A simple formula for the Picard number of K3 surfaces of BHK type

Abstract

The Berglund–Hübsch–Krawitz (BHK) mirror symmetry construction applies to certain types of Calabi–Yau varieties that are birational to finite quotients of Fermat varieties. Their definition involves a matrix $A$ and a certain finite abelian group $G$, and we denote the corresponding Calabi–Yau variety by $Z_{A,G}$. The transpose matrix $A^T$ and the so-called dual group $G^T$ give rise to the BHK mirror variety $Z_{A^T,G^T}$. In the case of dimension 2, the surface $Z_{A,G}$ is a K3 surface of BHK type. Let $Z_{A,G}$ be a K3 surface of BHK type, with BHK mirror $Z_{A^T,G^T}$. Using work of Shioda, Kelly has shown that the geometric Picard number $\rho \left(Z_{A,G}\right)$ of $Z_{A,G}$ may be expressed in terms of a certain subset of the dual group $G^T$. We simplify this formula significantly to show that $\rho \left(Z_{A,G}\right)$ depends only upon the degree of the mirror polynomial $F_{A^T}$.