Some applications of Projective Logarithmic Potentials

TL;DR

This paper explores applications of projective logarithmic potentials, comparing capacities, defining transfinite diameter, and establishing analogs of classical theorems in complex geometry.

Contribution

It introduces notions of projective logarithmic energy and capacity, compares them with Monge-Ampère capacity, and proves an Evans-type theorem in this context.

Findings

Projective logarithmic capacity is quantitatively compared with Monge-Ampère capacity.

The transfinite diameter of a set coincides with its logarithmic capacity.

An Evans-type theorem for sets of zero projective logarithmic capacity is established.

Abstract

We continue the study in \cite{As18, AAZ18} by giving a multitude of applications of projective logarithmic potentials. First we introduce the notions of projective logarithmic energy and capacity associated to projective kernel that was introduced and studied in \cite{As18, AAZ18}. We compare quantitatively the projective logarithmic capacity with the complex Monge-Amp\`ere capacity on $\mathbb P^n$ and we deduce that the set of zero logarithmic capacity is of Monge-Amp\`ere capacity zero. Further, we define transfinite diameter of a compact set and we show that it coincides with logarithmic capacity. Finally we deduce that there is an analogous of classical Evans's theorem that for any compact set $K$ of zero projective logarithmic capacity shows the existence of Probability measure whose potential admits $K$ as polar set.

Equations

$$\kappa (E):=\sup \{e^{-I(\mu )};\mu \in \text{Prob}({\mathbb{P}}^n),\mathrm{Supp}\mu \subset E\}$$

$$I(\mu )$$

$${\mathrm{Cap}}_{FS}^{\ast }(E)\le \frac{\sqrt{s_n}}{\sqrt{-\log \kappa (E)}},$$

$$E=\{\zeta \in {\mathbb{P}}^n;G_{\mu }(\zeta )=-\infty \}$$

$$E=\{\zeta \in {\mathbb{P}}^n/G_{\mu }(\zeta )=-\infty \}.$$

$$\underset{j\to +\infty }{\mathrm{limsup}}{G}_{{\mu }_j}(\zeta )\le G_{\mu }(\zeta ),\forall \zeta \in {\mathbb{P}}^n,$$

$$\underset{j\to +\infty }{\mathrm{limsup}}{\int }_{{\mathbb{P}}^n}{G}_{{\mu }_j}(\zeta )d{\mu }_j(\zeta )\le {\int }_{{\mathbb{P}}^n}{G}_{\mu }(\zeta )d\mu (\zeta ).$$

$$\underset{j}{\mathrm{limsup}}{G}_{{\mu }_j}(\zeta )\le {\mathbb{G}}_{\mu }(\zeta ),$$

$$\underset{\zeta \to {\zeta }_0}{\mathrm{liminf}}{G}_{\mu }(\zeta )=\underset{\underset{{\zeta }^{\prime }\in K}{{\zeta }^{\prime }\to {\zeta }_0}}{\mathrm{liminf}}{G}_{\mu }({\zeta }^{\prime }).$$

$$\underset{\underset{{\zeta }^{\prime }\in K}{{\zeta }^{\prime }\to {\zeta }_0}}{\lim }{G}_{\mu }({\zeta }^{\prime })=G_{\mu }({\zeta }_0),$$

$$\underset{\zeta \to {\zeta }_0}{\lim }{G}_{\mu }(\zeta )=G_{\mu }({\zeta }_0).$$

$$G_{\mu }(\zeta )={\int }_{{\mathbb{P}}^n}\log \sin \frac{d(\zeta ,\eta )}{\sqrt{2}}d\mu (\eta ),\zeta \in {\mathbb{P}}^n.$$

$$\underset{\zeta \to {\zeta }_0}{\lim }{G}_{\mu }(\zeta )$$

$$\underset{\zeta \to {\zeta }_0}{\lim }{G}_{\mu }(\zeta )=-\infty ,$$

$$\underset{{\zeta }^{\prime }\to {\zeta }_0}{\lim }{G}_{\mu }({\zeta }^{\prime })$$

$$\underset{{\zeta }^{\prime }\to {\zeta }_0}{\lim }{G}_{\mu }({\zeta }^{\prime })=-\infty ,$$

$$\underset{\zeta \to {\zeta }_0}{\mathrm{liminf}}{G}_{\mu }(\zeta )=\underset{\underset{{\zeta }^{\prime }\in K}{{\zeta }^{\prime }\to {\zeta }_0}}{\mathrm{liminf}}{G}_{\mu }({\zeta }^{\prime }).$$

$$\sin\Big{(}\frac{d(\zeta^{\prime},\zeta)}{\sqrt{2}}\Big{)}=\inf_{\eta\in\mathbb{P}^{n}}\sin\Big{(}\frac{d(\eta,\zeta)}{\sqrt{2}}\Big{)}.$$

$$\displaystyle\frac{\sin\Big{(}\frac{d(\zeta^{\prime},\eta)}{\sqrt{2}}\Big{)}}{\sin\Big{(}\frac{d(\zeta,\eta)}{\sqrt{2}}\Big{)}}$$

$$G_{\mu }(\zeta )-G_{\mu }({\zeta }^{\prime })$$

$$\underset{\zeta \to {\zeta }_0}{\mathrm{liminf}}{G}_{\mu }(\zeta )\ge \underset{\underset{{\zeta }^{\prime }\in K}{{\zeta }^{\prime }\to {\zeta }_0}}{\mathrm{liminf}}{G}_{\mu }({\zeta }^{\prime })-ϵ\log (2).$$

$$\underset{\zeta \to {\zeta }_0}{\mathrm{liminf}}{G}_{\mu }(\zeta )=G_{\mu }({\zeta }_0).$$

$$\underset{\zeta \to {\zeta }_0}{\mathrm{limsup}}{G}_{\mu }(\zeta )\le G_{\mu }({\zeta }_0).$$

$$\underset{\zeta \to {\zeta }_0}{\lim }{G}_{\mu }(\zeta )=G_{\mu }({\zeta }_0),$$

$$I_G({\mu }_1,{\mu }_2)$$

$$=$$

$$I_G(\mu )$$

$$I_G(\mu )\le \underset{j\to +\infty }{\mathrm{liminf}}{I}_G({\mu }_j).$$

$$I_G(\mu +\nu )=I_G(\mu )+I_G(\nu )+2I_G(\mu ,\nu ).$$

$$I_Q(\mu ,\nu ):=-{\int }_{{\mathbb{C}}^n}{\int }_{{\mathbb{C}}^n}Q(z,w)d\mu (z)d\nu (w).$$

$$K_{\alpha }(z,w)=R_{\alpha }(z,w):=-|z-w{|}^{\alpha -2n},0<\alpha <2n,$$

Full text

1. 1 Introduction
2. 2 Semi-continuity of the potentials
3. 3 Projective logarithmic energy
4. 4 Projective logarithmic capacity and proof of Theorem A
5. 5 Logarithmic capacity and Hausdorff measure
6. 6 Projective transfinite diameter and proof of Theorem B

