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On the monodromy and mixed Hodge structure on cohomology of the cyclic infinite covering of the complement to a plane algebraic curve. (English. Russian original) Zbl 0909.14004

Izv. Math. 59, No. 2, 367-386 (1995); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 2, 143-162 (1995).
Summary: The semisimplicity of the Alexander automorphism (the monodromy operator) is proved on the cohomology \(H^1(X_\infty)_{\neq 1}\) of the infinite cyclic covering of the complement to a plane non-reduced algebraic curve, and, in particular, the semi-simplicity of \(H^1 (X_\infty)\) in the case of an irreducible curve. A natural mixed Hodge structure on \(H^1 (X_\infty)\) is introduced and the irregularity of cyclic coverings of \(\mathbb{P}^2\) is calculated in terms of the number of roots of the Alexander polynomial of the branch curve.

MSC:

14E20 Coverings in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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