# Preprint 46/2005

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

### Jürgen Jost, Yi-Hu Yang, and Kang Zuo

**Contact the author:** Please use for correspondence this email.**Submission date: **09. May. 2005**Pages: 21****Bibtex****Download full preprint:** PDF (221 kB), PS ziped (197 kB)**Abstract:**

Let be a compact Riemann surface (holomorphic curve) of genus *g*. Let be *s*>0 points on it; these points define a divisor, and we denote the open Riemann surface
by *S*.
When 3*g*-3+*s*>0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points 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 . Denote the
inclusion map of *S* in by *j*. Let be a semisimple linear representation of
which is unipotent near the cusps (for the precise
definition, cf. §2.1). Corresponding to such a representation
, one has a local system over *S* and a
-equivariant harmonic map
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 --harmonic metric--so that we have a
so-called harmonic bundle [13]. Such a
bundle carries interesting structures, e.g. a Higgs bundle structure
, where , and it has a
-singularity at the divisor.

The purpose of this note is to investigate various cohomologies of
with degenerating coefficients (considered as a local system -- a flat vector bundle, a Higgs
bundle, or a -module, depending on the context): **the Cech cohomology of **
(note that in the higher dimensional case, one needs to consider
the corresponding intersection cohomology [3]), **the
-cohomology, the -Dolbeault cohomology, and the
-Higgs cohomology, and the relationships between them**. Here,
is defined by using the Poincaré(-like) metric and
the harmonic metric *h*. We want to generalize the results [15] valid for the case of variations of Hodge structures (VHS)
to the case of harmonic bundles, as was suggested by Simpson
[13]; in principle, in view of our assumption on the representations in
question being unipotent, the situation should be similar to the
case of VHS.

This paper is meant to be a part of the general program of studying cohomologies with degenerating coefficients on quasiprojective varieties and their Kählerian 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 VHSs; for the case of VHSs, the various cohomologies have been considered by various authors [1, 10, 14, 9] and are well understood by now.