Nonnegative curvature on vector bundles over ${CP}^3$ and its blow-up

Every real vector bundle of rank at least seven over $C{P}^3$, or every real vector bundle of rank at least eight over $C{P}^3\mathrm{\#}C{P}^3$, admits a complete metric of nonnegative sectional curvature on its total space. The corresponding unit sphere bundles also admit nonnegative sectional curvature. These are fixed-rank results: no additional trivial summand is required, and there is no restriction on any characteristic class. The proof combines low-dimensional bundle classification with the Grove–Ziller lifting theory over $S^4$ and $C{P}^2$.