Groups acting freely on Calabi–Yau threefolds embedded in a product of del Pezzo surfaces

Abstract

In this paper, we investigate quotients of Calabi-Yau manifolds Y embedded in Fano varieties X, which are products of two del Pezzo surfaces -with respect to groups G that act freely on Y . In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups G are subgroups of the automorphism groups of X, which is described in terms of the automorphism group of the two del Pezzo surfaces. e-print archive: http://lanl.arXiv.org/abs/1104.0247v1

Hodge numbers for CICYs with symmetries of order divisible by 4

“…More generally many of the special CICY's can be thought of as hypersurfaces in an embedding space S × S where S and S are del Pezzo surfaces. The importance of this class of Calabi-Yau manifolds was recognised by Bini and Favale [41]. Since P 2 and P 1 × P 1 are also del Pezzo surfaces this class contains also the configuration that follows, as well as the tetraquadric.…”

Calabi–Yau quotients with terminal singularities

“…Set X = P 1 × P 1 × P 1 × P 1 with (x i : y i ) that are projective coordinates on the i−th P 1 . In [BF11] and [BFNP13] the authors study the automorphisms of Calabi-Yau manifolds embedded in X that have empty fixed locus. The authors produce a classification of all the admissible pairs in X, i.e.…”

“…Every g ∈ Aut(X) acts on the 4 factors (see, for instance, [BF11]) giving a surjective homomorphism π : Aut(X) → S 4 with kernel PGL(2) ×4 . On the other hand the permutations of the factors give an inclusion S 4 ֒→ Aut(X) splitting π and therefore giving a structure of semidirect product…”

New Examples of Calabi–yau 3-Folds and Genus Zero Surfaces

“…In particular, for the product X of four complex projective lines, there exists a Calabi-Yau Y with Hodge numbers (h 1,1 , h 1,2 ) = (1, 5) and fundamental group isomorphic to Z 8 ⊕ Z 2 . In [BF11] an upper bound on the order of G -depending on X -has been found. This bound is maximal (equal to 16) if and only if X = P 1 × P 1 × P 1 × P 1 .…”

“…On the Calabi-Yau side, physicists have focused recently on Calabi-Yau's with small Hodge numbers (h 1,1 , h 1,2 ): see, for instance, [Bra11], [CD10], [BCD10], [Dav11], [FS11] and [BDS12]. In [BF11], the authors describe some new examples of Calabi-Yau varieties. They are given as quotients of anticanonical sections of Fano varieties by finite groups G acting freely.…”