Oka manifolds
Oka manifolds
复制标题
奥卡歧管
DOI：
10.1016/j.crma.2009.07.005
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
F. Forstnerič
中科院分区：
文献类型：
--
作者：
F. Forstnerič
这是本书两个核心章节中的第一章。我们开始的历史介绍奥卡-格劳尔特原则说，拓扑分类的主要纤维丛在斯坦空间同意全纯分类。接下来，我们描述了一个减少的问题，变形家庭的连续部分家庭的全纯部分在主纤维丛Stein空间。这自然导致奥卡流形的理论。一个复流形称为Oka流形，如果欧氏空间中任一凸集的全纯映射都能用整映射逼近。主要结果是，任何分层全纯纤维丛的Oka纤维在减少斯坦空间享受所有形式的Oka原则。由于每个复齐次流形都是Oka，这推广了经典的Oka-Grauert理论。我们给出了一个完整的证明，从最简单的步骤进行到最一般的情况。我们还讨论了Oka流形的性质和例子，并给出了这类流形的几个非平凡等价刻画。
This is the first of the two core chapters of the book. We begin with a historical introduction to the Oka-Grauert principle which says that the topological classification of principal fibre bundles over Stein spaces agrees with the holomorphic classification. Next we describe a reduction to the problem of deforming families of continuous sections to families of holomorphic sections in principal fibre bundles over Stein spaces. This naturally leads to the theory of Oka manifolds. A complex manifoldis said to be an Oka manifold if every holomorphic map from any convex set in a Euclidean spacecan be approximated by entire maps. The main result is that sections of any stratified holomorphic fibre bundle with Oka fibres over a reduced Stein space enjoy all forms of the Oka principle. Since every complex homogeneous manifold is Oka, this generalizes the classical Oka-Grauert theory. We give a complete proof, proceeding in steps from the simplest to the most general case. We also discuss properties and examples of Oka manifold and give several nontrivially equivalent characterizations of this class.