Harder-Narasimhan filtrations of decorated vector bundles
Emanuel Roth, Florent Schaffhauser
TL;DR
This work develops Harder-Narasimhan theory for decorated vector bundles, with a focus on symplectic and special-orthogonal cases. It constructs explicit HN filtrations and canonical reductions, linking Atiyah–Bott's and Biswas–Holla's perspectives, and proves the equivalence between Ramanathan stability and slope stability in decorated settings. The paper also builds an obstruction-theoretic framework to define Harder-Narasimhan types of principal G-bundles, tying these types to topological data via fundamental groups and lattices, and uses them to stratify the moduli stack Bun(G). Together these results provide a coherent approach to stability and stratification for decorated bundles and their principal counterparts, with implications for moduli theory and gauge-theoretic applications.
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 as done by 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.