Norming sets on a compact complex manifold

TL;DR

This paper characterizes norming sets for spaces of global holomorphic sections of powers of positive line bundles on compact complex manifolds, providing metric conditions for these sets to control the sections' norms.

Contribution

It offers a metric characterization of norming sets for holomorphic sections of line bundle powers on compact complex manifolds, extending understanding of their geometric properties.

Findings

Provides a criterion for norming sets in terms of measurable subsets.

Establishes bounds relating global sections and integrals over these subsets.

Characterizes the sequences of subsets that serve as norming sets for all sections.

Abstract

We describe the norming sets for the space of global holomorphic sections to a $k$-power of a positive holomorphic line bundle on a compact complex manifold $X$. We characterize in metric terms the sequence of measurable subsets $\{G_{k}\}_{k}$ of $X$ such that there is a constant $C > 0$ where $$\|s\|^{2}\leq C \int_{G_{k}} |s(z)|^{2}\ dV(z)$$ for every $s\in H^{0}(L^{k})$ and for all $k\in\mathbb{N}$.

Equations

$$\Vert s{\Vert }^2\le C{\int }_{G_k}|s(z){|}^{2}dV(z)$$

$${\int }_{{\mathbb{R}}^n}|f{|}^{2}dm\le C{\int }_G|f{|}^{2}dm$$

$$m((Q+x)\cap G)\ge \gamma$$

$${\int }_{\mathbb{D}}|f{|}^{p}dm\le C{\int }_G|f{|}^{p}dm$$

$$m(G\cap D(a,R))>\delta m(D(a,R))$$

$$D(a,R)=\{z\in \mathbb{D}:|z-a|<R(1-|a|)\},$$

$${\int }_{{\mathbb{C}}^n}|f{|}^2{e}^{-2|z{|}^2}dm(z)\le C{\int }_G|f{|}^2{e}^{-2|z{|}^2}dm(z)$$

$$m(G\cap G(z,R))\ge \delta m(B(a,R))$$

$${\int }_X|s(x){|}^{2}dV(x)\le C{\int }_{G_k}|s(x){|}^{2}dV(x)$$

$$V\left(G_k\cap B\left(a,\frac{R}{\sqrt{k}}\right)\right)>\delta V\left(B\left(a,\frac{R}{\sqrt{k}}\right)\right)$$

$$⟨u,v⟩={\int }_X⟨u(x),v(x)⟩,u,v\in H^0(L)$$

$$\Pi (x,y)=\sum _{j=1}^N{s}_j(x)\otimes \overline{s_j(y)}$$

$$s(x)={\int }_X⟨s(y),\Pi (x,y)⟩dV(y)$$

$$|\Pi (x,x)|={\int }_X|\Pi (x,y){|}^{2}dV(y).$$

$$|{\Pi }_k(x,y)|\lesssim k^n{e}^{-c\sqrt{k}d(x,y)}$$

$$|{\Pi }_k(x,x)|\asymp k^n.$$

$$\dim H^0(L^k)\asymp k^n.$$

$${\Pi }_k(x,y)=\sum _{j=1}^N{s}_j(x)\otimes \overline{s_j(y)}$$

$$|{\Phi }_{k,y}(x)|=|{\Pi }_k(x,y)|,x\in X,$$

$$g_{k,y}(x)=\frac{{\Phi }_{k,y}(x)}{\sqrt{|{\Pi }_k(y,y)|}},x\in X,$$

$${\int }_X|g_{k,y}(x){|}^{2}dV(x)=\frac{1}{|{\Pi }_k(y,y)|}{\int }_X|{\Pi }_k(x,y){|}^{2}dV(x)=\frac{|{\Pi }_k(y,y)|}{|{\Pi }_k(y,y)|}=1.$$

