# Random systems of holomorphic sections of a sequence of Line bundles on
compact Kähler manifolds

afrim bojnik and ozan günyüz

###### Abstract.

This paper primarily provides an equidistribution result of zeros of systems
of random holomorphic sections associated with a sequence of positive
Hermitian holomorphic line bundles on a compact Kähler manifold $X$ by using
variance estimate and expected distribution of random zero currents of
integration in any codimension $k$. The probability measures taken into
consideration in this paper are only subject to a moment condition that is
sufficiently general to include a wide range of the measures commonly
encountered in the literature. In particular, we also give equidistribution
results for several known measures such as Gaussian, Fubini-Study measures,
area measure of spheres, probability measures whose distributions have bounded
densities with logarithmically decaying tails and locally moderate measures
among others.

###### Key words and phrases:

Random holomorphic section, equidistribution of zeros, variance

###### 2020 Mathematics Subject Classification:

Primary: 32A60, 60D05 Secondary: 32L10, 32C20, 32U40, 81Q50

A. Bojnik is partially supported by TÜBİTAK grant ARDEB-3501/118F049

##  1\. Introduction

In recent years, equidistribution and statistical properties of zeros of
random holomorphic sections have been progressed heavily. There are numerous
results as to the distribution of zeros of holomorphic sections in diverse
probabilistic frameworks. Amongst these, what has been more largely focused on
is the tensor powers of a given positive Hermitian line bundle over a
compact(or non-compact) Kähler manifold within a Gaussian setting. In this
background, [SZ99] is one of the first papers considering the equidistribution
problem of (Gaussian) random holomorphic sections. Additionally, in [SZ08,
SZ10], asymptotic variance formulas are obtained not only for linear
statistics but also for their smooth analogs, all within the same geometric
and probabilistic context. One of the most recent results, proved via the
techniques of [SZ10] in [Shif1], is the asymptotic expansion of the variance
for the codimension $1$ case in the aforementioned setting. This asymptotic
expansion shows also that the coefficient of the first term in the expansion,
which also appeared as the leading-order term in the asymptotic formula
obtained in [SZ10], is sharp. In this framework, Dinh and Sibony [DS06]
innovated a method from complex dynamics for analyzing zero distribution, and
set convergence speed bounds in the compact case, enhancing Shiffman and
Zelditch’s initial results, namely [SZ99]. Alongside the Gaussian setting, in
the papers [Bay16], [BCHM18] [BL15], [CLMM], [CM1], [CMM] and [BCM], more
general scenarios are investigated, including the Gaussian case as a
particular instance. For example, in [BL15], the authors focus on the complex
random variables that posseses bounded distribution functions on the whole
complex plane $\mathbb{C}$ and outside of a very large disk with radius
$\rho$, its integral with respect to the two-dimensional Lebesgue measure has
an upper bound depending on $\rho$, the latter condition is called the tail-
end estimate. Meanwhile, in [BCM, CMM] the authors expand their research to
the equidistribution problem within a wider context, involving a sequence of
Hermitian line bundles over a normal reduced complex Kähler space with
singularities.

It is essential to highlight that global holomorphic sections are natural
generalizations of polynomials. In relation to the general setting, there has
been a great deal of interest in the statistical problems related to the zero
sets of random polynomials of several variables in both real and complex
domains. For a comprehensive overview of results in this direction, interested
readers can refer to [Bay17b, BL15, BS07, BL05, BD18, EDKO, ROJ, SHSM, HN08,
IZ13](and references cited therein). These sources cover a wide range of
results encompassing both Gaussian and non-Gaussian cases along with
historical developments of the polynomial theory. Long before these
developments, it is worth acknowledging the pioneering work of mathematicians
such as Littlewood-Offord, Kac, Hammersley, and Erdös-Turan, who were among
the first to investigate the distribution of roots of random algebraic
equations in a single real variable. For more insights into these foundational
studies, interested readers can consult the papers [Kac43, LO43, HAM56, ET50].
On the other hand, there is also a growing physics literature dealing with the
equidistribution and probabilistic problems of zeros of complex random
polynomials. For studies of fundamental importance in this direction, see,
e.g., [BBLE, FH, Hann, NV98].