1. Introduction

In Potential Theory different notions of energies (depending on the problem under study) associated to Borel measures, were introduced with respect to theirs potentials ([C65, Ran95, Pa04]). They allow to define capacities as a set functions which characterize small sets. On the other hand, the kernel allows to define the transfinite diameter and chebyshev constant notions. These help to give another approach to capacity which has revealed so useful in the local theory.

In the complex plane ([Ran95]), capacities enjoy several natural properties. This is due to the fact that logarithmic kernel satisfies the maximum principle. The importance of the maximum principle lies in the fact that from local assumptions it derives global conclusions. Such results are usually very powerful, and the maximum principle is no exception. Carleson [C65] gave an exceptional class of examples satisfying the maximum principle in the euclidean space $\mathbb{R}^{d},\;\;d\geq 3$. The situation is different for the logarithmic potential in higher complex dimension, since there is no maximum principle.

The aim of this paper is to investigate which properties of classical potentials remain true in the context of complex projective space. More precisely, the projective logarithmic potential $G_{\mu}$ which is defined earlier in [As18, AAZ18] enjoys several interesting properties like in the local setting. For applicatons: we study the projective logarithmic energy of a Probability measure on the complex projective space $\mathbb{P}^{n}$ that allows to define the projective logarithmic capacity $\kappa$ and $\kappa-$polar sets.

We also introduce and study the projective logarithmic transfinite diameter, the projective chebyshev constant and we show that there is a relationship between them and capacity $\kappa$. We conclude by proving our main result which is a strong generalization of classical Evans’s theorem.

We briefly describe the main results and ideas of this paper: let $\mu$ be a Probability measure on the complex projective space $\mathbb{P}^{n}$. Then its projective logarithmic capacity is defined on $Prob(\mathbb{P}^{n})$ as follows:

[TABLE]

where

[TABLE]

is the projective logarithmic energy.

Using results and tools in [As18, AAZ18], we show that $\kappa-$polars sets (i.e a set has zero projective logarithmic capacity), are pluripolars with a precise quantitative estimate of the Monge-Ampère Capacity in term of the logarithmic capacity :

Theorem A. Let $E\subset\mathbb{P}^{n}$ be a Borel set. Then

[TABLE]

*where $s_{n}$ is a constant of lemma 4.4. In particular, every Borel $\kappa$-polar set is pluripolar.

*If $\mu\in\text{Prob}(\mathbb{P}^{n})$ be a measure with $I(\mu)<+\infty$, then the Borel set

[TABLE]

is $\kappa-$polar. The reverse of this result constitutes the main theorem of this paper, which generalizes and improves Evans’s theorem in local setting.

The second fundamental objects that we are going to prove in this paper can be stated as follows :

Theorem B. Let $E\subset\mathbb{P}^{n}$ be a closed set such that $\kappa(E)=0$. Then there exists a Probability measure $\mu$ in $\mathbb{P}^{n}$ with support in $E$ such that

[TABLE]

In particular $E$ is a complete pluripolar set of $\mathbb{P}^{n}$.

2. Semi-continuity of the potentials

We study the upper semi-continuity property of the operator $\mu\longmapsto\mathbb{G}_{\mu}$. We use the weak-topology on the compact space $\text{Prob}(\mathbb{P}^{n})$ of Probability measures on $\mathbb{P}^{n}$. We recall that this space is compact.

Proposition 2.1**.**

Let $(\mu_{j})$ be a sequence converging to $\mu$ in $\text{Prob}(\mathbb{P}^{n})$. Then

[TABLE]

and

[TABLE]

Proof.

Observe that by definition, the integrand is $\leq 0$. Hence by upper semi-continuity and Fatou’s lemma, we have

[TABLE]

which proves the inequality (2.1). Integrating and applying Fatou’s again lemma yields the inequality(2.2). ∎

The potential $G_{\mu}$ enjoys several other interesting properties, we now prove one of these: the continuity principle.

Theorem 2.2** (Continuity Principle).**

*Let $\mu$ be a Probability measure on $\mathbb{P}^{n}$ and let $K:=\text{Supp}\,\mu\subset\mathbb{P}^{n}$.

1. If $\zeta_{0}\in K$, then

[TABLE]

2. If further

[TABLE]

then

[TABLE]

We recall that the potential $G_{\mu}$ can expressed in terms of the geodesic distance $d$ ( see [As18]) on the kähler manifold $(\mathbb{P}^{n},\omega_{FS})$, namely we have

[TABLE]

Proof.

1. If $G_{\mu}(\zeta_{0})=-\infty$, then we have by using upper semi-continuity that

[TABLE]

this implies that

[TABLE]

also

[TABLE]

this implies that

[TABLE]

and we have

[TABLE]

Thus we can suppose that $G_{\mu}(\zeta_{0})>-\infty$. Then we have $\mu(\{\zeta_{0}\})=0$. Given $\epsilon>0$, we can thus find $0<r<<1$, suth that $\mu(B(\zeta_{0},r))<\epsilon$.

For a given $\zeta\in\mathbb{P}^{n}$. Choose $\zeta^{\prime}\in K$ such that,

[TABLE]

Since $0\leq d(\zeta,\eta)\leq\frac{\pi}{\sqrt{2}}$, for all $\eta\in K$.

[TABLE]

Therefore,

[TABLE]

Observe that for $\zeta$ close to $\zeta_{0}$ in $\mathbb{P}^{n}$, we can choose the corresponding projection $\zeta^{\prime}$ close to $\zeta_{0}$ in $K$, and hence

[TABLE]

Since $\epsilon$ is arbitrary, the result follows.

