By Christian Rohde

ISBN-10: 3642006388

ISBN-13: 9783642006388

The major aim of this e-book is the development of households of Calabi-Yau 3-manifolds with dense units of complicated multiplication fibers. the hot households are made up our minds by way of combining and generalizing methods.

Firstly, the strategy of E. Viehweg and okay. Zuo, who've developed a deformation of the Fermat quintic with a dense set of CM fibers by way of a tower of cyclic coverings. utilizing this technique, new households of K3 surfaces with dense units of CM fibers and involutions are received.

Secondly, the development approach to the Borcea-Voisin reflect relatives, which in relation to the author's examples yields households of Calabi-Yau 3-manifolds with dense units of CM fibers, is usually applied. furthermore fibers with complicated multiplication of those new households also are determined.

This e-book used to be written for younger mathematicians, physicists and likewise for specialists who're drawn to complicated multiplication and types with advanced multiplication. The reader is brought to widely used Mumford-Tate teams and Shimura information, that are one of the major instruments used right here. The prevalent Mumford-Tate teams of households of cyclic covers of the projective line are computed for a large variety of examples.

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**Additional info for Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication **

**Sample text**

We have two cases. Assume that G is a Q-algebraic torus. In this case G(R)/K consists of one point. The fact that we have a closed embedding G → GL(W ) implies that the Hodge group of the Hodge structure over this point is a subtorus of the torus G. In the other case G is not a Q-algebraic torus.

The group G semisimple, if R(G) = {e}. • The group G is simple, if {G} and {e} are the only normal connected subgroups of G. There exists an alternative description of semisimple groups. By the following proposition, one sees that a semisimple algebraic group G is isogeneous to the ﬁberproduct of simple groups. 3. Let {e} = G be a semisimple algebraic group. Then G is isogeneous to the ﬁberproduct of its minimal nontrivial normal subgroups. Proof. (follows from [10], IV. 10) By comparing the deﬁnition of reductive algebraic groups and semisimple algebraic groups, one sees easily that semisimple groups are reductive.

From this fact one concludes that hJ(X) (S 1 ) is contained in the centralizer of T . Since GL(H 2n−1 (X, Q)) is reductive, the maximal torus T is its own centralizer. This follows from the fact that the centralizers of the maximal tori (i. e. the Cartan subgroups) of a reductive group are the maximal tori (see [10], IV. 17). Hence one concludes that hJ(X) (S 1 ) ⊂ T , which implies that J(X) is of CM type. 1. Let D be a complex manifold and V be a holomorphic variation of rational Hodge structures on D.

### Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication by Christian Rohde

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