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

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      https://www.ias.ac.in/article/fulltext/pmsc/123/03/0361-0363

    • Keywords

       

      Frobenius splitting; Borel–Weil–Bott theorem; strong Harder–Narasimhan Filtrations.

    • Abstract

       

      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.

    • Author Affiliations

       

      Saurav Bhaumik1 Vikram Mehta2

      1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
      2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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