×

zbMATH — the first resource for mathematics

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.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
PDF BibTeX XML Cite
Full Text: DOI