3.1. Global extremal functions [033C]

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3.1. Global extremal functions

Definition 3.1.

Let $K$ be a Borel subset of $X$ . We set

V_{K,\omega}:=\sup\left\{\varphi(x)\,/\,\varphi\in PSH(X,\omega),\,\varphi\leq 0\text{ on }K\right\}.

This definition mimics the definition of the so-called ”Siciak’s extremal function” usually defined for Borel subset of $X=\mathbb{C}\mathbb{P}^{n}$ that are bounded in $\mathbb{C}^{n}=X\setminus H_{\infty}$ , where $H_{\infty}$ denotes some hyperplane at infinity. This function was introduced and studied by Siciak in [35],[36] (see also [38]). One can indeed check that this definition coincides with the classical one if one chooses $\omega=[H_{\infty}]$ to be the current of integration along the hyperplane $H_{\infty}$ . Similarly one could consider the case where $\omega=[D]$ is the current of integration along a positive divisor $D$ on $X$ and let $D$ play the role of infinity. This approach has been used by some authors working in Arakelov geometry to define capacities on projective varieties (see [30],[9] and references therein). However this forces them to consider only compact subsets of $X\setminus D$ and leads to less intrinsic notions of capacities.

In this article we always assume that the currents $\omega$ involved admit continuous potentials. This insures that the Monge-Ampère operator $\omega_{\varphi}^{n}$ is well-defined on extremal functions $V_{K,\omega}$ . If $\varphi\in L^{1}(X)$ , we shall denote by $\varphi^{*}$ its upper-semi-continuous regularization.

Theorem 3.2.

Let $K$ be a Borel subset of $X$ .

1) $K$ is $PSH(X,\omega)$ -polar iff $\sup_{X}V_{K,\omega}^{*}=+\infty$ iff $V_{K,\omega}^{*}\equiv+\infty$ .

2) If $K$ is not $PSH(X,\omega)$ -polar then $V_{K,\omega}^{*}\in PSH(X,\omega)$ and satisfies $V_{K,\omega}^{*}\equiv 0$ in the interior of ${K}$ , $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $X\setminus\overline{K}$ and

\int_{\overline{K}}(\omega_{V_{K,\omega}^{*}})^{n}=\int_{X}\omega^{n}=Vol_{\omega}(X).

Proof.

Assume $\sup_{X}V_{K,\omega}^{*}=+\infty$ . By a lemma of Choquet (see lemma 4.23 in [15], chapter 1), we can find an increasing sequence of functions $\varphi_{j}\in PSH(X,\omega)$ such that $\varphi_{j}=0$ on $K$ and $V_{K,\omega}^{*}=(\lim\nearrow\varphi_{j})^{*}$ . Extracting a subsequence if necessary, we can assume $\sup_{X}\varphi_{j}\geq 2^{j}$ . Set $\psi_{j}=\varphi_{j}-\sup_{X}\varphi_{j}$ . These functions belong to ${\mathcal{F}}_{0}$ which is a compact subfamily of $PSH(X,\omega)$ (corollary 1.7). Recall that if $\mu$ is a smooth volume form on $X$ then there exists $C_{\mu}$ such that $\int\psi_{j}d\mu\geq-C_{\mu}$ for all $j$ . Set $\psi:=\sum_{j\geq 1}2^{-j}\psi_{j}$ . Then $\psi\in PSH(X,\omega)$ as a decreasing limit of functions in $PSH(X,\omega)$ with $\int_{X}\psi d\mu\geq-C_{\mu}>-\infty$ . Now for every $x\in K$ we get $\psi(x)=-\sum_{j\geq 1}2^{-j}\sup_{X}\varphi_{j}=-\infty$ hence $K\subset\{\psi=-\infty\}$ , i.e. $K$ is $PSH(X,\omega)$ -polar.

Conversely assume $K$ is $PSH(X,\omega)$ -polar, $K\subset\{\psi=-\infty\}$ for some $\psi\in PSH(X,\omega)$ . Then for all $c\in\mathbb{R}$ , $\psi+c\in PSH(X,\omega)$ and $\psi+c\leq 0$ on $K$ . Therefore $V_{K,\omega}\geq\psi+c$ , $\forall c\in\mathbb{R}$ . This yields $V_{K,\omega}=+\infty$ on $X\setminus\{\psi=-\infty\}$ hence $V_{K,\omega}^{*}\equiv+\infty$ on $X$ since $\{\psi=-\infty\}$ has zero volume. We have thus shown the following circle of implications: $K\text{ is }PSH(X,\omega)-\text{polar}\Rightarrow V_{K,\omega}^{*}\equiv+\infty\Rightarrow\sup_{X}V_{K,\omega}^{*}=+\infty\Rightarrow K\text{ is }PSH(X,\omega)-\text{polar}$ .

