Regularizations of products of residue and principal value currents

Let f1 and f2 be two functions on some complex n-manifold and let ϕ be a test form of bidegree (n, n− 2). Assume that (f1,f2) defines a complete intersection. The integral of ϕ/(f1f2) on {|f1|2 = 1, |f2|2 = 2} is the residue integral I ϕ f1,f2 ( 1, 2). It is in general discontinuous at the origin. Let χ1 and χ2 be smooth functions on [0,∞] such that χj (0) = 0 and χj (∞) = 1. We prove that the regularized residue integral defined as the integral of ¯∂χ1 ∧ ¯∂χ2 ∧ ϕ/(f1f2), where χj = χj (|fj |2/ j ), is Hölder continuous on the closed first quarter and that the value at zero is the Coleff–Herrera residue current acting on ϕ. In fact, we prove that if ϕ is a test form of bidegree (n, n − 1) then the integral of χ1 ¯∂χ2 ∧ ϕ/(f1f2) is Hölder continuous and tends to the ¯∂-potential [(1/f1) ∧ ¯∂(1/f2)] of the Coleff– Herrera current, acting on ϕ. More generally, let f1 and f2 be sections of some vector bundles and assume that f1 ⊕f2 defines a complete intersection. There are associated principal value currents Uf and Ug and residue currents Rf and Rg. The residue currents equal the Coleff–Herrera residue currents locally. One can give meaning to formal expressions such as e.g. Uf ∧Rg in such a way that formal Leibnitz rules hold. Our results generalize to products of these currents as well. © 2006 Elsevier Inc. All rights reserved.