2. If $G_{\mu}$ satisfies the promise of 2., then by part 1 of theorem we have,

[TABLE]

As $G_{\mu}$ is upper semicontinuous,

[TABLE]

Combining these two results yields,

[TABLE]

as required. ∎

3. Projective logarithmic energy

Let us define the mutual (projective logarithmic) energy of two Borel measures $\mu_{1},\mu_{2}$ in $\text{Prob}(\mathbb{P}^{n})$ by

[TABLE]

which is either a non-negative number or $+\infty$.

In case $\mu_{1}=\mu_{2}=\mu$, we call $I_{G}(\mu):=I_{G}(\mu,\mu)$ the projective logarithmic energy of $\mu$ i.e.

[TABLE]

It is clear that $I(\mu)\in[0,+\infty]$.

It follows from the upper semi-continuity of the potentials that the energy is upper semi-continuous i.e.

Proposition 3.1**.**

The operator $I:\mathcal{M}^{+}(\mathbb{P}^{n})\longrightarrow\mathbb{R}^{+}\cup\{+\infty\}$ is lower semi-continuous: If $(\mu_{j})$ be a sequence weakly converging to $\mu$ in $\text{Prob}(\mathbb{P}^{n})$. Then

[TABLE]

Proof.

Is an immediate consequence of the semi-continuity property of $G$ proposition 2.1 and Fatou’s lemma. ∎

Definition 3.2**.**

We say that a positive Borel measure $\mu$ on $\mathbb{P}^{n}$ is a finite $G$-energy if $0\leq I_{G}(\mu)<+\infty$.

In the sequel, ${\mathcal{M}_{G}}^{+}(\mathbb{P}^{n})$ will denote the space of positives Borel measures of finite energy.

We now establish an important lemma which expresses polarization identity for the projective logarithmic energy:

Lemma 3.3**.**

Let $\mu,\nu\in{\mathcal{M}_{G}}^{+}(\mathbb{P}^{n})$ be a positive Borel measures of finite energy. Then $0\leq I_{G}(\mu,\nu)<+\infty$ and we have the following polarization identity:

[TABLE]

In particular $\mu+\nu\in{\mathcal{M}_{G}}^{+}(\mathbb{P}^{n})$.

Proof.

The lemma would be an easy exercice of linear algebra if we know how to show that the kernel $G$ is positive definite i.e., the mutual energy associated $I_{G}(\mu,\nu)$ is symmetric bilinear form, positive definite in the real vector space $\mathcal{M}^{0}_{G}(\mathbb{P}^{n})$ of the signed Borel measures in $\mathbb{P}^{n}$ of zero total mass and finite energy. It is not obvious and constitutes an interesting question which is still open.

We will localize the problem and use the classical results of potential theory in $\mathbb{R}^{2n}$. Recall first some necessary notations from ([La72], Chapter 1).

Let $Q(z,w)$ be a kernel on $\mathbb{C}^{n}\times\mathbb{C}^{n}$ i.e. a locally bounded (from) above Borel function with values in $\mathbb{R}\cup\{-\infty\}$. We define the energy associated to $Q$ on $\mathcal{M}(\mathbb{C}^{n})$ the set of signed Borel measures with compact support by the following formula: If $\mu,\nu\in\mathcal{M}(\mathbb{C}^{n})$ are compactly supported, we put

[TABLE]

If $\mu=\nu$ we put $I_{Q}(\mu):=I_{Q}(\mu,\mu)$. By hypothesis, $I_{Q}(\mu,\nu)\in\mathbb{R}\cup\{+\infty\}$. We will say that $\mu$ is of finite energy (with respect to $Q$) if $I_{Q}(|\mu|)<\infty.$

Note that $\mathcal{M}_{Q}(\mathbb{C}^{n})$ is the set of signed Borel measures with compact support of finite energy and $\mathcal{M}^{0}_{Q}(\mathbb{C}^{n})$ the subset formed of which are zero total mass.

It is well known that the Riesz kernel $R_{\alpha}$ defined on $\mathbb{C}^{n}\simeq\mathbb{R}^{2n}$ by

[TABLE]

is a positive definite in the sens that in the mutual energy $I_{K_{\alpha}}$ associated to Riesz’s kernel is scalar product on the space $\mathcal{M}_{K_{\alpha}}(\mathbb{C}^{n})$ of signed Borel measures with compact support and finite energy.

It follows that by letting $\alpha\to 2n$ that the mutual energy $I_{L}$ associated to the logarithmic potential $L$ defined by

[TABLE]

possess the same property provided to reduce on the space of signed Borel measures with compact support, of zero total mass and of finite energy (see [La72], page 50).

More precisely, the mutual energy $I_{L}$ associated to the kernel $L$ on $\mathbb{C}^{n}$ defined for two signed measures $\mu,\nu$ of finite energy by

[TABLE]

is positive definite on the vector space ${\mathcal{M}}_{L}^{0}(\mathbb{C}^{n})$ of the signed measures with zero mass on $\mathbb{C}^{n}\simeq\mathbb{R}^{2n}$ with compact support of finite energy (see [La72]).

It follows that: If $\mu$ and $\nu$ are the Probability measures with compact supports, then the signed measure (i.e. any difference of two non-negative Borel measures at least one of which is finite) $\mu-\nu$ with compact support of zero mass total of finite energy, we have then

[TABLE]

Hence we deduce the following fundamental inequality: If $\mu$ and $\nu$ are two Probability measures with compact supports in $\mathbb{C}^{n}$, we have

[TABLE]

Consider the normalized projective logarithmic kernel studied recently in ([As18, AAZ18]) and defined for $(z,w)\in\mathbb{C}^{n}\times\mathbb{C}^{n}$ as follows

[TABLE]

It is not known whether the kernel $\mathcal{N}$ possess the property (3.5).

