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MiS 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

Abstract

Let $\overline S$ be a compact Riemann surface (holomorphic curve) of genus $g$. Let $p_1, p_2, \cdots, p_s$ be $s>0$ points on it; these points define a divisor, and we denote the open Riemann surface ${\overline S}\setminus\{p_1, \dots, p_s\}$ by $S$.

When $3g-3+s>0$, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points $p_1, p_2, \cdots, p_s$ 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 $\omega$. Denote the inclusion map of $S$ in $\overline S$ by $j$. Let $\rho: \pi_1(S)\to Sl(n, \mathbb{C})$ be a semisimple linear representation of $\pi_1(S)$ which is unipotent near the cusps (for the precise definition, cf. \S2.1). Corresponding to such a representation $\rho$, one has a local system $L_{\rho}$ over $S$ and a $\rho$-equivariant harmonic map $h: S\to Sl(n, \mathbb{C})/SU(n)$ 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 \S2.2). This harmonic map can be considered as a Hermitian metric on $L_{\rho}$---harmonic metric---so that we have a so-called harmonic bundle $(L_{\rho}, h)$ [13]. Such a bundle carries interesting structures, e.g. a Higgs bundle structure $(E, \theta)$, where $\theta=\partial h$, and it has a $\log$-singularity at the divisor.

The purpose of this note is to investigate various cohomologies of $\overline S$ 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 $j_*L_{\rho}$ (note that in the higher dimensional case, one needs to consider the corresponding intersection cohomology [3]), the $L^2$-cohomology, the $L^2$-Dolbeault cohomology, and the $L^2$-Higgs cohomology, and the relationships between them. Here, $L^2$ is defined by using the Poincaré(-like) metric $\omega$ 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.