A simple formula for the Picard number of \(K3\) surfaces of BHK type. (English) Zbl 1460.14081
Summary: 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 \(\rho (Z_{A,G})\) 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 \(\rho (Z_{A,G})\) depends only upon the degree of the mirror polynomial \(F_{A^T}\).
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14C22 | Picard groups |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
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