Observe that for $(z,w)\in\mathbb{C}^{n}\times\mathbb{C}^{n}$, the function

[TABLE]

is a locally bounded negative function in $\mathbb{C}^{n}\times\mathbb{C}^{n}$. It follows that the Borel measures with compact support of finite energy are the same for the two kernel $L$ and $\mathcal{N}$ and that

[TABLE]

where $I_{S}(\mu,\nu)$ is a bounded function whose bound depend only upon the bound of $S$ on the support of $\mu\otimes\nu$. It follows that if $\mu$ and $\nu$ are two Probability measures with compact supports in $\mathbb{C}^{n}$ of finite energy, $0\leq I_{L}(\mu-\nu)<+\infty$ and we have

[TABLE]

Therefore we deduce that if $\mu$ and $\nu$ are two Probability measures with compact supports in $\mathbb{C}^{n}$, of finite energy, we have

[TABLE]

In particular $I_{\mathcal{N}}(\mu,\nu)<+\infty.$

The first assertion of the lemma will follow namely, if $\mu,\nu\in\mathcal{M}_{G}(\mathbb{P}^{n})$ are two Probability measures in $\mathbb{P}^{n}$ of finite energy, then

[TABLE]

Recall that $\mathbb{P}^{n}=\cup_{0\leq j\leq n}\,\,\mathcal{U}_{j}$, where     $\mathcal{U}_{j}:=\{\zeta\in\mathbb{P}^{n};\zeta_{j}\neq 0\}$.

In each open $\mathcal{U}_{j}$, we have the coordinates system $z=\phi_{j}(\zeta)$ so that in $\mathcal{U}_{j}\times\mathcal{U}_{j}$, we have

[TABLE]

1. Suppose now that $\mu$, $\nu$ are supported in the same domain of chart $\mathcal{U}_{j}$. Then

[TABLE]

where $\tilde{\mu}=(\phi^{j})_{*}\mu$ and $\tilde{\nu}=(\phi^{j})_{*}\nu$ are two prabability measures with compact support in $\mathbb{C}^{n}$ of finite energy with respect to $\mathcal{N}$. Which proves $0\leq I_{G}(\mu,\nu)<+\infty$ according to the foregoing.

2. Suppose $\mu$ is supported on $\mathcal{U}_{j}$ and $\nu$ is supported on $\mathcal{U}_{k}$ with $j\neq k$. Consider the transformation $\psi_{j,k}:\mathcal{U}_{k}\longrightarrow\mathcal{U}_{j}$ which consists to switch the coordinates $\zeta_{j}$ and $\zeta_{k}$. Then, putting $\hat{\nu}:=(\psi_{j,k})_{*}\nu$, here we obtain a Probability measure with compact measure in $\mathcal{U}_{j}$ such that

[TABLE]

according to the first step.

3. If $\mu$ and $\nu$ are two Probability measures in $\mathbb{P}^{n}$ of finite energy, we can decompose $\mu=\sum_{j=0}^{n}\alpha_{j}\mu_{j}$ and $\nu=\sum_{k=0}^{n}\beta_{k}\nu_{k}$ into convex combination Probability measures with finite energy such that $\mu_{j}$ and $\nu_{j}$ be supported in $\mathcal{U}_{j}$ for $j=0,\cdots,n$.

It follows by bilinearity that

[TABLE]

The polarization identity is a consequence of the bilinearity and the symmetry of $I_{G}(\mu,\nu)$ in the space $\mathcal{M}_{G}(\mathbb{P}^{n})$. Which finishes the proof of the lemma. ∎

In the sequel we write $I=I_{G}$ to simplify notations.

3.1. Calculus the energy of some measures

In this subsection we will discuss about energy of two extreme examples of measures : Dirac measure and Fubini-Study measure.

Example 3.4**.**

We first consider, Dirac measure $\mu:=\delta_{a}$ at the point $a\in\mathbb{P}^{n}$. We have

[TABLE]

More generally if $a,b\in\mathbb{P}^{n}$ such that $a\neq b$, we have

[TABLE]

Observe that polarization identity (3.4) is valid for all Borel measures in $\mathbb{P}^{n}$. It follows that $I(\delta_{a}+\delta_{b})=+\infty$.

Example 3.5**.**

We now consider Fubini$-$Study measure $\mu_{FS}:=\frac{\omega^{n}_{FS}}{n!}=dV_{FS}$, we have $0\leq d(\zeta,\eta)\leq\frac{\pi}{\sqrt{2}}$ for all $\zeta,\eta\in\mathbb{P}^{n}$.

[TABLE]

where $A(r):=c_{n}\sin^{2n-2}(r/\sqrt{2})\sin(\sqrt{2}r)$ is the area of the sphere of center $\eta$ and radius $r$ of $\mathbb{P}^{n}$ and $c_{n}$ a positive constant.

$\kappa-$Polar Set:

The $\kappa-$polar set play the role of negligible sets in projective logarithmic potentials, much as sets of measure zero do in potentials theory and measure theory. It follows from the projective logarithmic energy the following definition.

Definition 3.6**.**

A subset $E$ of $\mathbb{P}^{n}$ is called $\kappa-$polar if $I(\mu)=+\infty$ for every Borel measure $\mu\in\mathrm{Prob}(\mathbb{P}^{n})$ for which $\mathrm{Supp}\mu\,\,\subset E$.

Example 3.7**.**

It follows from previous example that every finite or countable set of $\mathbb{P}^{n}$ is $\kappa$-polar.

The following proposition of [[Ran95]] generalized to $\mathbb{P}^{n}$ shows in particular that the measures of finite energy do not charge $\kappa-$polar sets.

Proposition 3.8**.**

Let $\mu$ be a finite Borel measure in $Prob(\mathbb{P}^{n})$, and suppose that $I(\mu)<+\infty$. Then $\mu(E)=0$ for every Borel $\kappa-$polar set $E$ of $\mathbb{P}^{n}$.

Proof.

Suppose $E$ be a Borel set of $\mathbb{P}^{n}$ such that $\mu(E)>0$. There exists a compact subset $K\subset E$ such that $\mu(K)>0$. Set $\nu=1_{K}\frac{\mu}{\mu(K)}$. Then $\nu$ is a Probability measure whose support in $E$ and its energy checks:

[TABLE]

It follows that $E$ is not $\kappa-$polar. ∎

4. Projective logarithmic capacity and proof of Theorem A

Recall the definition of the projective logarithmic energy of $\mu$ in $\text{Prob}(\mathbb{P}^{n})$ from the previous section,

[TABLE]

We define for a subset $E\subset\mathbb{P}^{n}$, the Robin constant

[TABLE]

and its projective logarithmic capacity as

[TABLE]

Observe that $\kappa(E)>0$ if and only if there exists $\mu\in\text{Prob}(\mathbb{P}^{n}),Supp\mu\subset E$ such that $I(\mu)<+\infty$.

By duality, we obtain the following:

Proposition 4.1**.**

For all Borel $E\subset\mathbb{P}^{n}$, we have

[TABLE]

In particular, $E$ is $\kappa$-polar if and only if $\nu(E)=0$ for all Borel measure $\nu\in\mathcal{M}^{+}(\mathbb{P}^{n})$ such that $I(\nu)<+\infty$.

