Maximal Families of CMCY Type

Abstract

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.