Complete Kähler metrics with positive holomorphic sectional curvatures on certain line bundles (related to a cohomogeneity one point of view on a Yau conjecture)

In this article, we study Kähler metrics on a certain line bundle over some compact Kähler manifolds to find complete Kähler metrics with positive holomorphic sectional (or bisectional) curvatures. Thus, we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.