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arXiv · 2208.01465

Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes

Tomonao Matsumura·Atsuhira Nagano
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Abstract

We determine the Néron-Severi lattices of $K3$ hypersurfaces with large Picard number in toric three-folds derived from Fano polytopes. On each $K3$ surface, we introduce a particular elliptic fibration. In the proof of the main theorem, we show that the Néron-Severi lattice of each $K3$ surface is generated by a general fibre, sections and appropriately selected components of the singular fibres of our elliptic fibration. Our argument gives a certain proof of the Dolgachev conjecture for Fano polytopes, which is a conjecture on mirror symmetry for $K3$ surfaces.

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Tomonao Matsumura, Atsuhira Nagano. 2025-05-20. Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes. https://doi.org/10.1007/s10711-025-01014-5

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