Assume now that $K$ is not $PSH(X,\omega)$ -polar. Then $V_{K,\omega}^{*}\in PSH(X,\omega)$ (see proposition 1.6.2) and clearly satisfies $V_{K,\omega}^{*}=0$ in the interior of ${K}$ . If we show that $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $X\setminus\overline{K}$ then

\int_{\overline{K}}(\omega_{V_{K,\omega}^{*}})^{n}=\int_{X}(\omega_{V_{K,\omega}^{*}})^{n}=\int_{X}\omega^{n},

as follows from Stokes theorem. Let $\varphi_{j}\in PSH(X,\omega)$ be an increasing sequence such that $\varphi_{j}=0$ on $K$ and $V_{K,\omega}^{*}=(\lim\nearrow\varphi_{j})^{*}$ . Fix $B$ a small ball in $X\setminus\overline{K}$ . Let $\hat{\varphi_{j}}$ be the solution of the Dirichlet problem with boundary values $\varphi_{j}$ . Then $\hat{\varphi_{j}}\in PSH(X,\omega)$ , $\hat{\varphi_{j}}=\varphi_{j}$ in $X\setminus B$ (in particular $\hat{\varphi_{j}}=0$ on $K$ hence $\hat{\varphi_{j}}\leq V_{K,\omega}$ ) and the sequence $(\hat{\varphi_{j}})$ is again increasing (theorem 2.12). Since $(\omega_{\hat{\varphi_{j}}})^{n}=0$ in $B$ and $(\lim\nearrow\hat{\varphi_{j}})=V_{K,\omega}^{*}$ , it follows from the continuity of the complex Monge-Ampère on increasing sequences that $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $B$ . As $B$ was an arbitrarily small ball in $X\setminus\overline{K}$ we infer $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $X\setminus\overline{K}$ . ∎

The following corollary has to be related to proposition 1.7.

Corollary 3.3.

Let $K$ be a Borel subset of $X$ and set

{\mathcal{F}}_{K}:=\{\varphi\in PSH(X,\omega)\,/\,\sup_{K}\varphi=0\}.

Then ${\mathcal{F}}_{K}$ is relatively compact iff $K$ is not $PSH(X,\omega)$ -polar.

Proof.

Observe that $V_{K,\omega}(x)=\sup\{\varphi(x)\,/\,\varphi\in{\mathcal{F}}_{K}\}$ . Thus if ${\mathcal{F}}_{K}$ is relatively compact then it is uniformly bounded from above, hence $\sup_{X}V_{K,\omega}<+\infty$ , i.e. $K$ is not $PSH(X,\omega)$ -polar.

Assume conversely that $K$ is not $PSH(X,\omega)$ -polar. Let $(\varphi_{j})\in{\mathcal{F}}_{K}^{\mathbb{N}}$ . Then $\varphi_{j}\leq V_{K,\omega}\leq\sup_{X}V_{K,\omega}<+\infty$ hence $(\varphi_{j})$ is uniformly bounded from above. It follows from proposition 1.6 that $(\varphi_{j})$ is relatively compact. Indeed it can not converge uniformly to $-\infty$ since $\sup_{K}\varphi_{j}=0$ (see proposition 1.6). ∎

Proposition 3.4.

Let $K$ be a Borel subset of $X$ .

1) If $K^{\prime}\subset K$ then $V_{X,\omega}\leq V_{K,\omega}\leq V_{K^{\prime},\omega}$ and $\sup_{X}V_{X,\omega}=0$ . Furthermore $V_{X,\omega}\equiv 0$ when $\omega\geq 0$ .

2) If $\omega_{1}\leq\omega_{2}$ then $V_{K,\omega_{1}}\leq V_{K,\omega_{2}}$ .

3) For all $A>0$ , $V_{K,A\omega}=A\cdot V_{K,\omega}$ .

4) If $\omega^{\prime}=\omega+dd^{c}\chi$ then

-\chi+\inf_{X}\chi+V_{K,\omega}\leq V_{K,\omega^{\prime}}\leq V_{K,\omega}+\sup_{X}\chi-\chi.

