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

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/124/03/0315-0332

• # Keywords

Principal 𝐺-bundle; Harder–Narasimhan type; canonical reduction.

• # Abstract

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.

• # Author Affiliations

1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019