3.1. Proof of Theorem 3.1 [01A2]

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3.1. Proof of Theorem 3.1

We adapt to our setting the Bedford-Taylor approach as explained, for instance, in [Dem, Theorem 3.7, p.188].

Fix $0\leq p\leq n$ and $\theta$ -psh model functions $\varphi_{p+1}^{\prime},\dots,\varphi_{n}^{\prime}$ . Consider the following statement.

Assertion A(p).

To any $p$ -tuple $\varphi_{1},\dots,\varphi_{p}$ of bounded $\theta$ -psh functions is associated a positive Radon measure $\MAC(\varphi_{1},\dots,\varphi_{p})$ of mass $\{\theta\}^{n}$ such that:

•

if $\varphi_{1},\dots,\varphi_{p}$ are model functions then

(3.2)

\MAC(\varphi_{1},\dots,\varphi_{p})=(\theta+dd^{c}\varphi_{1})\wedge\dots\wedge(\theta+dd^{c}\varphi_{p})\wedge(\theta+dd^{c}\varphi_{p+1}^{\prime})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n}^{\prime})

•

the mapping

$(\psi,\varphi_{1},\dots,\varphi_{p})\mapsto\int\psi\MAC(\varphi_{1},\dots,\varphi_{p})$

is continuous along decreasing nets of bounded $\theta$ -psh functions.

We shall prove A $(p)$ by induction on $p$ . Observe that for $p=n$ , this proves Theorem 3.1.

The assertion A $(0)$ is clear, since $\MAC(\varphi_{1}^{\prime},\dots,\varphi_{n}^{\prime})$ is a finite sum of Dirac masses at divisorial points of $X$ . Assume that A $(p-1)$ holds for any $(n-p+1)$ -tuple of $\theta$ -psh model functions and let $\varphi_{p+1}^{\prime},\dots,\varphi_{n}^{\prime}$ be $\theta$ -psh model functions.

Given bounded $\theta$ -psh functions $\varphi_{1},\dots,\varphi_{p}$ , we define $\MAC(\varphi_{1},\dots,\varphi_{p})$ by forcing the integration by parts formula

\int\psi\MAC(\varphi_{1},\dots,\varphi_{p-1},\varphi_{p}):=\\ \int\varphi_{p}\,(\theta+dd^{c}\varphi_{1})\wedge\dots\wedge(\theta+dd^{c}\varphi_{p-1})\wedge(\theta+dd^{c}\psi)\wedge(\theta+\varphi^{\prime}_{p+1})\wedge\dots\wedge(\theta+dd^{c}\varphi^{\prime}_{n})\\ +\int(\psi-\varphi_{p})(\theta+dd^{c}\varphi_{1})\wedge\dots\wedge(\theta+dd^{c}\varphi_{p-1})\wedge\theta\wedge(\theta+\varphi^{\prime}_{p+1})\wedge\dots\wedge(\theta+dd^{c}\varphi^{\prime}_{n})

for every model function $\psi$ .

Observe that the right-hand side is continuous along decreasing nets as a function of $(\varphi_{1},\dots,\varphi_{p})$ by the induction hypothesis. Since equality holds in (3.1) when all the $\varphi_{i}$ are model functions and since $\MAC(\varphi_{1},\dots,\varphi_{p})$ is a positive measure of mass $\{\theta\}^{n}$ , it follows by regularization (Theorem 2.11) that the right-hand side is also linear in $\psi$ , and non-negative when $\psi\geq 0$ .

Now the space of model functions is spanned by $\theta$ -psh model functions by Proposition 2.6; hence $\MAC(\varphi_{1},\dots,\varphi_{p})$ is well-defined as a positive measure of mass $\{\theta\}^{n}$ and is continuous along decreasing nets as a function of $(\varphi_{1},\dots,\varphi_{p})$ . It remains to show that

(\psi,\varphi_{1},\dots,\varphi_{p})\mapsto\int\psi\MAC(\varphi_{1},\dots,\varphi_{p})

is continuous along decreasing nets of bounded $\theta$ -psh functions. Let thus $(\varphi_{i}^{j})_{j}$ , $i=1,\dots,p$ and $\psi^{j}$ be decreasing nets of $\theta$ -psh functions converging, respectively, to bounded $\theta$ -psh functions $\varphi_{i}$ and $\psi$ . Set

\mu^{j}:=\MAC(\varphi_{1}^{j},\dots,\varphi_{p}^{j}).

We already know that $\mu^{j}$ converges weakly to $\mu:=\MAC(\varphi_{1},\dots,\varphi_{p})$ . Since $\psi^{j}$ is usc for each $j$ , Corollary 2.25 yields

\limsup_{j}\int\psi^{j}\mu^{j}\leq\int\psi\mu.

For the reverse estimate, we rely on the following approximate monotonicity property:

Lemma 3.4.

Let $\psi$ and $\chi_{i}\geq\varphi_{i}$ , $i=1,\dots,p$ be bounded $\theta$ -psh functions. Then we have

	$\displaystyle\int\psi\MAC(\chi_{1},\dots,\chi_{p})$	$\displaystyle\geq\int\psi\MAC(\varphi_{1},\dots,\varphi_{p})$
		$\displaystyle+\sum_{i=1}^{p}\int(\varphi_{i}-\chi_{i})\MAC(\varphi_{1},\dots,\varphi_{i-1},0,\chi_{i+1},\dots,\chi_{p}).$

The lemma implies that, for each $j$ :

\int\psi^{j}\mu^{j}\geq\int\psi\mu^{j}\geq\int\psi\mu+\sum_{i=1}^{p}\int(\varphi_{i}-\varphi_{i}^{j})\MAC(\varphi_{1},\dots,\varphi_{i-1},0,\varphi_{i+1}^{j},\dots,\varphi_{p}^{j}).

