Linking numbers and boundaries of varieties

Abstract

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.