Pome

Theorem 13.14.18AEE72111FEDA

Theorem 13.14.18.AEE72111FEDA If $v_+(\omega )=+\infty $, there is $U\in \mathrm{PSH}(X,\omega )$, $U\leq 0$, such that

\[ \mathrm{MA}^{\mathrm{ext}}_\omega (U)(X)=+\infty , \qquad \int _ X\mathrm{e}^{-pU}\, \mathrm{d}V<+\infty \quad \hbox{for every }p>0. \]

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

\[ M_ j:=\int _ X\omega _{u_ j}^ n\geq 2^{j(n+1)}. \]

The partial sums $U_ N=\sum _{j\leq N}a_ ju_ j$ decrease and satisfy

\[ \omega +\mathrm{dd}^{\mathrm{c}}U_ N =\sum _{j\leq N}a_ j\omega _{u_ j} +\left(1-\sum _{j\leq N}a_ j\right)\omega \geq 0. \]

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

\[ v_{j,t}=\max (v_ j,-t),\qquad W_{j,t}=a_ ju_ j+v_{j,t}. \]

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

\begin{align*} \mathbf1_{D_ t}\omega _{\max (U,-t)}^ n & =\mathbf1_{D_ t} \bigl (a_ j\omega _{u_ j} +(1-a_ j)\omega +\mathrm{dd}^{\mathrm{c}}v_{j,t}\bigr )^ n\\ & \geq a_ j^ n\mathbf1_{D_ t}\omega _{u_ j}^ n. \end{align*}

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

\[ \mathrm{MA}^{\mathrm{ext}}_\omega (U)(X) \geq a_ j^ nM_ j\geq 2^ j. \]

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

\begin{align*} \exp \! \left(-p\sum _{N<j\leq m}a_ ju_ j\right) & \leq \sum _{N<j\leq m}\frac{a_ j}{\delta } \mathrm{e}^{-p\delta u_ j} +1-\sum _{N<j\leq m}\frac{a_ j}{\delta }. \end{align*}

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. $\square$

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