Communications in Number Theory and Physics

Volume 10 (2016)

Number 2

Calabi–Yau threefolds of type K (II): mirror symmetry

Pages: 157 – 192



Kenji Hashimoto (Graduate School of Mathematical Sciences, University of Tokyo, Maguro-ku, Tokyo, Japan)

Atsushi Kanazawa (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


A Calabi–Yau threefold is called of type K if it admits an étale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper, based on Oguiso–Sakurai’s fundamental work, we have provided the full classification of Calabi– Yau threefolds of type K and have studied some basic properties thereof. In the present paper, we continue the study, investigating them from the viewpoint of mirror symmetry. It is shown that mirror symmetry relies on duality of certain sublattices in the second cohomology of the K3 surface appearing in the minimal splitting covering. The duality may be thought of as a version of the lattice duality of the anti-symplectic involution on K3 surfaces discovered by Nikulin. Based on the duality, we obtain several results parallel to what is known for Borcea–Voisin threefolds. Along the way, we also investigate the Brauer groups of Calabi–Yau threefolds of type K.

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