PomeLemma 16.28.9.5C02C050C6B5 The map $u\mapsto u-\delta \chi /2$, followed by the canonical extension across the hyperplane at infinity, is a bijection from $\mathcal L_\delta $ onto $\mathrm{PSH}(\mathbb P^ n,\theta _\delta )$.
16.28.8 Affine growth and projective envelopes6EE0CD52DB46
The contact property concerns the regularized envelope. For the classical weighted extremal functions on $\mathbb C^ n$, the obstacle is imposed at every point of a compact set, and one may ask when the regularized envelope still touches the obstacle there. Compactifying $\mathbb C^ n$ to $\mathbb P^ n$ turns the logarithmic growth allowance into a semipositive background. This correspondence distinguishes contact of the raw supremum with the obstacle from contact of its upper semicontinuous regularization, and it shows how both depend on the growth allowance.
In the rest of this section we work on $\mathbb C^ n$ and on $\mathbb P^ n$ rather than on $X$. Write $\chi (z)=\log (1+\lVert z\rVert ^2)$ on $\mathbb C^ n$, and let $\omega _{\mathrm{FS}}$ be the Fubini–Study form on $\mathbb P^ n$ with this local potential, so that $\int _{\mathbb P^ n}\omega _{\mathrm{FS}}^ n=1$ with our normalization of $\mathrm{dd}^{\mathrm{c}}$. For $\delta >0$ put
\[ \theta _\delta =\tfrac \delta 2\omega _{\mathrm{FS}},\qquad \mathcal L_\delta = \{ u\in \mathrm{PSH}(\mathbb C^ n): u(z)\leq \delta \log ^+\lVert z\rVert +C_ u\} . \]
Here $\log ^+=\max (\log ,0)$, and $C_ u$ is a constant depending on $u$. Thus $\mathcal L_\delta $ is the class of plurisubharmonic functions of logarithmic growth at most $\delta $. Let $\mathcal L_\delta ^+$ consist of the members of $\mathcal L_\delta $ that are also bounded below by $\delta \log ^+\lVert z\rVert -C$ for some constant $C$.
Proof. Use homogeneous coordinates $[Z_0:\cdots :Z_ n]$ and the affine chart $Z_0\ne 0$. On a chart $Z_ i\ne 0$ meeting $H=\{ Z_0=0\} $, the local plurisubharmonic representative of $p=u-\delta \chi /2$ is
\[ p+\frac\delta 2\log \frac{\sum _{j=0}^ n|Z_ j|^2}{|Z_ i|^2} =u+\delta \log |Z_0/Z_ i|. \]
This function is locally bounded above near $H$, by the growth bound on $u$, because the norm of the affine coordinates is at most a constant, depending on the chart, divided by $|Z_0/Z_ i|$. By lemma 5.24.7, it therefore extends canonically across $H$. Since the local potentials of $\theta _\delta $ differ by smooth functions on overlaps, the extensions agree there.
Conversely, every $p\in \mathrm{PSH}(\mathbb P^ n,\theta _\delta )$ is bounded above, since $\mathbb P^ n$ is compact. The function $u=p+\delta \chi /2$ is plurisubharmonic on the affine chart, and
\[ 0\leq \tfrac 12\chi -\log ^+\lVert z\rVert \leq \tfrac 12\log 2 \]
gives the required growth bound. Neither function is identically $-\infty $ on the affine chart. Uniqueness of the canonical extension shows that the two maps are inverse to each other. $\square$
Fix a non-pluripolar compact set $K\subset \mathbb C^ n$ and a bounded real function $\psi $ on $K$. Define the raw and the regularized envelope
\[ V_\delta =\sup \{ u\in \mathcal L_\delta :u|_ K\leq \psi \} , \qquad F_\delta =V_\delta ^*, \]
where the obstacle condition is imposed at every point of $K$. On $\mathbb P^ n$ put
\[ G_\delta =\sup \{ p\in \mathrm{PSH}(\mathbb P^ n,\theta _\delta ): p|_ K\leq \psi -\delta \chi /2\} . \]
Proposition 16.28.10.B1F342A8772A The regularized envelope $G_\delta ^*$ on $\mathbb P^ n$ is bounded and $\theta _\delta $-plurisubharmonic. On the affine chart,
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16.28.11
\begin{equation} V_\delta =\tfrac \delta 2\chi +G_\delta ,\qquad F_\delta =\tfrac \delta 2\chi +G_\delta ^*\in \mathcal L_\delta ^+. \end{equation}
Moreover, $V_\delta =F_\delta $ almost everywhere.
