Harmonic mappings of Kähler manifolds to locally symmetric spaces (Q910070)

Define the following extension of Gromov ordering for manifolds M and N not necessarily of the same dimension: \(M\geq N\) means the existence of a continuous map f: \(M\to N\) which is surjective in homology. Now let M be a compact Kähler manifold and N be a compact locally symmetric space of the form \(\Gamma\) \(\setminus G/K\), where G is semisimple Lie group with compact factors, K is a maximal compact subgroup and \(\Gamma\) is a cocompact discrete subgroup. The main result is to show that \(M\geq N\) is impossible unless N is already Kähler in an obvious way, i.e., is locally Hermitian symmetric.