Abstract. We study the deformations of twisted harmonic maps f with respect to the representation ρ. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of f in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional E coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kähler form of the "Betti" moduli space; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of E at critical points.
Abstract. We investigate representations of Kähler groups Γ " π 1 pXq to a semisimple non-compact Hermitian Lie group G that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor-Wood inequality similar to those found by Burger-Iozzi and Koziarz-Maubon. Thanks to the study of the case of equality in Royden's version of the Ahlfors-Schwarz Lemma, we can completely describe the case of maximal holomorphic representations. If dim C X ě 2, these appear if and only if X is a ball quotient, and essentially reduce to the diagonal embedding Γ ă SUpn, 1q Ñ SUpnq, qq ãÑ SUpp, qq. If X is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, that thus appear as preferred elements of the respective maximal connected components.
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