PomeCorollary 11.17.5.E29138267596 For every $e\geq 0$, there is an ample rank-two bundle $E$ on the Hirzebruch surface $X=\mathbb F_ e$ such that $\mathbb P(\mathcal O_ X\oplus E)\setminus \mathbb P(\mathcal O_ X)$ is not $2$-convex. The bundle may be chosen with the slope-stability and Chern classes of theorem 11.15.5.
Corollary 11.17.5E29138267596
Proof. Use theorem 11.15.5 to choose $E$ whose dual Dolbeault operator lies in the $C^4$ neighborhood of theorem 11.9.4; the split dual bundle satisfies the hypotheses of that theorem by proposition 11.13.4. Thus $E^*$ admits no smooth complex norm with plurisubharmonic square, and corollary 11.17.4 with $r=2$ gives the assertion. $\square$
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