Representations of fundamental groups and logarithmic symmetric differential forms

. — In this paper we develop some non-abelian Hodge techniques over complex quasi-projective manifolds X in both archimedean and non-archimedean con-texts. In the non-archimedean case, we first generalize a theorem by Gromov-Schoen: for any Zariski dense representation ρ : π 1 ( X ) → G ( K ) , where G is a semisimple algebraic group defined over some non-archimedean local field K , we construct a ρ equivariant harmonic map from X into the Bruhat-Tits building ∆( G ) of G with some suitable asymptotic behavior. We then construct logarithmic symmetric differential forms over X when the image of such ρ is unbounded. Our main result in the archimedean case is that any semisimple representation σ : π 1 ( X ) → GL N ( C ) is rigid provided that X does not admit logarithmic symmetric differential forms. Fur-thermore, such representation σ is conjugate to σ ′ : π 1 ( X ) → GL N ( O L ) where O L is the ring of integer of some number field L , so that σ ′ is a complex direct factor of a Z -variation of Hodge structures. As an application we prove that a complex quasi-projective manifold X has nonzero global logarithmic symmetric differential forms if there is linear representation π 1 ( X ) → GL N ( C ) with infinite images.