Numerically Flat Bundles on Curves and Strong Semistability

Time: 10:30–11:30, 11 September, 2026

Venue: Room 612, Building A6

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

Abstract: This talk explains the relationship between positivity and strong semistability on a smooth projective curve. After recalling nef and numerically flat vector bundles, we describe nefness in terms of slopes after pullback to curves. The central result is the curve case of Langer’s theorem: a vector bundle is numerically flat if and only if it is strongly semistable of degree zero. The degree-zero hypothesis is essential and will be illustrated by simple examples. We also discuss the characterization of numerical flatness by pullback along arbitrary morphisms from smooth projective curves, together with examples involving line bundles and extensions of the trivial bundle. Finally, we briefly explain the tensor-categorical significance of numerically flat bundles and their relation with the (S)-fundamental group scheme.