Cohomologies of unipotent harmonic bundles over quasi-projective varieties I: The case of noncompact curves

Let $S$ be a Riemann surface obtained by deleting a finite number of points, called cusps, from a compact Riemann surface. Let $\rho: \pi_1(S)\to Sl(n, $\mathbb{C}$)$beasemisimplelinearrepresentationof$\pi_1(S)$ which is unipotent near the cusps. We investigate various cohomologies associated to $\rho$ of $\bar{S}$ with degenerating coefficients $L_{\rho }$ (considered as a local system – a flat vector bundle, a Higgs bundle, or a $\mathcal{D}$-module, depending on the context): the \v{C}ech cohomology of $j_{\ast }{L}_{\rho }$, the $L^2$-cohomology, the $L^2$-Dolbeault cohomology, and the $L^2$-Higgs cohomology, and the relationships between them. This paper is meant to be a part of the general program of studying cohomologies with degenerating coefficients on quasiprojective varieties and their K"ahlerian generalizations. The general aim here is not restricted to the case of curves nor to the one of representations that are unipotent near the divisor. The purpose of this note therefore is to illuminate at this particular case where many of the (analytic and geometric) difficulties of the general case are not present what differences will appear when we consider unipotent harmonic bundles instead of Variations of Hodge Structures where the results are known.