$${\int }_{X\setminus B\left(y,\frac{R}{\sqrt{k}}\right)}|g_{k,y}(x){|}^{2}dV(x)=\frac{1}{|{\Pi }_k(y,y)|}{\int }_{X\setminus B\left(y,\frac{R}{\sqrt{k}}\right)}|{\Pi }_k(x,y){|}^{2}dV(x)$$

$$=\frac{1}{|{\Pi }_k(y,y)|}{\int }_0^{\infty }V\left(\left\{x:|{\Pi }_k(x,y)|>\lambda \right\}\setminus B\left(y,\frac{R}{\sqrt{k}}\right)\right)2\lambda d\lambda$$

$$\le \frac{1}{|{\Pi }_k(y,y)|}{\int }_0^{k^{n}D}V\left(\left\{x:D{e}^{-c\sqrt{k}d(x,y)}\ge k^{-n}\lambda \right\}\setminus B\left(y,\frac{R}{\sqrt{k}}\right)\right)2\lambda d\lambda .$$

$$\frac{1}{|{\Pi }_k(y,y)|}{\int }_0^{\infty }V\left(B\left(y,\frac{u}{\sqrt{k}}\right)\setminus B\left(y,\frac{R}{\sqrt{k}}\right)\right)(2k^{n}D{e}^{-cu})|-c{k}^{n}D{e}^{-cu}|du$$

$$\lesssim \frac{1}{k^n}{\int }_R^{\infty }{\left(\frac{u}{\sqrt{k}}\right)}^{2n}(2k^{n}D{e}^{-cu})|-c{k}^{n}D{e}^{-cu}|du$$

$$\lesssim {\int }_R^{\infty }{u}^{2n}{e}^{-2cu}du=e^{-2cR}{\int }_0^{\infty }(\mu +R{)}^{2n}{e}^{-2c\mu }d\mu$$

$$=e^{-2cR}\sum _{j=0}^{2n}\left(\genfrac{}{}{0ex}{}{2n}{k}\right)R^{2n-j}{\int }_0^{\infty }{\mu }^j{e}^{-2c\mu }d\mu$$

$$\lesssim e^{-2cR}\sum _{j=0}^{2n}\left(\genfrac{}{}{0ex}{}{2n}{k}\right)\frac{R^{2n-j}}{(2c{)}^j}j!\lesssim e^{-2cR}\sum _{j=0}^{2n}\left(\genfrac{}{}{0ex}{}{2n}{k}\right)(2cR{)}^{2n-j}=e^{-2cR}(1+2cR{)}^{2n}.$$

$${\int }_{X\setminus B\left(y,\frac{R}{\sqrt{k}}\right)}|g_{k,y}(x){|}^{2}dV(x)$$

$$|s(x){|}^2=|⟨s(y),{\Pi }_k(x,y)⟩{|}^2\le \Vert s{\Vert }^2|\Pi (x,x)|\lesssim k^n.$$

$${\int }_X|s(x){|}^{2}dV(x)\le C{\int }_{G_k}|s(x){|}^{2}dV(x)$$

$$V\left(G_k\cap B\left(a,\frac{R}{\sqrt{k}}\right)\right)>\delta V\left(B\left(a,\frac{R}{\sqrt{k}}\right)\right)$$

Full text

[TABLE]

for every $s\in H^{0}(L^{k})$ and for all $k\in\mathbb{N}$.

1. Introduction

Let $\mathcal{L}$ be a certain space of functions. The problem lies in giving a metric characterization of the set $G$ that verifies that the integral over $G$ is comparable to the norm for every function of $\mathcal{L}$. As we will see below this problem has already proved for some space of functions.

In [LS74] it appears the proof of the Logvinenko-Sereda theorem for functions of the Paley-Wiener space $PW_{K}$ for a fixed $K$:

Theorem 1.1**.**

For a measurable subset $G\subset\mathbb{R}^{n}$ the following are equivalent:

1. (1)

There is a constant $C>0$ such that

[TABLE]

for every $f\in PW_{K}$. We will say that $G$ is a norming set for the Paley-Wiener space. 2. (2)

There is a cube $Q\subset\mathbb{R}^{n}$ ans a constant $\gamma>0$ such that

[TABLE]

for all $x\in\mathbb{R}^{n}$. We will say that $G$ is relatively dense in $\mathbb{R}^{n}$.

