PomeCorollary 11.29.10.A86A6F25FF07 If $E$ is globally generated and $m_0=0$, the level-zero test succeeds exactly when the complete evaluation map $\Phi :F\to H^0(X,E)^*$ is an immersion off the zero section.
Corollary 11.29.10A86A6F25FF07
Proof. At $m=0$, one has $\Psi _0=\mathrm d\Phi (\xi )\mathrm d\Phi (\xi )^*$, where $\Phi $ is written in the coordinates given by the chosen basis. If $\mathrm d\Phi $ is injective off the zero section, $B=\operatorname {Id}$ gives a strictly positive Levi form. If $\mathrm d\Phi $ has a nonzero kernel vector there, fiber dilation and normalization of the tangent vector place such a vector in $K$, where $\Psi _0=0$, and no Hermitian matrix can separate the image from zero. $\square$
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