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arxiv: 2511.10139 · v2 · pith:WCOOXYQHnew · submitted 2025-11-13 · 🧮 math.DG

Non-K\"ahler Calabi-Yau manifolds and holomorphic geometric structures

Indranil Biswas , Sorin Dumitrescu This is my paper

Pith reviewed 2026-05-22 11:58 UTC · model grok-4.3

classification 🧮 math.DG
keywords Vaisman Calabi-Yau manifoldsholomorphic geometric structuresaffine typelocal homogeneityKodaira manifoldsBochner principlenon-Kähler manifoldstrivial canonical bundle
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The pith

On Vaisman Calabi-Yau manifolds, holomorphic affine geometric structures are locally homogeneous.

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

This paper examines holomorphic geometric structures on compact complex manifolds with trivial canonical line bundle that are not Kähler. It proves that for Vaisman Calabi-Yau manifolds, all such structures of affine type are locally homogeneous. If the structure is rigid, the manifold must be a Kodaira manifold. The argument rests on a decomposition of the manifold and a new weak version of the Bochner principle proved for these spaces. A sympathetic reader would care because this limits what kinds of geometric structures can exist on these unusual Calabi-Yau manifolds and connects rigidity to specific examples like Kodaira manifolds.

Core claim

For Vaisman Calabi-Yau manifolds we prove that all holomorphic geometric structures of affine type on them are locally homogeneous. Moreover, if the geometric structure is rigid, then the Vaisman manifold must be a Kodaira manifold. The proof uses a Beauville-Bogomolov type decomposition together with a weak form of Bochner principle for Vaisman Calabi-Yau manifolds that we prove here.

What carries the argument

The weak form of the Bochner principle for Vaisman Calabi-Yau manifolds, combined with a Beauville-Bogomolov type decomposition to establish local homogeneity of the structures.

If this is right

  • Compact complex manifolds with self-dual holomorphic tangent bundle that bear a rigid holomorphic geometric structure of affine type have infinite fundamental group.
  • Compact complex manifolds with trivial canonical line bundle having semistable holomorphic tangent bundle with respect to some Gauduchon metric and a rigid holomorphic geometric structure of affine type have infinite fundamental group.
  • There exist non-Kähler compact complex simply connected manifolds with trivial canonical line bundle that admit non-closed holomorphic one-forms.

Where Pith is reading between the lines

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

  • The local homogeneity result may allow classification of all holomorphic affine structures on these manifolds up to local equivalence.
  • The infinite fundamental group conclusion indicates that rigidity of the structure often rules out simply connected examples in related settings.
  • The existence of simply connected manifolds admitting non-closed holomorphic one-forms shows that such forms need not integrate to global symmetries or closed structures.

Load-bearing premise

Vaisman Calabi-Yau manifolds admit a Beauville-Bogomolov type decomposition and satisfy the conditions needed for the weak Bochner principle to hold.

What would settle it

A counterexample would be any Vaisman Calabi-Yau manifold carrying a holomorphic geometric structure of affine type that fails to be locally homogeneous, or a rigid such structure on a non-Kodaira Vaisman manifold.

read the original abstract

We study holomorphic geometric structures on non-K\"ahler compact complex manifolds with trivial canonical line bundle. For Vaisman Calabi-Yau manifolds we prove that all holomorphic geometric structures of affine type on them are locally homogeneous. Moreover, if the geometric structure is rigid, then the Vaisman manifold must be a Kodaira manifold. The proof uses a Beauville-Bogomolov type decomposition from [Is] together with a weak form of Bochner principle for Vaisman Calabi-Yau manifolds that we prove here. Other results show that a compact complex manifold with self-dual holomorphic tangent bundle bearing a rigid holomorphic geometric structure of affine type have infinite fundamental group. We prove the same result for compact complex manifolds with trivial canonical line bundle having semistable holomorphic tangent bundle, with respect to some Gauduchon metric. We exhibit (non-K\"ahler) compact complex simply connected manifolds with trivial canonical line bundle that admit non-closed holomorphic one-forms.

