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Observations on harmonic maps and singular varieties. (English) Zbl 0855.58018
Using facts about harmonic maps from Kähler manifolds to symmetric spaces, to buildings, and to Hilbert spaces, the authors prove results about the homomorphisms of fundamental groups induced by certain kinds of morphisms of Kähler manifolds. The obtained results are stated in terms of linear representations of these fundamental groups. The main result from the paper, given by 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. The present work is also related to Shafarevich’s question: “Is perhaps the universal covering of a complete algebraic variety holomorphically convex?” [I. R. Shafarevich, ‘Basic algebraic geometry’, translated by K. A. Hirsch, Springer-Verlag, Berlin (1977; Zbl 0362.14001)].

MSC:
58E20 Harmonic maps, etc.
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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