Theorem 11.10.43FA7CB52F1E5

Proof. Fix a smooth Hermitian squared norm $G$ on $F$, and let $M=\mathbb P(F)$ be the bundle of complex lines in $F$. The quotient $m=Q/G$ is a positive continuous function on the compact manifold $M$. Define a sheaf $\mathcal S$ on $M$ by requiring that the corresponding function $Gm$ on the cone be positive, strictly plurisubharmonic, and locally strongly convex in the fibers.

This sheaf is closed under maxima. Indeed, multiplication by $G>0$ commutes with maxima. Near a point of the cone, subtract a common positive quadratic barrier from both functions; their maximum remains plurisubharmonic, and on sufficiently small fiber balls it remains convex. Positivity is also preserved.

For stability on a compact subset $K$ of $M$, restrict to vectors of norm one lying over $K$, and to a fixed compact annulus around them. Finitely many bundle charts provide positive common lower bounds for both barriers. If $u$ is a smooth function on $M$ that is sufficiently small in $C^2$ near $K$, the Levi form and the real fiber Hessian of $Gu$ are smaller than half these bounds on that annulus. The perturbed function is therefore still positive and has both strict margins. Homogeneity transports the local conditions to the rest of the cone over $K$. This proves the stability required in lemma 11.10.2.

On a sufficiently small projective coordinate chart, lemma 11.10.3 gives local approximation, including derivative control on compact subsets where $m$ is smooth; dividing its regularizations by $G$ gives local smooth sections of $\mathcal S$. By lemma 11.10.2, there are smooth positive global sections $m_ j\to m$ uniformly. Put $Q_ j=Gm_ j$ off the zero section and $Q_ j=0$ on the zero section. Homogeneity gives continuity at zero, uniformly over the compact base, as well as the asserted convergence on a fixed unit sphere bundle.

It remains to see that $p_ j$ is a norm. Local convexity in the fibers away from zero implies convexity on the whole fiber. Indeed, the restriction of $Q_ j$ to a real line avoiding zero is locally convex and hence convex, while on a real line through zero, homogeneity gives $Q_ j(tv)=t^2Q_ j(v)$ for real $t$, which is convex including at zero. Thus the set $B_ x=\{ v:Q_ j(v)\leq 1\} $ is convex, balanced and absorbing, and its gauge is $\sqrt{Q_ j}$. If $v,w$ are nonzero, convexity applied to $v/p_ j(v)$ and $w/p_ j(w)$, with weights proportional to $p_ j(v)$ and $p_ j(w)$, gives the triangle inequality; the cases with a zero vector follow directly. Positivity, smoothness off zero and complex homogeneity give the remaining properties of a smooth complex norm. $\square$