Extending Hodge bundles for abelian variations. (English) Zbl 0857.14007

Let \(B\) be a sufficiently small polydisc and let \(D\) be a divisor in \(B\). The asymptotic behaviour of a polarized variation of Hodge structure (VHS) over \(B \backslash D\) was studied by W. Schmid [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] in the case where \(D\) has normal crossing singularities at 0. In this case \(B \backslash D\) is a product of discs and punctured discs and the fundamental group \(\pi_1(B \backslash D)\) is abelian, hence the monodromy group of any local system over \(B \backslash D\) is abelian.
The authors consider the case of an arbitrary divisor \(D\) and a VHS over \(B \backslash D\) with abelian monodromy group, i.e., the monodromy representation of the VHS factors through \(H_1(B \backslash D)\). They extend results of E. Cattani and A. Kaplan [Invent. Math. 67, 101-115 (1982; Zbl 0516.14005)] and of these two authors and W. Schmid [Ann. Math., II. Ser. 123, 457-535 (1986; Zbl 0617.14005)] to this more general situation. Specifically, they prove the nilpotent orbit theorem and show the existence of a canonical limit mixed Hodge structure.
Reviewer: J.Nagel (Leiden)


14D07 Variation of Hodge structures (algebro-geometric aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K10 Algebraic moduli of abelian varieties, classification
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