Proposition 11.10.51263BBAE3998

Proof. For $\tau >0$, set $Q_\tau =Q+\tau H$; it is positive, complex homogeneous and strictly plurisubharmonic. On a compact annulus of vectors of norm close to one, local strong convexity of $Q$ in the fibers gives a positive uniform margin in finitely many frames, while the real fiber Hessian of $H$ is uniformly bounded there. Choose $\tau $ so small that the possible negative part of $\tau $ times this Hessian consumes less than half the margin. Homogeneity then gives local strong convexity of $Q_\tau $ in the fibers everywhere off zero. Apply theorem 11.10.4 to $Q_\tau $ and take one of its smooth regularizations; its square root is the required smooth complex norm. $\square$