Corollary 11.21.3F0FD4C3E5239

Proof. In parallel holomorphic unitary frames, the squared norm is $G(v)=\sum _ j|v_ j|^2$, which is plurisubharmonic on the total space. Since the transition matrices are constant and unitary, $G$ is globally defined; it is smooth, positive on $F^\times $ and homogeneous of degree two under complex scalars. Apply proposition 11.21.1. $\square$