Proof.

By definition, we have

[TABLE]

We denote by $\sigma(E)$ the second member of the equality (4.1). Let $\mu\in\text{Prob}(\mathbb{P}^{n}),\mathrm{Supp}\mu\subset E$ and $I(\mu)<+\infty$. Then for all $\varepsilon>0$ small enough, $\nu:=\mu/\sqrt{I(\mu)+\varepsilon}\in\mathcal{M}^{+}(\mathbb{P}^{n}),$ and checks $I(\nu)\leq 1$. We infer that $\nu(E)\leq\sigma(E)$. Since $\mu(E)=1$, it follows that

[TABLE]

Taking the upper bound on $\mu$ and tends $\varepsilon$ to [math], yields the first inequality in (4.1). Conversely, let $\nu\in\mathcal{M}^{+}(\mathbb{P}^{n}),I(\nu)\leq 1$ such that $\nu(E)>0$. Then $\mu:={\bf 1}_{E}\nu/\nu(E)$ is a Probability measure with compact support in $E$ such that $I(\mu)\leq I(\nu)/(\nu(E))^{2}\leq 1/(\nu(E))^{2}$. This ensures that $\gamma(E)\leq 1/(\nu(E))^{2}$. Therefore $\nu(E)\leq 1/\sqrt{\gamma(E)}$. Taking the supremum over $\nu$, we obtain the second required inequality $\geq$ in (4.1). ∎

As a consequence we state the following results:

Proposition 4.2**.**

The set function $E\longmapsto\frac{1}{\sqrt{-\log\kappa(E)}}$ is subadditive on the $\sigma-$Algebra of Borel subsets $\mathcal{B}(\mathbb{P}^{n})$ of $\mathbb{P}^{n}$ i.e., if $(E_{j})$ is a sequence of Borel sets of $\mathbb{P}^{n}$ and $E:=\cup_{j}E_{j}$, then

[TABLE]

Proof.

A consequence of the formula of duality and by subadditivity of Borel measures on $\mathbb{P}^{n}$. ∎

Corollary 4.3**.**

For all Borel $E\subset\mathbb{P}^{n}$, we have

[TABLE]

where $a_{n}>0$ is an absolute constant.

Proof.

It follows from Example 3.5 that there exists a constant $b_{n}>0$ such that $I(\mathrm{Vol})\leq b_{n}$. It follows that $\nu:=\mathrm{Vol}_{FS}/\sqrt{b_{n}}\in\mathcal{M}^{+}(\mathbb{P}^{n})$ and checks $I(\nu)\leq 1$. It follows from proposition 4.1 that $\nu(E)\leq\frac{1}{\sqrt{-\log\kappa(E)}}$, which yields the desired inequality with $a_{n}:=\sqrt{b_{n}}$. ∎

Lemma 4.4**.**

There exists a constant $s_{n}>0$ such that if $u:\mathbb{P}^{n}\longrightarrow[-\infty,+\infty[$ is $\omega$-plurisubharmonic on $\mathbb{P}^{n}$ suth that $-1\leq u\leq 0$, the measure $\mu=(\omega+dd^{c}u)^{n}$ verifies $I(\mu_{u})\leq s_{n}$.

Proof.

Let $\mu=(\omega+dd^{c}u)^{n}$ with $u\in PSH(\mathbb{P}^{n},\omega)\cap L^{\infty}(\mathbb{P}^{n})$. We have then

[TABLE]

Integrating by parts thanks to Stokes formula, we infer

[TABLE]

It follows from Chern$-$Levine$-$Nirenberg inequalities for $G_{\mu}\in PSH(\mathbb{P}^{n},\omega)\cap L^{1}(\mathbb{P}^{n})$ and $u\in PSH(\mathbb{P}^{n},\omega)\cap L^{\infty}(\mathbb{P}^{n})$ that the first integral is bounded by a constant depending only on $n$.

[TABLE]

Also for the last integral we have,

[TABLE]

Then if $-1\leq u\leq 0$, we have

[TABLE]

∎

4.1. Proof of Theorem A

Proof.

We can assume that $\mathrm{Cap}_{FS}(E)>0$. Then by [GZ05], the Monge-Ampère capacity of $E$ is given by the following formula:

[TABLE]

Let then $u\in PSH(\mathbb{P}^{n},\omega)$, $-1\leq u\leq 0$, and let $\mu=(\omega+dd^{c}u)^{n}$. By lemma 4.4, we have $I(\mu)\leq s_{n},$ where $s_{n}>0$ is an absolute constant. Let $\nu=\frac{\mu}{\sqrt{s_{n}}}$, then $\nu\in\mathcal{M}^{+}(\mathbb{P}^{n})$ and verifies $I(\nu)\leq 1$. The proposition 4.1 implies that

[TABLE]

So

[TABLE]

Since $-1\leq h_{E}^{\ast}\leq 0$, we also have

[TABLE]

∎

For the converse of the proposition A is not true. We have the following example which confirms this guess.

Example 4.5**.**

The set $\mathbb{D}\times\{0\}\subset\mathbb{C}^{2}$ is pluripolar but non- $\kappa-$polar. Indeed, the set

[TABLE]

is pluripolar in $\mathbb{C}^{2}$. Consider the normalized Lebesgue measure

[TABLE]

on the disk $\mathbb{D}\times\{0\}\subset\mathbb{C}^{2}$. As the support of the measure $\mu$ is contained in $\mathcal{U}_{0}$, we have the logarithmic potential $G_{\mu}$ in affine coordinates(see [AAZ18]): $G_{\mu}(\zeta)=\mathcal{N}_{\mu}(z)$, where

[TABLE]

then,

[TABLE]

We calculate $I(\mu)$ as follows:

[TABLE]

which proves that $\mathbb{D}\times\{0\}$ is non- $\kappa-$polar.

Our purpose here is to give an important property of logarithmic capacity.

Theorem 4.6**.**

Let $E\subset\mathbb{P}^{n}$ be a Borel set. Suppose that there exist $\mu\in\text{Prob}(\mathbb{P}^{n})$ such that $E=\{G_{\mu}=-\infty\}$ and $I(\mu)<\infty$. Then the set $E$ is $\kappa-$polar.

Proof.