See also [HJ94, pag 112-115].

The previous theorem has been extended to functions in the Bergman space $A^{p}$ in the unit disk $\mathbb{D}$ in [Lue81]:

Theorem 1.2**.**

For a measurable subset $G\subset\mathbb{D}$ the following are equivalent:

1. (1)

There is a constant $C>0$ such that

[TABLE]

for every $f\in A^{p}$. We will say that $G$ is a norming set for the Bergman space. 2. (2)

There is a constant $\delta>0$ and radius $R\in(0,1)$ such that

[TABLE]

for all $a\in\mathbb{D}$. We will say that $G$ is relatively dense in the disc $\mathbb{D}$.

Remark 1.3**.**

In [Lue81] a different geometry than usual will be used. The disks $D(a,R)$ will be of the form

[TABLE]

where $a\in\mathbb{D}$ and $R\in(0,1)$.

It should be noted that using similar arguments to those used in [Lue81] one can prove the Logvinenko-Sereda theorem for the classical Fock space $\mathcal{F}^{2}_{2|z|^{2}}(\mathbb{C}^{n})$:

Theorem 1.4**.**

For a a measurable subset $G\subset\mathbb{C}^{n}$ the following are equivalent:

1. (1)

There is a constant $C>0$ such that

[TABLE]

for every $f\in\mathcal{F}^{2}_{2|z|^{2}}(\mathbb{C}^{n})$. We will say that $G$ is a norming set for the classical Fock space. 2. (2)

There is a constant $\delta>0$ and a radius $R>0$ such that

[TABLE]

for all $z\in\mathbb{C}^{n}$. We will say that $G$ is relatively dense in $\mathbb{C}^{n}.$

Our main goal will be to prove the Logvinenko-Sereda theorem for the space of global holomorphic sections to a $k$-power of a positive line bundle on a compact complex manifold $X$. Before giving the statement of the theorem, we will present the convenient definitions.

First of all, we consider a compact complex manifold $X$ which will be endowed with a smooth Hermitian metric $\omega$. As we know, this metric induces a distance function $d(x,y)$ on $X$, which will be used to define the balls, and a volume form $V$, which will be used to integrate over the manifold.

We assume that the holomorphic line bundle $L$ on a compact complex manifold $X$ is endowed with a smooth hermitian metric $\phi$.

We will denote by $H^{0}(L)$ the space of global holomorphic sections to $L$. Moreover, we will consider that the line bundle $(L,\phi)$ is positive.

As $\phi$ is a Hermitian metric on $L$, then $\partial\overline{\partial}\phi$ is a globally defined (1,1)-form on $X$, which is called the curvature form of the metric $\phi$. Moreover, the line bundle $L$ with the metric $\phi$ is called positive if $i\partial\overline{\partial}\phi$ is a positive form. The statement of the Logvinenko-Sereda theorem is the following:

Theorem 1.5**.**

For a sequence of measurable subsets $\{G_{k}\}$ in $X$ the following are equivalent:

1. (1)

There is a constant $C>0$ such that for all $k\in\mathbb{N}$,

[TABLE]

for every $s\in H^{0}(L^{k})$. We will say that $G_{k}$ is a norming set in $H^{0}(L^{k})$. 2. (2)

There is a constant $\delta>0$ and a radius $R$ such that

[TABLE]

for all $a\in X$ and $k\in\mathbb{N}$. We will say that $G_{k}$ is relatively dense in $X$.

Remark 1.6** (Bergman kernel).**

The space $H^{0}(L)$ admits a Hilbert space structure endowed with the scalar product

[TABLE]

where the integration is taken with respect to the volume form $V$.

