• Saurav Bhaumik

      Articles written in Proceedings – Mathematical Sciences

    • Characteristic Classes for $GO(2n)$ in étale Cohomology

      Saurav Bhaumik

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      Let $GO(2n)$ be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic $\neq 2$. 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.

    • Harder-Narasimhan Filtrations which are not Split by the Frobenius Maps

      Saurav Bhaumik Vikram Mehta

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      We will produce a smooth projective scheme 𝑋 over $\mathbb{Z}$, 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 $L_p\subset V_p$ be the restriction of $L\subset V$. Then for almost all primes 𝑝, and forall $t\geq 0,(F^∗_p)^t L_P\subset (F^∗_p)^t V_p$ 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 $G\geq 2$. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic 𝑝.

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