Suppose that $E=\{G_{\mu}=-\infty\}$, where $\mu\in\text{Prob}(\mathbb{P}^{n})$ such that $I(\mu)<+\infty$. Then $G_{\mu}$ is $\omega$-plurisubharmonic in $\mathbb{P}^{n}$.

To show that $E$ is $\kappa$-polar, it is enough by the proposition 4.1 to show that for positive Borel measure $\nu$ such that $I(\nu)<+\infty$, we have $\nu(E)=0$.

Indeed, put $E_{s}:=\{G_{\mu}<-s\}$ for $s>0$. Then for all $s>0$, we have $-G_{\mu}\geq s$ on $E_{s}$ and then

[TABLE]

By definition, we have $\int_{\mathbb{P}^{n}}(-G_{\mu})d\nu=I(\mu,\nu)$. By lemma 3.3 we have $0\leq I(\mu,\nu)<+\infty$. Hence: $\forall s>0$,

[TABLE]

It follows that $\nu(E)=0$. ∎

Remark 4.7**.**

It would be interesting to establish the theorem in the case $I(\mu)=+\infty$.

In the following proposition, we will be interested in the measures that minimizes the energy inspired by the work of [ST97].

Proposition 4.8**.**

Let $E\subset\mathbb{P}^{n}$ be a closed subset such that $\kappa(E)>0$. Then, there exists a measure $\mu_{E}\in\mathrm{Prob}(\mathbb{P}^{n})$ such that $\mathrm{Supp}\mu_{E}\subset E$ and $\gamma(E)=I(\mu_{E}),$ i.e

[TABLE]

Proof.

By definition there exists a minimizing sequence $(\mu_{j})$ in $\text{Prob}(\mathbb{P}^{n})$ with $Supp\mu_{j}\subset E$ such that

[TABLE]

By compactness of the space $\mathrm{Prob}(\mathbb{P}^{n})$, taking a susbsequence if necessary, we can assume that $\mu_{j}\to^{*}\mu$, where $\mu\in\mathrm{Prob}(\mathbb{P}^{n})$ with $\mathrm{Supp}\mu\subset E$.

By lower semi-continuity of the energy ( see 3.1), we get

[TABLE]

By definition, $\gamma(E)\leq I(\mu)$, this yields the equality $\gamma(E)=I(\mu)$. ∎

The measure $\mu_{E}$ is called an equilibrium measure for $E$.

Now, we prove that the projective logarithmic capacity $\kappa$ defined before is a capacity in the sense of Choquet. We start by recalling the definition of the outer and inner projective logarithmic capacities:

Definition 4.9**.**

Let $E$ be a subset of $\mathbb{P}^{n}$. We define the outer and inner projective logarithmic capacities of $E$ by:

[TABLE]

[TABLE]

The set $E$ is called capacitable if $\kappa^{*}(E)=\kappa_{*}(E)$.

Observe that by the proposition 4.1, we have $\kappa_{*}(E)=\kappa(E)$ when $E$ is Borel.

Theorem 4.10**.**

The set function $\kappa^{*}$ is a capacity on $\mathbb{P}^{n}$. In particular, every Borel set is capacitable.

Proof.

To prove that $\kappa^{*}$ is a capacity on $\mathbb{P}^{n}$, we verify first that $\kappa$ defined on Borel sets $\mathcal{B}(\mathbb{P}^{n})\longrightarrow\mathbb{R}^{+}$ is precapacity i.e., it is enough to check the following conditions:

1. (1)

$\kappa(\emptyset)=0$ and if $A\subset B$ are the Borels of $\mathbb{P}^{n}$ then

[TABLE] 2. (2)

If $K_{1}\supset K_{2}\supset K_{3}\supset...$ are compact subsets of $\mathbb{P}^{n}$, and $K=\displaystyle\cap_{n}K_{n}$ then

[TABLE] 3. (3)

If $B_{1}\subset B_{2}\subset...$ are Borel subsets of $\mathbb{P}^{n}$, and $B=\cup_{n}B_{n}$, then

[TABLE]

The proof of $1)$: we have

[TABLE]

and

[TABLE]

Since

[TABLE]

then

[TABLE]

we obtain

[TABLE]

We now prove $2)$. Applying $1)$ we get that

[TABLE]

For each $n\geq 1$ let $\nu_{n}$ be an equilibrium measure for $K_{n}$. Then $\nu_{n}\in\text{Prob}(\mathbb{P}^{n})$ for all $n$, by compactness of the space $\mathrm{Prob}(\mathbb{P}^{n})$, taking a subsequence $(\nu_{n_{k}})$, we can assume that $\nu_{n_{k}}\to^{*}\nu$, where $\nu\in\mathrm{Prob}(K_{1})$. By lower semi-continuity of the energy (Proposition 3.1), we get

[TABLE]

Moreover, since $\mathrm{Supp}\nu_{n}\subset K_{n}$ for all $n$, it follows that $\mathrm{Supp}\nu\subset K$, and so $e^{-I(\nu)}\leq\kappa(K)$. thus we obtain

[TABLE]

[TABLE]

and combining with $(\star)$ we get the desired conclusion.

We now prove $3)$. Using again $1)$, we get

[TABLE]

let $K$ be a compact subset of $B$, and let $\nu$ be an equilibrium measure for $K$. Since $\nu(B_{n}\cap K)\rightarrow\nu(K)=1$ as $n\rightarrow\infty$, we can produce compact sets $K_{n}\subset B_{n}\cap K$ such that $K_{1}\subset K_{2}\subset...$ and $\nu(K_{n})\rightarrow 1$. For $n$ sufficiently large, we have $\nu(K_{n})>0$, we define

[TABLE]

Then $\mu_{n}$ is a Borel Probability measure on $K_{n}$ and

[TABLE]

as $n\rightarrow\infty$, we have $\nu(K_{n})\rightarrow 1$ and $1_{K_{n}}\uparrow 1_{K}$ for $\nu-$almost everywhere, so

[TABLE]

Since each $\mu_{n}$ is supported on a compact subset $B_{n}$, we have $\kappa(B_{n})\geq e^{I(\mu_{n})}$, and it follows that

[TABLE]

Finally, as $K$ is an arbitrary subset of $B$, by the proposition 3.1, we have

[TABLE]

Using $(\star\star)$ we obtain the desired conclusion.

