6.3. Non-pluripolar Monge-Ampère measures [01B7]

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6.3. Non-pluripolar Monge-Ampère measures

Let us introduce the class of $\omega$ -psh functions with finite energy

\mathcal{E}^{1}(X,\omega):=\left\{\varphi\in\PSH(X,\omega)\mid E_{\omega}(\varphi)>-\infty\right\}.

This is a convex set which contains all bounded $\omega$ -psh functions.

In this section and its sequel, we explain how to extend the Monge-Ampère operator to $\mathcal{E}^{1}(X,\omega)$ and prove that its basic properties continue to hold in this more general setting.

Consider an arbitrary $\omega$ -psh function $\varphi$ . In the sequel we shall use the notation

\varphi^{\langle t\rangle}:=\max\{\varphi,-t\}.

Note that for $s>t\geq 1$ , $\{\varphi>-t\}=\{\varphi^{\langle s\rangle}>-t\}$ and $\max\{\varphi^{\langle s\rangle},-t\}=\varphi^{\langle t\rangle}$ ; hence Theorem 5.1 implies

\one_{\{\varphi>-t\}}\MA(\varphi^{\langle s\rangle})=\one_{\{\varphi^{\langle s\rangle}>-t\}}\MA(\varphi^{\langle s\rangle})=\one_{\{\varphi^{\langle s\rangle}>-t\}}\MA(\varphi^{\langle t\rangle})=\one_{\{\varphi>-t\}}\MA(\varphi^{\langle t\rangle}).

This equation allows us to introduce

Definition 6.4.

[BT87, GZ07] The non-pluripolar Monge-Ampère measure $\MA(\varphi)$ of any $\omega$ -psh function $\varphi$ is the increasing limit of the measures $\one_{\{\varphi>-t\}}\MA(\varphi^{\langle t\rangle})$ as $t\to\infty$ .

Here the limit exists in a very strong sense: we have

(6.5)

\lim_{t\to\infty}\one_{\{\varphi>-t\}}\MA(\varphi^{\langle t\rangle})(E)=\MA(\varphi)(E)

for any Borel set $E$ .

Remark 6.5.

The terminology ”non-pluripolar” comes from the fact that $\MA(\varphi)$ does not put mass on pluripolar sets. This in turn follows from Proposition 3.7 applied to the bounded $\omega$ -psh function $\varphi^{\langle t\rangle}$ and from (6.5).

The measure $\MA(\varphi)$ is always defined and supported on the set $\{\varphi>-\infty\}$ , but its total mass may be strictly less than one.

Definition 6.6.

A $\omega$ -psh function $\varphi$ has full Monge-Ampère mass when $\MA(\varphi)$ is a probability measure.

This is the case iff $\MA(\varphi^{\langle t\rangle})\{\varphi\leq-t\}\to 0$ as $t\to\infty$ , and implies that $\MA(\varphi^{\langle t\rangle})$ converges weakly to $\MA(\varphi)$ .

Lemma 6.7.

If $\varphi\in\mathcal{E}^{1}(X,\omega)$ , then $\MA(\varphi^{\langle t\rangle})\{\varphi\leq-t\}=o(t^{-1})$ as $t\to\infty$ ; hence $\varphi$ has full Monge-Ampère mass.

Proof.

We may assume $\varphi\leq 0$ . Set $\mu_{t}:=\MA(\varphi^{\langle t\rangle})$ . Since (6.2) applies to bounded $\omega$ -psh functions by Proposition 6.3, we get

E_{\omega}(\varphi^{\langle t/2\rangle})-E_{\omega}(\varphi^{\langle t\rangle})\geq\frac{1}{n+1}\int(\varphi^{\langle t/2\rangle}-\varphi^{\langle t\rangle})\mu_{t}=\frac{1}{n+1}\int_{0}^{t/2}\mu_{t}\left\{\varphi^{\langle t/2\rangle}-\varphi^{\langle t\rangle}\geq s\right\}\,ds\\ \geq\frac{1}{n+1}\int_{0}^{t/2}\mu_{t}\left\{\varphi^{\langle t/2\rangle}-\varphi^{\langle t\rangle}\geq t/2\right\}\,ds=\frac{t}{2(n+1)}\mu_{t}\left\{\varphi\leq-t\right\},

where $\mu_{t}=\MA(\varphi^{\langle t\rangle})$ . Since $\lim_{t\to\infty}E_{\omega}(\varphi^{\langle t/2\rangle})=\lim_{t\to\infty}E_{\omega}(\varphi^{\langle t\rangle})=E_{\omega}(\varphi)$ by the continuity of $E_{\omega}$ along decreasing sequences, the proof is complete. ∎

Lemma 6.8.

