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Some Preliminaries for Families of Cyclic Covers

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1975)

In this chapter we collect the remaining preparations for the computations concerning the VHS of our families π : C → P n of cyclic covering of P1, which we construct in this chapter.

Let V denote the VHS of the family X → Y of curves and Mon0(V) denote the identity component of the Zariski closure of the monodromy group of V. In Section 3.1 we introduce the generic Hodge group Hg(V), which is the maximum of the Hodge groups of all occurring Hodge structures in V. Moreover Hg(V) coincides with the Hodge groups of the Hodge structures in V over the complement of a unification of countably many submanifolds of Y . Our families π : C → Pn are constructed in Section 3.2. We will also make some general remarks about the monodromy representation of V including the fact that the Galois group action yields an eigenspace decomposition in Section 3.2. In Section 3.3 we make some explicit computations of the monodromy representations of these eigenspaces. These computations are motivated from the fact that Mon0(V) is a normal subgroup of the derived group Hgder(V) of the generic Hodge group! as we see in Section 3.1.

Keywords

  • Galois Group
  • Braid Group
  • Hodge Structure
  • Monodromy Group
  • Dehn Twist

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Correspondence to Jan Christian Rohde .

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© 2009 Springer-Verlag Berlin Heidelberg

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Rohde, J.C. (2009). Some Preliminaries for Families of Cyclic Covers. In: Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication. Lecture Notes in Mathematics(), vol 1975. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00639-5_4

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