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Picard-Fuchs equations, integrable systems and higher algebraic $$K$$-theory. (English) Zbl 1081.14503
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 43-55 (2003).
Summary: This paper continues the work done in [Duke Math. Journal 112, No. 3, 581–598 (2002; Zbl 1060.14011)] and is an attempt to establish a conceptual framework which generalizes the work of Yu. I. Manin [in: Geometry of differential equations. Am. Math. Soc. Transl. Ser. 186, 131–151 (1998; Zbl 0948.14025)] on the relation between nonlinear second-order ordinary differential equation of type Painlevé VI and integrable systems. The principle behind everything is a strong interaction between $$K$$-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of R. Donagi and E. Markman [Lect. Notes Math. 1620, 1–119 (1996; Zbl 0853.35100)].
For the entire collection see [Zbl 1022.00014].

##### MSC:
 14C25 Algebraic cycles 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 32G20 Period matrices, variation of Hodge structure; degenerations 34A26 Geometric methods in ordinary differential equations 14C15 (Equivariant) Chow groups and rings; motives 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects)