Theorem 18.13.1.6529F0A74D79 For every nonempty finite or countably infinite subset $\{ x_ j\} \subset X$, there are nonpositive $u,w\in \mathrm{PSH}(X,h)$ such that
\[ h+\mathrm{dd}^{\mathrm{c}}u,\ h+\mathrm{dd}^{\mathrm{c}}w\geq \tfrac 34h, \qquad u(x_ j)=w(x_ j)=-\infty . \]
The same $w$ satisfies $tw\preceq u$ for every $0<t\leq 1$, and
\[ u\in \mathcal E_{\mathrm{env}}(X,h,0),\qquad \mathrm e^{-su},\ \mathrm e^{-sw}\in L^1(X,\mathrm dV) \quad (s>0). \]
Thus $\mathcal I(su)=\mathcal I(sw)=\mathcal O_ X$ for every $s>0$. If the prescribed set is dense, both potentials are unbounded on every nonempty open set.
Proof.
Fix $0<\beta <\beta '<1$. Choose a finite atlas of relatively compact charts with larger coordinate neighborhoods, on which the coefficients of $\mathrm dV$ are uniformly bounded. For each $x_ j$, choose a coordinate ball of radius $r_ j\leq 1$ centered at $z_ j$, contained in one of the larger charts. Let $\chi $ be a smooth cutoff supported inside the unit ball, with $0\leq \chi \leq 1$ and $\chi =1$ near zero. On this ball set
\[ \chi _ j(z)=\chi ((z-z_ j)/r_ j),\qquad L_ j=1-\log (|z-z_ j|^2/r_ j^2). \]
The functions $-\chi _ jL_ j^\sigma $, for $\sigma =\beta ,\beta '$, extend by zero outside the ball. Near the center they are plurisubharmonic: for $s=\log (|z-z_ j|^2/r_ j^2)$ the profile $q_\sigma (s)=-(1-s)^\sigma $ has
\[ q_\sigma '=\sigma (1-s)^{\sigma -1}>0, \qquad q_\sigma ''=\sigma (1-\sigma )(1-s)^{\sigma -2}>0. \]
To justify the extension across the center, take a smooth nondecreasing $\zeta $ equal to zero on $(-\infty ,-2]$ and one on $[-1,+\infty )$. For large $A$ put
\[ q_{\sigma ,A}'(s)=q_\sigma '(s)\zeta (s/A),\qquad q_{\sigma ,A}(s)=q_\sigma (0)-\int _ s^0q_{\sigma ,A}'(v)\, \mathrm dv. \]
These profiles are convex and increasing, constant near the center, and agree with $q_\sigma $ on $s\geq -A$. As $A$ increases they decrease to $q_\sigma $, including its value $-\infty $ at the center. Radial integrability gives convergence in $L^1$. The cutoff annulus is unchanged for large $A$, and its Hessian has a finite lower bound. After a fixed local quadratic correction, the decreasing-limit theorem gives the canonical quasi-plurisubharmonic extension; its current bound passes to the limit.
Put $e_ j=2^{-j-2}$. Scale both profiles by the same $0<\delta _ j\leq 2^{-j}$ so that the nonpositive functions
F52CC5DA3364
18.13.2
\begin{equation} a_ j=-\delta _ j\chi _ jL_ j^\beta , \qquad c_ j=-\delta _ j\chi _ jL_ j^{\beta '}, \qquad \mathrm{dd}^{\mathrm{c}}a_ j,\ \mathrm{dd}^{\mathrm{c}}c_ j\geq -e_ jh \end{equation}
have these lower bounds globally. This is possible because only the smooth cutoff annulus can contribute a negative Hessian. Coordinate rescaling and the finite atlas give
\[ \| a_ j\| _{L^1}+\| c_ j\| _{L^1}\leq C\delta _ j. \]
Consequently the nonpositive partial sums decrease with uniformly bounded $L^1$ norms. By corollary 5.25.3, they define potentials
\[ u=\sum _ j a_ j,\qquad w=\sum _ j c_ j \]
with the stated canonical values and with currents at least $(1-\sum _ j e_ j)h\geq 3h/4$. Each prescribed point is a pole of its own summand. This also proves the assertion when the prescribed set is dense.
For $b\geq 1$ let
\[ S=\sum _ j\delta _ j>0,\qquad M_ b=\max \{ 0,\sup _{L\geq 1}(bL^\beta -L^{\beta '})\} <\infty . \]
Multiplying the defining inequality by $\delta _ j\chi _ j$ and summing gives
1696AB66FF1D
18.13.3
\begin{equation} w\leq bu+M_ bS. \end{equation}
The inequality first holds for finite sums and then for their decreasing limits, including common poles. Taking $b=1/t$ gives $tw\preceq u$. Since $h>0$, convexity with zero makes $tw$ $h$-plurisubharmonic. Also $w-M_ bS\leq bu$ is an $h$-plurisubharmonic stretching competitor for every $b\geq 1$, so definition 16.28.61 gives $u\in \mathcal E_{\mathrm{env}}(X,h,0)$.
It remains to prove integrability at every positive exponential scale. Fix $s>0$ and use Jensen's inequality with weights $\delta _ j/S$, assigning the unused weight in each finite sum to zero:
\[ \mathrm e^{-s\sum _{j\leq N}a_ j} \leq 1-\sum _{j\leq N}\frac{\delta _ j}{S} +\sum _{j\leq N}\frac{\delta _ j}{S} \mathrm e^{sS\chi _ jL_ j^\beta }. \]
The integrals on the right are bounded uniformly in $j$, since coordinate rescaling reduces the contribution inside the ball to a constant times
\[ \int _0^1\exp \bigl (sS(1-\log R^2)^\beta \bigr )R^{2n-1}\, \mathrm dR. \]
With $x=-\log R^2$, its exponent is $-nx+sS(1+x)^\beta $, which is at most $-nx/2$ for large $x$. Integrating the Jensen inequality and using monotone convergence proves $\mathrm e^{-su}\in L^1$. The same calculation with $\beta '$ proves the assertion for $w$. Every holomorphic germ is bounded on a smaller coordinate neighborhood, so definition 6.8.5 gives the stated trivial ideals.
$\square$
Pome
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