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?" [14].

Theorem 4.1 shows that the universal covering of a complete algebraic variety is holomorphically convex, proving a non-abelian generalization of the fact that functorial mixed Hodge structures exist on complex cohomology of complex algebraic varieties.

Observations on harmonic maps and singular varieties

Key Takeaway: Theorem 4.1 shows that the universal covering of a complete algebraic variety is holomorphically convex, proving a non-abelian generalization of the fact that functorial mixed Hodge structures exist on complex cohomology of complex algebraic varieties.