## Hodge theory with degenerating coefficients: $$L_2$$ cohomology in the Poincaré 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)

Zbl 0329.14008
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