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Kodaira dimension and Chern hyperbolicity of the Shafarevich maps for representations of \(\pi_ 1\) of compact Kähler manifolds. (English) Zbl 0838.14017

We prove that the Shafarevich variety is of general type and Chern hyperbolic for any Zariski dense representation of \(\pi_1\) of a compact Kähler manifold into an almost simple algebraic group. As a consequence we show that any reductive representation of \(\pi_1\) of a compact Kähler manifold of algebraic dimension or Kodaira dimension equal to zero always splits into a direct sum of one-dimensional representations after passing to a finite étale covering.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14J45 Fano varieties
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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