5) If $f:X\rightarrow X$ is holomorphic then

V_{f(K),\omega}\circ f\leq V_{K,f^{*}\omega}.

In particular if $f$ is a $\omega$ -isometry then $V_{f(K),\omega}=V_{K,\omega}$ .

Proof.

That $K\mapsto V_{K,\omega}$ is decreasing follows straightforwardly from the definition. Observe that $0\in PSH(X,\omega)$ when $\omega\geq 0$ , hence $V_{X,\omega}\equiv 0$ in this case. When $\omega$ is smooth (but not positive), considering $V_{X,\omega}^{*}$ will be a useful way of constructing a positive closed current $\omega_{V_{X,\omega}^{*}}\sim\omega$ with minimal singularities (see section 4).

Assertions 2,3,4 are simple consequences of proposition 1.3. The last assertion results from the following observation: if $\varphi\in PSH(X,\omega)$ is such that $\varphi\leq 0$ on $f(K)$ , then $\varphi\circ f$ belongs to $PSH(X,f^{*}\omega)$ and satisfies $\varphi\circ f\leq 0$ on $K$ . ∎

Example 3.5.

Assume $X=\mathbb{C}\mathbb{P}^{n}$ , $\omega$ is the Fubini-Study Kähler form and let $B_{R}$ denote the euclidean ball centered at the origin and of radius $R$ in $\mathbb{C}^{n}\subset\mathbb{C}\mathbb{P}^{n}$ . Then for $x\in\mathbb{C}^{n}$ ,

V_{B_{R},\omega}(x)=\max\left(\log\frac{||x||}{R}+\frac{1}{2}\log[1+R^{2}]-\frac{1}{2}\log[1+||x||^{2}];0\right).

Indeed set $\psi_{R}:=\max(\frac{1}{2}\log[1+||x||^{2}],\varphi_{R})$ , where $\varphi_{R}=\frac{1}{2}\log[1+R^{2}]+\log\frac{||x||}{R}$ . Recall that the usual Siciak’s extremal function of $B_{R}$ is $\log^{+}\frac{||x||}{R}$ . Therefore $\frac{1}{2}\log[1+||x||^{2}]\leq\varphi_{R}=\psi_{R}$ for $||x||\geq R$ . On the other hand if $||x||<R$ then $1+||x||^{2}>(1+R^{2})\frac{||x||^{2}}{R^{2}}$ hence $\frac{1}{2}\log[1+||x||^{2}]>\varphi_{R}$ in $B_{R}$ .

Now let $u\in PSH(\mathbb{C}\mathbb{P}^{n},\omega)$ such that $u\leq 0$ in $B_{R}$ . Then $v=u+\frac{1}{2}\log[1+||x||^{2}]\in{\mathcal{L}}(\mathbb{C}^{n})$ . Since $v\leq\frac{1}{2}\log[1+R^{2}]$ in $B_{R}$ we infer $v\leq\frac{1}{2}\log[1+R^{2}]+\log^{+}\frac{||x||}{R}=\psi_{R}$ in $\mathbb{C}^{n}\setminus B_{R}$ . Moreover $v\leq\frac{1}{2}\log[1+||x||^{2}]=\psi_{R}$ in $B_{R}$ hence $v\leq\psi_{R}$ in $\mathbb{C}^{n}$ . This shows $V_{B_{R},\omega}=\psi_{R}-\frac{1}{2}\log[1+||x||^{2}]$ on $\mathbb{C}\mathbb{P}^{n}$ .

Proposition 3.6.

1) If $E$ is an open subset, then $V_{E}=V_{E}^{*}$ .

2) Let $E$ be a Borel subset and $P$ a $PSH(X,\omega)$ -polar set. Then

V_{E\cup P}^{*}\equiv V_{E}^{*}.

3) Let $(E_{j})$ be an increasing sequence of Borel subsets and set $E=\cup E_{j}$ . Then $V_{E,\omega}^{*}=\lim\searrow V_{E_{j},\omega}^{*}$ if $\omega$ is Kähler.

4) Let $K_{j}$ be a decreasing sequence of compact subsets of $X$ and set $K=\cap K_{j}$ . Then $V_{K_{j},\omega}\nearrow V_{K,\omega}$ , hence $V_{K_{j},\omega}^{*}\nearrow V_{K,\omega}^{*}$ a.e.

