1. Quasiplurisubharmonic functions [032D]

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1. Quasiplurisubharmonic functions

In the sequel, unless otherwise specified, $L^{p}$ -norms will always be computed with respect to a fixed volume form on $X$ , which is a compact connected Kähler manifold. Let $\omega$ be a closed real current of bidegree $(1,1)$ on $X$ . We assume throughout the article that $\omega$ has continuous local potentials.

Definition 1.1.

Set

PSH(X,\omega):=\{\varphi\in L^{1}(X,\mathbb{R}\cup\{-\infty\})\,/\,dd^{c}\varphi\geq-\omega\text{ and }\varphi\text{ is }u.s.c.\}.

The set $PSH(X,\omega)$ is the set of ” $\omega$ -plurisubharmonic” functions.

Observe that $PSH(X,\omega)$ is non empty if and only if there exists a positive closed current of bidegree $(1,1)$ on $X$ which is cohomologous to $\omega$ . One then says that the cohomology class $[\omega]$ is pseudoeffective. In the sequel we always assume this property holds. We also always assume that $\omega$ has continuous local potentials. This guarantees that $\omega$ -plurisubharmonic functions ( $\omega$ -psh for short) are upper semi-continuous (u.s.c.), so they are locally hence globally bounded from above. We endow $PSH(X,\omega)$ with the $L^{1}$ -topology. Observe that $PSH(X,\omega)$ is a closed subspace of $L^{1}(X)$ .

Example 1.2.

The most fundamental example which may serve as a guideline to everything that follows is the case where $X=\mathbb{C}\mathbb{P}^{n}$ is the complex projective space and $\omega=\omega_{FS}$ is the Fubini-Study Kähler form. There is then a 1-to-1 correspondence between $PSH(\mathbb{C}\mathbb{P}^{n},\omega_{FS})$ and the Lelong class

{\mathcal{L}}(\mathbb{C}^{n}):=\left\{\psi\in PSH(\mathbb{C}^{n})\,/\,\psi(z)\leq\frac{1}{2}\log[1+|z|^{2}]+C_{\psi}\right\}

which is given by the natural mapping

\psi\in{\mathcal{L}}(\mathbb{C}^{n})\mapsto\varphi(x)=\left\{\begin{array}[]{ccc}\psi(x)-\frac{1}{2}\log[1+|x|^{2}]&\text{ if }&x\in\mathbb{C}^{n}\\ \overline{\lim}_{y\in\mathbb{C}^{n}\rightarrow x}(\psi(y)-\frac{1}{2}\log[1+|y|^{2}])&\text{ if }&x\in H_{\infty},\end{array}\right.

where $H_{\infty}$ denotes the hyperplane at infinity. One can easily show that this mapping is bicontinuous for the $L_{loc}^{1}$ topology.

The Lelong class ${\mathcal{L}}(\mathbb{C}^{n})$ of plurisubharmonic functions with logarithmic growth in $\mathbb{C}^{n}$ has been intensively studied in the last thirty years. It seems to us that the properties of ${\mathcal{L}}(\mathbb{C}^{n})$ are more easily seen when ${\mathcal{L}}(\mathbb{C}^{n})$ is viewed as $PSH(\mathbb{C}\mathbb{P}^{n},\omega_{FS})$ . Further we shall see hereafter that the class $PSH(X,\omega)$ of $\omega$ -psh functions enjoys several properties of ${\mathcal{L}}(\mathbb{C}^{n})$ when $\omega$ is Kähler. We start by observing (proposition 1.3.1 & 1.3.2 below) that $PSH(X,\omega)$ and $PSH(X,\omega^{\prime})$ are comparable if $\omega,\omega^{\prime}$ are both Kähler.

Proposition 1.3.

1) If $\omega_{1}\leq\omega_{2}$ then $PSH(X,\omega_{1})\subset PSH(X,\omega_{2})$ .

2) $\forall A\in\mathbb{R}_{+}^{*}$ , $PSH(X,A\omega)=A\cdot PSH(X,\omega)$ .

3) If $\omega^{\prime}$ is cohomologous to $\omega$ , $\omega^{\prime}=\omega+dd^{c}\chi$ , then

PSH(X,\omega^{\prime})=PSH(X,\omega)+\chi.

4) If $\varphi,\psi\in PSH(X,\omega)$ then

\max(\varphi,\psi)\;,\;\frac{\varphi+\psi}{2}\;,\;\log[e^{\varphi}+e^{\psi}]\in PSH(X,\omega)

Proof.