By the inductive hypothesis, the sum in the right-hand side tends to $0$ as $j\to\infty$ , so we infer as desired that $\liminf_{j}\int\psi^{j}\mu^{j}\geq\int\psi\mu$ .

Proof of Lemma 3.4.

Note first that $\psi$ may be assumed to be a model function by

Lemma 3.5.

Let $\nu$ be a positive Radon measure on $X$ and let $\varphi$ be a bounded $\theta$ -psh function. Then we have

\int\varphi\nu=\inf_{\psi\geq\varphi}\int\psi\nu

where $\psi$ ranges over all $\theta$ -psh model functions such that $\psi\geq\varphi$ .

Since we already know that $(\varphi_{1},\dots,\varphi_{p})\mapsto\MAC(\varphi_{1},\dots,\varphi_{p})$ is continuous along decreasing nets, we may by regularization assume that all $\varphi_{i}$ and $\chi_{i}$ are also model functions. Integration by parts (3.1) then yields

\int\psi\MAC(\chi_{1},\chi_{2},\dots,\chi_{p})-\int\psi\MAC(\varphi_{1},\chi_{2},\dots,\chi_{p})=\\ =\int(\chi_{1}-\varphi_{1})\MAC(\psi,\chi_{2},\dots,\chi_{p})-\int(\chi_{1}-\varphi_{1})\MAC(0,\chi_{2},\dots,\chi_{p})

hence

\int\psi\MAC(\chi_{1},\dots,\chi_{p})\geq\int\psi\MAC(\varphi_{1},\chi_{2},\dots,\chi_{p})+\int(\varphi_{1}-\chi_{1})\MAC(0,\chi_{2},\dots,\chi_{p}).

We similarly have

	$\displaystyle\int\psi\MAC(\varphi_{1},\chi_{2},\chi_{3},\dots,\chi_{p})$	$\displaystyle\geq\int\psi\MAC(\varphi_{1},\varphi_{2},\chi_{3},\dots,\chi_{p})$
		$\displaystyle+\int(\varphi_{2}-\chi_{2})\MAC(\varphi_{1},0,\chi_{3},\dots,\chi_{p}).$

Iterating this argument and summing up then yields the desired result. ∎

Proof of Lemma 3.5.

Let $\varepsilon>0$ . Since $\varphi$ is usc, Lemma 2.24 shows that there exists a continuous function $v$ on $X$ such that $v\geq\varphi$ and $\int v\nu\leq\int\varphi\nu+\varepsilon$ . The result now follows since [BFJ11, Corollary 8.6] yields an $\theta$ -psh model function $\psi$ such that $\varphi\leq\psi\leq v+\varepsilon$ . ∎

Definition 3.6.

A pluripolar set is a subset of $\{\psi=-\infty\}$ for some $\psi\in\PSH(X,\theta)$ .

Proposition 3.7.

Let $\varphi_{1},...,\varphi_{n}$ be bounded $\theta$ -psh functions. Then any $\psi\in\PSH(X,\theta)$ is integrable with respect to the measure $\mu:=(\theta+dd^{c}\varphi_{1})\wedge\cdots\wedge(\theta+dd^{c}\varphi_{n})$ . In particular, $\mu$ does not put mass on pluripolar sets.

Proof.

Pick $\varphi_{0}\in\PSH(X,\theta)\cap\mathcal{D}(X)$ . Upon replacing $\theta$ , $\varphi_{i}$ , and $\psi$ with $\theta+dd^{c}\varphi_{0}$ , $\varphi_{i}-\varphi_{0}$ and $\psi-\varphi_{0}$ respectively, we may assume that $\theta$ is semipositive and that $\varphi_{i}\leq 0$ for all $i$ . Adding a constant to $\psi$ we may also assume $\sup_{X}\psi=0$ . Set $M:=\max_{i}\sup|\varphi_{i}|$ . First assume that $\psi$ is also bounded. We claim that $\int-\psi\mu$ is bounded by a constant depending only on $M$ (but not on $\sup_{X}|\psi|$ ). Integrating by parts we have

0\leq\int(-\psi)\mu=\int(-\psi)\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})\\ +\int(-\varphi_{1})(\theta+dd^{c}\psi)\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})+\int\varphi_{1}\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n}).

Here the second to last integral is bounded by $M\{\theta\}^{n}$ , while the last integral to the right is non-positive since $\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})$ is a positive measure. Hence

\int(-\psi)\mu\leq\int(-\psi)\theta\wedge(\theta+dd^{c}\varphi_{2})\wedge\dots\wedge(\theta+dd^{c}\varphi_{n})+M\{\theta\}^{n}.

Iterating this argument yields

0\leq\int(-\psi)\mu\leq\int(-\psi)\theta^{n}+nM\{\theta\}^{n}.

Now $\int(-\psi)\theta^{n}$ is bounded above by some $C>0$ only depending on $\theta$ , by compactness of $\{\psi\in\PSH(X,\theta)\mid\sup_{X}\psi=0\}$ and the fact that $\theta^{n}$ is an atomic measure supported at finitely many divisorial points. We conclude that

(3.3)

0\leq\int(-\psi)\mu\leq C+nM\{\theta\}^{n}

for some constant $C>0$ only depending on $\theta$ , as long as $\psi$ is a bounded $\theta$ -psh function with $\sup_{X}\psi=0$ . If $\psi$ is now a possibly unbounded $\theta$ -psh function normalized by $\sup_{X}\psi=0$ , $\psi$ is the decreasing limit of the bounded $\theta$ -psh functions $\psi_{m}:=\max\{\psi,-m\}$ , so that (3.3) continues to hold, by monotone convergence. ∎