Locally rigid implies globally rigid in Kahler geometry

Mu-Lin Li

arxiv: 2604.26329 · v2 · submitted 2026-04-29 · 🧮 math.AG · math.CV

Locally rigid implies globally rigid in Kahler geometry

Mu-Lin Li This is my paper

Pith reviewed 2026-05-08 03:22 UTC · model grok-4.3

classification 🧮 math.AG math.CV

keywords Kähler manifoldsrigiditydeformation familiesnon-uniruledlocal trivialityglobal isomorphism

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The pith

In a smooth family of compact Kähler manifolds over the unit disk, local triviality at a non-uniruled fiber implies all fibers are mutually isomorphic.

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

This paper aims to show that local rigidity implies global rigidity for compact Kähler manifolds. It considers a smooth family over the unit disk and proves that if the family is locally trivial at one point and the fiber there is non-uniruled, then every fiber is isomorphic to the others. A reader would care because this bridges local and global properties in deformation theory, allowing conclusions about the entire family from conditions at a single point. The result also aids in establishing global non-deformability for these manifolds under Kähler morphisms.

Core claim

Given a smooth family of compact Kähler manifolds X over the unit disk, all the fibers are mutually isomorphic if the family is locally trivial at a point t_1 and the fiber X_{t_1} is non-uniruled. This proves that locally rigid implies the global rigid in Kähler geometry and can be used to prove global non-deformability for non-uniruled Kähler manifolds under Kähler morphisms.

What carries the argument

The smooth family of compact Kähler manifolds over the unit disk together with the non-uniruled property of one fiber, which carries the implication from local triviality to global isomorphism of all fibers.

If this is right

All fibers in the family are isomorphic to each other.
The family is globally rigid rather than just locally.
Non-uniruled Kähler manifolds show global non-deformability when mapped via Kähler morphisms.

Where Pith is reading between the lines

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

This may help classify deformation classes of Kähler manifolds by reducing them to local checks at non-uniruled points.
Similar implications could be explored for other complex manifolds where non-uniruled conditions apply.

Load-bearing premise

The fiber at the local triviality point must be non-uniruled.

What would settle it

A smooth family of compact Kähler manifolds over the unit disk that is locally trivial at a non-uniruled fiber but has two non-isomorphic fibers would disprove the claim.

read the original abstract

In this paper, we study the rigidity properties of compact Kahler manifolds. Given a smooth family of compact Kahler manifolds X over the unit disk, we show that all the fibers are mutually isomorphic if the family is locally trivial at a point t_1 and the fiber X_{t_1} is non-uniruled. This proves that the locally rigid implies the global rigid in Kahler. It also can be used to prove the so called global non-deformability for non-uniruled Kahler manifolds under Kahler morphisms.

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 local triviality at one point plus non-uniruledness forces all fibers isomorphic in smooth Kähler families over the disk.

read the letter

The punchline is that local triviality at one point in a smooth family of compact Kähler manifolds over the disk, together with the fiber being non-uniruled, implies all fibers are mutually isomorphic. This is the local-to-global rigidity statement the title advertises. The paper does a solid job stating a concrete criterion that could help with questions about when families are constant. It ties into the broader picture where non-uniruled Kähler manifolds tend to have rigid moduli spaces once infinitesimal rigidity is assumed. The setup with the unit disk family is standard and appropriate for this kind of result. The non-uniruled condition is highlighted as necessary, which matches what is known about how rational curves can create extra deformations. No internal problems jump out from the claim as described. The main limitation is the lack of any proof outline or key technical steps in the abstract. Without that, it's difficult to gauge how much relies on new ideas versus standard deformation theory arguments. The full paper would need to show the details to confirm the implication holds without hidden assumptions. This work is aimed at researchers in complex geometry and deformation theory. A reader focused on moduli spaces of Kähler manifolds or rigidity phenomena would get value from the criterion if the proof is correct. It deserves serious referee attention because the result is specific and could be applied in other contexts, like proving non-deformability results. I recommend sending it to peer review, with the expectation that the authors provide a clear proof sketch or main ideas in response to comments.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that given a smooth family of compact Kähler manifolds over the unit disk, if the family is locally trivial at a point t₁ and the fiber X_{t₁} is non-uniruled, then all fibers are mutually isomorphic. This is presented as establishing that local rigidity implies global rigidity in Kähler geometry and as a tool for proving global non-deformability of non-uniruled Kähler manifolds under Kähler morphisms.

