Normal functions, Picard-Fuchs equations, and elliptic fibrations on K3 surfaces

Details Normal functions, Picard-Fuchs equations, and elliptic fibrations on K3 surfaces Normal functions, Picard-Fuchs equations, and elliptic fibrations on K3 surfaces

2011-08-10

arxiv.org/abs/1108.2223v2

Mathematics

Algebraic Geometry

32 pages. Typographical revisions. The only mathematical parts affected are the displayed formula at the top of page 5, Remark

Scientific paper

Using Gauss-Manin derivatives of normal functions, we arrive at some remarkable results on the non-triviality of the transcendental regulator for $K_m$ of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for $K_1$ of a very general $K3$ surface. We also construct an explicit family of $K_1$ cycles on $H \oplus E_8 \oplus E_8$-polarized $K3$ surfaces, and show they are indecomposable by a direct evaluation of the real regulator. Critical use is made of natural elliptic fibrations, hypersurface normal forms, and an explicit parametrization by modular functions.

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