PomeB1D39F6DC8D4A positive matrix-valued measure on $W\to S$ is a linear functional $T$ on the space of continuous Hermitian forms $\Phi $ on $W$ such that $\langle T,\Phi \rangle \geq 0$ whenever $\Phi \geq 0$. Its trace mass relative to $b$ is $\langle T,b\rangle $. We write $D^*T=0$ if $\langle T,Dh\rangle =0$ for every smooth Hermitian form $h$ on $F$.
Theorem 11.29.9.C334EF3CAF72 With the fixed projective data above, the following are equivalent:
$E$ admits a smooth strictly Griffiths-positive Hermitian metric.
For some $m\geq m_0$, a Hermitian matrix $B$ makes $G_{m,B}$ strictly plurisubharmonic off the zero section.
For some $m\geq m_0$, $0\notin \operatorname {conv}(\Psi _ m(K))$.
There is no nonzero positive matrix-valued measure $T$ on $W\to S$ satisfying $D^*T=0$.
Proof. Given a metric as in the first assertion, its dual metric $h$ on $F$ satisfies $Dh\geq \delta b$ on $S$ for some $\delta >0$, by lemma 11.29.1 and compactness. The $C^2$ density in lemma 11.29.7 and the continuity of $D$ give a form $h_{m,B}$ with $D h_{m,B}\geq \delta b/2$, which proves the second assertion. Conversely, if $G_{m,B}$ is strictly plurisubharmonic, its vertical Levi form shows that $h_{m,B}$ is positive, and lemma 11.29.1 shows that its dual metric is strictly Griffiths-positive. The second and third assertions are equivalent by proposition 11.29.4.
If the first assertion holds, every nonzero positive matrix-valued measure $T$ has positive trace mass, and
so such a measure cannot satisfy $D^*T=0$. Conversely, suppose that the third assertion fails at every level. By proposition 11.29.4, choose at each level $m$ an atomic probability measure $\mu _ m$ on $K$ with $\int _ K\Psi _ m\, \mathrm d\mu _ m=0$, so that $\mu _ m$ annihilates $Dh$ for every $h\in \mathcal H_ m$. A subsequence converges weakly to a probability measure $\mu $ on the compact set $K$. Since the spaces $\mathcal H_ m$ are nested, the limit annihilates $Dh$ for every $h$ in each fixed $\mathcal H_ j$, and the $C^2$ density and the continuity of $D$ extend this to every smooth Hermitian form.
Define
This is a positive matrix-valued measure on $W\to S$ with $\langle T,b\rangle =1$ and $D^*T=0$. Its entries in local frames are represented by scalar measures: positivity bounds the functional on Hermitian forms of uniform norm at most one by the trace mass, so the Riesz representation theorem applies to the coordinate functionals, and a finite partition of unity combines these local representations. This contradicts the fourth assertion and completes the proof. $\square$
The equation $D^*T=0$ concerns a bundle over the sphere bundle $S\subset F=E^*$; it should not be confused with the equation $\mathrm{dd}^{\mathrm{c}}T=0$ for currents on $X$. The theorem gives no bound for the first successful level, and no finite procedure for deciding that all levels fail.
Corollary 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.
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. $\square$
When this untwisted test fails, the higher twists remain available. The finite alternatives use all Hermitian coefficient matrices and the complete spaces of sections; testing only positive coefficient matrices or a smaller collection of sections gives a different criterion.
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