On automorphism groups of compact Kähler manifolds
On automorphism groups of compact Kähler manifolds
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DOI：
10.1007/bf01403162
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发表时间：
1978-10
影响因子：
3.1
通讯作者：
A. Fujiki
中科院分区：
文献类型：
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作者：
A. Fujiki
设X是一个紧复流形。那么X的生物全纯自同构群Aut X具有生物全纯作用于X的复李群的自然结构（cf.[11]）。设Aut o X是Aut X的恒等式的连通分量。如果X是射影代数，则Aut o X具有代数作用于X的代数群的自然结构。因此，特别地，利用Chevalley关于代数群的基本结构定理[23]，可以得到Aut o X是线性代数群对阿贝变换的扩展。另一方面，有一些迹象表明，即使X是紧化Kfihler流形，类似结构定理仍然成立（参见[1,2,5,17,19,26,27]）。本文的目的是从解析几何的角度研究这一问题，推广代数几何问题，并以似乎是适当的形式证明预期结构定理（定理5.5）。即我们证明如下：设X是紧化的Kdhler流形。因此，AutoX具有作用于X的亚纯群的天然结构，并且存在着AutoX的唯一亚纯子群L (X)，它与一个线性代数群亚纯同构，使得商T (X)= AutoX/L (X)是复环面。这是亚纯的
Let X be a compact complex manifold. Then the group of biholomorphic automorphisms of X, Aut X, has the natural structure of a complex Lie group acting biholomorphically on X (cf.[11]). Let Aut o X be the connected component of the identity of Aut X. If X is projective algebraic, then Aut o X has the natural structure of an algebraic group acting algebraically on X. Hence in particular by the fundamental structure theorem of Chevalley on algebraic groups [23] it is obtained as an extension of an abelian variety by a linear algebraic group. On the other hand, there are some indications that the analogous structure theorem still holds even if X is a compact Kfihler manifold (cf.[1, 2, 5, 17, 19, 26, 27]). The purpose of the present paper is to study this problem from analyticgeometric point of view, generalizing the algebro-geometric one, and to prove the expected structure theorem in what seems to be a proper form (Theorem 5.5).Namely we show the following: Let X be a compact Kdhler manifold. Then Aut o X has the natural structure of a meromorphic group acting biholomorphically and meromorphically on X. Further there exists a unique meromorphic subgroup, L (X), of Aut o X, which is meromorphically isomorphic to a linear algebraic group and such that the quotient T (X)= AutoX/L (X) is a complex torus. Here, a meromorphic