family M(k,c) is defined by requiring ⟨(Δ∂̄+Δμ)α,α⟩ ≥ c⟨(Δμ+Δμ̄)α,α⟩ on the L²-orthogonal complement of ⊕_{p+q=k}H^{p,q}_d. This subspace is well-defined independently of the conclusion H^k_d = ⊕_{p+q=k}H^{p,q}_d: H^{p,q}_d is the kernel of Δd restricted to Ω^{p,q}, computable without knowing H^k_d. The hypothesis is a coercivity condition on the complement, not an assertion of the equality it helps prove. The proof of Theorem 1.1 is a genuine estimate (Proposition 3.4, yielding the explicit threshold c>20 in equation (3.4)) followed by a kernel argument (Lemma 3.6); it is not a restatement of the definition. Similarly, fM(k,c̃) is defined on the complement of H^k_{∂̄+μ}, which is the target space H^k_d∩H^k_dΛ of Theorem 1.4; again this is a spectral-gap assumption, not a circular equivalence. Theorem 1.7 combines Theorem 1.1 with an external vanishing result of Cirici–Wilson ([6, Lemma 5.6]) and the Lefschetz decomposition; no fitted parameter is renamed as a prediction. The self-citations [15] and [27] support auxiliary identities in Section 4, but those results are also proved in the text or are standard in the surrounding literature, and they are not load-bearing for the main Hodge-decomposition theorem. The paper's real weakness is external rather than circular: no non-integrable manifold satisfying M(k,20) is exhibited, and Proposition 5.6 establishes only a manifold-dependent constant c>1/2, not a uniform gap approaching 20. Those are matters of domain of applicability, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; all constants in the proofs are derived explicitly. The central claim rests on standard elliptic Hodge theory, the Cirici-Wilson almost Kähler identities, the Lefschetz decomposition, Poincaré duality, the Newlander-Nirenberg theorem, and a pointwise curvature bound on ∇J from the literature. No new geometric entities are introduced.

assumptions (6)
  • standard math Closed manifold Hodge theory: Ω^k = ker Δ ⊕ Im Δ for self-adjoint elliptic operators, and H^k_d ≅ H^k_{dR}.
    Used throughout Sections 3 and 4 for decompositions of k-forms and the identification of harmonic spaces with cohomology.
  • domain assumption Cirici-Wilson algebraic identities for almost Kähler manifolds (Proposition 2.4-2.6): commutator identities and Laplacian relations.
    Cited from [6,7]; these identities underlie all Laplacian decompositions and bracket estimates in Propositions 3.4, 4.9, and Lemma 5.1.
  • standard math Lefschetz decomposition of forms and Poincaré duality for closed manifolds.
    Used in Section 4 for the HLC equivalences and in Theorem 1.7 to compute b+_2 = 1 + 2h^{2,0}.
  • standard math Newlander-Nirenberg theorem: J integrable if and only if the Nijenhuis tensor vanishes, equivalently μ = 0.
    Invoked in Lemma 2.1 and in the rigidity theorem 1.7 to conclude integrability from the vanishing of H^{2,0}_d.
  • domain assumption Pointwise bound |∇J|² ≤ C(n)∥Rm(g)∥_{C^0} (Lemma 2.3).
    Cited from [31,22,23]; used in Corollary 3.7 to convert spectral constants into curvature bounds.
  • domain assumption Hodge cancellation statements: H^{1,0}_{∂̄} = H^{1,0}_d and H^{0,1}_∂ = H^{0,1}_d for 1-forms on almost Kähler manifolds.
    Used in Proposition 5.6; attributed to [6, Corollary 4.6] and Lemma 5.3.

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Pith. "Pith review of Hodge decomposition and Hard Lefschetz Condition on almost K\"{a}hler manifolds." pith.science (2026). https://pith.science/paper/Z5NSTAEK

@misc{pith2026250606402,
  author       = {Pith},
  title        = {Pith review of: Hodge decomposition and Hard Lefschetz Condition on almost K\"ahler manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5NSTAEK}},
  note         = {Machine review of arXiv:2506.06402}
}
abstract

In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost K\"{a}hler manifold $(X, J,\omega)$. We show that if the almost complex structure $J$ on the almost K\"{a}hler manifold $X$ is not too non-integrable in some sense, then the spaces $\mathcal{H}^{\bullet}_{d}$ have the Hodge decomposition $\mathcal{H}^{k}_{d}=\oplus_{p+q=k}\mathcal{H}^{p,q}_{d}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\infty}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on $\mathcal{H}^{\bullet}_{d}$ is satisfied. Moreover, we can prove a rigidity result for the closed $4$-dimensional almost K\"{a}hler manifold with $b^{+}_{2}(X)\geq2$.

