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Hodge theory with local coefficients on compact varieties. (English) Zbl 0755.14001

A functorial mixed Hodge structure is constructed on the cohomology groups \(H^ i(X,L)\) of compact analytic spaces \(X\) which are bimeromorphic to Kähler manifolds with coefficients in local systems \(L\) of real vector spaces arising from orthogonal representations of \(\pi_ 1(X)\). Some striking applications are given. Here are two of them:
(1) if \(X\) is smooth and \(f:X\to C\) is a surjective holomorphic map to a smooth curve then \(R^ if_ *{\mathcal L}\) is locally free for any line bundle \({\mathcal L}\) which is algebraically equivalent to zero; and
(2) if \(X\) is a compact Kähler manifold which maps holomorphically onto a smooth curve of genus at least 2 then \(\pi_ 1(X)'/\pi_ 1(X)''\) is not finitely generated.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F35 Homotopy theory and fundamental groups in algebraic geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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