Pome54BB8D4D734DIf the bounded mass property fails, there are smooth $\omega $-plurisubharmonic functions $u_ j$ whose masses $M_ j$ tend to infinity. In a convex combination $U=\sum _ ja_ ju_ j$, positivity of mixed products shows that each summand contributes at least $a_ j^ nM_ j$ to the extended measure of $U$. Choosing the masses $M_ j$ to outgrow $a_ j^{-n}$ therefore forces infinite mass. On the other hand, each tail $\sum _{j>N}a_ ju_ j$ is a small multiple of a convex combination of normalized potentials, and the uniform exponential integrability of lemma 5.26.2 applies to it at every exponent.
Theorem 13.14.18.AEE72111FEDA If $v_+(\omega )=+\infty $, there is $U\in \mathrm{PSH}(X,\omega )$, $U\leq 0$, such that
In particular, $\mathcal I(pU)=\mathcal O_ X$ for every $p>0$.
Proof. Set $a_ j=2^{-j}$ for $j\geq 1$. By proposition 13.14.15, we can choose smooth $\omega $-plurisubharmonic functions $u_ j$ with $\sup _ Xu_ j=0$ and
The partial sums $U_ N=\sum _{j\leq N}a_ ju_ j$ decrease and satisfy
Their $L^1$ norms are bounded by (5.26.1), so their limit $U$ is $\omega $-plurisubharmonic and nonpositive.
We first prove that the mass is infinite. Fix $j$ and write $v_ j=U-a_ ju_ j$. Passing to the limit in the remaining partial sums gives $(1-a_ j)\omega +\mathrm{dd}^{\mathrm{c}}v_ j\geq 0$. For $t>0$, put
The function $v_{j,t}$ is a bounded potential for the positive background $(1-a_ j)\omega $, so $W_{j,t}$ is bounded and $\omega $-plurisubharmonic.
On $D_ t=\{ U>-t\} $, the inequality $u_ j\leq 0$ gives $v_ j=U-a_ ju_ j\geq U>-t$, and hence $W_{j,t}=U=\max (U,-t)$ there. Plurifine locality and positivity of bounded mixed products yield
The sets $D_ t$ exhaust $X\setminus \{ U=-\infty \} $. This pole set is pluripolar, hence has zero smooth volume, so the smooth measure $\omega _{u_ j}^ n$ gives it no mass. Letting $t\to +\infty $, we obtain
Since $j$ is arbitrary, the mass is infinite.
For the integrability, fix $p>0$, let $\alpha $ be the exponent of lemma 5.26.2, and take $N$ so large that $p\delta \leq \alpha $, where $\delta =\sum _{j>N}a_ j=2^{-N}$. Write $U=A_ N+V_ N$ with $A_ N=\sum _{j\leq N}a_ ju_ j$ smooth. For a finite tail ending at $m>N$, Jensen's inequality for the convex exponential function, with the unused weight assigned to the value zero, gives
Because $u_ j\leq 0$ and $p\delta \leq \alpha $, the integrals of the exponentials on the right are uniformly bounded. Integrating and letting $m\to +\infty $ by monotone convergence, we obtain $\int _ X\mathrm{e}^{-pV_ N}\mathrm{d}V<+\infty $. Multiplication by the bounded smooth function $\mathrm{e}^{-pA_ N}$ proves $\int _ X\mathrm{e}^{-pU}\mathrm{d}V<+\infty $.
Consequently, the constant germ $1$ is locally integrable against each weight $\mathrm{e}^{-pU}$, and every multiplier ideal $\mathcal I(pU)$ is the whole structure sheaf. $\square$
Corollary 13.14.19.60F41A586D0F The bounded mass property holds if and only if $\mathrm{MA}^{\mathrm{ext}}_\omega (u)(X)<+\infty $ for every $u\in \mathrm{PSH}(X,\omega )$. If the bounded mass property fails, there are potentials with infinite extended mass and trivial multiplier ideals $\mathcal I(pu)=\mathcal O_ X$, $p>0$, arbitrarily close to zero in $L^1$.
Proof. The first assertion follows from proposition 13.14.16, theorem 13.14.18. For the second, take $\varepsilon U$, $0<\varepsilon \leq 1$, with $U$ as in theorem 13.14.18. It is $\omega $-plurisubharmonic and has the required exponential integrability. On $D_ t=\{ U>-t\} $, its truncation at level $\varepsilon t$ is $\varepsilon \max (U,-t)$, and
Positivity of mixed products bounds the restricted measure of $\varepsilon U$ below by $\varepsilon ^ n$ times that of $U$. Its limiting mass is therefore infinite. Finally, $\| \varepsilon U\| _{L^1}\to 0$ as $\varepsilon \to 0$. $\square$
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