Stable ample vector bundles without semipositive Kobayashi curvature

We construct slope-stable ample rank-two vector bundles on every Hirzebruch surface ${\mathbb{F}}_e$ that admit no smooth strongly pseudoconvex Finsler metric with semipositive Kobayashi curvature. These examples give a negative answer to Kobayashi’s problem. The same bundles give counterexamples to the smooth strongly pseudoconvex Finsler formulation of Fulton’s problem and to the Hartshorne–Schneider conjecture on the convexity of complements. For $e=1$, these counterexamples have a simply connected Fano base that is not a product.