Higher dimensional Calabi-Yau manifolds of Kummer type

Based on Cynk-Hulek method we construct complex Calabi-Yau varieties of arbitrary dimensions using elliptic curves with automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize result of Katsura and Sch\"utt to obtain arbitrarily dimensional Calabi-Yau manifolds which are Zariski in any characteristic $p\not\equiv 1\pmod{12}.$Comment: 13 pages, 2 figure

Generalized Borcea-Voisin Construction

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost.Comment: 15 pages, 2 figures. v2: typos corrected & references adde

Comparison of normalization methods for differential gene expression analysis in RNA-Seq experiments: A matter of relative size of studied transcriptomes

In recent years, RNA-Seq technologies became a powerful tool for transcriptome studies. However, computational methods dedicated to the analysis of high-throughput sequencing data are yet to be standardized. In particular, it is known that the choice of a normalization procedure leads to a great variability in results of differential gene expression analysis. The present study compares the most widespread normalization procedures and proposes a novel one aiming at removing an inherent bias of studied transcriptomes related to their relative size. Comparisons of the normalization procedures are performed on real and simulated data sets. Real RNA-Seq data sets analyses, performed with all the different normalization methods, show that only 50% of significantly differentially expressed genes are common. This result highlights the influence of the normalization step on the differential expression analysis. Real and simulated data sets analyses give similar results showing 3 different groups of procedures having the same behavior. The group including the novel method named “Median Ratio Normalization” (MR N) gives the lower number of false discoveries. Within this group the MR N method is less sensitive to the modification of parameters related to the relative size of transcriptomes such as the number of down- and upregulated genes and the gene expression levels. The newly proposed MR N method efficiently deals with intrinsic bias resulting from relative size of studied transcriptomes. Validation with real and simulated data sets confirmed that MR N is more consistent and robust than existing methods

On orbifolds and free fermion constructions

This work develops the correspondence between orbifolds and free fermion models. A complete classification is obtained for orbifolds X/G with X the product of three elliptic curves and G an abelian extension of a group (Z_2)^2 of twists acting on X. Each such quotient X/G is shown to give a geometric interpretation to an appropriate free fermion model, including the geometric NAHE+ model. However, the semi-realistic NAHE free fermion model is proved to be non-geometric: its Hodge numbers are not reproduced by any orbifold X/G. In particular cases it is shown that X/G can agree with some Borcea-Voisin threefolds, an orbifold limit of the Schoen threefold, and several further orbifolds thereof. This yields free fermion models with geometric interpretations on such special threefolds.Comment: 46 pages; typos corrected and references adde