PomeA211FB2FDF28We now take the Hermitian background $\theta =\omega $ and the reference $\varphi =0$. If $\omega $ does not have the bounded mass property, that is, if the upper mass $v_+(\omega )$ of definition 13.14.14 is infinite, then the series constructed in theorem 13.14.18 has infinite mass. It splits into a smooth finite sum and a tail whose curvature is bounded below by an arbitrarily small multiple of $-\omega $. Multiplying a sufficiently late tail by the stretching factor gives a competitor for that stretch, so the series lies in the relative envelope class of zero.
Proposition 16.28.72.AA248AFFA815 If $v_+(\omega )=+\infty $, the potential $U$ in theorem 13.14.18 can be chosen so that, in addition,
The same conclusions hold for $\varepsilon U$, $0<\varepsilon \leq 1$. Each of these potentials has infinite extended mass and trivial multiplier ideals at every positive weight.
Proof. Use the smooth nonpositive $\omega $-plurisubharmonic functions $u_ j$, $j\geq 1$, of theorem 13.14.18 and their convergent series $U=\sum _{j\geq 1}2^{-j}u_ j$. For each integer $N\geq 1$, write
The head $A_ N$ is smooth, and since the finite partial sums of the tail decrease, $V_ N\in \mathrm{PSH}(X,\delta _ N\omega )$; the tail is not identically $-\infty $, because $U=A_ N+V_ N$ is not.
Fix $b>1$ and choose $N$ so large that $b\delta _ N\leq 1$. Then
is $\omega $-plurisubharmonic, because $\mathrm{dd}^{\mathrm{c}}h_ b\geq -b\delta _ N\omega \geq -\omega $. Moreover, $h_ b\leq bU$. Thus $h_ b\in \mathcal A_ b(U,0)$ for every $b$, and since $U\leq 0$, this proves $U\in \mathcal E_{\mathrm{env}}(X,\omega ,0)$. For $\varepsilon U$, use the tails $\varepsilon V_ N$, and choose $N$ with $b\varepsilon \delta _ N\leq 1$. By proposition 16.28.65, the singularity envelope of each of these potentials is $P_\omega [0]=0$. Their infinite mass and trivial multiplier ideals are given by corollary 13.14.19. $\square$
The bound on the mass in proposition 16.28.7 concerns the envelope fixed point itself. This example shows that potentials with the same singularity envelope as a fixed point, even with trivial multiplier ideals at every positive weight, need not have finite mass.
Corollary 16.28.73.944620474A50 The following conditions on a Hermitian metric $\omega $ are equivalent:
$v_+(\omega )<+\infty $;
every member of $\mathcal E_{\mathrm{env}}(X,\omega ,0)$ has finite extended mass;
every $u\in \mathrm{PSH}(X,\omega )$ with $P_\omega [u]=0$ has finite extended mass.
Proof. If $v_+(\omega )<+\infty $, all $\omega $-plurisubharmonic functions have finite extended mass by proposition 13.14.16. If $v_+(\omega )=+\infty $, proposition 16.28.72 contradicts each of the other two conditions. $\square$
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