Time: 9:00-12:00, Saturday, October 11, 2025
Venue: E4-201
Host: Xin Fu, ITS
Speaker: Shiyu Zhang, USTC
Title: Differential-Geometric Curvature Positivity and Rational Connectedness
Abstract: Recently, we proved that a compact Kähler manifold has rational dimension at least n-k+1 if its tangent bundle is BC-p positive for every p≥k. This curvature positivity, introduced by L. Ni, can be guaranteed by various differential-geometric curvature positivity of the tangent bundle, such as positive holomorphic sectional curvature, mean curvature positivity, uniformly RC-positivity and etc. We demonstrate that this positivity naturally arises in a Bochner-type formula associated with the MRC fibration. As a new application in a broader context, we answer a question posed by F. Zheng, Q. Wang, and L. Ni, namely, that any compact Kähler manifold with positive orthogonal Ricci curvature must be rationally connected. Additionally, I will introduce our earlier work, which generalizes Yau's conjecture on positive holomorphic sectional curvature to the quasi-positive case via a Bochner-type integral inequality. These works are joint with Xi Zhang.
