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

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Submission date: 09. May. 2005
Pages: 21
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Let formula19 be a compact Riemann surface (holomorphic curve) of genus g. Let formula23 be s>0 points on it; these points define a divisor, and we denote the open Riemann surface formula27 by S. When 3g-3+s>0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points formula23 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 formula39. Denote the inclusion map of S in formula19 by j. Let formula47 be a semisimple linear representation of formula49 which is unipotent near the cusps (for the precise definition, cf. §2.1). Corresponding to such a representation formula51, one has a local system formula53 over S and a formula51-equivariant harmonic map formula59 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 formula53--harmonic metric--so that we have a so-called harmonic bundle formula63 [13]. Such a bundle carries interesting structures, e.g. a Higgs bundle structure formula65, where formula67, and it has a formula69-singularity at the divisor.

The purpose of this note is to investigate various cohomologies of formula19 with degenerating coefficients formula53 (considered as a local system -- a flat vector bundle, a Higgs bundle, or a formula75-module, depending on the context): the Cech cohomology of formula77 (note that in the higher dimensional case, one needs to consider the corresponding intersection cohomology [3]), the formula79-cohomology, the formula79-Dolbeault cohomology, and the formula79-Higgs cohomology, and the relationships between them. Here, formula79 is defined by using the Poincaré(-like) metric formula39 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.

04.09.2019, 14:40