Proof. By lemma 16.28.9, the map $u\mapsto u-\delta \chi /2$ is a bijection between the two families of competitors, which proves the identity for $V_\delta $. Constants below $\inf _ K(\psi -\delta \chi /2)$ are competitors for $G_\delta $.
Since all competitors for $G_\delta $ lie below the constant $\sup _ K(\psi -\delta \chi /2)$ on $K$, lemma 5.40.1 gives a common upper bound for them. Together with a constant competitor, the upper-envelope theorem shows that $G_\delta ^*$ is bounded and $\theta _\delta $-plurisubharmonic. Addition of the smooth function $\delta \chi /2$ in (16.28.11) commutes with regularization, and the two-sided bounds for $G_\delta ^*$ give $F_\delta \in \mathcal L_\delta ^+$.
Locally, adding a potential of $\theta _\delta $ makes the competitors plurisubharmonic. The upper-envelope theorem recalled in section 5.21 shows that regularization changes their supremum only on a pluripolar set. Thus $G_\delta =G_\delta ^*$ almost everywhere, and adding $\delta \chi /2$ gives the same assertion for $V_\delta $. $\square$
The regularized envelope does not change if the obstacle is imposed only outside a pluripolar subset of $K$. Within the growth class this can be seen directly: a function in $\mathcal L_\delta $ with poles on the exceptional set repairs each competitor.
Lemma 16.28.12.7A591F1F87FA For every pluripolar set $E\subseteq \mathbb C^ n$ and every $\delta >0$, there is $b\in \mathcal L_\delta $ with $b=-\infty $ on $E$. Given a non-pluripolar compact set $K$, it can be chosen with $b\leq 0$ on $K$.
Proof. Put $E_ j=E\cap \overline B(0,j)$. Each $E_ j$ is pluripolar as a subset of $\mathbb P^ n$: it is locally pluripolar near every affine point, and a neighborhood of the hyperplane at infinity does not meet this bounded set. By Josefson's theorem on compact manifolds, [Theorem 1.1.5, SGPT], there is a quasi-plurisubharmonic function on $\mathbb P^ n$, not identically $-\infty $, that equals $-\infty $ on $E_ j$. Rescaling it so that it becomes $\theta _1$-plurisubharmonic and subtracting its supremum gives
\[ p_ j\in \mathrm{PSH}(\mathbb P^ n,\theta _1),\qquad p_ j\leq 0,\qquad p_ j|_{E_ j}=-\infty . \]
The $L^1$ norms of these normalized potentials are uniformly bounded by theorem 5.25.2. The partial sums
\[ p_ N=\sum _{j=1}^ N2^{-j}p_ j \]
decrease, have uniformly bounded $L^1$ norms, and remain $\theta _1$-plurisubharmonic because $\theta _1\geq 0$ and the coefficients sum to at most one. By corollary 5.25.3, their limit $p$ belongs to $\mathrm{PSH}(\mathbb P^ n,\theta _1)$. Every point of $E$ is a pole of one of the summands, hence $p=-\infty $ on $E$. By lemma 16.28.9,
\[ b=\delta (p+\chi /2)\in \mathcal L_\delta \]
has the required properties. Its supremum on $K$ is finite, since $b$ is bounded above on the compact set $K$ and is not identically $-\infty $ on $K$, which is non-pluripolar. Subtracting this supremum gives $b\leq 0$ on $K$. $\square$
Corollary 16.28.13.0C3E3295B18F The envelope $F_\delta $ is unchanged if $u|_ K\leq \psi $ is imposed only quasi-everywhere, or only on $K\setminus E$ for any pluripolar $E\subseteq K$.
Proof. Let $u\in \mathcal L_\delta $ satisfy $u\leq \psi $ on $K\setminus E$, where $E$ is pluripolar. Choose $b$ as in the lemma, with $b=-\infty $ on $E$ and $b\leq 0$ on $K$, and put $m=\inf _ K\psi $. For $\epsilon >0$ define
\[ u_\epsilon =\frac{u+\epsilon b+\epsilon m}{1+\epsilon }. \]
This function belongs to $\mathcal L_\delta $. On $K\setminus E$, the inequalities $u\leq \psi $, $b\leq 0$ and $m\leq \psi $ give $u_\epsilon \leq \psi $, and on $E$ the pole of $b$ gives $u_\epsilon =-\infty $. Thus $u_\epsilon $ is a competitor for $V_\delta $, so $u_\epsilon \leq F_\delta $; moreover, $u_\epsilon \to u$ almost everywhere as $\epsilon \downarrow 0$. By lemma 5.24.3, $u\leq F_\delta $ everywhere. Taking the supremum over $u$ and regularizing proves the asserted equalities.
