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arxiv: 2606.23996 · v1 · pith:SKLZV2KLnew · submitted 2026-06-22 · 🧮 math.SG · math.DG

Symplectic non-K\"ahler manifolds with and without the Hard Lefschetz Condition

Richard Hind , Adriano Tomassini This is my paper

Pith reviewed 2026-06-26 05:45 UTC · model grok-4.3

classification 🧮 math.SG math.DG
keywords symplectic manifoldsHard Lefschetz conditionnon-Kähler manifoldssymplectic blowuporbifold resolutionsymplectic formsconnected components
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The pith

A non-Kähler manifold admits both Hard Lefschetz and non-Hard Lefschetz symplectic forms in one connected component.

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

The paper constructs compact manifolds without Kähler structures that carry a connected family of symplectic forms in which the Hard Lefschetz Condition holds for most members but fails at one point. The construction begins with a one-parameter family of symplectic forms on a known symplectic orbifold, resolves the orbifold symplectically to obtain manifolds that fail the condition for every parameter, and then applies a symplectic blowup along a union of tori. After the blowup the condition is restored precisely when the parameter is nonzero, yielding the mixed behavior inside a single connected component of the space of symplectic forms. A sympathetic reader would care because the result shows that the Hard Lefschetz Condition can vary inside a deformation class of symplectic forms even when no Kähler structure exists on the underlying manifold.

Core claim

The authors exhibit a smooth manifold with no Kähler structure whose space of symplectic forms contains both HLC and non-HLC structures in the same connected component. This is achieved by taking a one-parameter family of symplectic forms on the Fernández-Muñoz orbifold, resolving it symplectically so that the resulting manifolds fail the Hard Lefschetz Condition for every parameter, and then performing a symplectic blowup along a union of tori so that the Hard Lefschetz Condition holds for all nonzero parameters but fails at the central parameter.

What carries the argument

The symplectic blowup along a union of tori performed on the resolved one-parameter family of symplectic forms, which selectively restores the Hard Lefschetz isomorphism away from the central parameter value.

If this is right

  • The Hard Lefschetz Condition is not invariant under continuous deformations inside the space of symplectic forms on these manifolds.
  • A single connected component of the space of symplectic forms can contain both structures that satisfy and structures that violate the Hard Lefschetz Condition.
  • These manifolds supply new examples in which the Hard Lefschetz property is independent of the existence of a Kähler structure.

Where Pith is reading between the lines

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

  • The construction suggests that parameter-dependent restoration of the Hard Lefschetz Condition might be arranged in other symplectic resolutions of orbifolds by choosing different loci for the blowup.
  • One could test whether the failure of the condition at an isolated parameter persists under further symplectic operations such as additional blowups or fiber sums.
  • The result leaves open whether the single-point failure of the Hard Lefschetz Condition imposes any further restriction on the cohomology ring beyond what is already visible from the orbifold resolution.

Load-bearing premise

The blowup along the union of tori restores the Hard Lefschetz Condition for all nonzero parameters while preserving its failure at the central value, without introducing a Kähler structure or changing the connectedness of the symplectic-form space.

What would settle it

An explicit computation of the Lefschetz map induced by a nonzero-parameter symplectic class on the cohomology of one of the blown-up manifolds that shows the map fails to be an isomorphism would falsify the claimed separation of HLC and non-HLC forms.

read the original abstract

In this paper we construct compact manifolds without K\"ahler structures that admit both a symplectic form satisfying the Hard Lefschetz Condition (HLC) and another symplectic form that does not. Our construction builds upon the orbifold introduced by Fern\'andez and Mu\~noz and its symplectic resolution studied by Cavalcanti, Fern\'andez, and Mu\~noz. By considering a one-parameter family of symplectic forms on the orbifold, we show that the corresponding resolved manifolds fail to satisfy the HLC for all parameters. However, after performing a suitable symplectic blowup along a union of tori, we obtain a family of symplectic manifolds for which the HLC holds for all non-zero parameters but fails at the central parameter. As a consequence, we exhibit a smooth manifold with no K\"ahler structure whose space of symplectic forms contains both HLC and non-HLC structures in the same connected component. This provides new examples of the subtle interplay between symplectic topology and the Hard Lefschetz property.

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 constructs compact non-Kähler manifolds admitting both HLC and non-HLC symplectic forms in the same connected component of the symplectic form space. Starting from a one-parameter family of symplectic forms on the Fernández-Muñoz orbifold, the symplectic resolutions fail HLC for all parameters; a subsequent symplectic blowup along a union of tori then yields a family of smooth manifolds on which HLC holds precisely for nonzero parameters while failing at the central value.

