Brendon Lasell, M. Ramachandran
Annales Scientifiques De L Ecole Normale Superieure
Using facts about harmonic maps from Kahler manifolds to symmetric spaces, to buildings, and to Hilbert spaces, we prove results about the homomorphisms of fundamental groups induced by certain kinds of morphisms of Kahler manifolds. Our results are stated in terms of linear representations of these fundamental groups. Our main result, theorem 4.1, may be viewed as a non-abelian generalization of a fact following from the existence of functorial mixed Hodge structures on the complex cohomology of complex algebraic varieties. Our work is also related to Shafarevich's question: "is perhaps the universal covering of a complete algebraic variety holomorphically convex?" .