Harmonic maps of compact Kähler manifolds to exceptional local symmetric spaces of Hodge type and holomorphic liftings to complex homogeneous fibrations

Abstract

Let $M$ be a compact $K\ddot{a}hler$ manifold and $G/K$ be a non-Hermitian Riemannian symmetric space of Hodge type. Certain harmonic maps $f : M \rightarrow \Gamma \setminus G / K$ will be proved to admit holomorphic liftings $F_p : M \rightarrow \Gamma \setminus G / G \cap P$ to complex homogeneous fibrations, where $P$ are parabolic subgroups of $G^\mathbb{C}$. The work studies whether the images $F_P(M)=\Gamma_h \setminus G_h/K_h$ are local equivariantly embedded Hermitian symmetric subspaces of $\Gamma \setminus G / G \cap P$. For each of the cases examples of harmonic maps $f$ which do not holomorphic liftings are supplied.