Significance. If the cohomology calculations and blowup formulas are correct, the result supplies new examples of the subtle dependence of the Hard Lefschetz property on the choice of symplectic form, even within a single connected component and on a fixed non-Kähler manifold. It builds directly on the orbifold and resolution constructions of Fernández-Muñoz and Cavalcanti-Fernández-Muñoz, adding an independent parameter and a targeted blowup step that separates HLC behavior without restoring Kählerness.

major comments (2)
  1. [Blowup construction section] Blowup construction (the section following the resolution of the orbifold): the claim that the symplectic blowup along the union of tori restores the Hard Lefschetz isomorphism for all nonzero parameters while preserving its failure at the central parameter is load-bearing for the main theorem. The manuscript must supply the explicit post-blowup computation of the Lefschetz operators L^{n-k} on the cohomology ring (including the new exceptional classes) to verify that the kernel dimension changes exactly at the zero parameter and that no Kähler structure is introduced.
  2. [Final section / abstract] Connected-component claim (abstract and final section): it is asserted that the post-blowup forms remain in the same connected component as the original family. The argument that the blowup does not disconnect the space of symplectic forms or alter the component structure needs a precise reference to the deformation theory or path-connectedness argument used.
minor comments (2)
  1. Notation for the one-parameter family and the exceptional divisors should be introduced once and used consistently; some symbols appear to be redefined without cross-reference.
  2. The statement that the resolved manifolds fail HLC for all parameters would benefit from a short table summarizing the Betti numbers or the rank of the Lefschetz maps before and after blowup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Blowup construction section] Blowup construction (the section following the resolution of the orbifold): the claim that the symplectic blowup along the union of tori restores the Hard Lefschetz isomorphism for all nonzero parameters while preserving its failure at the central parameter is load-bearing for the main theorem. The manuscript must supply the explicit post-blowup computation of the Lefschetz operators L^{n-k} on the cohomology ring (including the new exceptional classes) to verify that the kernel dimension changes exactly at the zero parameter and that no Kähler structure is introduced.

    Authors: We agree that an explicit post-blowup computation of the Lefschetz operators is required for full verification. In the revised manuscript we will add a dedicated subsection computing the action of L^{n-k} on the cohomology ring of the blown-up manifold, explicitly incorporating the new exceptional classes arising from the symplectic blowup along the union of tori. The calculation will track the kernel dimensions parameter-by-parameter and confirm that the change occurs precisely at the central value. We will also include a short argument, based on the existence of a non-HLC symplectic form in the family, showing that the resulting manifold remains non-Kähler. revision: yes

  2. Referee: [Final section / abstract] Connected-component claim (abstract and final section): it is asserted that the post-blowup forms remain in the same connected component as the original family. The argument that the blowup does not disconnect the space of symplectic forms or alter the component structure needs a precise reference to the deformation theory or path-connectedness argument used.

    Authors: The family is obtained by performing the symplectic blowup uniformly and continuously with respect to the one-parameter family on the resolved orbifold. In the revision we will supply a precise reference to the standard deformation-theoretic fact that a symplectic blowup performed in a smooth family of symplectic forms preserves path-connectedness of the parameter space (citing the relevant statements in McDuff-Salamon, J-holomorphic curves and symplectic topology, §7.2, together with the continuity of the blowup construction in the space of symplectic forms). This reference will be inserted both in the abstract discussion and in the final section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is independent of inputs.

full rationale

The paper's derivation starts from known orbifold examples (Fernández-Muñoz) and their resolutions (Cavalcanti-Fernández-Muñoz), then introduces an explicit one-parameter family of symplectic forms on the orbifold. The resolved manifolds are shown to fail HLC uniformly, after which a new symplectic blowup along a union of tori is performed. The post-blowup family is claimed to restore HLC for nonzero parameters while preserving failure at the central value. This outcome is obtained by direct computation of the Lefschetz operators on the modified cohomology ring and is not equivalent to the input data by definition or by any fitted parameter. No self-citations by the present authors are load-bearing, no ansatz is smuggled, and no prediction reduces to a renaming or self-definition. The central claim therefore rests on an independent geometric construction rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard facts about symplectic resolutions and blowups from the cited literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the summary.

axioms (1)
  • domain assumption Symplectic resolutions and blowups preserve the symplectic category and act in a controlled way on cohomology
    Invoked implicitly when the authors pass from the orbifold to the resolved manifold and then to the blowup.

pith-pipeline@v0.9.1-grok · 5707 in / 1285 out tokens · 39526 ms · 2026-06-26T05:45:22.101294+00:00 · methodology

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Reference graph

Works this paper leans on

23 extracted references · 3 canonical work pages

  1. [1]

    A.AndradaandA.Garrone, Constructionofsymplecticsolvmanifoldssatisfyingthehard-Lefschetzcondition,Linear Algebra Appl.706(2025), 70–100

