The Galois Group Decomposition of the Hodge Structure

In this chapter we make some general observations about the VHS of C → P _n and its generic Hodge group. Moreover we will give an upper bound for the generic Hodge group and a sufficient criterion for dense sets of complex multiplication fibers.

Let ζ be a primitive m-th. root of unity and r < m be a divisor of m. Recall that a fiber C of one of our families π : C → P _n is given by

$$y^m = (x - a_1)^{d_1}..... (x - a_{n+3})^{d_n+3}.$$

By the Galois group action we have a decomposition of H ¹(C,Q) into subspaces N ¹(C _r ,Q) such that the Galois group action endows N1(C _r ,Q) with the structure of a Q(ζ^r)-vector space as we see in Section 4.1. In Section 4.2 we see that this decomposition is also a decomposition into sub-Hodge structures, which are closely related to the quotients C _r of C. By the centralizer of the Galois group action, we obtain an upper bound for the generic Hodge group in Section 4.3. The real valued points of the centralizer are given by the direct product of the unitary groups with respect to the Hermitian forms on the eigenspaces Lj with j ≤ m/2 . By using this description of the centralizer, we define pure (1, n) variations of Hodge structures and show that a family C with a pure (1, n) – VHS has a dense set of CM fibers in Section 4.4.