On Kummer-like surfaces attached to singularity and modular forms

TL;DR

This paper investigates a special family of lattice polarized K3 surfaces, extending Kummer surfaces, characterized by singularities and parametrized by Hermitian modular forms, with detailed lattice and covering structure analysis.

Contribution

It identifies the transcendental and Néron-Severi lattices of generic members and describes the double covering structure of these K3 surfaces.

Findings

Determined the transcendental lattice of generic K3 surfaces in the family.

Identified the Néron-Severi lattice structure.

Described the double covering structure associated with the surfaces.

Abstract

We study a family of lattice polarized $K3$ surfaces which is an extension of the family of Kummer surfaces derived from principally polarized Abelian surfaces. Our family has two special properties. First, it is coming from a resolution of a simple $K3$ singularity. Second, it has a natural parametrization by Hermitian modular forms of four complex variables. In this paper, we show two results: (1) We determine the transcendental lattice and the N\'eron-Severi lattice of a generic member of our family. (2) We give a detailed description of the double covering structure associated with our $K3$ surfaces.