    2025

  2. [2]

    Andrada and A

    A. Andrada and A. Garrone, Symplectic solvmanifolds not satisfying the hard-Lefschetz condition,arXiv:2505.08113

    work page arXiv

  3. [3]

    W. L. Baily, The decomposition theorem forV-manifolds,Amer. J. Math.78(1956), no. 4, 862–888

    1956

  4. [4]

    Brylinski, A differential complex for Poisson manifolds,J

    J.-L. Brylinski, A differential complex for Poisson manifolds,J. Differential Geom.28(1988), no. 1, 93–114

    1988

  5. [5]

    G. R. Cavalcanti, New aspects of theddc-lemma, PhD Thesis, University of Oxford, (2004)

    2004

  6. [6]

    G. R. Cavalcanti, The Lefschetz property, formality and blowing up in symplectic geometry,Trans. Amer. Math. Soc.359 (2007), no. 1, 333–348

    2007

  7. [7]

    G. R. Cavalcanti, M. Fernández, V. Muñoz, Symplectic resolutions, Lefschetz property and formality,Adv. Math.218(2008) 576–599

    2008

  8. [8]

    Cho, Hard Lefschetz property of symplectic structures on Compact Kähler manifolds,Trans

    Y. Cho, Hard Lefschetz property of symplectic structures on Compact Kähler manifolds,Trans. Amer. Math. Soc.,368 (2016), 8223–8248

    2016

  9. [9]

    de Bartolomeis and A

    P. de Bartolomeis and A. Tomassini, On solvable generalized Calabi-Yau manifolds,Ann. Inst. Fourier56(2006), 1281–1296

    2006

  10. [10]

    Deligne, P

    P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real Homotopy Theory of Kähler Manifolds,Inventiones Math.29(1975) pp. 245–274

    1975

  11. [11]

    Fernández, V

    M. Fernández, V. Muñoz, An8-dimensional non-formal simply connected symplectic manifold,Ann. of Math. (2)167(2008), 1045–1054

    2008

  12. [12]

    Joyce,Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000

    D.D. Joyce,Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000

    2000

  13. [13]

    Griffiths, J

    P. Griffiths, J. Harris,Principles of Algebraic Geometry, Interscience Publ., New York (1978)

    1978

  14. [14]

    Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems,J

    H. Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems,J. Differential Geom.93(2013), 269–297 https://doi.org/10.4310/jdg/1361800867

    work page doi:10.4310/jdg/1361800867 2013

  15. [15]

    Kasuya, Formality and hard Lefschetz property of aspherical manifolds,Osaka J

    H. Kasuya, Formality and hard Lefschetz property of aspherical manifolds,Osaka J. Math.50(2) (2013), 439– 455

    2013

  16. [16]

    Lusetti, A

    F. Lusetti, A. Tomassini, Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds,Math. Z.311, 74 (2025) https://doi.org/10.1007/s00209-025-03878-5

    work page doi:10.1007/s00209-025-03878-5 2025

  17. [17]

    Manin,Three constructions of Frobenius manifolds: a comparative study, Survey in Diff

    Y.I. Manin,Three constructions of Frobenius manifolds: a comparative study, Survey in Diff. Geom. 2000, International Press Vol. 7 2000, 497–554

    2000

  18. [18]

    Mathieu, Harmonic cohomology classes of symplectic manifolds,Comment

    O. Mathieu, Harmonic cohomology classes of symplectic manifolds,Comment. Math. Helv.,70(1995), 1–9

    1995

  19. [19]

    McDuff, Examples of symplectic simply connected manifolds with no Kähler structure,J

    D. McDuff, Examples of symplectic simply connected manifolds with no Kähler structure,J. Differential Geom.20(1984), 267–277

    1984

  20. [20]

    McDuff, D

    D. McDuff, D. Salamon,Introduction to symplectic topology, Oxford Mathematical Monographs, Oxford University Press, 1995

    1995

  21. [21]

    S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds,Internat. Math. Res. Notices.14 (1998), 727–733

    1998

  22. [22]

    Satake, On a generalization of the notion of manifold,Proc

    I. Satake, On a generalization of the notion of manifold,Proc. Nat. Acad. Sci. U.S.A.42(1956), n.6, p. 359–363

    1956

  23. [23]

    Yan, Hodge structure on symplectic manifolds,Adv

    D. Yan, Hodge structure on symplectic manifolds,Adv. Math.120(1996), no. 1, 143–154. 10 RICHARD HIND AND ADRIANO TOMASSINI (Richard Hind)Department of Mathematics, University of Notre Dame, 255 Hurley Bldg, Notre Dame, IN 46556 Email address:hind.1@nd.edu (Adriano Tomassini)Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e...

    1996