The present article follows a general framework akin to papers [CLMM] and
[CMM]. In the recent work presented by Coman-Lu-Ma-Marinescu [CLMM], they
establish an equidistribution result for a sequence of line bundles
$(L_{p},h_{p})$ instead of the tensor powers of a single line bundle $L$,
i.e., $(L^{\otimes p},h^{\otimes p})$ by imposing a natural convergence
condition on the Chern curvature forms $c_{1}(L_{p},h_{p}).$ As a probability
measure they consider the Fubini-Study measures and use the standard formalism
of meromorphic transforms from complex dynamics, as introduced by Dinh and
Sibony in [DS06]. In contrast to their work, the novelty of the current paper
lies in its reliance on the variance estimate of zero currents of integration
over the zero sets of systems of random holomorphic sections. This classical
but efficient approach allows for the extension of the previous results to a
broader spectrum of probability distributions, generalizing Theorem 0.4
considered in [CLMM] as well. It is also worthwile that, in the context of
sequences of line bundles, the variance estimate is established for the first
time in this work setting it apart from previous related results in the
literature. Furthermore, our findings can be regarded as a form of
universality result within a wide range of probability measures studied in the
literature, including those with continous distributions.

###  1.1. Current Setting:

Let $(L_{p},h_{p})_{p\geq 1}$ be a sequence of holomorphic line bundles on a
compact Kähler manifold $(X,\omega)$ of $\textup{dim}_{\mathbb{C}}\,X=n$ with
a fixed Kähler form $\omega$ and $\mathscr{C}^{2}$ Hermitian metrics $h_{p}$
(see Section 2 below) such that the curvature forms $c_{1}(L_{p},h_{p})$
satisfy the following so called “prequantization condition”:

(1.1)
$\left\|\frac{1}{A_{p}}c_{1}(L_{p},h_{p})-\,\omega\right\|_{\mathscr{C}^{0}}=O(A^{-a}_{p})$

where $A_{p}>0$, $a>0$ and $\lim_{p\rightarrow\infty}{A_{p}}=+\infty.$

Riemannian $k$-volume form $\mathrm{Vol}_{k}$ is expressed as
$\mathrm{Vol}_{k}(A)=\int_{A}{\omega^{n-k}}$. We suppress the subindex because
it will be clear from the context which codimension is meant. We also remark
that, unlike [CLMM], for the sake of simplifying our notation, we will not
utilize additional volume form on the base manifold $X$ besides $\omega$. Even
though employing a different form than $\omega$ might alter the notation, it
will not affect the equidistribution results of this paper. The adjustment
mainly involves substituting the appropriate powers of $\omega$ with another
form $\vartheta$ in the relevant parts, such as basic cohomology arguments and
the total variation of the signed measure $dd^{c}\phi$ for a test form $\phi$.

In the standard setup of geometric quantization, we work with a compact Kähler
manifold notated as $(X,\omega)$, equipped with a Hermitian holomorphic line
bundle $(L,h)$, known as the prequantum line bundle, fulfilling the
prequantization condition given by

(1.2) $\omega=\frac{\sqrt{-1}}{2\pi}R^{L}=c_{1}(L,h).$

Here $R^{L}$ is the curvature of the Chern connection on $L$, and $c_{1}(L,h)$
is the Chern curvature form of $(L,h)$. The existence of the prequantum line
bundle $(L,h)$ allows the investigation into the Hilbert space $H^{0}(X,L)$ of
holomorphic sections and to establish a mapping between classical observables
on $X$ and quantum operators on $H^{0}(X,L)$ in the setting where the Planck
constant approaches zero. The modification of Planck’s constant corresponds to
scaling the Kähler form via tensor powers $L^{\otimes p}$, and the curvature
of the line bundle is thus described by $\hbar=\frac{1}{p}$.

The stipulation in (1.2) is recognized as an integrality condition. The
existence of a prequantum holomorphic line bundle is strongly connected with
the integral nature of the de Rham cohomology class $[\omega]$, i.e.,
$[\omega]\in H^{2}(X,\mathbb{Z})$. When dealt with a Kähler form $\omega$ that
is not integral, one can construct an associated family of positive line
bundles $(L_{p},h_{p})$. The curvatures of these bundles approximate integer
multiples of $\omega$, thus serving as a prequantization of the non-integral
Kähler form $\omega$.