The Bergman kernel $\Pi(x,y)$ associated to this space is a section to the line bundle $L\boxtimes\overline{L}$ over the manifold $X\times X$, defined by

[TABLE]

where $s_{1},\dots,s_{N}$ is an orthonormal basis for $H^{0}(L)$. Moreover, this definition does not depend on the choice of te orthonormal basis. Notice that $L\boxtimes\overline{L}=\pi^{\ast}_{1}(L)\otimes\pi^{\ast}_{2}(\overline{L})$ where $\pi_{i}=X\times X\rightarrow X$ is the projection onto the $i$-factor.

The Bergman kernel $\Pi(x,y)$ is in a sense the reproducing kernel for the space $H^{0}(L)$, satisfying the reproducing formula

[TABLE]

for $s(y)\in H^{0}(L)$.

The pointwise norm of the Bergman kernel in symmetric, $|\Pi(x,y)|=|\Pi(y,x)|$. Moreover, it satisfies

[TABLE]

Lemma 1.7**.**

Let $(L,\phi)$ be a positive line bundle. We have the off-diagonal estimate

[TABLE]

where $c$ is an appropriate positive constant and the diagonal estimate

[TABLE]

Moreover, from this we obtain the estimate of the dimension of $H^{0}(L^{k})$

[TABLE]

See [Ber03] where this lemma is proved using a method that appears in [Lin01].

2. Previous result

The main goal of the lemma is to prove for each ball the existence of a normalized peak-section, that is, a section such that most of the mass of the function is in such ball. For this we will use the reproducing kernel of $H^{0}(L^{k})$.

Lemma 2.1**.**

Given $\varepsilon>0$, there is a radius $R>0$ such that for all ball $B\left(\xi,\frac{R}{\sqrt{k}}\right)$, there is a section $s=s_{\xi,k}\in H^{0}(L^{k})$ such that

• •

$\|s\|^{2}={\int_{X}|s(x)|^{2}\ dV(x)=1}$,

• •

${\int_{X\setminus B\left(\xi,\frac{R}{\sqrt{k}}\right)}|s(x)|^{2}\ dV(x)<\varepsilon}$,

Proof.

Given $\varepsilon>0$. We consider the reproducing kernel of the Hilbert space $H^{0}(L^{k})$

[TABLE]

where $s_{1},\dots,s_{N}$ is an orthonormal basis for $H^{0}(L^{k})$.

If we fix $y\in X$, there is a section $\Phi_{k,y}\in H^{0}(L^{k})$ such that

[TABLE]

by the lemma in [LOC12, pag. 431].

We will consider the section

[TABLE]

where $\Pi_{k}$ denotes the Bergman kernel for the $k$’th power $L^{k}$ of the line bundle $L$.

Let us see the first property.

[TABLE]

Now, we check the second property

[TABLE]

where we use that $|\Pi_{k}(x,y)|\leq k^{n}De^{-c\sqrt{k}\ d(x,y)}$ for some constants $c,D>0$, as we see in Lemma 1.7. Applying the change of variable $\lambda=k^{n}De^{-cu}$.

[TABLE]

Finally, for $R$ large enough we obtain that

[TABLE]

∎

Remark 2.2**.**

Notice that taking the same section of the previous lemma and using the Cauchy-Schwarz inequality we obtain that

[TABLE]

3. Main results

Theorem 3.1**.**

For a sequence of measurable subset $\{G_{k}\}_{k}$ in $X$ the following are equivalent:

1. (1)

There is a constant $C>0$ such that for all $k\in\mathbb{N}$,

[TABLE]

for every $s\in H^{0}(L^{k})$. We will say that $G_{k}$ is a norming set of $H^{0}(L^{k})$. 2. (2)

There is a constant $\delta>0$ and a radius $R$ such that

[TABLE]

for all $a\in X$ and $k\in\mathbb{N}$. We will say that $G_{k}$ is relatively dense in X.

