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Harmonic mappings of Kähler manifolds to locally symmetric spaces. (English) Zbl 0695.58010
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.
Reviewer: U.D’Ambrosio

##### MSC:
 58E20 Harmonic maps, etc. 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58A14 Hodge theory in global analysis 53C35 Differential geometry of symmetric spaces
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