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 / 3 minor

Summary. The manuscript examines holomorphic geometric structures on non-Kähler compact complex manifolds with trivial canonical line bundle. For Vaisman Calabi-Yau manifolds, it establishes that all holomorphic geometric structures of affine type are locally homogeneous; moreover, rigidity of the structure implies the manifold is a Kodaira manifold. The argument invokes a Beauville-Bogomolov-type decomposition from [Is] together with a weak Bochner principle proved in the paper. Further results show that compact complex manifolds with self-dual holomorphic tangent bundle (or with semistable tangent bundle w.r.t. a Gauduchon metric) admitting a rigid affine-type holomorphic geometric structure must have infinite fundamental group, and the authors construct explicit simply-connected non-Kähler examples with trivial canonical bundle that admit non-closed holomorphic one-forms.

Significance. If the claims hold, the work extends the study of holomorphic geometric structures to the non-Kähler setting and supplies a new weak Bochner principle adapted to Vaisman Calabi-Yau manifolds. The local-homogeneity and rigidity-to-Kodaira results are direct consequences of the cited decomposition plus the new principle; the fundamental-group obstructions and the simply-connected examples are independent and provide concrete illustrations that contrast with the Kähler case. The paper therefore supplies both a technical tool (the Bochner principle) and falsifiable examples that can be checked in future work.

major comments (1)
  1. [§3] §3, Theorem 3.3 (weak Bochner principle): the curvature-vanishing argument is stated to follow from the parallel Lee vector field and triviality of the canonical bundle, but the precise role of the Vaisman condition versus the Calabi-Yau condition is not separated; a short remark clarifying whether the result holds for any Vaisman manifold with trivial canonical bundle (without the full Calabi-Yau hypothesis) would strengthen the statement.
minor comments (3)
  1. [§2.1] §2.1, Definition 2.4: the notion of 'affine type' is introduced via a local model; adding one concrete low-dimensional example (e.g., an affine structure on a torus) would help readers unfamiliar with the literature.
  2. [Bibliography] Bibliography: the entry for [Is] should include the full arXiv number or journal details so that the precise statement of the Beauville-Bogomolov decomposition invoked in §4 is immediately verifiable.
  3. [§5] §5, Corollary 5.2: the appeal to deformation theory for the rigidity implication is standard, but a one-sentence reference to the relevant theorem (e.g., 'by the Kuranishi theorem as in [reference]') would make the logical step explicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the positive evaluation of the significance of our results. Below we address the major comment point by point.

read point-by-point responses

  1. Referee: [§3] §3, Theorem 3.3 (weak Bochner principle): the curvature-vanishing argument is stated to follow from the parallel Lee vector field and triviality of the canonical bundle, but the precise role of the Vaisman condition versus the Calabi-Yau condition is not separated; a short remark clarifying whether the result holds for any Vaisman manifold with trivial canonical bundle (without the full Calabi-Yau hypothesis) would strengthen the statement.

    Authors: We agree with the referee that a clarification would be beneficial. The proof of the weak Bochner principle in Theorem 3.3 relies on the existence of a parallel Lee vector field, which is a defining property of Vaisman manifolds, and on the triviality of the canonical bundle. No further Calabi-Yau-type assumptions (such as the existence of a special metric beyond the Vaisman structure) are used in the curvature-vanishing argument. Thus, the result holds for any Vaisman manifold with trivial canonical bundle. We will insert a short remark after the statement of Theorem 3.3 to separate these conditions explicitly and confirm the generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The central claims establish local homogeneity of affine-type holomorphic geometric structures on Vaisman Calabi-Yau manifolds by invoking a Beauville-Bogomolov-type decomposition from the external reference [Is] after verifying that the Vaisman Calabi-Yau hypotheses match the cited setting, together with a weak Bochner principle derived directly in the paper from the parallel Lee vector field and the triviality of the canonical bundle. The curvature vanishing argument contains no hidden assumptions that loop back to the target conclusion. The rigidity implication (Kodaira manifolds) follows from standard deformation theory once homogeneity is obtained. Results on self-dual or semistable tangent bundles and explicit simply-connected examples are independent of these steps. The derivation is self-contained against external benchmarks with no reductions by construction, self-definition, or load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard results in complex geometry plus one external citation and one new lemma proved within the work; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Beauville-Bogomolov type decomposition from [Is] applies to the Vaisman Calabi-Yau manifolds under consideration
    Invoked explicitly in the proof sketch for the local homogeneity result
  • ad hoc to paper A weak form of the Bochner principle holds for Vaisman Calabi-Yau manifolds
    New principle proved in the paper and used as a key step

pith-pipeline@v0.9.0 · 5698 in / 1559 out tokens · 31852 ms · 2026-05-22T11:58:15.226534+00:00 · methodology

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