On projectivized vector bundles and positive holomorphic sectional curvature

We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M M with positive holomorphic sectional curvature and any holomorphic vector bundle E E over M M , the projectivized vector bundle P ( E ) { P}(E) admits a Kähler metric with positive holomorphic sectional curvature.