If $0\geq\varphi\in\mathcal{E}^{1}(X,\omega)$ and $f\in\mathcal{D}(X)$ , then

\left|\int f\MA(\varphi^{\langle t\rangle})-\int f\MA(\varphi)\right|\leq\frac{2(n+1)}{t}|E_{\omega}(\varphi)|\sup_{X}|f|

for any $t>0$ .

Proof.

We may assume $\sup_{X}|f|=1$ . Pick $s\geq t$ . The probability measures $\mu_{t}:=\MA(\varphi^{\langle t\rangle})$ and $\mu_{s}$ agree on $\{\varphi>-t\}$ . Hence

\left|\int f\mu_{t}-\int f\mu_{s}\right|\leq(\mu_{t}+\mu_{s})\{\varphi\leq-t\}\leq\frac{1}{t}\left(\int-\varphi^{\langle t\rangle}\mu_{t}+\int-\varphi^{\langle s\rangle}\mu_{s}\right)\\ \leq\frac{n+1}{t}(|E_{\omega}(\varphi^{\langle t\rangle})|+|E_{\omega}(\varphi^{\langle s\rangle})|)\leq\frac{2(n+1)}{t}|E_{\omega}(\varphi)|.

The result follows by letting $s\to\infty$ . ∎

Proposition 6.9.

If $\varphi\in\mathcal{E}^{1}(X,\omega)$ and $(\varphi_{j})_{j}$ is a decreasing net of $\omega$ -psh functions converging to $\varphi$ , then $\varphi_{j}\in\mathcal{E}^{1}(X,\omega)$ for all $j$ and $\MA(\varphi_{j})\to\MA(\varphi)$ as $j\to\infty$ in the weak sense of measures.

Proof.

Given $f\in\mathcal{D}(X)$ , we have by definition that $\int f\MA(\varphi^{\langle t\rangle})\to\int f\MA(\varphi)$ as $t\to\infty$ and $\int f\MA(\varphi^{\langle t\rangle}_{j})\to\int f\MA(\varphi_{j})$ as $t\to\infty$ for every $j$ . Moreover, Lemma 6.8 shows that the latter convergence is uniform in $j$ . Since for each $t$ we have $\int f\MA(\varphi^{\langle t\rangle}_{j})\to\int f\MA(\varphi^{\langle t\rangle})$ as $j\to\infty$ by Theorem 3.1, the result follows. ∎

Lemma 6.10.

If $\varphi,\psi\in\mathcal{E}^{1}(X,\omega)$ and $\varphi,\psi\leq 0$ , then we have the estimate

-\infty<E\left(\frac{\varphi+\psi}{2}\right)\leq\frac{2^{-(n+1)}}{n+1}\int(-\psi)\MA(\varphi).

Proof.

Pick $s,t>0$ . Since (6.1) holds for bounded $\omega$ -psh functions, we see using (3.1) that

-\infty<E\left(\frac{\varphi+\psi}{2}\right)\leq E\left(\frac{\varphi^{\langle t\rangle}+\psi^{\langle s\rangle}}{2}\right)\leq\frac{2^{-(n+1)}}{n+1}\int\psi^{\langle s\rangle}\MA(\varphi^{\langle t\rangle}).

Since $\psi^{\langle s\rangle}$ decreases to $\psi$ at any point of $X$ , the right hand side converges to

\frac{2^{-(n+1)}}{n+1}\int\psi\MA(\varphi^{\langle t\rangle})\leq\frac{2^{-(n+1)}}{n+1}\int_{\{\varphi>-t\}}\psi\MA(\varphi^{\langle t\rangle})=\frac{2^{-(n+1)}}{n+1}\int_{\{\varphi>-t\}}\psi\MA(\varphi)

by monotone convergence. We obtain the desired estimate by letting $t\to\infty$ . ∎