A simple formula for the Picard number of K3 surfaces of BHK type
| Autor: | Lyons, Christopher, Olcken, Bora |
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| Rok vydání: | 2017 |
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| Zdroj: | Kyoto J. Math. 60, no. 3 (2020), 941-964 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1215/21562261-2019-0051 |
| Popis: | The BHK mirror symmetry construction stems from work Berglund and Huebsch, and 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 shows that the geometric Picard number 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 this Picard number depends only upon the degree of the mirror polynomial $F_{A^T}$. Comment: 17 pages |
| Databáze: | arXiv |
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