The approximation condition (1.1) is derived naturally as follows: Starting
with a Kähler form $\omega$, one may initially approximate the associated
cohomology class $[\omega]\in H^{2}(X,\mathbb{R})$ with integral classes in
$H^{2}(X,\mathbb{Z})$ through diophantine approximation, as described by
Kronecker’s lemma. Subsequently, one can construct smooth forms corresponding
to these integral approximations. The paper [CLMM] provides a full asymptotic
expansion of the Bergman kernel restricted to the diagonal in case there is
such a good diophantine approximation of $\omega$. Using the same
approximation, we will have an equidistribution result on the zeros of systems
of random holomorphic sections. For the equidistribution problem, following
[CLMM], we work in a setting as general as possible, where metrics are of
class $\mathscr{C}^{2}$ and the convergence of Chern forms in (1.1) is in the
$\mathscr{C}^{0}$-topology.

It is also important to highlight that, within the examples of sequences of
line bundles $(L_{p},h_{p})$ fulfilling the condition (1.1), one natural
instance is $(L_{p},h_{p})=(L^{\otimes p},h^{\otimes p})$ for some fixed
prequantum line bundle $(L,h).$ Other examples include cases where
$(L_{p},h_{p})=(L^{\otimes p},h_{p})$ but here, $h_{p}$ is not necessarily the
product metric $h^{p},$ e.g. $h_{p}=h^{p}e^{-\varphi_{p}},$ with appropriate
weights $\varphi_{p}.$ Additionally, there are examples involving tensor
powers of several line bundles, for more details see [CLMM].

We denote the vector space of global holomorphic sections of $L_{p}$ by
$H^{0}(X,L_{p})$. We take into consideration the following inner product on
the space of smooth sections $\mathscr{C}^{\infty}(X,L_{p})$ with respect to
the metric $h_{p}$ and the volume form $\omega^{n}$ on $X$:

$\langle
s_{1},s_{2}\rangle_{p}:=\int_{X}{\big{\langle}s_{1}(x),s_{2}(x)\big{\rangle}_{h_{p}}\omega^{n}}\,\,\,\,\text{and}\,\,\,\|s\|^{2}_{p}:=\langle
s,s\rangle_{p}.$

By virtue of Cartan-Serre finiteness theorem (see, e.g., chapter 6, [Grar3]),
the space $H^{0}(X,L_{p})$ is finite dimensional (since every line bundle can
be seen as a coherent sheaf) and we will write $d_{p}:=\dim{H^{0}(X,L_{p})}$.
The space $\mathcal{L}^{2}(X,L_{p})$ represents the completion of
$\mathscr{C}^{\infty}(X,L_{p})$ with respect to this norm, forming the Hilbert
space of square integrable sections of $L_{p}$. Within this framework,
consider the orthogonal projection operator
$\Pi_{p}:\mathcal{L}^{2}(X,L_{p})\rightarrow H^{0}(X,L_{p}).$ The Bergman
kernel, denoted as $K_{p}(x,y)$ is defined as the integral kernel associated
to this projection (for details see, e.g., [MM1], section $1.4$). If
$\\{S^{p}_{j}\\}_{j=1}^{d_{p}}$ is an orthonormal basis for $H^{0}(X,L_{p}),$
by employing the orthonormal representation of $K_{p}(x,y)$ with respect to
the basis $\\{S^{p}_{j}\\}_{j=1}^{d_{p}}$ and making use of the reproducing
property on the space $H^{0}(X,L_{p})$ we have:

$K_{p}(x,y)=\sum_{j=1}^{d_{p}}{S^{p}_{j}(x)\otimes S^{p}_{j}(y)^{*}}\in
L_{p,x}\otimes L_{p,y}^{*},$

where $S^{p}_{j}(y)^{*}=\langle\,.\,,S^{p}_{j}(y)\rangle_{h_{p}}\in
L^{*}_{p,y}$. The restriction of the Bergman kernel to the diagonal of $X$ is
called the Bergman kernel function of $H^{0}(X,L_{p})$ which we denote by
$K_{p}(x):=K_{p}(x,x),$ and it has dimensional density property, namely
$d_{p}=\int_{X}K_{p}(x)\omega^{n}.$ Now we make the following assumption about
the behaviour of $K_{p}(x):$ There exists a constant $M_{0}>1$ and
$p_{0}\in\mathbb{N}$ such that

(1.3) $\frac{A_{p}^{n}}{M_{0}}\leq K_{p}(x)\leq M_{0}A_{p}^{n}$

holds for every $x\in X$ and $p\geq p_{0}.$ Consequently, this leads to the
following estimates on the dimension $d_{p}$ which will be useful in the
subsequent analysis:

