Hermitian symmetric space, flat bundle and holomorphicity criterion

Abstract

Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $\Gamma \subset G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kähler manifold and $\rho :{\pi }_1(X,x_0)⟶\Gamma$ a homomorphism such that the adjoint action of $\rho ({\pi }_1(X,x_0))$ on $\text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $\rho$ a harmonic map $X⟶K\backslash G/\Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.

Final version