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Calculation of mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations. (English) Zbl 1117.14014

Brasselet, Jean-Paul (ed.) et al., Real and complex singularities, São Carlos workshop 2004. Papers of the 8th workshop, Marseille, France, July 19–23, 2004. Basel: Birkhäuser (ISBN 3-7643-7775-5/hbk). Trends in Mathematics, 247-262 (2007).
The paper introduces algorithms for computing iterations of Gauss-Manin connections, Picard-Fuchs equations of abelian integrals and mixed Hodge structures of affine varieties of dimension \(n\) in terms of differential forms. For \(n=1\) such computations can be applied to counting the limit cycles of planar systems of polynomial differential equations. For \(n>3\) they give explicit definitions of Hodge cycles.
For the entire collection see [Zbl 1108.14001].

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32S65 Singularities of holomorphic vector fields and foliations
14D07 Variation of Hodge structures (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)

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