Title of the course: On the geometry of holomorphic sectional curvature and rational curves
Instructor: Professor Shin-ichi Matsumura
Institution: Tohoku University (Japan)
Dates: 27 January – 1 February, 2020
Prerequisites: Complex manifolds , (holomorphic) vector bundles(not a must but preferable)
Level: Graduate and Advanced undergraduate
Abstract: In this lecture, we discuss the notation of holomorphic sectional curvatures after we review hermitian metrics, Chern curvatures, and their properties of holomorohic vector bundle over complex manifolds. In particular, I explain a realtion between positivity of holomorphic sectional curvatures and the geometry of rationtal curves (that is, 1-dim projective space embedded in manifolds). The goal of this talk is to give a structure theorem for rationally connected fibrations of projective manifolds with non-negative holomorphic sectional curvature.
Textbook or/and course webpage:
1. Foundations of Differential Geometry, written by Shoshichi Kobayashi and Katsumi Nomizu, Wiley Classics Library.
2. RC-positivity, rational connectedness and Yau’s conjecture, written by Xiaokui Yang, Camb. J. Math. 6 (2018), no. 2, 183–212
3. On projective manifolds with semi-positive holomorphic sectional curvature,
4. On morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature written Shin-ichi Matsumura, available at arXiv.
Language: EN