The previous properties show that the sets function defined on the Borel $\sigma-$Algebra subsets

[TABLE]

is a precapacity on $\mathbb{P}^{n}$. By the Proposition 4.1, it is sub-additive. It follows by Choquet Theory that the set functions

[TABLE]

is a capacity on $\mathbb{P}^{n}$. Using the same reasoning for $\kappa^{*}$ as (see [Ch55], [GZ17]).

The last assertion is consequence of version Choquet’s capacitability theorem (see [Ch55], [GZ17]). ∎

We will say that a subset $E\subset\mathbb{P}^{n}$ is $\kappa^{*}$-polar if $\kappa^{*}(E)=0$.

We have also the subadditivity property which follows from proposition 4.1.

Proposition 4.11**.**

The set function $E\longmapsto\frac{1}{\sqrt{-\log\kappa^{*}(E)}}$ is subadditive i.e. if $(E_{j})$ is a sequence of subsets of $\mathbb{P}^{n}$ and $E:=\cup_{j}E_{j}$, then

[TABLE]

In particular a countable union of $\kappa^{*}$-polar sets is $\kappa^{*}$-polar.

5. Logarithmic capacity and Hausdorff measure

5.1. Hausdorff measure

Let $f(r)$ be a continuous function defined for $r\geq 0$ with the properties:

$f(0)=0$,   $f(r)$ increasing.

Let $E$ be a subset of $\mathbb{P}^{n}$, and $\rho\geq 0$ a real number. Define

[TABLE]

where the infimum is taken over all countable covers of $E$, and $\mathcal{S}_{i}$ sphere with radii $r_{i}<\rho$.

If we allow $\rho$ to approach zero, we have

[TABLE]

we call this limit Hausdorff measure of $E$ associated to $f$.

We now give some measures whose potentials have finite energy:

Theorem 5.1**.**

Let $h:[0.1]\longrightarrow\mathbb{R}^{+}$, such that  $h(0)=0$ and

[TABLE]

If

[TABLE]

Then $G_{\mu}$ is bounded and $I(\mu)<+\infty$.

Proof.

For all Probability measure satisfying 5.2 we have:

[TABLE]

[TABLE]

The change of variables: $s=\sqrt{2}\arcsin(e^{-t})$, we have $ds=\sqrt{2}\frac{-e^{-t}}{\sqrt{1-e^{-2t}}}$ yields $t=-\log\sin(\frac{s}{\sqrt{2}})$. We conclude that

[TABLE]

so that the right hand side of the above inequality is finite ( this is possible since the function $\coth s=\frac{1}{s}$ near 0). As $\int\frac{h(s)}{s}ds<+\infty$, there exists $C>0$ such that for all $\zeta\in\mathbb{P}^{n}$,

[TABLE]

Which implies that

[TABLE]

as desired. ∎

Using proposition 3.8 and theorem 5.1 we get the following consequence:

Corollary 5.2**.**

Let $0<m\leq 2n$ be a fixed real number. Let $\mathcal{H}_{m}$ denote the Hausdorff measure (associated to $h$ 5.3) of dimension $m$ on $\mathbb{C}^{n}\simeq\mathbb{R}^{2n}$. Let $F\subset\mathbb{P}^{n}$ be a closed set such that $0<\mathcal{H}_{m}(F)<+\infty$. Then $\mathcal{H}_{m}(E)=0$, for all $\kappa-$polar subset $E\subset F$.

Proof.

The measure $\mu:={\bf 1}_{F}\mathcal{H}_{m}$ is Borel measure satisfying the condition 5.1 for an increasing function $h$ defined by:

[TABLE]

By using theorem 5.1, we conclude that $I(\mu)<+\infty$ and then $\mathcal{H}_{m}(E)=0$ for all $\kappa-$polar set $E\subset F$, by proposition 3.8. ∎

6. Projective transfinite diameter and proof of Theorem B

Like in classic setting ([Ran95, ST97, Pa04]), we introduce the associated (projective logarithmic) transfinite diameter and (projective logarithmic) chebychev constant notions and we connect them to projective logarithmic capacity $\kappa$.

We recall that

[TABLE]

For $s\in\mathbb{N},s\geq 2$, we define the (projective logarithmic) diameter of order $s\in\mathbb{N}^{*}$ of $E\subset\mathbb{P}^{n}$ by

[TABLE]

where

[TABLE]

Proposition 6.1**.**

The sequence $s\longmapsto D_{s}:=D_{s}(E)$ is decreasing and then the limit

[TABLE]

exists in $\mathbb{R}^{+}$ and we have the following estimate:

[TABLE]

where $\mathrm{diam}(E)$ is diameter of $E$ of $\mathbb{P}^{n}$ with respect to the Fubini-Study metric.

Proof.

Note that $\theta_{s}:=-\log D_{s}(E)$. Choose $\zeta_{1},\cdots,\zeta_{s}\in E$ arbitrary. If we leave out any index $i=1,\cdots,s$ then for the remaining $s-1$ points, we obtain by the definition of $\theta_{s-1}$ that

[TABLE]

After summing up for $i=1,2,\cdots,s$ this yields

[TABLE]

for each term $(-G(\zeta_{j},\zeta_{l}))$ occurs exactly $s-2$ times. Now taking the infimum for all possible $\zeta_{1},\cdots,\zeta_{s}\in E$, we obtain $s.\theta_{s}\geq s.\theta_{s-1}$, as claimed. The last property follows from the fact that $D(E)\leq D_{2}(E)=\sin\left(\frac{\mathrm{diam}(E)}{\sqrt{2}}\right)$. ∎

Definition 6.2**.**

If $E\subset\mathbb{P}^{n}$ is a closed set, then the number

[TABLE]

is called (projective logarithmic) tranfinite diameter of $E$.

Theorem 6.3**.**

Let $E\subset\mathbb{P}^{n}$ be compact set. Then, we have

[TABLE]

The proof of the theorem 6.3 relies the following results.

Lemma 6.4**.**

Let $E\subset\mathbb{P}^{n}$ be a closed set. Then

[TABLE]

Proof.

Let $\mu\in Prob(E)$ be a arbitrary measure and define $\nu:=\mu^{\otimes s}$ a product measure on $\mathbb{P}^{n}\times\cdots\times\mathbb{P}^{n}$. Le us consider the following lower semi-continuous function $h$ defined by

[TABLE]

Since by definition $0\leq\theta_{s}\leq h$, it follows that

[TABLE]

Taking infimum over $\mu$, yields

[TABLE]

Taking the limit in $s$, we obtain

[TABLE]

∎

Lemma 6.5**.**

For all closed set $E\subset\mathbb{P}^{n}$, we have

[TABLE]

Proof.

