2000/08/03 by H. Alexander, John Wermer
Mathematics · #math.CV #msc:32S #msc:57M
published as Ann. of Math. (2) 151 (2000), no. 1, 125--150 · 26 pages
arxiv created 2000/08/03 · arxiv updated 2009/11/30
The intersection index at a common point of two analytic varieties of complementary dimensions in \Bbb Cn 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 C3. 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 C3 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) ≥ 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) ≥ 0 if M bounds a positive holomorphic 2-chain. Recall that a \it holomorphic k-chain in Ω⊆ \Bbb Cn is a sum ∑ nj [Vj] where \Vj\ is a locally finite family of irreducible k-dimensional subvarieties of Ω and nj ∈ \Bbb Z and that the holomorphic 2-chain is \it positive if nj >0 for all j. Our first result is that, conversely, the nonnegativity of the linking number characterizes boundaries of positive holomorphic 2-chains.