PomeLemma 11.11.1.40D29F4EF95B A smooth Hermitian metric $h$ on $L$ determines a positive smooth complex homogeneous function $H$ of degree two on $F^\times $: the squared norm of $v$ in the dual metric on the tautological line $\mathcal O_{\mathbb P(F)}(-1)_{[v]}$. The function $H$ is strictly plurisubharmonic if and only if $R_ h$ is strictly positive on $\mathbb P(F)$.
Lemma 11.11.140D29F4EF95B
Proof. Take local holomorphic coordinates $y$ on $\mathbb P(F)$ and a holomorphic frame $e$ of the tautological line bundle, and write the metric of the dual frame as $h=\mathrm e^{-\varphi }$. The dual metric gives
\[ H(y,\lambda e)=|\lambda |^2\mathrm e^{\varphi (y)},\qquad R_ h=\mathrm{dd}^{\mathrm{c}}\varphi . \]
Let $\mathcal L$ denote the complex Hessian, without the factor $\mathrm i/(2\pi )$ of the curvature normalization. Differentiating at a point with $\lambda \ne 0$ gives
\[ \mathcal L H(u,t) =H\left(\mathcal L\varphi (u)+ \left|\frac{t}{\lambda }+\partial \varphi (u)\right|^2\right), \qquad (u,t)\in T_ y^{1,0}\mathbb P(F)\oplus \mathbb C. \]
If $\mathcal L\varphi $ is positive definite, the right-hand side is positive for every nonzero $(u,t)$: the first term detects $u\ne 0$, and the second detects $t\ne 0$ when $u=0$. Conversely, for $u\ne 0$, set $t=-\lambda \partial \varphi (u)$; strict positivity of $\mathcal L H$ then forces $\mathcal L\varphi (u)>0$. This is exactly positivity of $R_ h$. The squared tautological norm is independent of the chosen frame and is complex homogeneous, so the local functions define the asserted global function $H$. $\square$
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