摘要: In this talk, we discuss positivity properties and Hodge--Riemann relations for nef vector bundles. We establish Bloch--Gieseker theorems for top Chern classes of nef vector bundles under a Hall--Rado-type condition, a condition that has been noticed and extensively used by Jiajun Hu and Jian Xiao. For Schur classes, we prove a positivity of intersection numbers and Hodge--Riemann relations on bidegree (1,1) under a Hall--Rado-type condition, extending Demailly--Peternell--Schneider's classical theorems and the recent works of Ross--Toma and Lu--Zheng. Time permitting, we also discuss generalizations of the descent lemma to Schur classes of vector bundles and applications to matroid theory. This is joint work with Weizhe Zheng, as an extension of a 2025 summer school program at AMSS.