5) Fix $E\subset X$ a non-pluripolar set. Then there exists $G_{j}$ a decreasing sequence of open subsets, $E\subset G_{j}$ , such that $V_{E}^{*}=\lim V_{G_{j}}^{*}$ .

Proof.

We write here $V_{E}$ for $V_{E,\omega}$ since $\omega$ is fixed and no confusion can arise.

Let $E$ be an open subset of $X$ . Observe that $V_{E}\leq 0$ on $E$ , hence $V_{E}^{*}\leq 0$ on $E$ which is open. Therefore $V_{E}^{*}\leq V_{E}$ , whence equality. This proves 1).

Let $w\in PSH(X,\omega)$ , $w\leq 0$ , and fix $P\subset\{w=-\infty\}$ . Fix $E$ a Borel subset of $X$ . Clearly $V_{E\cup P}\leq V_{E}$ hence $V_{E\cup P}^{*}\leq V_{E}^{*}$ . Conversely let $\varphi\in PSH(X,\omega)$ be such that $\varphi\leq 0$ on $E$ . Then $\forall\varepsilon>0$ , $\psi_{\varepsilon}:=(1-\varepsilon)\varphi+\varepsilon w\in PSH(X,\omega)$ satisfies $\psi_{\varepsilon}\leq 0$ on $E\cup P$ , hence $\psi_{\varepsilon}\leq V_{E\cup P}$ . Letting $\varepsilon\rightarrow 0$ we infer $\varphi\leq V_{E\cup P}$ on $X\setminus P$ , hence $\varphi\leq V_{E\cup P}^{*}$ on $X$ . Thus $V_{E}^{*}\leq V_{E\cup P}^{*}$ .

Let $E_{j}$ be an increasing sequence of subsets of $X$ and set $E=\cup_{j\geq 1}E_{j}$ . Let $v:=\lim\searrow V_{E_{j}}^{*}$ (the limit is decreasing by 3.4.1). If $E$ is $PSH(X,\omega)$ -polar then so are all the $E_{j}^{\prime}$ s, hence $V_{E}^{*}\equiv+\infty=\lim V_{E_{j}}^{*}$ . So let us assume $E$ is not $PSH(X,\omega)$ -polar. Then $v\in PSH(X,\omega)$ since $v\geq V_{E,\omega}^{*}\not\equiv-\infty$ (see proposition 1.6.3). Observe that $v=0$ on the set $E\setminus N$ , where $N=\cup_{j\geq 1}\{V_{E_{j}}<V_{E_{j}}^{*}\}$ . The latter is called a negligible set. It follows from the local theory [5] together with theorem 5.2 that $N$ is $PSH(X,\omega)$ -polar. Therefore $V_{E}^{*}\leq v\leq V_{E\setminus N}^{*}=V_{E}^{*}$ by 2).

Let $K_{j}$ be a decreasing sequence of compact subsets and set $K=\cap_{j}K_{j}$ . Clearly $\lim\nearrow V_{K_{j}}\leq V_{K}$ . Fix $\varepsilon>0$ and let $\varphi\in PSH(X,\omega)$ be such that $\varphi\leq 0$ on $K$ . Then $\{\varphi<\varepsilon\}$ is an open set which contains all $K_{j}^{\prime}s$ , for $j\geq j_{\varepsilon}$ large enough. Thus $\varphi-\varepsilon\leq 0$ on $K_{j}$ , hence $\varphi-\varepsilon\leq\lim\nearrow V_{K_{j}}$ . Taking the supremum over all such $\varphi^{\prime}$ s and letting $\varepsilon\rightarrow 0$ yields the reverse inequality $V_{K}\leq\lim\nearrow V_{K_{j}}$ . The conclusion on the convergence of the upper semi-continuous regularizations follows now from proposition 1.6.

It remains to prove 5). By Choquet’s lemma, there exists an increasing sequence $\varphi_{j}\in PSH(X,\omega)$ such that $\varphi_{j}\leq 0$ on $E$ and $V_{E}^{*}=(\sup_{j}\varphi_{j})^{*}$ . Set $G_{j}:=\{\varphi_{j}<1/j\}$ . This defines a decreasing sequence of open subsets containing $E$ . Observe that $\varphi_{j}-1/j\leq V_{G_{j}}\leq V_{E}$ , hence $\lim\varphi_{j}\leq\lim V_{G_{j}}\leq V_{E}$ . Therefore $V_{E}^{*}=\lim V_{G_{j}}^{*}$ . ∎