The fundamental group, rational connectedness and the positivity of Kähler manifolds

Lei Ni

doi:10.1515/crelle-2020-0040

Published by De Gruyter December 16, 2020

The fundamental group, rational connectedness and the positivity of Kähler manifolds

Lei Ni

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2020-0040

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Abstract

Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric ⊥ > 0 must be simply-connected by applying a new viscosity consideration to Whitney’s comass of ( p , 0 ) -forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition Ric k > 0 (for some k ∈ { 1 , … , n } , with Ric n being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of [L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, preprint 2020, https://arxiv.org/abs/1804.09696]. Thirdly, motivated by Ric ⊥ and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing H q ⁢ ( N , T ′ ⁢ N ) = { 0 } for any 1 ≤ q ≤ n and a deformation rigidity result are obtained under these new curvature conditions. In particular, they are verified for all classical Kähler C-spaces with b 2 = 1 . The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

Funding statement: The research is partially supported by “Capacity Building for Sci-Tech Innovation-Fundamental Research Funds”.

A Appendix: Estimates on the harmonic ( 1 , 1 ) -forms of low rank

We prove a vanishing theorem for harmonic ( 1 , 1 ) -forms of low rank related to the condition QB k > 0 introduced earlier. This is particularly relevant given that in [38] examples of arbitrary large b 2 was constructed with CQB > 0 (in particular with Ric ⊥ > 0 ). First recall that

QB R ⁢ ( A ) = ∑ α , β = 1 n R ⁢ ( A ⁢ ( E α ) , A ⁢ ( E α ) ¯ , E β , E ¯ β ) - R ⁢ ( E α , E ¯ β , A ⁢ ( E β ) , A ⁢ ( E α ) ¯ )

vanishes for A = λ ⁢ id . Hence when define QB k ⁢ ( A ) > 0 we require the above expression positive for A in S 2 ⁢ ( ℂ n ) ∖ { λ ⁢ id } , and that A has rank not greater than k. The space of harmonic ( 1 , 1 ) -forms ℋ ∂ ¯ 1 , 1 can be decomposed further. First we observe that an ( 1 , 1 ) -form

Ω = - 1 ⁢ A i ⁢ j ¯ ⁢ d ⁢ z i ∧ d ⁢ z j ¯

can be decomposed as

Ω = Ω 1 - - 1 ⁢ Ω 2 = - 1 2 ⁢ B i ⁢ j ¯ ⁢ d ⁢ z i ∧ d ⁢ z j ¯ - - 1 ⁢ ( - 1 2 ⁢ C i ⁢ j ¯ ⁢ d ⁢ z i ∧ d ⁢ z j ¯ )

with

B i ⁢ j ¯ = A i ⁢ j ¯ + A j ⁢ i ¯ ¯ , C i ⁢ j ¯ = - 1 ⁢ ( A i ⁢ j ¯ - A j ⁢ i ¯ ¯ ) .

If Ω is harmonic, then ∂ ⁡ Ω = ∂ ¯ ⁢ Ω = 0 . It can be verified that Ω 1 and Ω 2 are both harmonic (cf. [30, Theorem 5.4 in Chapter 3]). This shows that Ω can be decomposed into the sum of a Hermitian symmetric one with - - 1 times another Hermitian symmetric one. Namely,

ℋ ∂ ¯ 1 , 1 = ℋ ∂ ¯ , s 1 , 1 - - 1 ⁢ ℋ ∂ ¯ , s 1 , 1 ,

where ℋ ∂ ¯ , s 1 , 1 is the spaces of harmonic Ω with ( A i ⁢ j ¯ ) being Hermitian symmetric. Within ℋ ∂ ¯ , s 1 , 1 we shall consider ℋ ∂ ¯ , s 1 , 1 ∖ { ℂ ⁢ ω } . To prove b 2 = 1 under the assumption QB > 0 , it suffices to show that

ℋ ∂ ¯ , s 1 , 1 ∖ { ℂ ⁢ ω } = { 0 } .

We can stratify the space into ones with rank bounded from above. Let ℋ s , k 1 , 1 denote the subspace of ℋ ∂ ¯ , s 1 , 1 which consists of

Ω = - 1 2 ⁢ A i ⁢ j ¯ ⁢ d ⁢ z i ∧ d ⁢ z j ¯

with ( A i ⁢ j ¯ ) being Hermitian symmetric and of rank no greater than k everywhere on N. The following result can be shown.

Theorem 1.

Assume that ( N n , g ) is a compact Kähler manifold with quasi-positive QB k with k < n . Then H s , k 1 , 1 ⁢ ( N ) = { 0 } . In particular, Ric ⊥ > 0 implies that H s , 1 1 , 1 ⁢ ( N ) = { 0 } .