Assertions 1),2),3) follow straightforwardly from the definition. Observe that 1.3.4 says that $PSH(X,\omega)$ is a convex set which is stable under taking maximum and also under the operation $(\varphi,\psi)\mapsto\log[e^{\varphi}+e^{\psi}]$ . These are all consequences of the corresponding local properties of psh functions. We nevertheless give a proof, in the spirit of this article. That $(\varphi+\psi)/2\in PSH(X,\omega)$ follows by linearity. The latter assertion is a consequence of the following computation

dd^{c}\log[e^{\varphi}+e^{\psi}]=\frac{e^{\varphi}dd^{c}\varphi+e^{\psi}dd^{c}\psi}{e^{\varphi}+e^{\psi}}+\frac{e^{\varphi+\psi}d(\varphi-\psi)\wedge d^{c}(\varphi-\psi)}{[e^{\varphi}+e^{\psi}]^{2}},

using that $df\wedge d^{c}f\geq 0$ . This computation makes sense if for instance $\varphi,\psi$ are smooth. The general case follows then by regularizing $\varphi,\psi$ (see Appendix). Finally observe that $\max(\varphi,\psi)=\lim j^{-1}\log[e^{j\varphi}+e^{j\psi}]\in PSH(X,\omega)$ . ∎

It follows from 1.3.3 that $PSH(X,\omega)$ essentially depends on the cohomology class $[\omega]$ . In the same vein we have the following:

Proposition 1.4.

Let ${\mathcal{T}}_{[\omega]}(X)$ denote the set of positive closed currents $\omega^{\prime}$ of bidegree $(1,1)$ on $X$ which are cohomologous to $\omega$ . Then

PSH(X,\omega)\simeq{\mathcal{T}}_{[\omega]}(X)\oplus\mathbb{R}.

Proof.

The mapping

\Phi:\varphi\in PSH(X,\omega)\mapsto\omega_{\varphi}:=\omega+dd^{c}\varphi\in{\mathcal{T}}_{[\omega]}(X)

is a continuous affine mapping whose kernel consists of constants mappings: indeed $\omega_{\varphi}=\omega_{\psi}$ implies that $\varphi-\psi$ is pluriharmonic hence constant by the maximum principle. Moreover $\Phi$ is surjective: if $\omega^{\prime}\geq 0$ is cohomologous to $\omega$ then $\omega^{\prime}=\omega+dd^{c}\varphi$ for some $\varphi\in L^{1}(X,\mathbb{R})$ -this is the celebrated $dd^{c}$ -lemma on Kähler manifolds (see e.g. lemma 8.6, chapter VI in [15]). Thus $\varphi$ coincides almost everywhere with a function of $PSH(X,\omega)$ and $\omega^{\prime}=\Phi(\varphi)$ . ∎

Remark 1.5.

The size of $PSH(X,\omega)$ is therefore related to that of ${\mathcal{T}}_{[\omega]}(X)$ hence only depends on the positivity of the cohomology class $[\omega]$ . The more positive $[\omega]$ , the bigger $PSH(X,\omega)$ .

When $[\omega]$ is Kähler then $PSH(X,\omega)$ is large: if e.g. $\chi$ is any ${\mathcal{C}}^{2}$ -function on $X$ then $\varepsilon\chi\in PSH(X,\omega)$ for $\varepsilon>0$ small enough. We will see (theorem 5.2) that $PSH(X,\omega)$ characterizes locally pluripolar sets when $[\omega]$ is Kähler. It follows from proposition 1.3 that $PSH(X,\omega)$ and $PSH(X,\omega^{\prime})$ have the same ”size” if $\omega$ and $\omega^{\prime}$ are both Kähler.

Note on the other hand that $PSH(X,\omega)\simeq\mathbb{R}$ when $\omega$ is cohomologous to $[E]$ , the current of integration along the exceptional divisor of a smooth blow up. Indeed let $\pi:X\rightarrow\tilde{X}$ be a blow up with smooth center $Y$ , $codim_{\mathbb{C}}Y\geq 2$ (see e.g. chapter 2 of [15] for the definition of blow-ups). Let $E=\pi^{-1}(Y)$ denote the exceptional divisor and $\omega=[E]$ be the current of integration along $E$ . If $\varphi\in PSH(X,[E])$ then $dd^{c}(\varphi\circ\pi^{-1})\geq 0$ in $\tilde{X}\setminus Y$ . Since $codim_{\mathbb{C}}Y\geq 2$ , $\varphi\circ\pi^{-1}$ extends trivially through $Y$ has a global psh function on $\tilde{X}$ . By the maximum principle $\varphi\circ\pi^{-1}$ is constant hence so is $\varphi$ . Alternatively there is no positive closed current of bidegree $(1,1)$ on $X$ which is cohomologous to $[E]$ except $[E]$ itself.