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Works this paper leans on

32 extracted references · 31 canonical work pages

  1. [1]

    Angella, D., Tomassini, A., On cohomological decomposition of almost-complex manifolds and deformations. J. Sym- plectic Geom. 9 (2011), 403–428

    work page 2011

  2. [2]

    Angella, D., Tomassini, A., Inequalities `a la Fr¨ohlicher and cohomological decompositions. J. Noncommut. Geom. 9(2) (2015), 505–542

    work page 2015

  3. [3]

    Compact Complex Surfaces

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

    work page 2004

  4. [4]

    Differential Geom

    Brylinski, J.-L., A differential complex for Poisson manifolds.J. Differential Geom. 28 (1988), 93–114

    work page 1988

  5. [5]

    Chern, S.-S., On curvature and characteristic classes of a Riemann manifold. Abh. Math. Sem. Univ. Hamburg 20 (1955), 117–126

    work page 1955

  6. [6]

    Cirici, J., Wilson, S.O., Topological and geometric aspects of almost K¨ahler manifolds via harmonic theory. Sel. Math. New Ser. 26, 35 (2020)

    work page 2020

  7. [7]

    Cirici, J., Wilson, S.O., Dolbeault cohomology for almost K¨ahler manifolds Adv. Math. 391 (2021), 107970

    work page 2021

  8. [8]

    de Bartolomeis, P., Tomassini, A., On formality of some symplectic manifolds. Int. Math. Res. Not. 24 (2001), 1287– 1314

    work page 2001

Show all 32 references
  1. [9]

    Deligne, P., Griffiths, P., Morgan, J., Sullivan, D., Real Homotopy theory of K¨ahler Manifolds. Invent. Math. 29 (1975), 245–274

    1975

  2. [10]

    Complex analytic and differential geometry

    Demaill, J.P. Complex analytic and differential geometry. Grenoble: Universit ´e de Grenoble I, 1997

    1997

  3. [11]

    Draghici, T., Li, T.J., Zhang, W., Symplectic Forms and Cohomology Decomposition of almost Complex Four-Manifolds. Int. Math. Res. Not. 1 (2010), 1–17

    2010

  4. [12]

    Draghici, T., Li, T.J., Zhang, W., On the J-anti-invariant cohomology of almost complex 4-manifolds. Q. J. Math. 64 (2013), 83–111

    2013

  5. [13]

    Ehresmann, C., Libermann, P., Sur le probl`eme d´equivalence des formes diff ´erentielles ext´erieures quadratiques. C. R. Acad. Sci. Paris 229 (1949), 697–698

    1949

  6. [14]

    L2 vanishing theorem on some K¨ahler manifolds

    Huang, T. L2 vanishing theorem on some K¨ahler manifolds. Israel J. Math. 241 (2021), 147–186

    2021

  7. [15]

    Huang, T., On Euler number of symplectic hyperbolic manifold. Adv. Math. 437 (2024), No. 109445

    2024

  8. [16]

    Libermann, P., Sur les structures presque complexes et autres structures infinit ´esimales r ´eeguli`eres. Bull. Soc. Math. France 83 (1955), 195–224

    1955

  9. [17]

    Gompf, R.E., Mrowka, T.S., Irreducible 4-manifolds need not be complex. Ann. of Math. 138 (1993), 61–111

    1993

  10. [18]

    Springer Science and Business Media

    Huybrechts, D., Complex geometry: an introduction. Springer Science and Business Media. (2006)

    2006

  11. [19]

    Zhang, W., Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

    Li, T.-J. Zhang, W., Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds. Comm. Anal. Geom. 17(4) (2009), 651–684

    2009

  12. [20]

    Mathieu, O., Harmonic cohomology classes of symplectic manifolds. Comm. Math. Helv. 70 (1995), 723–733

    1995

  13. [21]

    Merkulov, S.A., Formality of canonical symplectic complexes and Frobenius manifolds.Int. Math. Res. Not. 14 (1998), 727–733

    1998

  14. [22]

    Oguro, T., Sekigawa, K., Four-dimensional almost K¨ahler Einstein and ∗-Einstein manifolds. Geom. Dedicata 69 (1998), no. 1, 91–112

    1998

  15. [23]

    Kodai Math

    Oguro, T., Sekigawa, K., On some four-dimensional almost K¨ahler Einstein manifolds. Kodai Math. J. 24 (2001), no. 2, 226–258

    2001

  16. [24]

    Sillari, L., Tomassini, A., On the spaces of (d + dc)-harmonic forms and (d + dΛ)-harmonic forms on almost Hermitian manifolds and complex surfaces. Rev. Mat. Iberoam. 40 (2024), no. 6, 2371–2398

    2024

  17. [25]

    Internat

    Tardini, N., Tomassini, A., On the cohomology of almost-complex and symplectic manifolds and proper surjective maps. Internat. J. Math.27 (2016), 1650103 (20 pages) 30 Teng Huang and Weiwei Wang

    2016

  18. [26]

    Complex Mani- folds 7 (2020), 106–128

    Tardini, N., Tomassini, A., Differential operators on almost-Hermitian manifolds and harmonic forms. Complex Mani- folds 7 (2020), 106–128

    2020

  19. [27]

    Tomassini, A., Wang, X., Some results on the Hard Lefschetz Condition.Internat. J. Math.29 (2018), 1850095 (30 pages)

    2018

  20. [28]

    Differential Geom

    Tsai, C.J., Tseng, L.S., Yau, S.T., Cohomology and Hodge theory on symplectic manifolds: III.J. Differential Geom. 103 (2016), 83–143

    2016

  21. [29]

    Tseng, L.S., Yau, S.T., Cohomology and Hodge theory on symplectic manifolds: I. J. Differential Geom. 91 (2012), 383–416

    2012

  22. [30]

    Tseng, L.S., Yau, S.T., Cohomology and Hodge theory on symplectic manifolds: II. J. Differential Geom. 91 (2012), 417–443

    2012

  23. [31]

    Weinkove, B., The Calabi-Yau equation on almost-K¨ahler four-manifolds. J. Differential Geom. 76 (2007), 317–349

    2007

  24. [32]

    Yan, D., Hodge structure on symplectic manifolds. Adv. Math. 20 (1996), 143–154

    1996

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