A Calculus for Finite Parts and Residues of some Divergent Complex Geometric Integrals

Abstract

We consider divergent integrals $\int_X ω$ of certain forms $ω$ on a reduced pure-dimensional complex space $X$. The forms $ω$ are singular along a subvariety defined by the zero set of a holomorphic section $s$ of some holomorphic vector bundle $E$. Equipping $E$ with a smooth Hermitian metric allows us to define a finite part $\mathrm{fp}\,\int_X ω$ of the divergent integral as the action of a certain current extension of $ω$. We introduce a current calculus to compute finite parts for a special class of $ω$. Our main result is a formula that decomposes the finite part of such an $ω$ into sums of products of explicit currents. Lastly, we show that, in principle, it is possible to reduce the computation of $\mathrm{fp}\,\int_X ω$ for a general $ω$ to this class.