PomeCorollary 11.20.3.F8F43BAF4F9A Suppose that each $\omega _ i$ is Gauduchon, that $E_ i$ is $\omega _ i$-slope-stable as in definition 11.18.3, and that $\deg _{\omega _ i}E_ i>0$. Then the complement $\mathbb P(\mathcal O_ X\oplus E)\setminus \mathbb P(\mathcal O_ X)$ is $(n+r-k)$-convex. For $k$ stable ample Hirzebruch examples as in corollary 11.18.5, with their polarizations, this gives $(2^ k+k)$-convexity.
Corollary 11.20.3F8F43BAF4F9A
Proof. As in the proof of corollary 11.18.4, choose a Hermitian–Einstein metric on each $E_ i$. Its mean curvature is $\lambda _ i\operatorname {Id}_{E_ i}$, with $\lambda _ i>0$ by the positivity of the degree and the integrated trace identity. Apply theorem 11.20.2. For the Hirzebruch factors, $n_ i=r_ i=2$, so $n+r-k=2k+2^ k-k=2^ k+k$. $\square$
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