Harder-Narasimhan filtrations of decorated vector bundles

TL;DR

This paper constructs and compares Harder-Narasimhan filtrations for decorated vector bundles, especially symplectic and orthogonal types, and develops the obstruction theory for stratifying moduli stacks of principal bundles.

Contribution

It provides explicit constructions of filtrations for specific decorated bundles and establishes a framework for stratifying moduli stacks using Harder-Narasimhan types.

Findings

Constructed Harder-Narasimhan filtrations for symplectic and orthogonal bundles.

Compared canonical reductions with existing methods.

Set up obstruction theory for principal bundle stratification.

Abstract

A decorated vector bundle is a vector bundle equipped with a reduction of structure group to a complex reductive subgroup $G \subseteq \mathbf{GL}(r,\mathbb{C})$. Examples include symplectic and special-orthogonal vector bundles, as well as vector bundles with trivial determinants. In this expository paper, we provide direct constructions of Harder-Narasimhan filtrations of symplectic and special-orthogonal vector bundles, and use them to construct canonical reductions in the sense of Atiyah and Bott. We compare these canonical reductions to those constructed by Biswas and Holla. Lastly, we set up the obstruction theory necessary to define Harder-Narasimhan types of principal bundles, and stratify the moduli stack of principal $G$-bundles.