On Berglund-Hübsch-Krawitz Mirror Symmetry
We provide various suites of results for the Calabi-Yau orbifolds that have Berglund-HÃ¼bsch-Krawitz (BHK) mirrors. These Calabi-Yau orbifolds are certain finite symplectic quotients of hypersurfaces in weighted-projective space. First, we will describe their birational geometry using Shioda maps and prove that so-called alternate mirrors are birational. Next, we will compute algebraic invariants of the orbifolds and their crepant resolutions in the case where they are orbifold K3 surfaces, both over the complex numbers and fields of positive characteristic. Finally, we provide a conjectural framework that unifies the toric mirror construction of Batyrev and Borisov with the BHK construction in the context of Kontsevich's Homological Mirror Symmetry Conjecture.