Regularizations of products of residue and principal value currents

Journal article, 2006

Let f 1 and f 2 be two functions on some complex n-manifold and let φ be a test form of bidegree (n, n - 2). Assume that (f 1 , f 2 ) defines a complete intersection. The integral of φ / (f 1 f 2 ) on {| f 1 | 2 = ε{lunate} 1 , | f 2 | 2 = ε{lunate} 2 } is the residue integral I f1 , f 2 φ (ε{lunate} 1 , ε{lunate} 2 ). It is in general discontinuous at the origin. Let χ 1 and χ 2 be smooth functions on [0, ∞] such that χ j (0) = 0 and χ j (∞) = 1. We prove that the regularized residue integral defined as the integral of over(∂, ̄) χ 1 ∧ over(∂, ̄) χ 2 ∧ φ / (f 1 f 2 ), where χ j = χ j (| f j | 2 / ε{lunate} 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 χ 1 over(∂, ̄) χ 2 ∧ φ / (f 1 f 2 ) is Hölder continuous and tends to the over(∂, ̄)-potential [(1 / f 1 ) ∧ over(∂, ̄) (1 / f 2 )] of the Coleff-Herrera current, acting on φ. More generally, let f 1 and f 2 be sections of some vector bundles and assume that f 1 ⊕ f 2 defines a complete intersection. There are associated principal value currents U f and U g and residue currents R f and R g . The residue currents equal the Coleff-Herrera residue currents locally. One can give meaning to formal expressions such as e.g. U f ∧ R g 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.