By Christian Rohde

The major target of this e-book is the development of households of Calabi-Yau 3-manifolds with dense units of advanced multiplication fibers. the recent households are decided via combining and generalizing methods.

Firstly, the strategy of E. Viehweg and ok. Zuo, who've developed a deformation of the Fermat quintic with a dense set of CM fibers via 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 with regards to the author's examples yields households of Calabi-Yau 3-manifolds with dense units of CM fibers, is additionally applied. furthermore fibers with complicated multiplication of those new households also are determined.

This e-book was once written for younger mathematicians, physicists and in addition for specialists who're attracted to complicated multiplication and forms with advanced multiplication. The reader is brought to established Mumford-Tate teams and Shimura facts, that are one of the major instruments used right here. The widely used Mumford-Tate teams of households of cyclic covers of the projective line are computed for a wide variety of examples.

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**Example text**

This yields the Torelli theorem for curves. 3 The deﬁnition of the Shimura datum We will endow the set of principally polarized abelian varieties with symplectic basis with the structure of a Hermitian symmetric domain. Such a domain can be obtained from a Shimura datum. Let G be a Q-algebraic 26 1 An introduction to Hodge structures and Shimura varieties reductive group. A Shimura datum is given by a homomorphism h : S → GR of algebraic groups, which satisﬁes some conditions, which we explain here.

3). Thus the endomorphism algebra of X given by End(X)⊗Z Q can be identiﬁed with EndQ (H 1 (X, Q), hX ). 4 tells us that an abelian variety X has a commutative endomorphism algebra of rank equal to dim H 1 (X, Q), if Hg(H 1 (X, Q), hX ) is a torus. Thus the endomorphism algebra EndQ (H 1 (X, Q), hX ) of the Hodge structure contains a commutative endomorphism algebra of rank equal to dim H 1 (X, Q), if Hg(H 1 (X, Q), hX ) is a torus algebraic group. 6. Let F denote a totally real number ﬁeld and (V, h) be an F -Hodge structure.

The Lie group GSp(V, Q)(R) has two connected components. One component consists of matrices with positive determinant and the other consists of matrices with negative determinant. Hence the corresponding homogeneous space D parametrizing the elements of the conjugacy class of a Hodge structure of an abelian variety given by h has two connected components. Note that GSp(V, Q)(R)+ is a product of Sp(V, Q)(R) + is contained in the stabilizers of all points, the and G+ m (R). Since Gm (R) corresponding connected homogeneous space coincides with hg such that the preceding Theorem endows hg with the structure of a Hermitian symmetric domain.