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

TL;DR

This paper investigates the transcendental regulator for K-theory on K3 surfaces using Gauss-Manin derivatives of normal functions, revealing new results on indecomposability and explicit cycle constructions.

Contribution

It introduces novel methods to analyze the transcendental regulator on K3 surfaces and constructs explicit indecomposable cycles using elliptic fibrations and modular functions.

Findings

Proves non-triviality of the transcendental regulator for K_1 of very general K3 surfaces.

Constructs explicit indecomposable K_1 cycles on polarized K3 surfaces.

Utilizes elliptic fibrations and modular functions for explicit cycle parametrization.

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.