Harder–Narasimhan Filtrations which are not split by the Frobenius maps

We will produce a smooth projective scheme X over ?, a rank 2 vector bundle V on X with a line subbundle L having the following property. For a prime p, let F _p be the absolute Fobenius of X _p , and let L _p ???V _p be the restriction of L???V. Then for almost all primes p, and for all t?≥?0, $(F_p^*)^t L_P \subset (F_p^*)^t V_p$ is a non-split Harder-Narasimhan filtration. In particular, $(F_p^*)^t V_p$ is not a direct sum of strongly semistable bundles for any t. This construction works for any full flag veriety G/B, with semisimple rank of G?≥?2. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic p.     

Characteristic classes for GO(2n) in étale cohomology

Let $GO(2n)$ be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic $\ne 2$ . We determine the smooth-étale cohomology ring with $\mathbb F_2$ coefficients of the algebraic stack $BGO(2n)$ . In the topological category, Holla and Nitsure determined the singular cohomology ring of the classifying space $BGO(2n)$ of the complex Lie group $GO(2n)$ in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are: (i) an extension to étale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the ${\mathbb G_m}$ -fibration $BO(2n)\to BGO(2n)$ of algebraic stacks.     

Unpolarized coupled DGLAP evolution equation at small-x

In this paper, we have obtained the solution of the unpolarized coupled Dokshitzer–Gribove–Lipatov–Alterelli–Parisi (DGLAP) evolution equation in leading order at the small-x limit. Here, we have used a Taylor series expansion, separation of functions and then the method of characteristics to solve the evolution equations. We have also calculated t-evolution of singlet and gluon distribution functions and the results are compared with E665 and NNPDF data for singlet structure function and GRV1998 and MRST2004 gluon parametrizations. It is shown that our results are in good agreement with the parametrizations especially at small-x and high-Q ² region. From global parametrizations and our results, we have seen that the singlet and gluon distribution functions increase when Q ² increases for fixed values of x.