  Let $M$ be a compact complex manifold equipped with a Gauduchon metric. If
$TM$ is holomorphically trivial, and (V, \theta) is a stable SL(r,{\mathbb
C})-Higgs bundle on $M$, then we show that $\theta= 0$. 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/\Gamma, where
$\Gamma$ is a discrete torsionfree cocompact subgroup of a complex semisimple
group $G$.
