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Hodge theory with degenerating coefficients: \(L_2\) cohomology in the PoincarĂ© metric. (English) Zbl 0446.14002
Let \(\bar S\) be a smooth algebraic curve over \(\mathbb C\) and \(S\) an open subset of \(\bar S\). Let there be a locally constant system \(V\) on \(S\) of complex vector spaces underlying a polarizable variation of Hodge structures (over \(\mathbb R)\) of weight \(m\). Such a system is a Hodge theory with degenerating coefficients.
The main result of the paper is that there exists a natural polarizable Hodge structure of weight \(m+i\) on \(H^i(\bar S, j_*V)\), \(j\colon S\to \bar S\) the inclusion. This Hodge structure is constructed by using \(L_2\)-differential forms with respect to a suitable metric on \(S\). The applications include: the author’s theorem on normal functions relating to the Hodge conjecture [Invent. Math. 33, 185–222 (1976; Zbl. 329.14008)]; the Hodge structures of Shimura on the cohomology of Fuchsian groups.
In the last section \(H^*(\bar S, j_*V)\) is compared with \(H^*(X,\mathbb C)\) for a smooth family \(f\colon X\to S\) of varieties giving rise to the local system \(V\).
Sections 13, 14 deal with some associated mixed Hodge structures and cohomology with compact support.

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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