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arXiv · 1108.2223

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

Xi Chen·Charles Doran·Matt Kerr·James Lewis
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Abstract

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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Xi Chen, Charles Doran, Matt Kerr, James Lewis. 2011-08-10. Normal functions, Picard-Fuchs equations, and elliptic fibrations on K3 surfaces. https://arxiv.org/abs/1108.2223

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