Abstract
In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension n in terms of differential forms. In the case n = 1 such computations have many applications in differential equations and counting their limit cycles. For n > 3, these computations give us an explicit definition of Hodge cycles.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Monographs in Mathematics, 83 Birkhäuser Boston, Inc., Boston, MA, 1988.
P. Deligne, Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math. 143(3) (2001), 449–497.
L. Gavrilov, H. Movasati. The infinitesimal 16th Hilbert problem in dimension zero. Preprint 2005 (math.CA/0507061).
P.A. Griffiths. Infinitesimal variations of Hodge structure. III. Determinantal varieties and the infinitesimal invariant of normal functions. Compositio Math. 50(2–3) (1983), 267–324.
Yu. Ilyashenko. Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. (N.S.), 39(3) (2002), 301–354.
V.S. Kulikov, P.F. Kurchanov. Complex algebraic varieties: periods of integrals and Hodge structures. Algebraic geometry, III, 1–217, 263–270, Encyclopaedia Math. Sci., 36, Springer, Berlin, 1998.
H. Movasati. Center conditions: rigidity of logarithmic differential equations. J. Differential Equations, 197(1) (2004), 197–217.
H. Movasati. Mixed Hodge structure of affine varieties. To appear in Ann. Ins. Four. (2005).
H. Movasati. Abelian integrals in holomorphic foliations. Revista Matemática Iberoamericana, 20(1) (2004), 183–204.
J. Steenbrink. Intersection form for quasi-homogeneous singularities. Compositio Math. 34(2) (1977) 211–223.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Movasati, H. (2006). Calculation of Mixed Hodge Structures, Gauss-Manin Connections and Picard-Fuchs Equations. In: Brasselet, JP., Ruas, M.A.S. (eds) Real and Complex Singularities. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7776-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7776-2_18
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7775-5
Online ISBN: 978-3-7643-7776-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)