It follows from previous proposition that any set of ”normalized” $\omega$ -psh functions is in 1-to-1 correspondence with ${\mathcal{T}}_{[\omega]}(X)$ which is compact for the weak topology of currents. This is the key to several results to follow: normalized $\omega$ -psh functions form a compact family in $L^{1}(X)$ .

Proposition 1.6.

Let $(\varphi_{j})\in PSH(X,\omega)^{\mathbb{N}}$ .

1) If $(\varphi_{j})$ is uniformly bounded from above on $X$ , then either $\varphi_{j}$ converges uniformly to $-\infty$ on $X$ or the sequence $(\varphi_{j})$ is relatively compact in $L^{1}(X)$ .

2) If $\varphi_{j}\rightarrow\varphi$ in $L^{1}(X)$ , then $\varphi$ coincides almost everywhere with a unique function $\varphi^{*}\in PSH(X,\omega)$ . Moreover

\sup_{X}\varphi^{*}=\lim_{j\rightarrow+\infty}\sup_{X}\varphi_{j}.

3) In particular if $\varphi_{j}$ is decreasing, then either $\varphi_{j}\rightarrow-\infty$ or $\varphi=\lim\varphi_{j}\in PSH(X,\omega)$ . Similarly, if $\varphi_{j}$ is increasing and uniformly bounded from above then $\varphi:=(\lim\varphi_{j})^{*}\in PSH(X,\omega)$ , where ${\cdot}^{*}$ denotes the upper-semi-continuous regularization.

Proof.

This is a straighforward consequence of the analogous local result for sequences of psh functions. We refer the reader to [15], chapter 1, for a proof. Note that 1.6.2 is a special case of a celebrated lemma attributed to Hartogs. ∎

The next result is quite useful (see [39], [40] for a systematic use).

Proposition 1.7.

The family

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

is a compact subset of $PSH(X,\omega)$ .

If $\mu$ is a probability measure such that $PSH(X,\omega)\subset L^{1}(\mu)$ then

{\mathcal{F}}_{\mu}:=\{\varphi\in PSH(X,\omega)\,/\,\int_{X}\varphi d\mu=0\}

is a relatively compact subset of $PSH(X,\omega)$ . In particular there exists $C_{\mu}$ such that $\forall\varphi\in PSH(X,\omega)$ ,

-C_{\mu}+\sup_{X}\varphi\leq\int_{X}\varphi d\mu\leq\sup_{X}\varphi.

Proof.

It follows straightforwardly from proposition 1.6.1 that ${\mathcal{F}}_{0}$ is a relatively compact subset of $PSH(X,\omega)$ . Moreover ${\mathcal{F}}_{0}$ is closed by Hartogs lemma (1.6.2).

Let $(\varphi_{j})\in{\mathcal{F}}_{\mu}^{\mathbb{N}}$ . Then $\psi_{j}:=\varphi_{j}-\sup_{X}\varphi_{j}\in{\mathcal{F}}_{0}$ which is relatively compact. Assume first $\mu$ is smooth. Then $(\int_{X}\psi_{j}d\mu)$ is bounded: this is because if $\psi_{j_{k}}\rightarrow\psi$ in $L^{1}(X)$ then $\psi_{j_{k}}\mu\rightarrow\psi\mu$ in the weak sense of (negative) measures hence $\int_{X}\psi_{j_{k}}d\mu\rightarrow\int_{X}\psi d\mu>-\infty$ . Now $\int_{X}\psi_{j}d\mu=\int_{X}\varphi_{j}d\mu-\int_{X}\sup_{X}\varphi_{j}d\mu=-\sup_{X}\varphi_{j}$ thus $(\sup_{X}\varphi_{j})$ is bounded and we can apply the previous proposition to conclude that $(\varphi_{j})$ is relatively compact ( it cannot converge uniformly to $-\infty$ since $\int_{X}\varphi_{j}d\mu=0$ ).

