Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensions

TL;DR

This paper establishes a universal schematic stratification of the parameter scheme for families of principal G-bundles, extending Harder-Narasimhan theory to higher dimensions and showing these strata form an algebraic stack stratification.

Contribution

It introduces a universal schematic Harder-Narasimhan stratification for principal bundles in higher dimensions, generalizing previous curve results and providing a new stack-theoretic perspective.

Findings

Each Harder-Narasimhan stratum is a locally closed subscheme of the parameter scheme.

Principal bundles of a fixed Harder-Narasimhan type form an Artin algebraic stack.

The stratification extends to the stack of all principal G-bundles, forming a locally closed substack stratification.

Abstract

For any family of principal bundles with a reductive structure group G on a family X/S of smooth projective varieties in characteristic zero, it is known that the parameter scheme S has a set theoretic stratification by locally closed subsets which correspond to the Harder-Narasimhan types of the restriction of the principal bundle to the various fibers of X/S. We show that each of these subsets has in fact the structure of a locally closed subscheme of the parameter scheme S, with the following universal property: Under any base change, the pullback family admits a relative Harder-Narasimhan filtration (defined appropriately) with a given Harder-Narasimhan type if and only if the base change factors via the schematic stratum corresponding to that Harder-Narasimhan type. It follows that principal bundles of any given Harder-Narasimhan type on X/S form an Artin algebraic stack over S, and as the Harder-Narasimhan type varies, these stacks define a stratification the stack of all principal G-bundles on X/S by locally closed substacks. This result extends to principal bundles in higher dimensions our earlier similar results which were proved for principal bundles on families of curves. The result is new even for vector bundles, that is, for G = GL(n).