paper

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

arXiv:1108.2223

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 $K 3$ surface. We also construct an explicit family of $K_{1}$ cycles on $H \oplus E_{8} \oplus E_{8}$ -polarized $K 3$ 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.

Some relatively minor corrections, and a rewrite of some sections. 45 pages