(1.4) ${M_{0}}^{-1}\mathrm{Vol}(X)A_{p}^{n}\leq d_{p}\leq
M_{0}\mathrm{Vol}(X)A_{p}^{n}$

for all $p>p_{0}.$

Let us fix an orthonormal basis $\\{S^{p}_{j}\\}^{d_{p}}_{j=1}$ of
$H^{0}(X,L_{p}).$ Then, every $s\in H^{0}(X,L_{p})$ is written in a unique way

(1.5) $s_{p}=\sum_{j=1}^{d_{p}}{a_{j}^{p}S^{p}_{j}}.$

Using the aforementioned representation, we identify the spaces
$H^{0}(X,L_{p})$ with $\mathbb{C}^{d_{p}}$ and equip them with the
$d_{p}$-fold probability measures $\sigma_{p}$ which does not put any mass on
pluripolar sets and satisfy the following slightly different moment condition
compared to the one given in [BCM] (see, e.g., p.3, assumption ($B$) in
[BCM]). This adjustment is made to ensure that the variance is well-defined:
there exist a constant $\alpha\geq 2$ and for every $p\geq 1$ constants
$C_{p}>0$ such that

(1.6) $\int_{\mathbb{C}^{d_{p}}}{\bigl{\lvert}\log{|\langle
a,v\rangle|\bigr{\rvert}^{\alpha}}d\sigma_{p}(a)}\leq C_{p}$

for every $v\in\mathbb{C}^{d_{p}}$ with $\|v\|=1.$ The probability space
$(H^{0}(X,L_{p}),\sigma_{p})$ depends, as is seen above, on the choice of the
orthonormal basis used in the identification of $H^{0}(X,L_{p})$ (unless
$\sigma_{p}$ is unitary invariant). Additionally, we consider the product
probability space
$(\mathcal{H}_{\infty},\sigma_{\infty})=\Big{(}\prod_{p=1}^{\infty}H^{0}(X,L_{p}),\prod_{p=1}^{\infty}\sigma_{p}\Big{)}$
comprising of independent random sequences of holomorphic sections of $L_{p}$
with increasing values of $p.$

Throughout $\mathcal{D}^{p,q}(X)$ denotes the space of test forms of bidegree
$(p,q)$ on the complex manifold $X$ and we will let
$\mathcal{D^{\prime}}_{p,q}(X)$ denote the space of currents of bidegree
$(p,q)$ on $X,$ so $\langle T,\varphi\rangle=T(\varphi)$ will mean the pairing
of $T\in\mathcal{D^{\prime}}_{p,q}(X)$ and
$\varphi\in\mathcal{D}^{n-p,n-q}(X).$ For a thorough investigation of currents
in the context of complex manifolds, see, e.g., [Dem12].

Given a generic system $\Sigma_{p}^{k}:=(s^{1}_{p},\ldots,s^{k}_{p})$ of $k$
holomorphic sections of $L_{p},$ where $1\leq k\leq\dim{X},$ we denote their
simultaneous zero locus by

$Z_{\Sigma_{p}^{k}}:=\\{x\in X:s_{p}^{1}(x)=\cdots=s_{p}^{k}(x)=0\\}$

Its current of integration (with multiplicities) along the analytic subvariety
$Z_{\Sigma_{p}^{k}}$ is defined as follows, given
$\phi\in\mathcal{D}^{n-k,n-k}(X)$

$\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle:=\int_{Reg\,(Z_{\Sigma_{p}^{k}})}{\phi}.$

where $Reg\,(Z_{\Sigma_{p}^{k}})$ denotes the regular points of the common
zero locus $Z_{\Sigma_{p}^{k}}.$ Since the base locus
$Bs(H^{0}(X,L_{p}))=\emptyset$ for $p\geq p_{0},$ Bertini’s theorem (see
Appendix) implies that the zero sets $Z_{s_{p}^{j}}$ are in general position.
In particular, the common zero locus $Z_{\Sigma_{p}^{k}}$ is a submanifold of
codimension $k$ and the current $[Z_{\Sigma_{p}^{k}}]$ is well-defined and is
represented by

(1.7) $[Z_{\Sigma_{p}^{k}}]=[Z_{s_{p}^{1}}]\wedge\cdots\wedge[Z_{s_{p}^{k}}].$

The expectation and the variance of the current valued random variable
$(H^{0}(X,L_{p})^{k},\sigma_{p}^{k})\ni\Sigma_{p}^{k}\longmapsto[Z_{\Sigma_{p}^{k}}]$
are defined by

