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

We consider divergent integrals ${\int }_X\omega$ of certain forms $\omega$ on a reduced pure-dimensional complex space $X$. The forms $\omega$ 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 $\mathrm{f}\mathrm{p}{\int }_X\omega$ of the divergent integral as the action of a certain current extension of $\omega$. We introduce a current calculus to compute finite parts for a special class of $\omega$. Our main result is a formula that decomposes the finite part of such an $\omega$ into sums of products of explicit currents. Lastly, we show that, in principle, it is possible to reduce the computation of $\mathrm{f}\mathrm{p}{\int }_X\omega$ for a general $\omega$ to this class.