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

    • Fulltext


        Click here to view fulltext PDF

      Permanent link:

    • 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

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.