Based on Cynk–Hulek method from [7] we construct complex Calabi–Yau varieties of arbitrary dimensions using elliptic curves with an automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize the result of [11] to obtain arbitrarily dimensional Calabi–Yau manifolds which are Zariski in any characteristic p≢1(mod12).

We shall reproof formulas for the Hodge numbers of Calabi-Yau threefolds of Borcea-Voisin type constructed by A. Cattaneo and A. Garbagnati, using the orbifold cohomology formula and the orbifold Euler characteristic.

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We construct examples of modular rigid Calabi-Yau threefolds, which give a realization of some new weight 4 cusp forms. Involutions of double octic Calabi-Yau threefoldsLet S ⊂ P 3 be an arrangement of eight planes such that no six intersect and no four contain a line. Then there exists a resolution of singularities X of the double covering of P 3 branched along S which is a Calabi-Yau threefold.