Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

TL;DR

This paper proves a correspondence between tame harmonic bundles and stable parabolic Higgs bundles, establishes a Bogomolov-Gieseker inequality, and applies these results to deformation theory and fundamental group constraints.

Contribution

It extends the Kobayashi-Hitchin correspondence to tame harmonic bundles and applies it to deformation and group-theoretic problems in algebraic geometry.

Findings

Established the Kobayashi-Hitchin correspondence for tame harmonic bundles.

Proved a Bogomolov-Gieseker type inequality for stable parabolic Higgs bundles.

Showed that certain groups cannot be fundamental groups of smooth quasi-projective varieties.

Abstract

We establish the correspondence between tame harmonic bundles and $\mu_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $\mu_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can conclude that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasi projective variety.