Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler
Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler
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DOI：
10.1007/bf01159161
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发表时间：
1985-03
影响因子：
0.8
通讯作者：
A. Balas;P. Gauduchon
中科院分区：
文献类型：
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作者：
A. Balas;P. Gauduchon
如果函数K在整个酉切丛上为常数，则称全纯截面曲率为常数。全纯截面曲率可以给出M的复结构信息。例如，假设K处处为负且从零有界，则复流形在S的意义下必须是双曲的。小林（[6]，第61页）。与K~ ihler情形相反，全纯截面曲率并不决定整个Hermite曲率，而只决定其Kghler部分（[1]，定理2.2）。另一方面，当K被假定为常数时，似乎可能性非常小。到目前为止，唯一已知的非K ~ thler例子是平坦的（在Hermitian意义下）（[1]，命题3.1）。本文的主要目的是证明：当M的复维数为2时，K常数为负或为零的Hermite度量实际上是K/ihler（定理1）。
The holomorphic sectional curvature is said to be constant if the function K is constant on the whole unitary tangent bundle. The holomorphic sectional curvature may give information about the complex structure of M. For example, if K is assumed to be everywhere negative and bounded from zero, the complex manifold has to be hyperbolic in the sense of S. Kobayashi ([6], p. 61). In contrast with the K~ ihler case the holomorphic sectional curvature doesn't determine the whole Hermitian curvature, but only its Kghler part ([1], Theorem 2.2). On the other hand, when K is assumed to be constant, it seems that the possibilities are very few. Up to now, the only known non-K~ thler examples are fiat (in the Hermitian sense)([1], Proposition 3.1). The main purpose of this paper is to prove that if the complex dimension of M is 2, a Hermitian metric with K constant negative or zero actually is K/ihler (Theorem 1).