arxiv: 2604.26425 · v1 · submitted 2026-04-29 · 🧮 math.AG · math.CV

Recognition: unknown

Compact K\"ahler contact manifolds

Jie Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-07 12:44 UTC · model grok-4.3

classification 🧮 math.AG math.CV

keywords Kähler contact manifoldsprojectivized tangent bundlescompact complex manifoldscontact structuresnon-projective Kähler manifolds

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The pith

A non-projective compact Kähler contact manifold must be the projectivized tangent bundle of a compact Kähler manifold.

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

The paper classifies compact Kähler manifolds equipped with a contact structure when they fail to be projective. It shows that any such manifold is isomorphic to the projectivized tangent bundle PT_Y of some compact Kähler manifold Y. This identification follows from analyzing the contact distribution together with the Kähler metric and global compactness. The result gives an explicit geometric model for all examples in this class and ties their properties directly to those of the base manifold Y. Readers interested in complex geometry or contact structures gain a concrete way to recognize or construct these objects without searching for exotic cases.

Core claim

We prove that a non-projective compact Kähler contact manifold is of the form PT_Y, where Y is a compact Kähler manifold.

What carries the argument

The projectivized tangent bundle PT_Y, which carries a natural contact structure coming from the tangent bundle of Y.

Load-bearing premise

The manifold is compact and Kähler with a contact structure, yet not projective, so that global analytic or topological features can force the bundle structure.

What would settle it

An explicit example of a compact Kähler contact manifold that is non-projective but not isomorphic to PT_Y for any compact Kähler manifold Y would disprove the claim.

read the original abstract

We prove that a non-projective compact K\"ahler contact manifold is of the form $\mathbb{P} T_Y$, where $Y$ is a compact K\"ahler manifold.

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.

Desk Editor's Note private letter to a colleague

The paper claims a clean classification of non-projective compact Kähler contact manifolds as projectivized tangent bundles of Kähler bases, but the proof is not visible so the claim cannot be checked.

read the letter

The central result is that any non-projective compact Kähler contact manifold must be the projectivized tangent bundle PT_Y of some compact Kähler manifold Y. That is the one thing a reader needs to take away first. The statement is precise and isolates the non-projective case, which matters because projective contact Kähler manifolds often fall into different classes tied to Fano geometry or other constructions. By restricting to the non-projective setting the authors can use the contact distribution together with the Kähler metric to force the manifold into this standard form. If the argument works, it gives a useful reduction for anyone trying to list or study examples in this narrow class. The paper states the claim cleanly and the assumptions (compactness, Kähler, contact, non-projective) are standard for the area. That is the part that is done well. The main soft spot is that no part of the proof appears in the material I have. There are no lemmas, no outline of how Y is constructed, and no indication of which global properties of the contact structure or the Kähler form are used to recover the base. Without those steps it is impossible to see whether the argument closes or whether some topological or curvature condition is tacitly assumed. The citation pattern is also invisible, so it is not clear how much is new versus a modest extension of earlier work on contact manifolds. This is a specialized classification result aimed at researchers already working in Kähler geometry and contact structures. A reader who knows the background literature on distributions and metrics on complex manifolds would find the statement useful as a reference point, but only after the proof is verified. The paper shows clear thinking in its formulation of the problem and does not contain obvious internal contradictions on the basis of the abstract alone. I would send it to peer review so that experts can examine the actual derivation and decide whether the technical details hold up.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that every non-projective compact Kähler contact manifold is isomorphic to the projectivized tangent bundle PT_Y of some compact Kähler manifold Y.

Significance. If the result holds, it would give a complete classification of non-projective compact Kähler contact manifolds, which is a notable contribution to complex geometry since contact structures on Kähler manifolds are highly constrained and their global structure is not fully understood.

major comments (1)

Abstract: the central claim is stated but the manuscript supplies no derivation, lemmas, propositions, or arguments supporting it. Without any visible proof steps (e.g., construction of Y, use of the contact distribution, or application of Hodge theory), the result cannot be verified or assessed for correctness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the detailed feedback. We address the major comment below.

read point-by-point responses

Referee: Abstract: the central claim is stated but the manuscript supplies no derivation, lemmas, propositions, or arguments supporting it. Without any visible proof steps (e.g., construction of Y, use of the contact distribution, or application of Hodge theory), the result cannot be verified or assessed for correctness.

Authors: We agree that the version of the manuscript under review states the main result in the abstract but does not include the supporting derivations, lemmas, propositions, or explicit proof steps. This is a genuine shortcoming of the submitted draft. We will revise the manuscript to provide a complete, self-contained argument. The revised proof will construct the compact Kähler manifold Y from the given non-projective compact Kähler contact manifold X by using the contact distribution to induce a suitable foliation or quotient, then apply Hodge theory to verify that X is isomorphic to the projectivized tangent bundle PT_Y. This will make the reasoning verifiable and directly address the concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The supplied abstract states a classification theorem for non-projective compact Kähler contact manifolds without any equations, derivations, fitted parameters, or self-citations. No load-bearing steps reduce to inputs by construction, and the full manuscript text (though referenced) yields no visible internal chain that could be inspected for self-definition, renaming, or ansatz smuggling. This matches the default case of an honest non-finding when no concrete reductions are exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects standard background assumptions in complex geometry rather than paper-specific details.

axioms (2)

domain assumption A Kähler manifold is a complex manifold equipped with a Kähler metric.
Standard definition invoked implicitly by the claim.
domain assumption A contact structure on an odd-dimensional manifold is a maximally non-integrable hyperplane distribution.
Core definition of contact manifold used in the statement.

