A finiteness theorem for monodromy. (Un théorème de finitude pour la monodromie.) (French) Zbl 0656.14010
Discrete groups in geometry and analysis, Pap. Hon. G. D. Mostow 60th Birthday, Prog. Math. 67, 1-19 (1987).
[For the entire collection see Zbl 0632.00015.]
If \(S\) is a smooth complex algebraic variety, there are only finitely many local systems of \(\mathbb Q\)-vector spaces of a fixed dimension \(N\) that are a direct factor of a local system underlying a polarizable variation of Hodge structures. As a consequence, there are only finitely many \(\mathbb Q\)-representations of dimension \(N\) that are a direct factor of a monodromy representation in the cohomology of an algebraic family parametrized by \(S\). This is extended to representations over \(\mathbb Z\) and related to a paper by G. Faltings [Invent. Math. 73, 337–347 (1983; Zbl 0588.14025)] on Arakelov’s theorem for abelian varieties.
If \(S\) is a smooth complex algebraic variety, there are only finitely many local systems of \(\mathbb Q\)-vector spaces of a fixed dimension \(N\) that are a direct factor of a local system underlying a polarizable variation of Hodge structures. As a consequence, there are only finitely many \(\mathbb Q\)-representations of dimension \(N\) that are a direct factor of a monodromy representation in the cohomology of an algebraic family parametrized by \(S\). This is extended to representations over \(\mathbb Z\) and related to a paper by G. Faltings [Invent. Math. 73, 337–347 (1983; Zbl 0588.14025)] on Arakelov’s theorem for abelian varieties.
Reviewer: J. H. de Boer
MSC:
| 14F25 | Classical real and complex (co)homology in algebraic geometry |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