(1.8) $\displaystyle\langle\mathbb{E}[Z_{\Sigma_{p}^{k}}],\phi\rangle$
$\displaystyle:=\mathbb{E}\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle=\int_{H^{0}(X,L_{p})^{k}}{\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle\,d\sigma_{p}^{k}(\Sigma_{p}^{k})}$
(1.9) $\displaystyle\langle\mathrm{Var}[Z_{\Sigma_{p}^{k}}],\,\phi\rangle$
$\displaystyle:=\mathrm{Var}\langle[Z_{\Sigma_{p}^{k}}],\,\phi\rangle=\mathbb{E}\langle[Z_{\Sigma_{p}^{k}}],\,\phi\rangle^{2}-(\mathbb{E}\langle[Z_{\Sigma_{p}^{k}}],\,\phi\rangle)^{2},$

where $\phi\in\mathcal{D}^{n-k,n-k}(X)$ and
$\sigma_{p}^{k}=\sigma_{p}\times\cdots\times\sigma_{p}$ is the $k-$fold
product measure on the product space $H^{0}(X,L_{p})^{k}$. Throughout this
paper, in equidistribution settings for codimensions greater than $1$, we will
examine sequences in $H^{0}(X,L_{p})^{k}$. For this purpose, we will employ
the infinite product probability space denoted as
$(\mathcal{H}^{k}_{\infty},\sigma^{k}_{\infty})=\left(\prod_{n=1}^{\infty}H^{0}(X,L_{p})^{k},\prod_{n=1}^{\infty}\sigma^{k}_{p}\right)$.
Additionally, we will consider normalized currents of integration along the
zero set $Z_{\Sigma_{p}^{k}}$ of the random system
$\Sigma_{p}^{k}=(s^{1}_{p},\ldots,s^{k}_{p})$, more precisely,

$\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}:=\frac{1}{A_{p}^{k}}\big{[}Z_{\Sigma_{p}^{k}}\big{]}$

Now, we are ready to give the primary results of this paper:

######  Theorem 1.1.

Let $(L_{p},h_{p})_{p\geq 1}$, $(X,\omega)$ and $\sigma_{p}$ be as defined
above. Assume that they satisfy the conditions (1.1), (1.3) and (1.6). Assume
further that the sections $s^{1}_{p},s^{2}_{p},\ldots,s^{k}_{p}$ are
independent random holomorphic sections for each $p$. Then there exists
$P\in\mathbb{N}$ such that for all $p\geq P$ and any
$\phi\in\mathcal{D}^{n-k,n-k}(X)$, one has the following estimate

$\langle\mathrm{Var}[\widehat{Z_{\Sigma_{p}^{k}}]},\phi\rangle\leq\frac{(C_{p})^{2/\alpha}}{A^{2}_{p}}\,(2^{k-1}\mathrm{Vol}(X)B_{\phi}\,)^{2}$

where $B_{\phi}$ is a positive constant depending on the form $\phi$.

######  Theorem 1.2.

Let $(L_{p},h_{p})_{p\geq 1}$, $(X,\omega)$ and $\sigma_{p}$ be as defined
above. Assume that they satisfy the conditions (1.1), (1.3) and (1.6).

  * (i)

If $\ \lim_{p\rightarrow\infty}\frac{C_{p}^{1/\alpha}}{A_{p}}=0,$ then for
$1\leq k\leq\textup{dim}_{\mathbb{C}}X$

$\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

  * (ii)

If $\ \sum_{p=1}^{\infty}\frac{C_{p}^{2/\alpha}}{A_{p}^{2}}<\infty,$ then for
$\sigma_{\infty}^{k}-$almost every sequence
$\\{\Sigma_{p}^{k}\\}\in\mathcal{H}_{k}^{\infty}$

$\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

For codimension one, a method that circumvents the use of variance and
expected distribution, such as the approach employed by Marinescu, Coman, and
Bayraktar ([BCM]) in the setting where the first Chern forms diverge, is
noteworthy. However, this method is not applicable for codimensions greater
than one. Our approach with one summability condition generalizes their
results for equidistribution in various codimensions. One should also
emphasize that if the measure $\sigma_{p}$ satisfy the moment codition (1.6)
with constants $C_{p}=\Lambda$ independent of $p,$ then the assumption in (i),
$\lim_{p\rightarrow\infty}C_{p}^{1/\alpha}A_{p}^{-1}=0,$ is automatically
satisfied. Moreover, the hypothesis (ii) transforms into $\
\sum_{p=1}^{\infty}A_{p}^{-2}<\infty.$

