A Calculus for Finite Parts and Residues of some Divergent Complex Geometric Integrals
Ludvig Svensson
TL;DR
This work develops a finite-part calculus for divergent complex geometric integrals on reduced spaces with singularities along a divisor defined by a holomorphic section $s$. It regularizes forms via $\|s\|^{2\lambda}$ to produce current-valued Laurent coefficients $\mu_j^{\|s\|}(\omega)$ and proves a decomposition (Theorem 1) expressing $\mu_\ell^{\|s\|}(\omega_1\wedge\cdots\wedge\omega_\kappa)$ as sums of products of explicit elementary currents, enabling reduction to tame, elementary pieces. An explicit finite-part algorithm is provided, using a tame decomposition $\omega = \omega^{\mathrm{tame}} + d\gamma$ and a recursive computation of $\mu_j^{\|s\|}$, with a detailed treatment of normal crossings divisors. The approach is illustrated on projective space $\mathbb{P}^n$, yielding concrete values for $\mathrm{fp}\int_{\mathbb{P}^n}\omega$ in terms of zeta-values and powers of $\pi$, and revealing connections to periods and $L$-functions via Beilinson/Deligne-type phenomena. Overall, the paper provides a principled, diagrammatic calculus to reduce divergent geometric integrals to computable current products and tame components.
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.