Compact Hermitian manifolds of constant holomorphic sectional curvature
Compact Hermitian manifolds of constant holomorphic sectional curvature
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DOI：
10.1007/bf01175044
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发表时间：
1985-06
影响因子：
0.8
通讯作者：
A. Balas
中科院分区：
文献类型：
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作者：
A. Balas
虽然常全纯截面曲率的紧致K~ihler流形已被分类，但对更一般的Hermite情形知之甚少。S. Kobayshi证明了具有负全纯截面曲率的紧致Hermitian流形在小林（小林[5]）的意义下是双曲的，这意味着从复平面到流形的每个全纯函数都是常数。但很少有其他结果是已知的。本文是对常全纯截面曲率的紧致厄米流形的一个尝试。第一个问题是，是否有任何这样的流形不是K~ihler。我们通过给出曲率为零的紧致非K~ihler流形的例子来肯定地回答这个问题，在2以上的每个维度上（命题（3.1））。这些流形的形式是G/F，其中G是一个非交换李群，F是一个离散子群。我们不知道是否有全纯截面曲率为非零常数的例子，但很容易证明一个常全纯截面曲率的厄米特流形具有s + g常数，其中s和s是厄米特几何的两个截面曲率（推论（2.4））。利用S.小林、H. Wu和P.Gauduchon，我们证明了一个紧致的Hermitian流形不能有s +~等于一个非负常数，除非它的plurigenus满足一定的条件。这就引出了本文的主要结果，推论（3.5）：设M是具有常全纯截面曲率= k的紧致埃尔米特流形。设Pm是M的第m个复亏格，Qm是M的第m个对偶复亏格.然后
Although compact K~ ihler manifolds of constant holomorphic sectional curvature have been classified, little is known of the more general Hermitian case. S. Kobayshi has shown that a compact Hermitian manifold with negative holomorphic sectional curvature bounded away from zero, is hyperbolic in the sense of Kobayashi (Kobayashi [5]), which implies that every holomorphic function from the complex plane to the manifold is constant. But few other results are known. This paper is an attempt, to extend our knowledge of compact Hermitian manifolds of constant holomorphic sectional curvature. The first question that arises is whether there are any such manifolds that are not K~ ihler. We answer this in the affirmative by giving examples of compact non-K~ ihler manifolds of vanishing curvature, in every dimension above 2 (Proposition (3.1)). These manifolds are of the form G/F, where G is a non-abelian Lie group, and F is a discrete subgroup. It is not known whether there are such examples where the holomorphic sectional curvature is a nonzero constant.It is easy to show that a Hermitian manifold of constant holomorphic sectional curvature has s+ g constant, where s and s are the two sectional curvatures of Hermitian geometry (Corollary (2.4)). Using the methods of S. Kobayashi, H. Wu, and P. Gauduchon, we show that a compact Hermitian manifold cannot have s+~ equal to a nonnegative constant unless its plurigenera meet certain conditions. This leads to the main result of this paper, Corollary (3.5): Let M be a compact Hermitian manifold of constant holomorphic sectional curvature= k. Let Pm be the mth plurigenus and Qm be the mth dual plurigenus of M. Then