• Sudarshan Gurjar

      Articles written in Proceedings – Mathematical Sciences

    • Schematic Harder–Narasimhan Stratification for Families of Principal Bundles and 𝛬-modules

      Sudarshan Gurjar Nitin Nitsure

      More Details Abstract Fulltext PDF

      Let 𝐺 be a reductive algebraic group over a field 𝑘 of characteristic zero, let $X\to S$ be a smooth projective family of curves over 𝑘, and let 𝐸 be a principal 𝐺 bundle on 𝑋. The main result of this note is that for each Harder–Narasimhan type 𝜏 there exists a locally closed subscheme $S^\tau (E)$ of 𝑆 which satisfies the following universal property. If $f:T\to S$ is any base-change, then 𝑓 factors via $S^\tau (E)$ if and only if the pullback family $f^∗E$ admits a relative canonical reduction of Harder–Narasimhan type 𝜏. As a consequence, all principal bundles of a fixed Harder–Narasimhan type form an Artin stack. We also show the existence of a schematic Harder–Narasimhan stratification for flat families of pure sheaves of 𝛬-modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of $\mathcal{O}$-modules proved earlier by Nitsure. This again has the implication that 𝛬-modules of a fixed Harder–Narasimhan type form an Artin stack.

  • Proceedings – Mathematical Sciences | News

© 2017-2019 Indian Academy of Sciences, Bengaluru.