Vector Bundles on Curves, Slope Stability, and the Harder–Narasimhan Filtration

Time: 08:30–10:30, 10 September, 2026

Venue: Room 612, Building A6

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

Abstract: This introductory talk develops the basic language of vector bundles on smooth projective algebraic curves that will be used throughout the seminar. We recall degree and slope, stable and semistable bundles, and basic properties of morphisms and extensions between semistable bundles. The Jordan–Hölder filtration will be briefly discussed in order to distinguish it from the Harder–Narasimhan filtration. The main result is the existence and uniqueness of the Harder–Narasimhan filtration of an arbitrary vector bundle, together with its interpretation in terms of the Harder–Narasimhan polygon. Explicit examples on the projective line will illustrate the general theory and prepare the ground for studying the effect of Frobenius pullback in positive characteristic.