Linking numbers and boundaries of varieties

Details Linking numbers and boundaries of varieties Linking numbers and boundaries of varieties

2000-08-03

arxiv.org/abs/math/0008033v1

Ann. of Math. (2) 151 (2000), no. 1, 125--150

Mathematics

Complex Variables

26 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Linking numbers and boundaries of varieties does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Linking numbers and boundaries of varieties, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Linking numbers and boundaries of varieties will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-560268