PomeCorollary 11.21.2.39C76A336DDA If $F$ admits a smooth Hermitian metric $h$ with
\[ h(R_ h(u,\bar u)v,v)\leq 0 \quad \text{for every }x\in B,\ u\in T_ x^{1,0}B,\ v\in F_ x, \]
then the least smooth convexity index of $U_ F$ is $n+r$.
Corollary 11.21.239C76A336DDA
Proof. Put $G(v)=h(v,v)$. At $x\in B$, take a holomorphic frame that is unitary and Chern-normal at $x$, and write a tangent vector to $F$ at $v\in F_ x$ as $(u,w)$, with $u\in T_ x^{1,0}B$ and $w\in F_ x$. The calculation in the proof of lemma 11.18.1 gives
\[ \mathcal L_ G(u,w)=-h(R_ h(u,\bar u)v,v)+|w|_ h^2\geq 0, \]
with zero mixed block. Thus $G$ is smooth, plurisubharmonic and positive off the zero section, and it satisfies $G(cv)=|c|^2G(v)$. Apply proposition 11.21.1. $\square$
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