Calabi–yau Coverings over Some Singular Varieties and New Calabi-yau 3-folds with Picard Number One

A Calabi–Yau manifold is a compact Kähler manifold with trivial canonical class such that the intermediate cohomologies of its structure sheaf are all trivial (h(X,OX ) = 0 for 0 < i < dim(X)). One handy way of constructing Calabi–Yau manifolds is by taking coverings of some smooth varieties such that some multiples of their anticanonical class have global sections. Indeed many of known examples of Calabi–Yau 3-folds with Picard number one are constructed in this way (see, for example, Table 1 in [EnSt]). In this note we show that singular varieties with some cyclic singularities also admit Calabi–Yau manifolds as their coverings (Theorem 1.1). We give some formula for calculating their invariants by using degeneration method (Theorem 2.1, Theorem 3.2). In his beautiful papers ([Ta1], [Ta2]), H. Takagi classified possible invariants of certain Q -Fano 3-folds of Gorenstein index 2 and constructed some exotic examples of Q -Fano 3-folds. We apply our theorem to construct Calabi–Yau 3-folds which are double coverings of Takagi’s Q -Fano 3-folds. It turns out that 18 of them are new Calabi–Yau 3-folds with Picard number one (Table 1). Although a huge number of Calabi–Yau 3-folds have been constructed, those with Picard number one are still quite rare (for example, see Table 1 in [EnSt]). Note that they are primitive and play an important role in the moduli spaces of all Calabi–Yau 3-folds ([Gr]). We show that some of them are connected by projective flat deformation although they are of different topological types (Theorem 3.5). It is interesting that three of them have the invariants which were predicted by C. van Enckevort and D. van Straten in their paper ([EnSt]). Let us recall a notation for certain singularities. Let a1, · · · , an be integers and let x1, · · · , xn be coordinates on C . Suppose that the cyclic group G acts on C via