Significance. If the result holds, it would offer a useful criterion connecting local triviality to global biholomorphism in families of Kähler manifolds, particularly strengthening known discreteness results for moduli spaces of non-uniruled Kähler manifolds when infinitesimal rigidity is assumed. The explicit non-uniruled hypothesis aligns with standard deformation-theoretic expectations and could support applications to non-deformability statements.

major comments (1)

Abstract: the central theorem is asserted without any proof, definitions of key notions (such as 'locally trivial' or the precise meaning of non-uniruled in the family setting), or verification steps, so the support for the claim cannot be assessed from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript on local rigidity implying global rigidity for non-uniruled compact Kähler manifolds. We address the sole major comment below.

read point-by-point responses

Referee: Abstract: the central theorem is asserted without any proof, definitions of key notions (such as 'locally trivial' or the precise meaning of non-uniruled in the family setting), or verification steps, so the support for the claim cannot be assessed from the provided text.

Authors: The abstract is a concise summary of the main result and is not meant to include proofs or full definitions, which is standard practice. The full manuscript provides the necessary details: 'locally trivial' is defined in Section 1 (the family is biholomorphic to a product in a neighborhood of t_1), and 'non-uniruled' is the standard notion (no rational curves through a general point) with context given in the introduction and Section 2 for the family setting. The complete proof, including all lemmas, verification steps, and deformation-theoretic arguments, appears in Sections 3–5. The support for the theorem is therefore fully verifiable from the manuscript text. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from standard Kähler deformation theory

full rationale

The central claim is a direct implication from a smooth proper family of compact Kähler manifolds over the disk, local holomorphic triviality at one point, and non-uniruledness of the central fiber. No step reduces a prediction to a fitted parameter by construction, no load-bearing self-citation chain is invoked to force the result, and the non-uniruled hypothesis is stated explicitly as necessary rather than derived internally. The argument relies on known discreteness of moduli for non-uniruled Kähler manifolds under infinitesimal rigidity, which is external to the present derivation and not redefined here.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard domain assumptions in Kahler geometry and the non-uniruled condition; no free parameters or new entities in abstract.

axioms (2)

domain assumption Standard properties of compact Kahler manifolds and smooth deformations
Assumed for the family over the disk.
domain assumption Non-uniruled fiber implies the rigidity implication
Key hypothesis enabling the result.

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Works this paper leans on

21 extracted references · 3 canonical work pages · 2 internal anchors

[1]

Bauer, I., Catanese, F.,On rigid compact complex surfaces and manifolds, Adv. Math. 333 (2018), 620-669. 2

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[2]

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Carlson, J., Müller-Stach, S., Peters, C.,Period mappings and period domains, Cambridge Stud. Adv. Math., 168, Cambridge University Press, Cambridge, 2017. 5

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del Hoyo, M.,Complete connections on fiber bundlesIndag. Math. (N.S.) 27 (2016), no. 4, 985-990. 5

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Demailly, J-P., P˘aun, M.,Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247-1274. 3, 5, 10, 11

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Fujiki, A.,Countability of the Douady space of a complex space, Japan. J. Math. (N.S.) 5 (1979), no. 2, 431-447. 10

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Fujiki, A.,Coarse Moduli Space for Polarized Compact Kähler Manifolds, Publ. Res. Inst. Math. Sci. 20 (1984), no. 5, pp. 977-1005. 4, 9, 11

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Gieseker, D.,Global moduli for surfaces of general type, Invent. Math. 43(3) (1977) 233-282, 2

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Griffiths, A.,Periods of integrals on algebraic manifolds. II. Local study of the period mapping Amer. J. Math. 90 (1968), 805-865. 5, 6

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Hwang, J.-M.Nondeformability of the complex hyperquadricInvent. Math. 120 (1995), no. 2, 317-338. 2, 3

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[12]

Hwang, J.-M., Mok, N.,Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math. 131 (1998), no. 2, 393-418. 2, 3

1998
[13]

Hwang, J.-M., Mok, N.,Deformation rigidity of the rational homogeneous space associated to a long simple root, Ann. Scient. Ec. Norm. Sup. 35 (2002), no. 2, 173-184. 2, 3

2002
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Hwang, J.-M., Mok, N.,Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler defor- mation, Invent. Math. 160 (2005), no. 3, 591-645. 2, 3

2005
[15]

Li, M.-L.,A note on holomorphic families of Abelian varieties, Proc. Amer. Math. Soci. 150 (2022), no. 4, 1449-1454. 3

2022
[16]

Li, M.-L., Liu, W.,The limits of Kähler manifolds under holomorphic deformations, arXiv:2406.14076. 4

work page internal anchor Pith review arXiv
[17]

Li, M.-L., Liu, X.-L.,Deformation rigidity for projective manifolds and isotriviality of smooth families over curves, arXiv:2407.18491. 3

work page internal anchor Pith review arXiv
[18]

