September 2020 A simple formula for the Picard number of K3 surfaces of BHK type
Christopher Lyons, Bora Olcken
Kyoto J. Math. 60(3): 941-964 (September 2020). DOI: 10.1215/21562261-2019-0051


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 ρ ( 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 ρ ( Z A , G ) depends only upon the degree of the mirror polynomial F A T .


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Christopher Lyons. Bora Olcken. "A simple formula for the Picard number of K3 surfaces of BHK type." Kyoto J. Math. 60 (3) 941 - 964, September 2020.


Received: 10 August 2017; Revised: 12 January 2018; Accepted: 1 February 2018; Published: September 2020
First available in Project Euclid: 12 August 2020

MathSciNet: MR4134354
Digital Object Identifier: 10.1215/21562261-2019-0051

Primary: 14J28
Secondary: 14C22 , 14J33

Keywords: K3 surfaces , mirror symmetry , Picard numbers

Rights: Copyright © 2020 Kyoto University


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Vol.60 • No. 3 • September 2020
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