A characterization of semisimple local system by tame pure imaginary pluri-harmonic metric

Details A characterization of semisimple local system by tame pure imaginary pluri-harmonic metric A characterization of semisimple local system by tame pure imaginary pluri-harmonic metric

2004-02-08

arxiv.org/abs/math/0402122v1

Mathematics

Differential Geometry

Scientific paper

Let $L$ be a local system on a smooth quasi projective variety over $\cnum$. We see that $L$ is semisimple if and only if there exists a tame pure imaginary pluri-harmonic metric on $L$. Although it is a rather minor refinement of a result of Jost and Zuo, it is significant for the study of harmonic bundles and pure twistor $D$-modules. As one of the application, we show that the semisimplicity of local systems are preserved by the pull back via a morphism of quasi projective varieties. This paper will be included as an appendix to our previous paper, ``Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules'', math.DG/0312230. Keywords: Higgs fields, harmonic bundle, semisimple local system.

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