3.2. Alexander capacity [033N]

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3.2. Alexander capacity

Definition 3.7.

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

T_{\omega}(K):=\exp(-\sup_{X}V_{K,\omega}^{*}).

This capacity characterizes again $PSH(X,\omega)$ -polar sets:

Proposition 3.8.

Let $P$ be a Borel subset. Then $T_{\omega}(P)=0$ iff $P$ is $PSH(X,\omega)$ -polar. Moreover if $\varphi\in PSH(X,\omega)$ then

T_{\omega}(\varphi<-t)\leq C_{\varphi}\exp(-t),\;\forall t\in\mathbb{R},

where $C_{\varphi}=\exp(-\sup_{X}\varphi)$ .

Proof.

The first assertion follows from theorem 3.2. Let $\varphi\in PSH(X,\omega)$ , $t\in\mathbb{R}$ and set $K_{t}=\{\varphi<-t\}$ . Then $\varphi+t\leq 0$ on $K_{t}$ hence $\varphi+t\leq V_{K_{t},\omega}^{*}$ . We infer $\sup_{X}\varphi+t\leq\sup_{X}V_{K_{t},\omega}^{*}$ which yields $T_{\omega}(K_{t})\leq\exp(-\sup_{X}\varphi)\exp(-t)$ . ∎

The following proposition is an immediate consequence of proposition 3.4. It shows that capacities $T_{\omega},T_{\omega^{\prime}}$ are comparable if $\omega,\omega^{\prime}$ are both Kähler. Further they enjoy nice invariance properties.

Proposition 3.9.

1) For all Borel subsets $K^{\prime}\subset K\subset X$ , $T_{\omega}(K^{\prime})\leq T_{\omega}(K)\leq T_{\omega}(X)=1$ .

2) If $\omega_{1}\leq\omega_{2}$ then $T_{\omega_{1}}(\cdot)\geq T_{\omega_{2}}(\cdot)$ . Forall $A>0$ , $T_{A\omega}(\cdot)=[T_{\omega}(\cdot)]^{A}$ . In particular if $\omega$ and $\omega^{\prime}$ are both Kähler then there exists $C\geq 1$ such that

[T_{\omega}(\cdot)]^{C}\leq T_{\omega^{\prime}}(\cdot)\leq[T_{\omega}(\cdot)]^{1/C}.

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

\frac{1}{C}T_{\omega}(\cdot)\leq T_{\omega^{\prime}}(\cdot)\leq C\cdot T_{\omega}(\cdot),

where $C=\exp(\sup_{X}\chi-\inf_{X}\chi)\geq 1$ .

4) If $f:X\rightarrow X$ is a holomorphic map then $T_{f^{*}\omega}(\cdot)\leq T_{\omega}\circ f(\cdot)$ . In particular if $f$ is a $\omega$ -isometry then $T_{\omega}\circ f=T_{\omega}$ .

Remark 3.10.

Following Zeriahi [40] one can prove that for all $\alpha<2/\nu(X,\omega)$ there exists $C_{\alpha}>0$ such that

\text{Vol}_{\omega}(\cdot)\leq C_{\alpha}T_{\omega}(\cdot)^{\alpha},

where $\nu(X,\omega)=\sup\{\nu(\varphi,x)\,/\,\varphi\in PSH(X,\omega),x\in X\}$ and $\nu(\varphi,x)$ denotes the Lelong number of $\varphi$ at point $x$ . In particular it follows from proposition 3.8 that $\forall\varphi\in PSH(X,\omega)$ with $\sup_{X}\varphi=0$ ,

\text{Vol}_{\omega}(\varphi<-t)\leq C_{\alpha}\exp(-\alpha t),\;\forall t\in\mathbb{R}.

Such inequalities are quite useful in complex dynamics [20],[22] and in the study of the complex Monge-Ampère operator [24], [28].

Example 3.11.

Assume $X=\mathbb{C}\mathbb{P}^{n}$ , $\omega$ is the Fubini-Study Kähler form and $B_{R}$ is the euclidean ball centered at the origin and of radius $R$ in a chart $\mathbb{C}^{n}\subset\mathbb{C}\mathbb{P}^{n}$ . We have explicitly computed the extremal function in this case (example 3.5). This yields

T_{\omega}(B_{R})=\frac{R}{\sqrt{1+R^{2}}}.

Observe that $T_{\omega}(B_{R})\sim R$ as $R\rightarrow 0$ . This shows the optimality of the rate of decreasing in proposition 3.8.

