On the cohomology of Stover Surface

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 $(\mathbb{C}\mathrm{/}\mathbb{Z}[\alpha ]{)}^7$, for $\alpha =e^{2i\pi \mathrm{/}3}$.