On Positive Semi-definite Holomorphic Sectional Curvature with Many Zeroes

In this thesis we address the existence problem of complete K"ahler metrics of semi-positive holomorphic sectional curvature with many zeroes. The amount of zeroes is measured in terms of the rank invariant introduced by Heier-Lu-Wong. Specifically, we use Calabi's Ansatz in the form due to Koiso-Sakane to produce such metrics on the total space of line bundles over the projective line, inspired by recent work of Yang-Zheng in the positive curvature case on compact Hirzebruch manifolds. We formulate a general conjecture regarding semi-positive holomorphic sectional curvature with many zeroes in the compact case and use a product approach to establish an analog result in the non-compact case. \par To better understand our newfound metrics, we also investigate the sign of the Ricci curvature and the positivity of sums of eigenvalues of the Ricci curvature.\par In the case of the product of the Riemann sphere and the complex plane, which was our first object of study, we construct a complete non-K"ahler metric by conformal change of a product metric without the use of Calabi's Ansatz. Finally, in the appendix, we describe the difficulties that preclude us from getting results from Calabi's Ansatz in the compact case.