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

Let S be a compact Riemann surface (holomorphic curve) of genus g. Let p1, p2, · · · , ps be s > 0 points on it; these points define a divisor, and we denote the open Riemann surface S \ {p1, . . . , ps} by S. When 3g− 3+ s > 0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points p1, p2, · · · , ps then become cusps at infinity. Even in the remaining cases, that is, for a once or twice punctured sphere, we can equip S with a metric that is hyperbolic in the vicinity of the cusp(s), and for our purposes, the behavior of the metric there is all what counts, and we call such a metric Poincaré-like. In any case, our metric on S is denoted by ω. Denote the inclusion map of S in S by j. Let ρ : π1(S) → Sl(n,C) be a semisimple linear representation of π1(S) which is unipotent near the cusps (for the precise definition, cf. §2.1). Corresponding to such a representation ρ, one has a local system Lρ over S and a ρ-equivariant harmonic map h : S → Sl(n,C)/SU(n) with a certain special growth condition near the divisor. For the present case of complex dimension 1, this is elementary; it also follows from the general result of [6], see also the remark in §2.2). This harmonic map can be considered as a Hermitian metric on Lρ—harmonic metric—so that we have a so-called harmonic bundle (Lρ, h) [13]. Such a bundle carries interesting structures, e.g. a Higgs bundle structure (E, θ), where θ = ∂h, and it has a log-singularity at the divisor. The purpose of this note is to investigate various cohomologies of S with degenerating coefficients Lρ (considered as a local system — a flat vector bundle, a Higgs bundle, or a D-module, depending on the context): the Čech cohomology of j∗Lρ (note that in the higher dimensional case, one needs to