If $D(E)=0$, then with lemma 6.4 we have $\gamma(E)=+\infty$, the result holds. Suppose that $D(E)>0$, and let $\epsilon>0$ fixed. Let $\zeta_{1},\cdots,\zeta_{s}$ be points of $E$ which realize the infimum of $\theta_{s}(E)$, called Fekete points. Put $\mu:=\mu_{s}=\frac{1}{s}\sum_{i=1}^{s}\delta_{\zeta_{i}}$ where $\delta_{\zeta_{i}}$ are Dirac measures at the points $\zeta_{i},\;\;i=1,\cdots,s$. For a continue function $h$ such that $0\leq h\leq-G$ and with support in $E$, we have

[TABLE]

We have used proposition 6.1 in last step. In fact, we obtain for $s\geq N=N(\|h\|,\epsilon)$ the inequality

[TABLE]

If we take subsequence $\mu_{s_{k}}$ of $\mu$ weakly converges to $\nu$, we have

[TABLE]

and thus,

[TABLE]

for all $\epsilon>0$. This shows that

[TABLE]

∎

Finally, combining the lemma 6.4 and 6.5 to conclude

[TABLE]

for all compact set $E$ of $\mathbb{P}^{n}$, and the proof of the theorem 6.3 is finished.

We now have the equidistricution theorem of Fekete points according to equilibrium measure.

Proposition 6.6**.**

If $K\subset\mathbb{P}^{n}$ is compact set with $\gamma(K)<+\infty$, $\zeta_{1},...,\zeta_{s}\in K$ such that

[TABLE]

and

[TABLE]

then

[TABLE]

weakly in $\mathbb{P}^{n}$. Moreover, $\nu$ is the equilibrium measure of $K$.

Proof.

Since $\theta_{s}\rightarrow\gamma(K)$, in the proof of lemma 6.5 we proved that, if $\nu$ is the limit of $\nu_{s}$ in the weak sense in $K$, then

[TABLE]

Thus,

[TABLE]

we conclude that $\nu$ is the equilibrium measure of $K$. ∎

Definition 6.7**.**

Let $K$ be a compact of $\mathbb{P}^{n}$. The Chebyshev constant of order $s$ of $K$ is defined by $\tau_{s}(K):=-\log M_{s}(K)$, where

[TABLE]

Proposition 6.8**.**

Let $K\subset\mathbb{P}^{n}$ be a compact set, the sequence of Chebyshev constants $(\tau_{s}(K))$ converges in $\mathbb{R}^{+}$.

Proof.

The sum of two log-polynomes, $p(w)=\sum_{i=1}^{s}(-G(w,\zeta_{i}))$ of degree $s$ and $q(w)=\sum_{j=1}^{t}(-G(w,\zeta_{j}))$ of degree $t$, is also a log-polynôme of degree $s+t$. Then

[TABLE]

for all $s,t$. If $M_{s}(K)$ is infinite for some $s$, then all terms follow $M_{s^{\prime}}(K)$, $s^{\prime}\geq s$ are infinite also, thus the convergence.

Assume now that $M_{s}(K)$ is a finite sequence. Let $r$, $s$ two integers fixed. Then there exists $l=l(s,t)$ and $r=r(s,t)$, $0\leq r<t$ the integers non-negatives such that $s=l.t+r$. Using 6.3 we have

[TABLE]

Fixing now a value of $t$, all the values possibles of $r$ are bounded by $t$.

Divided both sides of the inequality by $s$, and taking $\liminf_{s\longrightarrow+\infty}$,

[TABLE]

This holds for any fixed $t\in\mathbb{N}$, so taking $\limsup_{t\longrightarrow+\infty}$ we obtain

[TABLE]

Thus,

[TABLE]

this gives the result. ∎

The number

[TABLE]

is called the Chebyshev constant of $K$.

In the following, we investigate the connection between the transfinite diameter and the Chebyshev constant.

Lemma 6.9**.**

Let $K\subset\mathbb{P}^{n}$ be compact set, then for all $s\geq 2$, we have

[TABLE]

In particular $\tau(K)\leq D(K)$.

Proof.

We have

[TABLE]

Since $G$ is continuous in $(\mathbb{P}^{n}\times\mathbb{P}^{n})\setminus\{(\zeta,\zeta),\;\;\zeta\in\mathbb{P}^{n}\}$, we find $\{\zeta_{i}\}_{1}^{s+1}$ such that,

[TABLE]

Let $k\in\{1,...,s+1\}$ be given. We define for $\zeta\in K$

[TABLE]

Since $-G(\zeta,\zeta_{j})\geq 0,$ we have

[TABLE]

Hence

[TABLE]

Since the sequence $\theta_{s}$ is increasing, we have

[TABLE]

∎

6.1. Proof of Theorem B

We will now prove the analogous stronger form of the classical Evans’s theorem (see [Ran95]).

The proof relies on the definition 6.7 and the lemma 6.9.

Proof.

By the definition of $M_{s}:=M_{s}(E)$, there exists $\zeta_{1},...,\zeta_{s}\in E$ such that

[TABLE]

and let $\mu_{s}$ be a Probability measure

[TABLE]

Since $M_{s}\rightarrow+\infty,$ there is a sequence $\{s_{h}\}$ such that

[TABLE]

We define the Probability measure

[TABLE]

Then,

[TABLE]

We show now that

[TABLE]

Indeed, the classical inequality $\frac{2}{\pi}t\leq\sin t\leq t,\;\;\;t\in[0,\frac{\pi}{2}],$ applied to the kernel $G$ implies

[TABLE]

Then

[TABLE]

It follows that $G_{\mu}\not\equiv-\infty$ since $E\neq\mathbb{P}^{n}$ and that

[TABLE]

On the other hand, by the inequality (6.4) we have

[TABLE]

This proves the theorem. ∎

Aknowlegements : This paper is a part of the Phd thesis of the second author supervised by the first author and Ahmed Zeriahi. This work was completed when she was visiting the ′′Institut de Mathématiques de Toulouse ′′ in June 2016. She would like to think this institution for the invitation and the Professeur Vincent Guedj for useful discussions and suggestions.