Linking numbers and boundaries of varieties

The intersection index at a common point of two analytic varieties of complementary dimensions in 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 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 C3 which is disjoint from M. Consider the linking number 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 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 link(M,A) ≥ 0 if M bounds a positive holomorphic 2-chain. Recall that a holomorphic k-chain in Ω ⊆ Cn is a sum ∑nj [Vj ] where {Vj} is a locally finite family of irreducible k-dimensional subvarieties of Ω and nj ∈ Z and that the holomorphic 2-chain is 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.