The Berglund–Hübsch–Chiodo–Ruan mirror symmetry for K3 surfaces

Abstract

Abstract. We prove that the mirror symmetry of Berglund-Hübsch-ChiodoRuan, applied to K3 surfaces with a non-symplectic involution, coincides with the mirror symmetry described by Dolgachev and Voisin.

Borcea–Voisin Calabi–Yau threefolds and invertible potentials

“…By Theorem 2.4 the invariants of the fixed lattice of the involution σ S are (r, a, δ) = (19, 1, 1), i.e. S ∈ K L (19,1,1) . Thus Z E/ ι ,S ∈ Z 19,1,1 and its Hodge numbers are h 1,1 = 60, h 2,1 = 6.…”

“…Since the fixed locus of σ S is a smooth curve of genus 10, the invariants of its fixed lattice are (see Section 2.1) (r, a, δ) = (1, 1, 1), i.e. S ∈ K L (1,1,1) . It follows that Z E,S ∈ Z 1,1,1 and that its Hodge numbers are h 1,1 = 6, h 2,1 = 60 (see Section 3).…”

“…The present paper is the final step of a project that started with our previous paper [1], where we proved that lattice mirrors of K3 surfaces with a non-symplectic involution can be obtained by the BHCR construction (see Theorem 2.1). We give a positive answer to the above question in Theorem 4.10.…”

“…The definition of the transposed group is mainly due to Berglund-Henningson [4], Krawitz [17] and Ebeling-Takahashi [14] and has been reinterpreted by many authors, we refer to [1] for a discussion of equivalent definitions. By [1, Corollary 3.7] we have…”

“…This requires first to produce birational projective models of the varieties Z E,S to which the BHCR construction can be applied (see Section 4.1). This follows an original idea of Borcea but we have to be more precise on the choices of the equations: here we make use of the results in [1], where we provided Delsarte type equations for K3 surfaces with a non-symplectic involution. Secondly, we have to control the behavior of the transposed groups that occur in the BHCR construction under these birational equivalences, this is the main technical point (see Section 4.2).…”