Proof.

The proof that (1) implies (2) is the easiest. In this proof, we consider a section with the properties of Lemma 2.1.

So, given $\varepsilon\leq 1/2C$ and applying the Lemma 2.1 there is a radius $R$ such that for all balls $B\left(\xi,\frac{R}{\sqrt{k}}\right)$ there is a section $s$ verifying the properties of the lemma. Hence, applying the Remark 2.2 we obtain

[TABLE]

and using the other properties with (2) we have

[TABLE]

Therefore, we have proved that (1) implies (2).

The proof that (2) implies (1) is the difficult one. We will use a similar argument as in [MOC08, Chapter 4]. We will partition $X$ in two pieces. The first piece, denoted by $\mathcal{A}$, are the points where the sections is much smaller that its average and the second is the complementary $X\setminus\mathcal{A}$. The following lemma proves that the integral over $\mathcal{A}$ is irrelevant as most of the mass is carried by $X\setminus\mathcal{A}$.

Lemma 3.2**.**

Let $\varepsilon>0$ and $s\in H^{0}(L^{k})$. Define the set

[TABLE]

Then there is a constant $C$ depending on $R$ and $k$ such that

[TABLE]

Proof.

For $\xi\in\mathcal{A}$ we have

[TABLE]

Integrating respect to $a$ and applying the Fubini’s theorem

[TABLE]

Therefore, we obtain

[TABLE]

∎

Let $\mathcal{F}=X\setminus\mathcal{A}=\left\{a\in X\ :\ |s(a)|^{2}\geq\frac{\varepsilon}{V(B(a,R/\sqrt{k}))}\int_{B(a,R/\sqrt{k})}|s(z)|^{2}\ dV(z)\right\}$. If we choose $\varepsilon$ such that $\varepsilon C<1/2$ we have

[TABLE]

since

[TABLE]

Thus this is enough to show that

[TABLE]

All we need to prove is the existence of a constant $C>0$ such that for all $w\in\mathcal{F}$

[TABLE]

Indeed, if this is the case then

[TABLE]

To prove (3.1) we argue by contradiction. If (3.1) is false there are for any $q\in\mathbb{N}$ sections $s_{q}\in H^{0}(L^{k_{q}})$ and $w_{q}\in\mathcal{F}$ such that

[TABLE]

By compactness of $X$ we can choose a convergent subsequence $\{w_{q}^{\ast}\}$ of $w_{q}$ to some $w^{\ast}\in X$ such that there is a local chart $(U,\varphi)$ where

[TABLE]

and $\varphi(w^{\ast})=0$.

By the positivity of $i\partial\overline{\partial}\phi$, we can make a linear change of variable so that $i\partial\overline{\partial}\phi(\varphi^{-1}(0))=i\partial\overline{\partial}|z|^{2}$. So if we consider the Taylor series of $\phi$ in the local ball $B(0,\tau)$ we obtain

[TABLE]

where $h$ is a pluriharmonic polynomial of degree less than or equal to 2, ${h(0)=\phi(\varphi^{-1}(0))}$ and $q(z)=o(|z|^{2})$. Hence, we have that

[TABLE]

in $B(0,\tau)$ for nondecreasing continuous function $c(\tau)$ such that $c(0)=0$. Notice that we can use the local chart $(\varphi^{-1}(B(0,\tau{\ast})),\varphi)$ instead since $\exists\nu\in\mathbb{N}$, $\forall q\geq\nu$ : $B\left(w_{q}^{\ast},\frac{R}{\sqrt{k_{q}}}\right)\subset\varphi^{-1}(B(0,\tau^{\ast}))$ and $\tau^{\ast}\leq\tau$.

Now we consider the sequence of holomorphic functions

[TABLE]

where $\Re(H)=h$ and $|s_{q}|^{2}=|f_{q}|^{2}e^{-k\phi}$.