At the same time, our principal result, which has been established for
codimension $k$, is applicable to a multitude of frequently investigated
probability measures. These include the area measure of spheres, Gaussian,
Fubini-Study measures, and measures with bounded density having logarithmic
decaying tails. When our main result is applied to these measures in the
context of codimension $k$ and tensor powers of a fixed prequantum line bundle
, the required summability assumption can be dropped. This reduction indicates
that our results are consistent with the existing literature in this
particular scenario. In [Bay16], for homogeneous projective manifolds,
Bayraktar obtained, in a weighted pluripotential theory setting, an
equidistribution result by employing Kolmogorov’s law of large numbers in
synthesizing the variance estimation with the expected distribution. In doing
so, he utilizes the properties of positive closed currents with super-
potentials ([DS09] and [DS10]), as introduced and studied by Sibony-Dinh in
complex dynamics, to demonstrate that the limit of the average sequence of
zero currents associated with their super-potentials is, in fact, identical to
the limit of the sequence of these zero currents itself. Our methods extend
the results of Zelditch-Shiffman. Bayraktar’s results can also be regarded as
generalizations of Shiffman-Zelditch’s if one assumes the projective manifolds
considered in [SZ99] are homogeneous. Moreover, in [Bay16], when we take the
locally regular compact set $K$ to be the whole manifold $X$ and $q=0$, our
results in this paper generalize his results in terms of probability
distributions and sequences of line bundles. In the same paper, he posed the
question whether equidistribution result (Theorem 1.1 of [Bay16]) holds for
any projective manifold. We also answer this question affirmatively by Theorem
1.2.

##  2\. Background

Let $(X,\omega)$ be a compact Kähler manifold with $\dim_{\mathbb{C}}{X}=n$.
On this manifold, a holomorphic line bundle $L$ is defined by compiling
complex lines $\\{L_{x}\\}_{x\in X}$ and constructing a complex manifold of
dimension $1+\dim_{\mathbb{C}}{X}$ with a projection map $\pi:L\rightarrow X$
such that $\pi$ assigns each line (or fiber) $L_{x}$ to $x$ and is
holomorphic. Using an open cover $\\{U_{\alpha}\\}$ of $X$, we can locally
trivialize $L$ through biholomorphisms
$\Psi_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow U_{\alpha}\times\mathbb{C}$
which map $L_{x}=\pi^{-1}(x)$ isomorphically onto $\\{x\\}\times\mathbb{C}$.
The line bundle $L$ is then uniquely (i.e., up to isomorphism) determined by
these transition functions $g_{\alpha\beta}$, which are non-vanishing
holomorphic functions on $U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}$ defined
by
$g_{\alpha\beta}=\Psi_{\alpha}\circ\Psi^{-1}_{\beta}|_{\\{x\\}\times\mathbb{C}}$.
These functions $g_{\alpha\beta}$ satisfy the cocycle condition
$g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=1$.

The cocycle condition for the transition functions yields that they define a
cohomology class, denoted as $[g_{\alpha\beta}]\in H^{1}(X,\mathcal{O}^{*})$.
Here, $H^{1}(X,\mathcal{O}^{*})$ is the first sheaf cohomology group of the
manifold $X$ with coefficients in the sheaf of non-zero holomorphic functions,
denoted by $\mathcal{O}^{*}$. The exponential short exact sequence
$0\rightarrow\mathbb{Z}\rightarrow\mathcal{O}\rightarrow\mathcal{O}^{*}\rightarrow
0$ produces a mapping $c_{1}:H^{1}(X,\mathcal{O}^{*})\rightarrow
H^{2}(X,\mathbb{Z})$ and the first Chern class $c_{1}(L,h)$ is defined by the
image of $[g_{\alpha\beta}]$ under this mapping.

