AI 中文总结
研究费马型卡拉比 - 丘三流形的orbifolds，定义罗恩的霍奇数，证明其能正确计算弦欧拉数，还应用该霍奇数得到博尔恰 - 沃伊辛构造与贝格伦德 - 许布施 - 克拉维茨镜像对偶性的关系。
AI 中文摘要
首先，对于费马型卡拉比 - 丘三流形的orbifolds，我们定义了罗恩的霍奇数。通过定理5.2证明，对于所有此类orbifolds，由于瓦法公式，罗恩的霍奇数能正确计算弦欧拉数。其次，对于费马型卡拉比 - 丘三流形，我们应用罗恩的霍奇数得到博尔恰 - 沃伊辛构造与贝格伦德 - 许布施 - 克拉维茨镜像对偶性之间的关系。
英文摘要
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.
Comments37 pages, Appendix with deformations and Roan's pairs for orbifolds of K3 surfaces of Fermat type, and new Table of all orbifolds of 147 Fermat threefolds
