On Kähler manifolds with non-negative mixed curvature

Jianchun Chu; Man-Chun Lee; Jintian Zhu

doi:10.1515/crelle-2025-0054

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On Kähler manifolds with non-negative mixed curvature

Jianchun Chu , Man-Chun Lee und Jintian Zhu

Veröffentlicht/Copyright: 21. August 2025

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2025 Heft 827

Abstract

In this work, we investigate compact Kähler manifolds with non-negative or quasi-positive mixed curvature coming from a linear combination of the Ricci and holomorphic sectional curvature, which covers various notions of curvature considered in the literature. Specifically, we prove a splitting theorem, analogous to the Cheeger–Gromoll splitting theorem, for complete Kähler manifolds with non-negative mixed curvature containing a line, and then establish a structure theorem for compact Kähler manifolds with non-negative mixed curvature. We also show that the Hodge numbers of compact Kähler manifolds with quasi-positive mixed curvature must vanish. Both results are based on the conformal perturbation method.

Funding source: National Key Research and Development Program of China

Award Identifier / Grant number: 2024YFA1014800

Award Identifier / Grant number: 2023YFA1009900

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12471052

Award Identifier / Grant number: 12271008

Award Identifier / Grant number: 12222122

Funding source: Research Grants Council, University Grants Committee

Award Identifier / Grant number: 24304222

Award Identifier / Grant number: 14300623

Funding statement: The first-named author was partially supported by National Key R&D Program of China 2024YFA1014800 and 2023YFA1009900, and NSFC grants 12471052 and 12271008. The second-named author was partially supported by Hong Kong RGC grant (Early Career Scheme) of Hong Kong No. 24304222, No. 14300623, and an NSFC grant No. 12222122. The third-named author was partially supported by National Key R&D Program of China 2023YFA1009900 as well as the startup fund from Westlake University.

A Conformal calculation

Lemma A.1

Let ( M n , g , J ) be a Kähler manifold and suppose that g ̂ = e − 2 ⁢ h ⁢ g for some f ∈ C ∞ ⁢ ( M ) . Then the following holds.

| ∇ ̂ ⁢ J | g ̂ 2 ≤ C n ⁢ e 2 ⁢ h ⁢ | ∇ h | g 2 for some dimensional constant C n .
For 𝑔-unit vector 𝑣, define g ̂ -unit vector v ̂ = e h ⁢ v . Then
C α , β , g ̂ ⁢ ( v ̂ ) = e 2 ⁢ h ( C α , β , g ( v ) + α Δ h + ( 2 n α − 2 α + β ) ∇ 2 h ( v , v ) + β ∇ 2 h ( J v , J v ) + ( 2 n α − 2 α + β ) v ( h ) 2 + β J v ( h ) 2 − ( 2 n α − 2 α + β ) | ∇ h | 2 ) .

Here ∇ ̂ denotes the Levi-Civita connection of g ̂ and C α , β , g ̂ denotes the mixed curvature of g ̂ defined in Definition 2.1.

Proof

Denote the curvature and Ricci tensor of g ̂ by R ̂ and Ric ̂ . For any 𝑔-unit vector 𝑤, define the g ̂ -unit vector w ̂ = e h ⁢ w . Direct calculation shows

∇ ̂ v ⁢ w − ∇ v w = − v ⁢ ( h ) ⁢ w − w ⁢ ( h ) ⁢ v + g ⁢ ( v , w ) ⁢ ∇ h

and so

∇ ̂ v ̂ ⁢ w ̂ = ∇ ̂ e h ⁢ v ⁢ ( e h ⁢ w ) = e 2 ⁢ h ⁢ ( v ⁢ ( h ) ⁢ w + ∇ ̂ v ⁢ w ) = e 2 ⁢ h ⁢ ( ∇ v w − w ⁢ ( h ) ⁢ v + g ⁢ ( v , w ) ⁢ ∇ h ) .

Replacing w ̂ by J ⁢ w ̂ , ∇ ̂ v ̂ ⁢ ( J ⁢ w ̂ ) = e 2 ⁢ h ⁢ ( ∇ v ( J ⁢ w ) − J ⁢ w ⁢ ( h ) ⁢ v + g ⁢ ( v , J ⁢ w ) ⁢ ∇ h ) . We compute

( ∇ ̂ v ̂ ⁢ J ) ⁢ w ̂ = ∇ ̂ v ̂ ⁢ ( J ⁢ w ̂ ) − J ⁢ ( ∇ ̂ v ̂ ⁢ w ̂ ) = e 2 ⁢ h ⁢ ( ∇ v ( J ⁢ w ) − J ⁢ w ⁢ ( h ) ⁢ v + g ⁢ ( v , J ⁢ w ) ⁢ ∇ h ) − e 2 ⁢ h ⁢ J ⁢ ( ∇ v w − w ⁢ ( h ) ⁢ v + g ⁢ ( v , w ) ⁢ ∇ h ) = e 2 ⁢ h ⁢ ( ( ∇ v J ) ⁢ w − J ⁢ w ⁢ ( h ) ⁢ v + g ⁢ ( v , J ⁢ w ) ⁢ ∇ h + w ⁢ ( h ) ⁢ J ⁢ v − g ⁢ ( v , w ) ⁢ J ⁢ ∇ h ) .

Combining this with ∇ J = 0 (which follows from the Kählerity of 𝑔), we obtain (a).

For (b), when g ⁢ ( v , w ) = 0 , it is well known that

R ̂ ⁢ ( v ̂ , w ̂ , w ̂ , v ̂ ) = e 2 ⁢ h ⁢ ( R ⁢ ( v , w , w , v ) + ∇ 2 h ⁢ ( v , v ) + ∇ 2 h ⁢ ( w , w ) + v ⁢ ( h ) 2 + w ⁢ ( h ) 2 − | ∇ h | 2 ) .

It then follows that

Ric ̂ ⁢ ( v ̂ , v ̂ ) = e 2 ⁢ h ⁢ ( Ric ⁢ ( v , v ) + Δ ⁢ h + ( 2 ⁢ n − 2 ) ⁢ ∇ 2 h ⁢ ( v , v ) − ( 2 ⁢ n − 2 ) ⁢ | ∇ h | 2 + ( 2 ⁢ n − 2 ) ⁢ v ⁢ ( h ) 2 )

and

R ̂ ⁢ ( v ̂ , J ⁢ v ̂ , J ⁢ v ̂ , v ̂ ) = e 2 ⁢ h ( R ( v , J v , J v , v ) + ∇ 2 h ( v , v ) + ∇ 2 h ( J v , J v ) + v ( h ) 2 + J v ( h ) 2 − | ∇ h | 2 ) .

Combining the above, we obtain (b). ∎

Acknowledgements

The authors would like to thank Valentino Tosatti for discussion. The authors would also like to thank Xiaokui Yang for his interest. The authors would like to thank the referee for useful comments for improving the readability.

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Received: 2025-03-06

Revised: 2025-06-14

Published Online: 2025-08-21

Published in Print: 2025-10-01

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