Next we need obtain that

[TABLE]

where $\varphi(w_{q}^{\ast})=\eta_{q}$,

[TABLE]

for a certain ball and

[TABLE]

First, for the sequence $\{w_{q}^{\ast}\}$ we have

[TABLE]

Furthermore for $z\in B(0,\tau^{\ast})$ we obtain

[TABLE]

Finally we obtain the inequality that we need

[TABLE]

where $E_{k_{q}}^{\ast}=\varphi(E_{k_{q}})$ and there is a ball $B(\eta_{k_{q}},\delta_{k_{q}})\subset\varphi(w_{q}^{\ast},R/\sqrt{k_{q}})$ such that $m(B(\eta_{q},\delta_{k_{q}}))\asymp V\left(B\left(w_{q}^{\ast},\frac{R}{\sqrt{k_{q}}}\right)\right)$, because as $\varphi,\varphi^{-1}\in\mathcal{C}^{1}$ we have that $\varphi$ is bi-Lipschitz.

By means of a dilatation and a translation we send $\eta_{q}$ to the origin of $\mathbb{C}^{n}$, the ball $B(\eta_{q},\delta_{k_{n}})$ to $B(0,1)\subset\mathbb{C}^{n}$ and the set $E_{k_{q}}^{\ast}$ to $E_{q}^{\ast}$. We will denote the set $\overline{B(0,1)}\cap E_{q}^{\ast}$ by $H_{q}$.

Moreover, multiplying by a constant we can assume that

[TABLE]

The subharmoniticity of $|\tilde{G}_{q}|^{2}$ and the fact that $w_{q}^{\ast}\in\mathcal{F}$ tells us that

[TABLE]

Notice that we can modify the inequality of the definition of $\mathcal{F}$ in the same way as the expression (3.3).

And this property together with (3.3) yields

[TABLE]

We have that $\{\tilde{G}_{q}\}$ is a locally bounded sequence of holomorphic functions defined in $B(0,1)$ and therefore using Montel theorem there exist a subsequence converging locally uniformly on $B(0,1)$ to some holomorphic function $G$.

We observe that the relative dense hypothesis yields

[TABLE]

The Helly selection theorem guaranties the existence of a weak-$\ast$limit $\tau$ of a subsequence of $\tau_{q}=\chi_{H_{q}}m$ such that $\tau\not\equiv 0$. Condition (3.4) implies that $\tilde{G}=0$ $\tau$-a.e. and therefore $supp\tau\subset\{\tilde{G}=0\}$.

We want to show that $supp\tau$ cannot lie on a complex $(n-1)$-dimensional submanifold $S\subset\mathbb{C}^{n}$. We argue by contradiction.

By definition of $\tau_{n}$ we have $\tau_{n}(B(x,r))\leq m(B(x,r))\lesssim r^{2n}$. As this inequality is true for all $\tau_{q}$, the same holds for $\tau$. But, using the Frostman lemma we obtain that $\dim_{Haus}(supp\tau)\geq 2n>0$.

Therefore, we have proved that (2) implies (1).

∎

Now, we will generalize the previous sampling problem using a sequence of measures $\{\mu_{k}\}_{k}$ of $\mathfrak{M}(X)$ instead of the particular sequence of measures $\{\chi_{G_{k}}V\}_{k}$ of $\mathfrak{M}(X)$, where $\{G_{k}\}_{k}$ are measurable sets. We will solve part of this problem, the remaining part continues being an open problem.

Theorem 3.3**.**

For a sequence of measures $\{\mu_{k}\}$ the following are equivalent:

1. (1)

There exists a constant $C_{1}>0$ such that for all $k\in\mathbb{N}$,

[TABLE]

for every $s\in H^{0}(L^{k})$. 2. (2)

There exists a constant $C_{2}>0$ such that for all $k\in\mathbb{N}$,

[TABLE]

for every $z\in X$, that is, the sequence of Berezin transforms $\{T[\mu_{k}]\}_{k}$ are uniformly bounded in $X$. 3. (3)

There is a constant $C_{3}>0$ such that

[TABLE]

Proof.

