PomeCorollary 11.30.5.9F9E99FDE392 For the data $E=L$ and $A=L^ q$, with the Fubini–Study metric of a basis of $H^0(X,A)$, the criterion of theorem 11.29.9 applies with $m_0=0$. Its untwisted level fails, although $E$ carries a strictly Griffiths-positive metric, and a higher level succeeds. The atom witnessing the failure at level zero annihilates only the Levi forms of the level-zero test space, and does not give a measure with $D^*T=0$.
11.30.4 Relation to the fixed very ample criterion32E68898DD56
Proof. By lemma 11.30.1, $L$ is globally generated and $A$ is very ample, as the criterion requires. At a ramification point, the complete untwisted evaluation map has a nonzero tangent vector in its kernel, in the direction of the cover, as in the preceding proof; hence corollary 11.29.10 gives failure at level zero. The positive root metric and theorem 11.29.9 give success at some higher level. Moreover, the Levi form of the dual of this positive metric pairs strictly positively with every nonzero positive matrix-valued measure on $W\to S$, so the atom at level zero cannot annihilate the Levi forms of all smooth Hermitian forms. $\square$
The level $m$ of the criterion in theorem 11.29.9 corresponds to $j=qm$ in the root-weight family, since $E\otimes A^ m=L^{qm+1}$ and $m\phi _ A=qm\phi $. The first success at $j=q-1$ therefore gives no bound on the first successful level $m$ of that criterion. The root-weight family is nested in steps of $q$, because multiplication by the sections of $A$ uses $\sum _ i|\sigma _ i|^2=\mathrm e^{q\phi }$; we do not claim that it is nested in steps of one. As $q$ grows, the root-weight family has arbitrarily many failed initial levels, although the bundle carries a strictly positive metric.
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