The capacity $T_{\omega}$ in example 3.11 has to be related to the capacity $T_{\mathbb{B}^{n}}$ which measures compact subsets of the unit ball $\mathbb{B}^{n}$ of $\mathbb{C}^{n}$ . It is defined as follows: given $K$ a Borel subset of $\mathbb{C}^{n}$ , $T_{\mathbb{B}^{n}}(K):=\exp(-\sup_{\mathbb{B}^{n}}L_{K})$ , where

L_{K}(z)=\sup\{v(z)\,/\,v\in{\mathcal{L}}(\mathbb{C}^{n}),\;\sup_{K}v\leq 0\}

is the Siciak’s extremal function of $K$ and ${\mathcal{L}}(\mathbb{C}^{n})$ denotes the Lelong class of psh functions with logarithmic growth in $\mathbb{C}^{n}$ (see example 1.2). Let $\omega=\omega_{FS}$ denote the Fubini-Study Kähler form on $\mathbb{C}\mathbb{P}^{n}$ . One easily checks that

V_{K,\omega}-\log\sqrt{2}\leq L_{K}-\frac{1}{2}\log[1+|z|^{2}]\leq V_{K,\omega}\text{ in }\mathbb{C}^{n}.

We infer straightforwardly $\sup_{\mathbb{B}^{n}}L_{K}\leq\log\sqrt{2}+\sup_{\mathbb{C}\mathbb{P}^{n}}V_{K,\omega}$ hence $T_{\mathbb{B}^{n}}(K)\geq 2^{-1/2}T_{\omega}(K)$ . We also have a reverse inequality. Indeed $\forall\varphi\in PSH(\mathbb{C}\mathbb{P}^{n},\omega)$ , $\sup_{\mathbb{C}\mathbb{P}^{n}}\varphi\leq\sup_{\mathbb{B}^{n}}\varphi+C_{1}$ , where $C_{1}=\sup_{\mathbb{C}\mathbb{P}^{n}}V_{\mathbb{B}^{n},\omega}=\log\sqrt{2}$ . Therefore

\sup_{\mathbb{C}\mathbb{P}^{n}}V_{K,\omega}\leq\sup_{\mathbb{B}^{n}}V_{K,\omega}+\log\sqrt{2}\leq\sup_{\mathbb{B}^{n}}L_{K}+\log 2,

which yields

\frac{1}{\sqrt{2}}T_{\omega}(K)\leq T_{\mathbb{B}^{n}}(K)\leq 2T_{\omega}(K).

Example 3.12.

Assume again $X=\mathbb{C}\mathbb{P}^{n}$ and $\omega$ is the Fubini-Study Kähler form. Consider the totally real subspace $\mathbb{R}\mathbb{P}^{n}$ of points with real coordinates (the closure of $\mathbb{R}^{n}\subset\mathbb{C}^{n}$ in $\mathbb{C}\mathbb{P}^{n}$ ). Then

\frac{1}{2(1+\sqrt{2})}\leq T_{\omega}(\mathbb{R}\mathbb{P}^{n})\leq 1.

Indeed set $B_{\mathbb{R}^{n}}:=\mathbb{R}^{n}\cap\mathbb{B}^{n}$ . It follows from the discussion above that

T_{\omega}(\mathbb{R}\mathbb{P}^{n})\geq\frac{1}{2}T_{\mathbb{B}^{n}}(\mathbb{B}_{\mathbb{R}^{n}}).

Now there is an explicit formula for $L_{B_{\mathbb{R}^{n}}}^{*}$ (Lundin’s formula, see [27]),

L_{B_{\mathbb{R}^{n}}}^{*}(z)=\sup\{\log^{+}|h(<z,\xi>)|\,/||\xi||=1\},\;z\in\mathbb{C}^{n},

where $h(\zeta)=\zeta+\sqrt{\zeta^{2}-1}$ . A simple computation yields $|h(\zeta)|\leq\log[|z|+\sqrt{|z|^{2}+1}]$ for $\zeta=<z,\xi>$ with $||\xi||=1$ . We infer

L_{\mathbb{B}_{\mathbb{R}^{n}}}(z)\leq\log\left[|z|+\sqrt{|z|^{2}+1}\right]\leq\log[1+\sqrt{2}]\;\text{ in }\mathbb{B}^{n},

which yields the desired inequality.

Observe that the minorant is independent of the dimension $n$ . This has been used recently in complex dynamics by Dinh and Sibony [18].

Remark 3.13.

It follows from proposition 3.6 that $T_{\omega}$ is a generalized capacity in the sense of Choquet which is outer regular.