On compact splitting complex submanifolds of quotients of bounded symmetric domains
On compact splitting complex submanifolds of quotients of bounded symmetric domains
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关于有界对称域商的紧分裂复子流形
DOI：
10.1007/s11425-016-9033-0
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发表时间：
2017-02
期刊：
影响因子：
--
通讯作者：
Sui-Chung Ng
中科院分区：
文献类型：
--
作者：
Ngaiming Mok;Sui-Chung Ng
研究不可约有界商流形X = Ω/Γ的紧复子流形S。对称域由自同构的无扭离散格构成，我们感兴趣的是表征。紧分裂复子流形S∧X中的全测地线子流形，即在。假设S上的切序列是全纯分裂的。我们证明了两类结果。第一个。S∧X是具有特征的复子流形，也就是说，将Ω嵌入为一个开放的。它的紧偶流形M的子集通过Borel嵌入，非零(1;0)-向量正切。到S在泛覆盖映射\pi： Ω→X的局部逆下提升到M的极小有理切线。我们证明了紧化特征复子流形S∧X在S是分裂复子流形时必然是完全测地线。我们的证明推广了由Mok（2005）得到的复单位球Bn的商的完全测地线复子流形的性质。这里给出的证明是。它是基于厄米全纯矢量子束曲率的单调性。并利用切序列的分裂来识别。将全纯切束TS作为商束而不是作为约束的子束。全纯切束TX到s。第二类结果是关于总测地线的表征。紧分裂复子流形中的子流形S∧X由Aubin（1978）的结果推导出来。和Yau(1978)，它暗示了S∧x上存在柯赫勒-爱因斯坦度量，我们证明了这个紧化。分割出足够大维度（取决于Ω）的复子流形S∧X必然是完全测地线的。这个证明依赖于与TS相关的全纯向量束的埃尔米特-爱因斯坦性质。这意味着这些束的自同态是平行的，以及它们的自同态的构造。用切线序列在s上的分裂得到了向量束。对于秩为i型的定义域，在dim(S)上保证S∧X的总测地线的尖锐下界。2和iv型域的情况，并检验一个对这两个猜想都至关重要的情况，即在四维李球的商的紧复表面上，即四维i型域对偶。到C^4中2平面的格拉斯曼方程。
We study compact complex submanifolds S of quotient manifolds X = Ω/Γ of irreducible bounded.symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization.of the totally geodesic submanifolds among compact splitting complex submanifolds S ⊂ X, i.e., under the.assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first.type of results concerns S ⊂ X which are characteristic complex submanifolds, i.e., embedding Ω as an open.subset of its compact dual manifold M by means of the Borel embedding, the non-zero (1; 0)-vectors tangent.to S lift under a local inverse of the universal covering map \pi : Ω → X to minimal rational tangents of M..We prove that a compact characteristic complex submanifold S ⊂ X is necessarily totally geodesic whenever S.is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic.complex submanifolds of quotients of the complex unit ball Bn obtained by Mok (2005). The proof given here is.however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles.of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify.the holomorphic tangent bundle TS as a quotient bundle rather than as a subbundle of the restriction of the.holomorphic tangent bundle TX to S. The second type of results concerns characterization of total geodesic.submanifolds among compact splitting complex submanifolds S ⊂ X deduced from the results of Aubin (1978).and Yau (1978) which imply the existence of K¨ahler-Einstein metrics on S ⊂ X. We prove that compact.splitting complex submanifolds S ⊂ X of sufficiently large dimension (depending on Ω) are necessarily totally.geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to TS,.which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these.vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the.sharp lower bound on dim(S) guaranteeing total geodesy of S ⊂ X for the case of the type-I domains of rank.2 and the case of type-IV domains, and examine a case which is critical for both conjectures, viz. on compact.complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual.to the Grassmannian of 2-planes in C^4.
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影响因子：
4.9
作者：
E. Calabi;E. Vesentini
通讯作者：
E. Calabi;E. Vesentini
DOI：
10.1007/978-3-0348-7486-1
发表时间：
1987
期刊：
--
影响因子：
--
作者：
Y. Siu
通讯作者：
Y. Siu
影响因子：
3
作者：
S. Yau
通讯作者：
S. Yau
DOI：
--
发表时间：
2013-04
期刊：
--
影响因子：
--
作者：
J. Hano
通讯作者：
J. Hano
DOI：
10.1007/bf02884700
发表时间：
2005-12
期刊：
Science in China Series A: Mathematics
影响因子：
--
作者：
N. Mok
通讯作者：
N. Mok