Regularizations of products of residue and principal value currents

Abstract

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 1 , | f 2 | 2 = ϵ 2 } is the residue integral I f 1 , f 2 φ ( ϵ 1 , ϵ 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 ∂ ¯ χ 1 ∧ ∂ ¯ χ 2 ∧ φ / ( f 1 f 2 ) , where χ j = χ j ( | f j | 2 / ϵ 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 ∂ ¯ χ 2 ∧ φ / ( f 1 f 2 ) is Hölder continuous and tends to the ∂ ¯ -potential [ ( 1 / f 1 ) ∧ ∂ ¯ ( 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.