Proof.

Assume that Ω is a nonzero element in ℋ s , k 1 , 1 ⁢ ( N ) . Applying the Δ operator to ∥ Ω ∥ 2 , by the Kodaira–Bochner formula we have that

1 2 ⁢ ( ∇ γ ⁡ ∇ γ ¯ + ∇ γ ¯ ⁡ ∇ γ ) ⁢ ∥ Ω ∥ 2 ⁢ ( x ) = ∥ ∇ γ ⁡ Ω ∥ 2 ⁢ ( x ) + ∥ ∇ γ ¯ ⁡ Ω ∥ 2 ⁢ ( x ) + 2 ⁢ Q ⁢ B ⁢ ( Ω ) ⁢ ( x ) .

Integrating on N, we have that

0 = ∫ N [ ∥ ∇ γ ⁡ Ω ∥ 2 ⁢ ( x ) + ∥ ∇ γ ¯ ⁡ Ω ∥ 2 ⁢ ( x ) ] ⁢ 𝑑 μ ⁢ ( x ) + 2 ⁢ ∫ N QB ⁢ ( Ω ) ⁢ ( x ) ⁢ 𝑑 μ ⁢ ( x ) > 0 .

The last strictly inequality is due to the fact that by the unique continuation we know at a neighborhood U where QB k > 0 , and Ω is identically zero. The contradiction implies that Ω ≡ 0 . ∎

For any holomorphic line bundle L over N with a Hermitian metric a, its first Chern form

c 1 ⁢ ( L , a ) = - - 1 2 ⁢ ∂ ⁡ ∂ ¯ ⁢ log ⁡ a

is a Hermitian symmetric ( 1 , 1 ) -form. If η is the harmonic representative of c 1 ⁢ ( L , a ) , then η is Hermitian symmetric by the uniqueness of the Hodge decomposition and Kähler identities (cf. [30, Chapter 3]). The following is a simple observation towards possible topological meanings of the rank of η (the minimum k such that η ∈ ℋ ∂ ¯ , k 1 , 1 , denoted as rk ⁡ ( L ) ).

Proposition A.1.

Recall that the numerical dimension of L is defined as

nd ⁡ ( L ) = max ⁡ { k = 0 , … , n : c 1 ⁢ ( L ) k ≠ 0 } .

Then rk ⁡ ( L ) ≥ nd ⁡ ( L ) .

The proof of the above theorem also shows that if QB k ≥ 0 , then any element in ℋ s , k 1 , 1 ⁢ ( N ) must be parallel. Thus we have the dimension estimate

dim ⁡ ( ℋ s , k 1 , 1 ⁢ ( N ) ) ≤ k 2 .

In fact, the existence of a non-vanishing ( 1 , 1 ) -form of rank at most k has a strong implication due to the De Rham decomposition.

Corollary 2.

Assume that QB k ≥ 0 and H s , k 1 , 1 ⁢ ( N ) ≠ { 0 } . Then N must be locally reducible. In particular, if N is locally irreducible and Ric ⊥ ≥ 0 , then H s , 1 1 , 1 ⁢ ( N ) = { 0 } .

Proof.

By the above, we know that the nonzero Ω ∈ ℋ s , k 1 , 1 ⁢ ( N ) must be parallel. Its null space is invariant under the parallel transport. This provides a nontrivial parallel distribution, hence the local splitting. ∎

The product example ℙ 2 × ℙ 2 , which satisfies Ric ⊥ > 0 and supports nontrivial rank 2 harmonic ( 1 , 1 ) -forms, shows that the above result is sharp for Ric ⊥ > 0 . Irreducible examples of dimension greater than 4 were constructed via the projectivized bundles in [36].

Acknowledgements

The author would like to thank James McKernan for helpful discussions, L.-F. Tam and F. Zheng and for their interests. F. Zheng read the draft carefully, spotted a discrepancy and made helpful suggestions. The author also thank B. Wilking for the connection between ( 2 ⁢ k - 1 ) -Ricci and Ric k . The first version of the paper was completed during the author’s visit of CUHK, Fuzhou Normal University, Peking University, SUST and Xiamen University, in December 2018. He thanks these institutions for the hospitality and the referee for very helpful comments.

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Received: 2020-09-18

Revised: 2020-09-26

Published Online: 2020-12-16

Published in Print: 2021-05-01

The fundamental group, rational connectedness and the positivity of Kähler manifolds

Abstract

A Appendix: Estimates on the harmonic ( 1 , 1 ) -forms of low rank

Theorem 1.

Proof.

Proposition A.1.

Corollary 2.

Proof.

Acknowledgements

References

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