Proposition 12.20.1.EBFCF76A5A51 The limits
\[ \underline a(\theta )=\liminf _{\epsilon \downarrow 0} a_{\mu ,h}(\theta ;\epsilon ), \qquad \overline a(\theta )=\limsup _{\epsilon \downarrow 0} a_{\mu ,h}(\theta ;\epsilon ) \]
are independent of $\mu $ and $h$, and satisfy
\[ 0\leq \underline a(\theta )\leq \overline a(\theta )\leq n. \]
They are unchanged when $\theta $ is replaced by $\theta +\mathrm{dd}^{\mathrm{c}}f$, with $f$ smooth, or by $r\theta $, with $r>0$. If $\mathcal V_\mu (\theta )>0$, both limits are zero.
Proof.
Since $\theta $ is pseudoeffective, monotonicity and homogeneity give
\[ \epsilon \lambda _\mu (h) \leq \lambda _\mu (\theta +\epsilon h) \leq \lambda _\mu (\theta +h), \qquad 0<\epsilon \leq 1. \]
Thus there are constants $c,C>0$ such that
\[ c\epsilon ^ n\leq \mathcal V_\mu (\theta +\epsilon h)\leq C, \qquad 0<\epsilon \leq 1. \]
Taking logarithms and dividing by the negative number $\log \epsilon $ shows that the quotients $a_{\mu ,h}(\theta ;\epsilon )$ stay bounded as $\epsilon \downarrow 0$, and that their lower and upper limits lie in $[0,n]$.
By proposition 12.15.1, changing the smooth probability measure multiplies the volume by a factor between two fixed positive constants. The resulting error in the logarithm is bounded, and after division by $\log \epsilon $ it tends to zero.
For another Hermitian metric $h'$, choose constants $c_2\geq c_1>0$ with $c_1h\leq h'\leq c_2h$. Then
\[ \mathcal V_\mu (\theta +c_1\epsilon h) \leq \mathcal V_\mu (\theta +\epsilon h') \leq \mathcal V_\mu (\theta +c_2\epsilon h). \]
For any fixed $t>0$ and small $\epsilon $,
\[ \frac{\log \mathcal V_\mu (\theta +t\epsilon h)}{\log \epsilon } =a_{\mu ,h}(\theta ;t\epsilon ) \frac{\log (t\epsilon )}{\log \epsilon }. \]
The last factor tends to one and the first is bounded, while the rescaling $\epsilon \mapsto t\epsilon $ does not change lower and upper limits as $\epsilon \downarrow 0$. Hence the left side has the same lower and upper limits as $a_{\mu ,h}(\theta ;\epsilon )$. Dividing the preceding inequalities by $\log \epsilon <0$ reverses them, and taking lower and upper limits proves independence of the metric.
Invariance under smooth $\mathrm{dd}^{\mathrm{c}}$ changes follows from that of $\mathcal V_\mu $. For scaling, homogeneity gives
\[ \mathcal V_\mu (r\theta +\epsilon h) =r^ n\mathcal V_\mu (\theta +(\epsilon /r)h). \]
The bounded error $n\log r$ and the same rescaling argument prove scale invariance. Finally, if $\mathcal V_\mu (\theta )>0$, then by theorem 12.11.1 the perturbed volumes lie between $\mathcal V_\mu (\theta )$ and $\mathcal V_\mu (\theta +h)$ for $0<\epsilon \leq 1$, so both limits are zero.
$\square$
Pome
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