Positive holomorphic sectional curvature on rational surfaces

Abstract

In 1975, Hitchin proved that any compact complex surface admitting a Kähler metric with positive holomorphic sectional curvature $HSC>0$ is rational. Conversely, he constructed such metrics on all Hirzebruch surfaces $\mathbb{F}_k$, as a first step towards characterizing rational surfaces by the existence of a Kähler metric with suitable curvature positivity. In this paper, we prove that every projective manifold $X$ obtained from a projective toric manifold by a finite sequence of blow-ups at points admits a Kähler metric with $HSC>0$. This statement applies to all rational surfaces and therefore completes Hitchin's result, resolving the complex surface case of a problem of Yau listed in "Open Problems in Geometry". The proof has two main ingredients. First, we prove that the toric Kähler metric on a projective toric manifold arising from Delzant's construction has $HSC>0$. Second, via a one-parameter degeneration, we construct, for any such $X$, a smooth projective family $π:\mathcal X\to\mathbb C$ such that $\mathcal X_t\simeq X$ for $t\ne0$, while $\mathcal X_0$ is a projective toric manifold.