Regularized Integrals on Configuration Spaces of Riemann Surfaces and Cohomological Pairings

We extend the notion of regularized integrals introduced by Li-Zhou that aims to assign finite values to divergent integrals on configuration spaces of Riemann surfaces. We then give cohomological formulations for the extended notion using the tools of current cohomology and mixed Hodge structures. We also provide practical ways of constructing representatives of the corresponding cohomology classes in terms of smooth differential forms.