Cyclic Covers of the Projective Line

Recall that we will study variations of Hodge structures of families of cyclic coverings of the projective line. Moreover some families of such covers are suitable for the construction of families of Calabi-Yau manifolds with dense sets of complex multiplication fibers. In order to understand variations of Hodge structures of such families of cyclic coverings we need to understand the Hodge structure of a cyclic covering C → P¹.

A cyclic cover π : C → P¹ is given by

$$y^m = (x - a_1)^{d_1}..... (x - a_n)^{d_n}$$

((2.1))

where each d _k is an integer satisfying 1 ≤ d _k ≤ m −> 1. The numbers d _k are not uniquely determined by the isomorphism class of a cover. However, these numbers determine the isomorphism class of a cover and we will use them for the computation of the variation of Hodge structures in the following chapters.