On the cohomology of Stover Surface

TL;DR

This paper investigates the unique Stover surface, highlighting its maximal Picard number, automorphism group, and algebraic properties, including its Albanese variety and absence of higher genus fibrations.

Contribution

It provides a detailed analysis of the Stover surface's cohomological properties, its maximal Picard number, and explicit description of its Albanese variety.

Findings

Stover surface has maximal Picard number.

It has no higher genus fibrations.

Its Albanese variety is explicitly characterized.

Abstract

We study a surface discovered by Stover which is the surface with minimal Euler number and maximal automorphism group among smooth arithmetic ball quotient surfaces. We study the natural map $\wedge^{2}H^{1}(S,\mathbb{C})\to H^{2}(S,\mathbb{C})$ and we discuss the problem related to the so-called Lagrangian surfaces. We obtain that this surface $S$ has maximal Picard number and has no higher genus fibrations. We compute that its Albanese variety $A$ is isomorphic to $(\mathbb{C}/\mathbb{Z}[\alpha])^{7}$, for $\alpha=e^{2i\pi/3}$.