Proposition 11.11.268260DAF47D2

Proof. Choose $m\geq 1$ such that $L^ m$ is very ample. Its complete linear system gives an embedding $\iota :\mathbb P(F)\to \mathbb P^ N$ with $L^ m=\iota ^*\mathcal O_{\mathbb P^ N}(1)$. Pull back the Fubini–Study metric to a metric $k$ on $L^ m$, and take its positive $m$th root $h$ on $L$: in a local frame $h=k^{1/m}$, and the transformation law for $L^ m$ makes these local roots a metric on $L$. Its curvature is

because the differential of the embedding is injective. Apply lemma 11.11.1. Since the base is compact, the resulting $H$ is bounded on the unit sphere bundle of a fixed Hermitian metric. Homogeneity then gives $0<H(v)\leq C|v|^2$ off zero, which proves continuity of the extension by zero. $\square$