品种的环绕数和界限
Linking numbers and boundaries of varieties
摘要：
$\Bbb C^n$中两个互补维数解析簇在公共点处的交指数为正。这种观察，这已被称为“基石”的代数几何（[GH，p.~ 62]），是一个简单的后果，事实上，分析品种进行自然的方向。在环绕数方面进行重铸，这是我们的主要动机。这意味着：设$M$是$\Bbb C^3$中的光滑定向紧致3-流形。假设$M$有界于一个有界复2-簇$V$。在这里，“界限”的意思是，在斯托克斯定理的意义上，即，即${b[V]}={[M]}$作为电流。设$A$是$\Bbb C^3$中与M不相交的代数曲线。考虑$M$和$A$的链接号${\rm link}(M,A)$。由于该连接数等于$V$和$A$的交集数（即交集索引之和），通过这些交集索引的正性，我们得到${\rm link}(M,A) \geq 0$。如果$V$和$A$不相交，则链接数当然为0。(As $A$并不紧凑，"链接数"的用法将在后面说明。）这个推理更一般地表明，如果$M$有界于一个正的全纯2-链，则${\rm link}(M,A) \geq 0$。回想一下，在$Ω\subseteq \Bbb C^n$中的{ \it全纯$k$ -链}是一个和$\sum n_j [V_j]$，其中$\{V_j\}$是$Ω$和$n_j \in \Bbb Z$的不可约$k$ -维子簇的局部有限族，如果对所有$j$都是$n_j >0$，则全纯2-链是{ \it正的}.我们的第一个结果是，相反，非负的连接数的特征边界的正全纯2-链。
英文摘要：
The intersection index at a common point of two analytic varieties of complementary dimensions in $\Bbb C^n$ is positive. This observation, which has been called a ``cornerstone'' of algebraic geometry ([GH, p.~62]), is a simple consequence of the fact that analytic varieties carry a natural orientation. Recast in terms of linking numbers, it is our principal motivation. It implies the following: Let $M$ be a smooth oriented compact 3-manifold in $\Bbb C^3$. Suppose that $M$ bounds a bounded complex 2-variety $V$. Here ``bounds'' means, in the sense of Stokes' theorem, i.e., that ${b[V]}={[M]}$ as currents. Let $A$ be an algebraic curve in $\Bbb C^3$ which is disjoint from M. Consider the linking number ${\rm link}(M,A)$ of $M$ and $A$. Since this linking number is equal to the intersection number (i.e. the sum of the intersection indices) of $V$ and $A$, by the positivity of these intersection indices, we have ${\rm link}(M,A) \geq 0$. The linking number will of course be 0 if $V$ and $A$ are disjoint. (As $A$ is not compact, this usage of ``linking number'' will be clarified later.) This reasoning shows more generally that ${\rm link}(M,A) \geq 0$ if $M$ bounds a positive holomorphic 2-chain. Recall that a {\it holomorphic $k$-chain} in $Ω\subseteq \Bbb C^n$ is a sum $\sum n_j [V_j]$ where $\{V_j\}$ is a locally finite family of irreducible $k$-dimensional subvarieties of $Ω$ and $n_j \in \Bbb Z$ and that the holomorphic 2-chain is {\it positive} if $n_j >0$ for all $j$. Our first result is that, conversely, the nonnegativity of the linking number characterizes boundaries of positive holomorphic 2-chains.
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