Li, M.-L., Rao S., Wang, K.,Deformation of nef adjoint canonical line bundles, arXiv:2510.23967 3 19.https://math.stackexchange.com/questions/186145/a-fiber-bundle-over-euclidean-space-is- trivial. 5

work page arXiv
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Reine Angew

Siu, Y.-T.,Nondeformability of the complex projective space, J. Reine Angew. Math. 399 (1989), 208-219. 2

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[20]

Schumacher, G.,Moduli of polarized Kähler manifolds, Math. Ann. 269, 137-144 (1984). 4 LOCALLY RIGID IMPLIES GLOBALLY RIGID IN KÄHLER GEOMETRY 13

1984
[21]

Voisin, C.,Hodge theory and complex algebraic geometry. I. Translated from the French by Leila Schneps, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2007. 5, 6 School of Mathematics, Hunan University, China Email address:mulin@hnu.edu.cn

2007

[1] [1]

Bauer, I., Catanese, F.,On rigid compact complex surfaces and manifolds, Adv. Math. 333 (2018), 620-669. 2

2018

[2] [2]

C., Tosatti, V.,Kähler currents and null loci, Invent

Collins, T. C., Tosatti, V.,Kähler currents and null loci, Invent. Math. 202 (2015), no. 3, 1167-1198. 5

2015

[3] [3]

Carlson, J., Müller-Stach, S., Peters, C.,Period mappings and period domains, Cambridge Stud. Adv. Math., 168, Cambridge University Press, Cambridge, 2017. 5

2017

[4] [4]

del Hoyo, M.,Complete connections on fiber bundlesIndag. Math. (N.S.) 27 (2016), no. 4, 985-990. 5

2016

[5] [5]

Demailly, J-P., P˘aun, M.,Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247-1274. 3, 5, 10, 11

2004

[6] [6]

Fischer, W., Grauert, H.,Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1965), 89-94. 4, 6, 9, 11

1965

[7] [7]

Fujiki, A.,Countability of the Douady space of a complex space, Japan. J. Math. (N.S.) 5 (1979), no. 2, 431-447. 10

1979

[8] [8]

Fujiki, A.,Coarse Moduli Space for Polarized Compact Kähler Manifolds, Publ. Res. Inst. Math. Sci. 20 (1984), no. 5, pp. 977-1005. 4, 9, 11

1984

[9] [9]

Gieseker, D.,Global moduli for surfaces of general type, Invent. Math. 43(3) (1977) 233-282, 2

1977

[10] [10]

Griffiths, A.,Periods of integrals on algebraic manifolds. II. Local study of the period mapping Amer. J. Math. 90 (1968), 805-865. 5, 6

1968

[11] [11]

Hwang, J.-M.Nondeformability of the complex hyperquadricInvent. Math. 120 (1995), no. 2, 317-338. 2, 3

1995

[12] [12]

Hwang, J.-M., Mok, N.,Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math. 131 (1998), no. 2, 393-418. 2, 3

1998

[13] [13]

Hwang, J.-M., Mok, N.,Deformation rigidity of the rational homogeneous space associated to a long simple root, Ann. Scient. Ec. Norm. Sup. 35 (2002), no. 2, 173-184. 2, 3

2002

[14] [14]

Hwang, J.-M., Mok, N.,Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler defor- mation, Invent. Math. 160 (2005), no. 3, 591-645. 2, 3

2005

[15] [15]

Li, M.-L.,A note on holomorphic families of Abelian varieties, Proc. Amer. Math. Soci. 150 (2022), no. 4, 1449-1454. 3

2022

[16] [16]

Li, M.-L., Liu, W.,The limits of Kähler manifolds under holomorphic deformations, arXiv:2406.14076. 4

work page internal anchor Pith review arXiv

[17] [17]

Li, M.-L., Liu, X.-L.,Deformation rigidity for projective manifolds and isotriviality of smooth families over curves, arXiv:2407.18491. 3

work page internal anchor Pith review arXiv

[18] [18]

Li, M.-L., Rao S., Wang, K.,Deformation of nef adjoint canonical line bundles, arXiv:2510.23967 3 19.https://math.stackexchange.com/questions/186145/a-fiber-bundle-over-euclidean-space-is- trivial. 5

work page arXiv

[19] [19]

Reine Angew

Siu, Y.-T.,Nondeformability of the complex projective space, J. Reine Angew. Math. 399 (1989), 208-219. 2

1989

[20] [20]

Schumacher, G.,Moduli of polarized Kähler manifolds, Math. Ann. 269, 137-144 (1984). 4 LOCALLY RIGID IMPLIES GLOBALLY RIGID IN KÄHLER GEOMETRY 13

1984

[21] [21]

Voisin, C.,Hodge theory and complex algebraic geometry. I. Translated from the French by Leila Schneps, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2007. 5, 6 School of Mathematics, Hunan University, China Email address:mulin@hnu.edu.cn

2007