On projectivized vector bundles and positive holomorphic sectional   curvature
  We generalize a construction of Hitchin to prove that, given any compact K\"ahler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a K\"ahler metric with positive holomorphic sectional curvature. 
Angelynn Alvarez, Gordon Heier, Fangyang Zheng
math.DG math.AG math.CV