When $\mu$ is not smooth, it only remains to prove that $(\int_{X}\psi_{j}d\mu)$ is bounded. Assume on the contrary that $\int_{X}\psi_{j}d\mu\rightarrow-\infty$ . Extracting a subsequence if necessary we can assume $\int_{X}\psi_{j}d\mu\leq-2^{j}$ . Set $\psi=\sum_{j\geq 1}2^{-j}\psi_{j}$ . This is a decreasing sequence of $\omega$ -psh functions, hence $\psi\in PSH(X,\omega)$ or $\psi\equiv-\infty$ . Now it follows from the previous discussion that $\int_{X}\psi_{j}dV\geq-C$ if $dV$ denotes some smooth probability measure on $X$ . Thus $\int_{X}\psi dV>-\infty$ hence $\psi\in PSH(X,\omega)$ . We obtain a contradiction since by the Monotone convergence theorem, $\int_{X}\psi d\mu=\sum_{j\geq 1}2^{-j}\int_{X}\psi_{j}d\mu=-\infty$ . ∎

Example 1.8.

It was part of our definition 1.1 that $\omega$ -psh functions are integrable with respect to a fixed volume form. Therefore $PSH(X,\omega)\subset L^{1}(\mu)$ for every smooth probability measure $\mu$ on $X$ . More generally if $\mu$ is a probability measure on $X$ such that

(1)

\mu=\Theta+dd^{c}(S),

where $\Theta$ is smooth and $S$ is a positive current of bidimension $(1,1)$ on $X$ , then $PSH(X,\omega)\subset L^{1}(\mu)$ for any smooth $\omega$ . Indeed let $\varphi$ in $PSH(X,\omega)$ , $\varphi\leq 0$ . If $\varphi$ is smooth, it follows from Stokes theorem that

$\displaystyle 0\leq\int_{X}(-\varphi)d\mu$	$\displaystyle=$	$\displaystyle\int_{X}(-\varphi)\Theta+\int_{X}(-\varphi)dd^{c}S$
	$\displaystyle\leq$	$\displaystyle C_{\Theta}\|\|\varphi\|\|_{L^{1}}+\int_{X}S\wedge(-dd^{c}\varphi)$
	$\displaystyle\leq$	$\displaystyle C_{\Theta}\|\|\varphi\|\|_{L^{1}}+\int_{X}S\wedge\omega<+\infty,$

where the last inequality follows from $S\geq 0$ and $-dd^{c}\varphi\leq\omega$ . The general case follows by regularizing $\varphi$ (see Appendix).

Probability measures satisfying $(1)$ naturally arise in complex dynamics (see [23]). Observe also that Monge-Ampère measures arising from the local theory of Bedford and Taylor [5] do satisfy $(1)$ : if $u$ is psh and locally bounded near e.g. the unit ball $B$ of $\mathbb{C}^{n}$ , we can extend it to $\mathbb{C}^{n}$ as a global psh function with logarithmic growth considering

U(z):=\left\{\begin{array}[]{rl}u(z)&\text{ if }z\in B\\ \max(u(z),A\log^{+}|z|-\sup_{B}|u|-1)&\text{ if }z\in(1+\varepsilon)B\setminus B\\ A\log^{+}|z|-\sup_{B}|u|-1&\text{ if }z\in\mathbb{C}^{n}\setminus(1+\varepsilon)B\end{array}\right.

where $\log^{+}|z|:=\max(\log|z|,0)$ and with $A$ large enough. We assume $A=1$ for simplicity. Now $\varphi:=U-\frac{1}{2}\log[1+|z|^{2}]+C$ extends as a bounded function in $PSH(\mathbb{C}\mathbb{P}^{n},\omega)$ , where $\omega$ is the Fubini-Study Kähler form on $\mathbb{C}\mathbb{P}^{n}$ , so $\varphi\geq 0$ if $C>0$ is large enough. To conclude note that, setting $\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$ , we get $\omega_{\varphi}^{n}=(dd^{c}u)^{n}$ in $B$ and

\omega_{\varphi}^{n}=\omega^{n}+dd^{c}S,\text{ where }S=\varphi\sum_{j=0}^{n-1}\omega_{\varphi}^{j}\wedge\omega^{n-1-j}\geq 0.

The Monge-Ampère operator $\omega_{\varphi}^{n}$ will be defined in the next section.

Example 1.9.

If $\mu$ is a probability measure on $X=\mathbb{C}\mathbb{P}^{n}$ and $\omega$ denotes as before the Fubini-Study Kähler form, then

\varphi_{\mu}(x):=\int_{\mathbb{C}\mathbb{P}^{n}}\log\left(\frac{||x\wedge y||}{||x||\cdot||y||}\right)d\mu(y)

defines a $\omega$ -psh function on $\mathbb{C}\mathbb{P}^{n}$ . Such functions have been considered by Molzon, Shiffman and Sibony [32], [31] in order to define capacities on $\mathbb{C}\mathbb{P}^{n}$ . However they do not characterize pluripolar sets.