Finite Determinacy of Harder–Narasimhan Filtrations under Frobenius

Time: 08:30–10:30, 11 September, 2026

Venue: Room 612, Building A6

Online Zoom link:  https://zoom.us/j/99636681387?pwd=0WscBnehOJig68SqctGluVuA3RwraE.1

Abstract: Although successive Frobenius pullbacks may repeatedly destabilize a vector bundle, Langer proved that this process eventually acquires a rigid structure. We introduce the asymptotic maximal and minimal slopes and the normalized Harder–Narasimhan polygons associated with iterated Frobenius pullbacks. These polygons form an increasing bounded sequence and converge to a limiting polygon. The main theorem states that every vector bundle has the finite determinacy of Harder–Narasimhan property: after a sufficiently high Frobenius pullback, all Harder–Narasimhan graded pieces are strongly semistable, and all subsequent Harder–Narasimhan filtrations are obtained simply by Frobenius pullback. We will explain the main mechanism of Langer’s proof and derive the stabilization of normalized polygons and the rationality of asymptotic slopes.