Projektive Kurve/Vektorbündel/Harder-Narasimhan Filtration/Definition/en

Let $𝒮$ be a vector bundle on a smooth projective curve $C$ over an algebraically closed field $K$. Then the (uniquely determined) filtration

$0={𝒮}_0\subset {𝒮}_1\subset \dots \subset {𝒮}_{t-1}\subset {𝒮}_t=𝒮$

of subbundles such that all quotient bundles ${𝒮}_k/{𝒮}_{k-1}$ are semistable with decreasing slopes  ${\mu }_k=\mu ({𝒮}_k/{𝒮}_{k-1})$,  is called the Harder-Narasimhan filtration of $𝒮$.