Theorem 11.29.9C334EF3CAF72

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. $\square$