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 K\"ahler manifold and $\rho\, :\, \pi_1(X, x_0)\,\longrightarrow\, \Gamma$ 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 $X\, \longrightarrow\, K\backslash G/\Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.