On positive semi-definite holomorphic sectional curvature with many zeroes

TL;DR

This paper constructs complete Kähler metrics with semi-positive holomorphic sectional curvature that have many zeroes, using Calabi's Ansatz on line bundle total spaces, and explores conjectures in the compact case.

Contribution

It introduces new methods to produce such metrics on line bundle total spaces and formulates a conjecture for the compact case.

Findings

Existence of metrics with semi-positive holomorphic sectional curvature and many zeroes.

Application of Calabi's Ansatz to line bundle total spaces.

Formulation of a conjecture for the compact case.

Abstract

We establish the existence of complete K\"ahler metrics of semi-positive holomorphic sectional curvature with many zeroes in an interesting and natural geometric setting. Specifically, we use Calabi's Ansatz in the form due to Koiso-Sakane to produce such metrics on the total space of powers of the tautological line bundles over the projective line. We formulate a general conjecture regarding the compact case and use a product approach to obtain an analogous result in the non-compact case.