  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 $\Omega \subseteq \Bbb C^n$ is a sum $\sum n_j [V_j]$
where $\{V_j\}$ is a locally finite family of irreducible $k$-dimensional
subvarieties of $\Omega$ 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.
