# BHK mirror symmetry for K3 surfaces with non-symplectic automorphism

## TL;DR

This paper demonstrates that for certain K3 surfaces with non-symplectic automorphisms, two different mirror symmetry constructions—Berglund-H"ubsch-Krawitz and Dolgachev—produce the same mirror surface, confirming their agreement.

## Contribution

It establishes the equivalence of two mirror symmetry constructions for a class of K3 surfaces with specific automorphisms, extending understanding of mirror symmetry.

## Key findings

- Both mirror constructions agree on the mirror K3 surface.
- The results apply to K3 surfaces in weighted projective spaces with non-symplectic automorphisms.
- Excludes automorphism orders 4, 8, and 12.

## Abstract

In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, and admitting a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-H\"ubsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface.