Holomorphic Chains and the Support Hypothesis Conjecture

Let Ω be a complex manifold and let V be a (holomorphic) subvariety of Ω of pure (complex) dimension k. Then integration over V defines a closed current of dimension 2k in Ω, denoted by [V ]. More generally, let {Vj} be a locally finite family of irreducible holomorphic subvarieties of Ω of pure dimension k and let {nj} be integers. Then T = ∑ nj [Vj ] is a 2kcurrent in Ω—these are the holomorphic k-chains in Ω. Holomorphic chains are particular examples of the locally rectifiable currents of H. Federer [F]. A locally rectifiable s-current T in an open subset Ω of R can be described as follows. There is a locally (H, s)-rectifiable set B in Ω and an s-vector field η on B which is locallyH-integrable over B such that forH-almost all x ∈ B, η(x) is a simple s-vector that represents the approximate tangent space to B at x and ||η(x)|| is a positive integer—the multiplicity of T at x. Then the s-current T is given by T (φ) = ∫ B〈φ, η〉dH s for all s-forms φ with compact support in Ω. In C, this description can be refined. The s-forms can be decomposed into sums of (a, b) forms with a + b = s. A 2k-current has bidimension (k, k) if T (φ) = 0 for all forms φ of type (a, b) with (a, b) 6= (k, k). If T has bidimension (k, k) it follows that the approximate tangent space to B is for H-almost all x ∈ B a complex k-dimensional linear space. If the natural orientation of this complex linear space agrees with the orientation induced by η(x) for H-almost all x ∈ B, one says that the (k, k) current T is positive. The space of all locally rectifiable (k, k) currents on Ω is denoted by R (k,k)(Ω). Thus every holomorphic k-chain is a closed locally rectifiable (k, k) current. We shall consider the converse. The first result in this direction was due to King [K]. We denote by Ω an open subset of C—however, being local, all of the results discussed below hold on complex manifolds.