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Maximal Families of CMCY Type

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

In this chapter we use the classification of involutions on K3 surfaces S by V. V. Nikulin [51], which act by ?1 on H0(S, ωS). If the divisor of fixed points consists at most of rational curves, the Borcea-Voisin construction yields a maximal holomorphic CMCY family of 3-manifolds.

After we have recalled some basic facts in Section 11.1, we define a Shimura datum by using involutions on the integral lattice in Section 11.2. Each of the points of a dense open subset of the bounded symmetric domain obtained from this Shimura datum represents a marked K3 surface with involution. By using this fact, we obtain our examples of maximal holomorphic CMCY families of 3-manifolds in Section 11.3. For each n ε N with n ≤ 11 there is a holomorphic maximal CMCY family over a basis of dimension n.

Keywords

  • Elliptic Curf
  • Hyperbolic Plane
  • Rational Curf
  • Hodge Structure
  • Symmetric Domain

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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). Maximal Families of CMCY Type. 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_12

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