Critical Points of holomorphic sections of line bundles and a spherical Gauss-Lucas theorem

TL;DR

This paper investigates the critical points of holomorphic sections of line bundles over complex projective space, providing a complete description for quadrics, establishing a spherical Gauss-Lucas theorem in one dimension, and analyzing the nature of critical points in general cases.

Contribution

It offers a comprehensive analysis of critical points for holomorphic sections, including explicit descriptions for quadrics and a new spherical Gauss-Lucas theorem for one-dimensional cases.

Findings

Complete description of critical points for quadrics

Proof of spherical Gauss-Lucas theorem for n=1

Generic sections have isolated, non-degenerate critical points

Abstract

We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When $n=1$, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.