The proof that (1) implies (2) is trivial because $\Pi_{k}(z,w)\in H^{0}(L^{k})$ for each $w\in X$ and $k\in\mathbb{N}$.

To prove that (2) implies (3) is enough the following lemma.

Lemma 3.4**.**

There are constants $\varepsilon>0$, $M>0$ and $k_{0}\in\mathbb{N}$ such that

[TABLE]

for every $w\in B(z,\varepsilon/\sqrt{k})$ and $k>k_{0}$.

Proof.

To prove (3.5) we use that

[TABLE]

and by the bound of the derivatives in [LOC12, pag. 434]

[TABLE]

where

[TABLE]

together with the sub-mean value property in [LOC12, pag. 432 ], we obtain that

[TABLE]

By the diagonal estimate of Lemma 1.7 we have

[TABLE]

where $D>0$ is a certain constant.

Then, if we consider the ball $B(z,\varepsilon/\sqrt{k})$ where $\varepsilon<\frac{D}{2M^{\prime}\sqrt{n}}$ we have

[TABLE]

for $w\in B(z,\varepsilon/\sqrt{k})$. ∎

Applying Lemma 3.4 we have that there are constants $\varepsilon>0$, $M>0$ and $k_{0}\in\mathbb{N}$ such that

[TABLE]

for every $z\in X$ and $k>k_{0}$.

We take a covering of the ball $B(z,1/\sqrt{k})$ by a collection of balls $B(z_{j},\varepsilon/5\sqrt{k})$, $z_{j}\in B(z,1/\sqrt{k})$ and applying the Vitali covering lemma we obtain a sub-collection of these balls which are disjoint such that

[TABLE]

Moreover, as

[TABLE]

by means of the volume we have that

[TABLE]

that is, $N\lesssim\left(\frac{2}{\varepsilon}\right)^{2n}$ where $N$ is the number of balls of the sub-collection. Therefore, we conclude

[TABLE]

Notice that for $k\leq k_{0}$ (3) holds immediately since $\mu_{k}(B(z,1/\sqrt{k}))$ are bounded by $\mu_{k}(X)$.

All we need to prove that (3) implies (1) is the existence of a constant $Q>0$ such that for all $w\in X$

[TABLE]

This is proved by the sub-mean value property in [LOC12, pag. 432 ]. Indeed, if this is the case then

[TABLE]

for every $s\in H^{0}(L^{k})$.

∎

Example 3.5** (Tautological line bundle).**

We will apply the Theorem 3.1 to the complex projective space $\mathbb{CP}^{n}$ with the hyperplane bundle $\mathcal{O}(1)$, endowed with the Fubini-Study metric. Notice that the holomorphic sections to $\mathcal{O}(k)$, the k’th power of $\mathcal{O}(1)$, can be identified with the homogeneous polynomials of degree $k$ in homogeneous coordinates.

Using Theorem 3.1 we obtain that for a given sequence of measurable subsets $\{\Omega_{k}\}_{k}$ in $\mathbb{C}^{n}$, we have that the following statements are equivalent:

1. (1)

For all $k\in\mathbb{N}$, it is verified that

[TABLE]

for every polynomial $p_{k}$ of degree less or equal than $k$ in $n$ variables. 2. (2)

There is a radius $R$ such that

[TABLE]

for all $z\in\mathbb{C}^{n}$ and $k\in\mathbb{N}$, where

[TABLE]

Notice that the measure $V$ is the Lebesgue measure on $\mathbb{C}^{n}$.

Acknowledgement

I would like to express my deep appreciation to Joaquim Ortega Cerdà for his unconditional support and advices to improve in mathematics. In addition, I would like to show my gratitude to my family and friends of Canary Islands for their encouragement and help to fulfil my dreams.