Alternatively, apply proposition 5.40.6 on $\mathbb P^ n$ with the big background $\theta _\delta $ and the bounded obstacle $\psi -\delta \chi /2$, and then use (16.28.11). $\square$
These equalities concern regularized envelopes. The raw envelope $V_\delta $ is still defined with the obstacle imposed at every point of $K$.
Changing $\delta $ changes both the background $\theta _\delta $ and the obstacle $\psi -\delta \chi /2$. Nevertheless, the set of parameters for which contact holds has a simple structure. Put
\[ \Lambda =\{ \delta >0:V_\delta |_ K=\psi \} ,\qquad \Lambda _{\mathrm{reg}}(a)= \{ \delta \in \Lambda :F_\delta (a)=\psi (a)\} \quad (a\in K). \]
The parameters $\delta \in \Lambda $ are those for which the raw envelope touches the obstacle at every point of $K$; we say that raw contact holds for them. For $a\in K$, the parameters in $\Lambda _{\mathrm{reg}}(a)$ are those for which, in addition, the regularized envelope touches the obstacle at $a$; we say that regular contact holds at $a$.
Theorem 16.28.14.6D209F2CFC94 The set $\Lambda $ is empty, $(0,\infty )$, or $[\lambda ,\infty )$ for some $\lambda >0$. If $\Lambda =[\lambda ,\infty )$, each $\Lambda _{\mathrm{reg}}(a)$ is empty, $\{ \lambda \} $, or $\Lambda $. If $\Lambda =(0,\infty )$, each $\Lambda _{\mathrm{reg}}(a)$ is empty or $\Lambda $. The same alternatives hold for $\bigcap _{a\in K}\Lambda _{\mathrm{reg}}(a)$.
Proof. Let $m=\inf _ K\psi $. Inclusion of the growth classes and scaling of competitors give, for $0<\alpha <\beta $,
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\begin{equation} 0\leq V_\beta -V_\alpha \leq \frac{\beta -\alpha }{\alpha }(V_\alpha -m). \end{equation}
Indeed, if $u$ is a competitor for $V_\beta $, then $(\alpha /\beta )(u-m)+m$ is a competitor for $V_\alpha $, since $\psi \geq m$. The constant $m$ is also a competitor, so all terms are finite and the right-hand side is nonnegative. The estimate shows that $V_\delta (z)$ depends continuously on $\delta >0$ at every point $z$. Since $V_\delta \leq \psi $ on $K$ and $V_\delta $ increases with $\delta $, the set $\Lambda $ is closed upwards, and by continuity it is relatively closed in $(0,\infty )$. This proves the three possible forms of $\Lambda $.
For $0\leq t\leq 1$, convex combinations of competitors give
\[ V_{(1-t)\alpha +t\beta }\geq (1-t)V_\alpha +tV_\beta . \]
By the preceding proposition, the same inequality holds for the regularized envelopes almost everywhere, and hence everywhere by lemma 5.24.3. Thus $F_\delta (z)$ is a concave nondecreasing function of $\delta $ at every point $z$.
For $a\in K$ and $\delta \in \Lambda $, the defect $d_\delta =F_\delta (a)-\psi (a)$ is nonnegative, concave and nondecreasing in $\delta $. Suppose $d_\alpha =0$, and that there is $\eta \in \Lambda $ with $\eta <\alpha $. For every $\beta >\alpha $, write $\alpha $ as a convex combination of $\eta $ and $\beta $ with positive weights. Concavity then forces both nonnegative defects $d_\eta $ and $d_\beta $ to vanish, and monotonicity gives $d_\delta =0$ for all $\delta \in \Lambda $ with $\delta <\alpha $. Hence a zero of the defect at a parameter other than the smallest element of $\Lambda $ forces every defect to vanish. This proves the alternatives for each $\Lambda _{\mathrm{reg}}(a)$, and intersecting them gives the alternatives for $\bigcap _{a\in K}\Lambda _{\mathrm{reg}}(a)$. $\square$
Thus raw contact persists as the growth allowance increases, whereas regular contact at a point may hold only at the smallest contact parameter. The correspondence with $\mathbb P^ n$ keeps the background and the obstacle matched throughout this variation.
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