Hodge numbers for orbifolds of Calabi-Yau threefolds Fermat type and the Roan pairs

Abstract

First, for orbifolds of Calabi-Yau threefolds of Fermat type, we define Roan's Hodge numbers. We prove, Theorem \ref{main} , that for all orbifols of Calabi-Yau threefolds Fermat type, Roan's Hodge numbers correctly count the stringy Euler numbers due to Vafa formula. Second, for Calabi-Yau threefolds Fermat type, we apply Roan's Hodge numbers to get a relation between the Borcea-Voisin construction \cite{Borcea, Voisin} and Berglund-Hübsch-Krawits mirror symmetry (BHK mirror symmetry). Third, for orbifolds of K3 surfaces Fermat type, we establish relations twisted and untwisted parts of the second mixed cohomology to deformations and the Roan pairs. This allow us to confirm the BHK mirror symmetry for such orbifolds, since the Euler numbers computed by the Vafa formula are equal $24$ for all of them.