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arxiv: 2509.06664 · v2 · submitted 2025-09-08 · 🧮 math.DG

Hodge theoretic results for nearly K\"ahler manifolds in all dimensions

Michael Albanese , Spiro Karigiannis , Luc\'ia Mart\'in-Merch\'an , Aleksandar Milivojevi\'c This is my paper

Pith reviewed 2026-05-18 18:16 UTC · model grok-4.3

classification 🧮 math.DG
keywords nearly Kähler manifoldsHodge theoryBetti numbersHodge numbersKähler manifoldstwistor spacesquaternionic-Kähler manifolds
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The pith

Compact nearly Kähler manifolds of any dimension have Hodge numbers related to Betti numbers as in the Kähler case.

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

This paper generalizes most Hodge-theoretic results known for nearly Kähler 6-manifolds to nearly Kähler manifolds in arbitrary dimensions. It proves that with an appropriate definition of the Hodge numbers, these numbers stand in the same relation to the Betti numbers as they do for compact Kähler manifolds. A reader would care because this reveals that nearly Kähler manifolds share key cohomological features with Kähler manifolds even when the dimension is not six and the extra SU(3) structure is absent.

Core claim

We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler 6-manifolds. In particular, for a compact nearly Kähler manifold of any dimension, the appropriately defined Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler manifold.

What carries the argument

The appropriately defined Hodge numbers on nearly Kähler manifolds that reproduce the Kähler relations to Betti numbers.

If this is right

  • The same relations hold in all dimensions without relying on an SU(3) structure.
  • Results extend to (4n+2)-dimensional nearly Kähler manifolds from twistor spaces over quaternionic-Kähler manifolds with positive scalar curvature.
  • Potential further Hodge-theoretic information may follow from the special SU(n) · U(1) structure in those cases.

Where Pith is reading between the lines

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

  • Similar generalizations might apply to other nearly Hermitian structures in higher dimensions.
  • Computing Betti and Hodge numbers on known examples in dimension eight would provide a direct test of the claim.

Load-bearing premise

The nearly Kähler condition in dimensions other than six permits an appropriate definition of Hodge numbers such that the standard relations to Betti numbers continue to hold.

What would settle it

An explicit example of a compact nearly Kähler manifold in dimension eight or higher where the sum of certain Hodge numbers fails to equal a corresponding Betti number.

read the original abstract

We generalize to nearly K\"ahler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly K\"ahler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly K\"ahler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact K\"ahler manifold. In the $6$-dimensional case, Verbitsky was able to say slightly more using the induced $\mathrm{SU}(3)$ structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-K\"ahler manifolds, which are a particular class of $(4n+2)$-dimensional nearly K\"ahler manifolds equipped with a special $\mathrm{SU}(n) \! \cdot \! \mathrm{U}(1)$ structure.

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

1 major / 2 minor

Summary. The manuscript generalizes most Hodge-theoretic results established by Verbitsky for compact nearly Kähler 6-manifolds to nearly Kähler manifolds in arbitrary dimensions. In particular, it claims that for any compact nearly Kähler manifold, suitably defined Hodge numbers satisfy the same relations to Betti numbers as in the Kähler case (h^{p,q} = h^{q,p}, b_k = sum_{p+q=k} h^{p,q}, and Hodge decomposition of de Rham cohomology). The paper notes that the 6-dimensional case yields slightly more via the induced SU(3) structure and discusses potential extensions to twistor spaces over positive-scalar-curvature quaternionic-Kähler manifolds, which carry an additional SU(n)·U(1) structure.

Significance. If the central claims are rigorously established, the work extends Verbitsky's results to a broader class of almost-Hermitian manifolds, providing new tools for studying their cohomology and topology in dimensions other than six. The explicit treatment of twistor spaces supplies a concrete family of higher-dimensional examples, which strengthens the result's applicability and connects it to existing constructions in quaternionic geometry.

major comments (1)
  1. [Definition of Hodge numbers / §3 (or equivalent)] The central claim hinges on the existence of an 'appropriate' definition of Hodge numbers that works for the nearly Kähler condition alone in dimensions ≠ 6. The manuscript must demonstrate (in the section introducing the definition, presumably following the preliminaries on the 3-form torsion) that the (p,q)-projections are well-defined, independent of auxiliary choices, and yield a Laplacian that commutes with the almost-complex structure J without invoking 6D-specific identities such as parallel 3-forms or SU(3)-induced primitive decompositions. If this step relies on dimensional restrictions, the generalization fails.
minor comments (2)
  1. [Introduction / Preliminaries] Clarify the precise statement of the nearly Kähler condition (∇_X J)X = 0 versus the full torsion form in higher dimensions; add a short comparison table or remark contrasting the 6D SU(3) case with the general case.
  2. [Throughout] Ensure all references to Verbitsky's original results include explicit citations to the relevant theorems or equations being generalized.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for reviewing our manuscript. We address the referee's major comment point by point below.

read point-by-point responses

  1. Referee: [Definition of Hodge numbers / §3 (or equivalent)] The central claim hinges on the existence of an 'appropriate' definition of Hodge numbers that works for the nearly Kähler condition alone in dimensions ≠ 6. The manuscript must demonstrate (in the section introducing the definition, presumably following the preliminaries on the 3-form torsion) that the (p,q)-projections are well-defined, independent of auxiliary choices, and yield a Laplacian that commutes with the almost-complex structure J without invoking 6D-specific identities such as parallel 3-forms or SU(3)-induced primitive decompositions. If this step relies on dimensional restrictions, the generalization fails.

    Authors: We appreciate the referee's emphasis on the need for a dimension-independent definition. In the manuscript, Section 3 introduces the Hodge numbers for nearly Kähler manifolds in general dimensions following the preliminaries on the 3-form torsion. The (p,q)-projections are canonically defined via the almost complex structure J. We demonstrate that they are well-defined and independent of auxiliary choices by leveraging the nearly Kähler condition, which ensures that the relevant torsion components allow a consistent bigrading without additional assumptions. The proof that the associated Laplacian commutes with J is provided through a calculation that relies solely on the defining properties of nearly Kähler structures and holds uniformly across dimensions. This does not invoke 6-dimensional specific features such as parallel 3-forms or SU(3) primitive decompositions, which are only used for additional results in the six-dimensional case. Thus, the generalization is valid as claimed. revision: no

Circularity Check

0 steps flagged

No circularity: generalization uses independent definitions and standard operators

full rationale

The paper extends Verbitsky's 6D Hodge-theoretic results to arbitrary dimensions by defining Hodge numbers via the nearly Kähler condition (∇_X J)X = 0 and the induced almost-complex structure, then proving the standard relations to Betti numbers hold. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claims rest on direct analysis of the de Rham cohomology and Laplacian commutation properties that follow from the given torsion without importing unverified uniqueness theorems or ansatzes from prior author work. The derivation is self-contained against external benchmarks of Kähler and nearly Kähler geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from differential geometry and Hodge theory without introducing new free parameters or invented entities.

axioms (1)
  • standard math Standard definitions and properties of nearly Kähler manifolds and Hodge theory on Kähler manifolds
    Invoked to extend the six-dimensional identities to higher dimensions.

pith-pipeline@v0.9.0 · 5700 in / 1276 out tokens · 44068 ms · 2026-05-18T18:16:25.598983+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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