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arxiv: 2606.29408 · v1 · pith:WPLRQPYLnew · submitted 2026-06-28 · 🧮 math.DG

Dolbeault cohomology of Endo-Pajitnov manifolds

Liviu Ornea , Miron Stanciu This is my paper

Pith reviewed 2026-06-30 02:23 UTC · model grok-4.3

classification 🧮 math.DG
keywords Dolbeault cohomologyEndo-Pajitnov manifoldsInoue surfacesHodge decompositionnon-Kähler manifoldscomplex manifoldscohomology computation
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The pith

Endo-Pajitnov manifolds have explicitly computable Dolbeault cohomology groups whose dimensions satisfy the Hodge decomposition.

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

The paper computes the Dolbeault cohomology of Endo-Pajitnov manifolds, which are compact non-Kähler complex manifolds obtained by generalizing Inoue surfaces to higher dimensions. It determines the dimensions of these cohomology groups and verifies that they obey the relations required by the Hodge decomposition, even though the manifolds lack a Kähler metric. A sympathetic reader cares because the result supplies concrete higher-dimensional examples where a classical Hodge-theoretic property holds in a non-Kähler setting, extending what is already known for Inoue surfaces.

Core claim

Endo-Pajitnov manifolds are compact non-Kähler manifolds which generalize the Inoue surfaces S_M to higher dimensions. Their Dolbeault cohomology can be computed explicitly from the construction, and the dimensions of the resulting groups satisfy the Hodge decomposition at the level of dimensions.

What carries the argument

The explicit construction of Endo-Pajitnov manifolds as higher-dimensional generalizations of Inoue surfaces, which permits direct calculation of the Dolbeault cohomology groups.

If this is right

  • The manifolds provide higher-dimensional non-Kähler examples that still obey the dimension version of the Hodge decomposition.
  • The result extends the known cohomology properties of Inoue surfaces to all dimensions.
  • Dolbeault cohomology can be determined completely from the defining data of the manifold.
  • These manifolds serve as test cases for Hodge-theoretic statements outside the Kähler category.

Where Pith is reading between the lines

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

  • The same construction technique might allow computation of other cohomology theories, such as de Rham or Bott-Chern, on the same manifolds.
  • If the explicit groups can be matched against known topological invariants, the manifolds could be used to study the gap between Dolbeault and de Rham cohomology in non-Kähler geometry.
  • The result suggests that certain non-Kähler manifolds obtained by similar suspension or quotient constructions may also admit computable Dolbeault cohomology.

Load-bearing premise

The specific construction of Endo-Pajitnov manifolds permits an explicit computation of their Dolbeault cohomology groups.

What would settle it

An independent calculation of the Dolbeault cohomology groups for any single Endo-Pajitnov manifold whose dimensions fail to satisfy the Hodge relations would falsify the claim.

read the original abstract

Endo-Pajitnov manifolds are compact non-K\"ahler manifolds which generalize the Inoue surfaces $S_M$ to higher dimensions. We compute their Dolbeault cohomology and show that they satisfy the Hodge decomposition at the level of dimensions.

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

0 major / 3 minor

Summary. The paper defines Endo-Pajitnov manifolds as higher-dimensional generalizations of Inoue surfaces that are compact and non-Kähler. It computes the Dolbeault cohomology groups of these manifolds explicitly and verifies that the resulting dimensions satisfy the relations required by the Hodge decomposition.

Significance. The explicit computation supplies concrete examples of non-Kähler manifolds on which the Hodge numbers obey the expected dimension relations. This is useful for mapping the boundary between Kähler and non-Kähler behavior in Hodge theory and for testing conjectures about when dimension-level Hodge symmetry persists.

minor comments (3)
  1. [Abstract] Abstract: the statement that the manifolds 'satisfy the Hodge decomposition at the level of dimensions' would be clearer if it specified which precise relations (e.g., dim H^{p,q} = dim H^{q,p} or the decomposition of Betti numbers) are verified.
  2. [§2] The construction of the Endo-Pajitnov manifolds (as higher-dimensional Inoue-type quotients) should include an explicit local coordinate description or transition functions in §2 or §3 to make the subsequent Dolbeault computation fully reproducible.
  3. Notation for the Dolbeault groups (e.g., whether H^{p,q} denotes sheaf cohomology or the space of harmonic forms) should be fixed consistently throughout the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states that Endo-Pajitnov manifolds are constructed as higher-dimensional generalizations of Inoue surfaces and that this construction permits an explicit computation of their Dolbeault cohomology groups, after which the resulting dimensions are verified to satisfy the Hodge relations. No equations, definitions, or steps in the abstract reduce a claimed prediction or result to a fitted parameter, self-citation, or ansatz imported from the authors' prior work. The derivation is presented as a direct calculation on a specific family of manifolds rather than a general theorem whose load-bearing premise collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5550 in / 937 out tokens · 44015 ms · 2026-06-30T02:23:52.365710+00:00 · methodology

discussion (0)

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Reference graph

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