摘要：We will consider a class of (typically) singular Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of P^1 branched over 4 points. Two-fold covers of P^2 branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry, and discuss the Riemann-Hilbert problem for periods of these singular varieties. The new insight here is the idea of ‘abelian gauge fixing’ and ‘fractional complete intersections’ that leads to a new interpretation of those classical results. The idea further points to a construction of large class of Calabi-Yau mirror pairs. The lecture is based on recent and on-going joint work with S. Hosono, T.-J. Lee, M. Romo, L. Santilli, H. Takagi, S.-T. Yau.