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Compact locally conformal K\"ahler manifolds with constant Chern holomorphic sectional curvature

T0 review · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.

desk verdict This paper proves compact LCK manifolds with constant Chern holomorphic sectional curvature must be Kähler space forms by deriving a curvature identity on the universal cover. read the letter →

arxiv 2606.24425 v1 pith:JEXT6BSV submitted 2026-06-23 math.DG math.CV

Zhuzhu Huang , Xueyuan Wan This is my paper · ORCID
classification math.DGmath.CV
keywords locallyconformalKählermanifoldChernholomorphicsectionalcurvaturecomplexspaceformBochner-KählermetricKamishimauniformizationconstant
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

The paper establishes that any compact locally conformal Kähler manifold of complex dimension at least two whose Chern holomorphic sectional curvature is constant must in fact be Kähler. This means the metric is a complex space form of that constant curvature. The result removes the nonpositivity assumption that appeared in earlier work on the topic. It does so by lifting to the universal cover, deriving a curvature identity there, and applying rigidity theorems for Bochner-Kähler metrics along with uniformization to rule out the strictly locally conformal case.

What carries the argument

A curvature identity derived on the universal Kähler cover that forces the covering metric to be Bochner-Kähler, followed by rigidity results and Kamishima's uniformization theorem to exclude strict LCK cases.

What would settle it

The existence of a compact strict locally conformal Kähler manifold with constant Chern holomorphic sectional curvature that is not Kähler would falsify the claim.

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Extended reading notes

Core claim

Let (M^n, h) with n ≥ 2 be a compact locally conformal Kähler manifold with constant Chern holomorphic sectional curvature c. Then h is Kähler and is a complex space form metric of holomorphic sectional curvature c. In particular when c = 0 the metric is Kähler flat.

Load-bearing premise

The lifted metric on the universal Kähler cover satisfies the Bochner-Kähler condition, which then permits the application of known rigidity and uniformization theorems.

Editorial extensions

If this is right

  • The metric h must coincide with a Kähler metric.
  • It is a complex space form of curvature c.
  • When c=0 it is flat in the Kähler sense.
  • This extends previous results by dropping the nonpositivity condition on curvature.

Reading between the lines

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

  • Similar conclusions might hold for other curvature conditions on LCK manifolds if analogous identities can be derived.
  • The result suggests that constant Chern curvature is a strong rigidity condition that forces conformality to be global.
  • Non-compact LCK manifolds with the same curvature condition may behave differently and warrant separate study.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that any compact locally conformal Kähler manifold (M^n, h) with n ≥ 2 whose Chern holomorphic sectional curvature is constant equal to c must in fact be Kähler, hence a complex space form of holomorphic sectional curvature c (in particular Kähler-flat when c=0). The argument derives a curvature identity on the universal Kähler cover showing that the lifted metric is Bochner-Kähler, then invokes compact Bochner-Kähler rigidity for the globally conformally Kähler case and Kamishima's uniformization theorem (together with automorphy of the conformal factor) to exclude the strict LCK case. This removes the nonpositivity hypothesis from the earlier theorem of Chen-Chen-Nie.

Significance. If the curvature identity and its consequences hold, the result is a substantive contribution to the classification of compact LCK manifolds and to the constant-curvature problem in Hermitian geometry. It supplies the Chern-curvature analogue of known results for the Riemannian holomorphic sectional curvature and strengthens the rigidity theory for LCK structures. The reduction to the universal cover and the clean application of Kamishima's theorem are technically economical strengths.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for the positive assessment of the manuscript. We are pleased that the referee finds the result a substantive contribution and recommends acceptance.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on external theorems

full rationale

The paper derives a curvature identity on the universal Kähler cover to establish the Bochner-Kähler condition, then invokes Kamishima's uniformization theorem (distinct authors) and compact Bochner-Kähler rigidity to conclude the metric must be Kähler. No quoted step reduces the central claim to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain; the cited results are treated as independent external input with no reduction to the paper's own fitted quantities or ansatz. The argument is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The proof rests on standard background facts of Hermitian geometry together with two external theorems; no free parameters or new entities are introduced.

assumptions (2)
  • domain assumption Existence and properties of the universal Kähler cover for an LCK manifold
    Invoked in the proof sketch to lift the metric and derive the curvature identity.
  • standard math Kamishima's uniformization theorem for LCK manifolds
    Used to exclude the strict LCK case once the cover is shown Bochner-Kähler.

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Cite this review

Pith. "Pith review of Compact locally conformal K\"ahler manifolds with constant Chern holomorphic sectional curvature." pith.science (2026). https://pith.science/paper/JEXT6BSV

@misc{pith2026260624425,
  author       = {Pith},
  title        = {Pith review of: Compact locally conformal K\"ahler manifolds with constant Chern holomorphic sectional curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JEXT6BSV}},
  note         = {Machine review of arXiv:2606.24425}
}
abstract

We prove the Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal K\"ahler manifolds. More precisely, let $(M^n,h)$, $n\geq2$, be a compact locally conformal K\"ahler manifold whose Chern holomorphic sectional curvature is a constant $c$. We show that $h$ is necessarily K\"ahler and therefore is a complex space form metric of holomorphic sectional curvature $c$. In particular, when $c=0$, the metric is K\"ahler flat. This removes the nonpositivity assumption from a theorem of Chen, Chen, and Nie. The proof derives a curvature identity on the universal K\"ahler cover and shows that the covering metric is Bochner--K\"ahler. The globally conformally K\"ahler case is then treated by compact Bochner--K\"ahler rigidity, while the strict LCK case is excluded by Kamishima's uniformization theorem and the automorphy of the conformal factor.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature

    math.DG 2026-08 accept novelty 7.0 of 10

    Compact balanced threefolds with nonpositive constant Chern holomorphic sectional curvature are Chern flat (c=0) or Kähler (c<0), and constant-curvature LCK manifolds are Kähler or Hopf-covered.

  2. Locally conformally K\"ahler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature

    math.DG 2026-08 conditional novelty 6.0 of 10

    Compact locally conformally Kähler manifolds with constant Levi-Civita holomorphic sectional curvature are Kähler, and with constant Bismut holomorphic sectional curvature are Kähler or isosceles Hopf manifolds with z...

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