Stable Higgs bundles on compact Gauduchon manifolds

Let $M$ be a compact complex manifold equipped with a Gauduchon metric. If $TM$ is holomorphically trivial, and (V, θ) is a stable SL(r,{\mathbb C})-Higgs bundle on $M$, then we show that $θ= 0$. We show that the correspondence between Higgs bundles and representations of the fundamental group for a compact Kaehler manifold does not extend to compact Gauduchon manifolds. This is done by applying the above result to G/Γ, where $Γ$ is a discrete torsionfree cocompact subgroup of a complex semisimple group $G$.

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