Stable Higgs bundles on compact Gauduchon manifolds

Abstract

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 , 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 Kähler manifold does not extend to compact Gauduchon manifolds. This is done by applying the above result to Γ \\ G , where Γ is a discrete torsionfree cocompact subgroup of a complex semisimple group G.