pith-pipeline@v0.9.0 · 5297 in / 1194 out tokens · 43349 ms · 2026-05-07T12:44:55.851418+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 3 canonical work pages

[1]

Kov \'a cs

Carolina Araujo, St \'e phane Druel, and S \'a ndor J. Kov \'a cs. Cohomological characterizations of projective spaces and hyperquadrics. Invent. Math. , 174(2):233--253, 2008

2008
[2]

Rational curves of minimal degree and characterizations of projective spaces

Carolina Araujo. Rational curves of minimal degree and characterizations of projective spaces. Math. Ann. , 335(4):937--951, 2006

2006
[3]

Fano contact manifolds and nilpotent orbits

Arnaud Beauville. Fano contact manifolds and nilpotent orbits. Comment. Math. Helv. , 73(4):566--583, 1998

1998
[4]

Images directes de cycles compacts par un morphisme et application \`a l'espace des cycles des tores

Fr\'ed\'eric Campana. Images directes de cycles compacts par un morphisme et application \`a l'espace des cycles des tores. In Functions of several complex variables, IV ( S em. F ran cois N orguet, 1977--1979) ( F rench) , volume 807 of Lecture Notes in Math. , pages 25--65. Springer, Berlin, 1980

1977
[5]

H¨ oring, T

Beno \^ t Claudon and Andreas H \"o ring. Projectivity criteria for K \"a hler morphisms. arXiv preprint arXiv:2404.13927 , 2024

work page arXiv 2024
[6]

Shepherd-Barron

Koji Cho, Yoichi Miyaoka, and Nicholas I. Shepherd-Barron. Characterizations of projective space and applications to complex symplectic manifolds. In Higher dimensional birational geometry ( K yoto, 1997) , volume 35 of Adv. Stud. Pure Math. , pages 1--88. Math. Soc. Japan, Tokyo, 2002

1997
[7]

On the F robenius integrability of certain holomorphic p -forms

Jean-Pierre Demailly. On the F robenius integrability of certain holomorphic p -forms. In Complex geometry ( G \" o ttingen, 2000) , pages 93--98. Springer, Berlin, 2002

2000
[8]

Structures de contact sur les vari\'et\'es alg\'ebriques de dimension 5

St\'ephane Druel. Structures de contact sur les vari\'et\'es alg\'ebriques de dimension 5. C. R. Acad. Sci. Paris S\'er. I Math. , 327(4):365--368, 1998

1998
[9]

On the structure of polarized varieties with -genera zero

Takao Fujita. On the structure of polarized varieties with -genera zero. J. Fac. Sci. Univ. Tokyo Sect. IA Math. , 22:103--115, 1975

1975
[10]

Countability of the D ouady space of a complex space

Akira Fujiki. Countability of the D ouady space of a complex space. Japan. J. Math. (N.S.) , 5(2):431--447, 1979

1979
[11]

On the D ouady space of a compact complex space in the category C

Akira Fujiki. On the D ouady space of a compact complex space in the category C . Nagoya Math. J. , 85:189--211, 1982

1982
[12]

On the D ouady space of a compact complex space in the category C

Akira Fujiki. On the D ouady space of a compact complex space in the category C . II . Publ. Res. Inst. Math. Sci. , 20(3):461--489, 1984

1984
[13]

On polarized manifolds whose adjoint bundles are not semipositive

Takao Fujita. On polarized manifolds whose adjoint bundles are not semipositive. In Algebraic geometry, S endai, 1985 , volume 10 of Adv. Stud. Pure Math. , pages 167--178. North-Holland, Amsterdam, 1987

1985
[14]

Algebraic geometry

Robin Hartshorne. Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52

1977
[15]

o ring and Thomas Peternell. Minimal models for K \

Andreas H \"o ring and Thomas Peternell. Minimal models for K \"ahler threefolds. Invent. Math. , 203(1):217--264, 2016

2016
[16]

On the canonical bundle formula and adjunction for generalized K \"ahler pairs

Christopher Hacon and Mihai Paun. On the canonical bundle formula and adjunction for generalized K \"ahler pairs. arXiv preprint arXiv:2404.12007 , 2024

work page arXiv 2024
[17]

Families of singular rational curves

Stefan Kebekus. Families of singular rational curves. J. Algebraic Geom. , 11(2):245--256, 2002

2002
[18]

Rational curves on algebraic varieties , volume 32 of Ergeb

J \'a nos Koll \'a r. Rational curves on algebraic varieties , volume 32 of Ergeb. Math. Grenzgeb., 3. Folge . Berlin: Springer-Verlag, 1996

1996
[19]

Sommese, and Jaros aw A

Stefan Kebekus, Thomas Peternell, Andrew J. Sommese, and Jaros aw A. Wi\' s niewski. Projective contact manifolds. Invent. Math. , 142(1):1--15, 2000

2000
[20]

Characterization of projective spaces and P ^ r -bundles as ample divisors

Jie Liu. Characterization of projective spaces and P ^ r -bundles as ample divisors. Nagoya Math. J. , 233:155--169, 2019

2019
[21]

A characterization of uniruled compact K \"ahler manifolds

Wenhao Ou. A characterization of uniruled compact K \"ahler manifolds. arXiv preprint arXiv: 2501.18088 , 2025

work page arXiv 2025
[22]

Towards a M ori theory on compact K \"ahler threefolds

Thomas Peternell. Towards a M ori theory on compact K \"ahler threefolds. III . Bull. Soc. Math. France , 129(3):339--356, 2001

2001