Zariski-van Kampen Method and Transcendental Lattices of Certain Singular $K3$ Surfaces

Abstract

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular $K3$ surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over $\mathbf{Q}(\sqrt{5})$ and are conjugate to each other by the action of $\text{Gal}(\mathbf{Q}(\sqrt{5})/\mathbf{Q})$.