Algebra/Topology Seminar

Sudarshan Gurjar (Aarhus U): Schematic Harder-Narasimhan stratification for families of principal bundles

Abstract: Let G be a connected reductive group over an algebraically closed field k of characteristic zero. For a principal G-bundle on a family of smooth, projective curves parametrized by a noetherian k-scheme, it is known that the Harder-Narasimhan type of its restriction to each fiber varies upper-semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally-closed subsets, known as the Harder-Narasimhan stratification. We show that each of these subsets can be endowed with the structure of a locally-closed subscheme with the universal property that under any base-change, the pullback family admits a relative Harder-Narasimhan reduction with a given Harder-Narasimhan type if and only if the base-change factors through the schematic stratum corresponding to that Harder-Narasimhan type. As a consequence, we show that principal bundles with a given Harder-Narasimhan type form an Artin stack. In the end I will briefly mention our recent generalization of this result to the case of higher relative dimensions.

(Joint work with Nitin Nitsure)