The Kobayashi distance induces the standard topology
The Kobayashi distance induces the standard topology
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小林距离推导出标准拓扑
DOI：
10.1090/s0002-9939-1972-0306545-x
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发表时间：
1972
期刊：
影响因子：
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通讯作者：
Theodore J. Barth
中科院分区：
文献类型：
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作者：
Theodore J. Barth
连通复空间上的Kobayashi伪距离相对于标准拓扑是连续的。这个伪距离是一个实际距离，它诱导出标准拓扑。设X是连通(约简)复空间。X的点p和q之间的Kobayashi伪距离DX(p，q)定义如下([6，p.462]，[7，pp.97-98])。设p表示复平面中开单位圆盘D上由PoincareBergman度量定义的距离。选择点P，PO，扑通‘.Pk-1Pk=Q，点A1，**，Ak，bl，***，bk，以及从D到X的全纯映射fl，*，*fk，使得当j=1，k时，fj(Aj)=pj-L，fj(Bj)=pj，则dx(p，q)定义为数p(a，bl)+…+p(Ak，Bk)接管了所有连接p和q的有限链。显然，dx是X上的伪距离。如果dx是一个实际距离，我们将证明它诱导出标准拓扑，即X上给定复杂结构背后的拓扑。几位作者默许了这一事实；请参阅[1，命题3.8，第68-69页]，[4，命题1，第50-51页]，[5，定理2和3，第590-591页]，[6，定理3.4，第465-466页]，[8，定理1，第11-12页]。然而，唯一发表的证明似乎是H.L.罗伊登[9，定理2，第133-134页]给出的关于复流形的证明。我们的证明仅使用了关于小林伪距离的三个基本事实：如果f：Y-Z是全纯映射，则对Y[6，命题2.1，p.462]中的所有p和q，dz(f(P)，f(Q))<；dy(p，q)是距离递减的；如果q=Dn是多圆，则Dq关于Q[7，p.47]上的标准拓扑是连续的。特别地，我们避免了罗伊登的微分度量。定理。设X是连通复空间。则小林伪距DX是连续的。如果DX是一个实际距离，则它推导出X上的标准拓扑。编辑于1971年12月30日收到。AMS 1970学科分类。主要是32h15，32h20。
The Kobayashi pseudodistance on a connected complex space is continuous with respect to the standard topology. IC this pseudodistance is an actual distance, it induces the standard topology. Let X be a connected (reduced) complex space. The Kobayashi pseudodistance dx(p, q) between points p and q of X is defined as follows ([6, p. 462], [7, pp. 97-98]). Let p denote the distance defined by the PoincareBergman metric on the open unit disk D in the complex plane. Choose points P PO,PloP' . Pk-1 pk=q of X, points a1,, * * , ak, bl, * * *, bk of D, and holomorphic maps fl, *, *fk from D into X such that fj(aj)= Pj-l and fj(bj)=pj for j= 1, , k. Then dx(p, q) is defined to be the infimum of the numbers p(a,, bl)+... + p(ak, bk) taken over all such finite chains joining p and q. Clearly dx is a pseudodistance on X. In case dx is an actual distance we will show that it induces the standard topology, i.e., the topology underlying the given complex structure on X. Several authors have tacitly used this fact; cf. [1, Proposition 3.8, pp. 68-69], [4, Proposition 1, pp. 50-51], [5, Theorems 2 and 3, pp. 590-591], [6, Theorem 3.4, pp. 465-466], [8, Theorem 1, pp. 11-12]. The only published proof, however, seems to be the one given by H. L. Royden [9, Theorem 2, pp. 133-134] for complex manifolds. Our proof uses only three elementary facts about the Kobayashi pseudodistance: if f: Y--Z is a holomorphic map, it is distance decreasing in the sense that dz(f(p),f (q))<dy(p, q) for all p and q in Y [6, Proposition 2.1, p. 462]; dD= p [6, Proposition 2.2, p. 462]; if Q=Dn is a polydisk, then dQ is continuous with respect to the standard topology on Q [7, p. 47]. In particular, we avoid Royden's differential metric. THEOREM. Let X be a connected complex space. Then the Kobayashi pseudodistance dx is continuous. If dx is an actual distance, it induces the standard topology on X. Received by the editors December 30, 1971. AMS 1970 subject classifications. Primary 32H15, 32H20.