Harder–Narasimhan filtration

Statement

Let $E$ be a vector bundle on a smooth projective geometrically connected curve. Its slope is $\mu(E)=\deg(E)/\operatorname{rk}(E)$ . The Harder–Narasimhan filtration is the unique filtration by subbundles

0=E_0\subset E_1\subset\cdots\subset E_r=E

such that every quotient $E_i/E_{i-1}$ is semistable and

\mu(E_1/E_0)>\mu(E_2/E_1)>\cdots> \mu(E_r/E_{r-1}).

Its slopes and ranks determine the Harder–Narasimhan polygon. The bundle is semistable precisely when the filtration has one nonzero quotient.

Principal G-bundles

For an algebraic principal bundle under a reductive group $G$ , the analogue is a canonical reduction to a parabolic subgroup whose degree data define a dominant rational cocharacter, the HN type. For $G=\operatorname{GL}_n$ this recovers the filtration above.

Truncation of moduli

Bounding the HN polygon or HN type defines open substacks of the moduli stack of bundles that are of finite type. This Harder–Narasimhan truncation is used to control non-quasi-compact stacks of $G$ -bundles and shtukas. It is a geometric boundedness operation, distinct from Arthur's analytic truncation, although the two reflect parallel parabolic asymptotics.

References

G. Harder and M. S. Narasimhan, “On the cohomology groups of moduli spaces of vector bundles on curves,” Mathematische Annalen 212 (1975), 215–248.
Sudarshan Gurjar and Nitin Nitsure, “Harder–Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics,” 2016. arXiv.