The homotopy classification of four-dimensional toric orbifolds

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

Identification of Genomic Rare Variants by Whole Genome Sequencing in Primary Torsion Dystonia

Background:Primary torsion dystonia (PTD) is a group of related movement disorders characterized by abnormal repetitive, twisting postures due to the involuntary co-contraction of opposing muscle groups. The research is based on whole genome sequencing technology of PTD patients to analyze the pathogenic genes and mutation sites in patients with primary dystonia, the relationship among genotype, clinical phenotype and prognosis. Methods: In order to investigate the association between the familial disease and its molecular mechanisms, 100 normal Han Chinese donors were also examined. The DNA of all the samples was sequenced using whole genome sequencing technique.The participants was conducted and submitted to the Macrogen Group (Seoul, Korea) for analysis.Results: We had detected the data output of precursor is 112.91G, throughut mean depth is 39.50X, mappable mean depth is 35.70X, genome coverage ratio is 99.50%.A novel heterozygous missense variant of uncertain significance (VUS) in ANO3 of Primary Torsion Dystonia had be found, but not in healthy control groups. Conclusions: Together, our results report a new mutation that may be similar in phenotype to known pathogenic genes, which will lay the foundation for future work. More families will be sequenced to identify more informations, which can help us to make the correct molecular diagnosis of the disease and to provide better genetic information.

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

We consider the generalized Jacobian {\widetilde{J}} of the modular curve {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the {\mathbb{Q}} -rational torsion points on {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on {\widetilde{J}} and its Eisenstein property.

On the Structures of K2(ℤ[G]), G a Finite Abelian p-Group

Let G be a finite abelian p-group, Γ the maximal ℤ-order of ℤ[G]. We prove that the 2-primary torsion subgroups of K2(ℤ[G]) and K2(Γ) are isomorphic when p ≡ 3, 5, 7 (mod 8), and [Formula: see text] is isomorphic to [Formula: see text] when p ≡ 2, 3, 5, 7. As an application, we give the structure of K2(ℤ[G]) for G a cyclic p-group or an elementary abelian p-group.

Finiteness of Brauer groups of K3 surfaces in characteristic 2

For a [Formula: see text] surface over a field of characteristic [Formula: see text] which is finitely generated over its prime subfield, we prove that the cokernel of the natural map from the Brauer group of the base field to that of the [Formula: see text] surface is finite modulo the [Formula: see text]-primary torsion subgroup. In characteristic different from [Formula: see text], such results were previously proved by Skorobogatov and Zarhin. We basically follow their methods with an extra care in the case of superspecial [Formula: see text] surfaces using the recent results of Kim and Madapusi Pera on the Kuga-Satake construction and the Tate conjecture for [Formula: see text] surfaces in characteristic [Formula: see text].

Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over finite fields

In order to study$p$-adic étale cohomology of an open subvariety$U$of a smooth proper variety$X$over a perfect field of characteristic$p>0$, we introduce new$p$-primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of$X$depending on effective divisors$D$supported in$X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of$U$and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wildly ramified class field theory for the open subvariety$U$.