Let $\\{U_{\alpha}\\}$ be an open covering of $X$ and $e_{\alpha}$ be
holomorphic frames of $L$ on each $U_{\alpha}$. Then the metric is given by a
family of functions
$h_{\alpha}=|e_{\alpha}|^{2}_{h}:U_{\alpha}\rightarrow[0,\infty]$ with local
weights
$\varphi_{\alpha}=-\frac{1}{2}\log{h_{\alpha}}\in\mathscr{C}^{2}(U_{\alpha})$.
Given the transition functions
$g_{\alpha\beta}=\frac{e_{\beta}}{e_{\alpha}}\in\mathcal{O}^{*}_{X}(U_{\alpha}\cap
U_{\beta})$ of $L$, then on $U_{\alpha}\cap U_{\beta},$ we have that
$h_{\beta}=|g_{\alpha\beta}|^{2}h_{\alpha}$ and therefore
$\log{h_{\beta}}=\log{|g_{\alpha\beta}|^{2}}+\log{h_{\alpha}},$ which is
equivalent to the following

$\varphi_{\alpha}=\varphi_{\beta}+\log{|g_{\alpha\beta}|}\,\,\text{in}\,\,\mathscr{C}^{2}(U_{\alpha}\cap
U_{\beta}).$

In this paper, the notion of positivity we employ will be what is known in the
literature as Nakano and Griffiths positivity for vector bundles. These are
different positivities on vector bundles and Nakano positivity always implies
Griffiths positivity, but the converse is not necessarily true; for details,
refer to the source [Dem12]. However, for line bundles, these two positivity
concepts coincide. Therefore, the positivity in this context is defined as
follows: A holomorphic line bundle $(L,h),$ with a Hermitian metric (may not
be $\mathscr{C}^{\infty}$) is said to be positive, semi-positive if the local
$\mathscr{C}^{2}$ weight functions $\varphi$ corresponding to the metric $h$
are strictly plurisubharmonic and plurisubharmonic, respectively.

Since we will be dealing with positive line bundles, the local weight
functions $\varphi_{\alpha}$ above are strictly plurisubharmonic, which means
that, in addition to being of the class $\mathscr{C}^{2}$,
$dd^{c}\varphi_{\alpha}>0$ holds pointwisely. By a well-known characterization
of ampleness (see, e.g., Corollary 13.3 of [Dem12]), line bundles we consider
will be ample, which also implies, by Kodaira’s projectivity criterion, that
any compact Kähler manifold with a positive line bundle is projective.

The following fact (see section 4 of [BCM]) will be required in the next
section: There exists a constant $b>0$ such that for every
$\phi\in\mathcal{D}^{n-1,n-1}(X)$,

(2.1) $-b\,\|\phi\|_{\mathscr{C}^{2}}\,\omega^{n}\leq dd^{c}\phi\leq
b\,\|\phi\|_{\mathscr{C}^{2}}\,\omega^{n},$

which amounts to saying that the total variation of $dd^{c}\phi$ verifies the
inequality $|dd^{c}\phi|\leq b\,\|\phi\|_{\mathscr{C}^{2}}\,\omega^{n}$.

In our local analysis throughout this paper, we always consider trivializing
open neighborhoods of given point $x\in X$. Let $U_{p}$ be such a trivializing
neighborhood of $x,$ and let $\\{S_{1}^{p},\ldots,S^{p}_{d_{p}}\\}$ be an
orthonormal basis for $H^{0}(X,L_{p})$. For a local holomorphic frame $e_{p}$
of $L_{p}$ in $U_{p}$, we have $S^{p}_{j}=s^{p}_{j}e_{p},$ where $s^{p}_{j}$
is a holomorphic function in $U_{p}.$ Then the Bergman kernel functions and
the Fubini-Study currents are defined as follows:

(2.2)
$K_{p}(x)=\sum^{d_{p}}_{j=1}{|S^{p}_{j}(x)|^{2}_{h_{p}}}\,\,\,\text{and}\,\,\,\gamma_{p}|_{U_{p}}=\frac{1}{2}dd^{c}\log\big{(}\sum^{d_{p}}_{j=1}{|s^{p}_{j}|^{2}}\big{)},$

where $d=\partial+\overline{\partial},$ and $d^{c}=\frac{1}{2\pi
i}(\partial-\overline{\partial})$. It is important to emphasize that $K_{p}$
and $\gamma_{p}$ are independent of the chosen basis
$\\{{S_{1}^{p},\ldots,S^{p}_{d_{p}}}\\}$, (see [CM1], section 3). Moreover,
$\gamma_{p}$ is a positive closed current of bidegree $(1,1),$ smooth away
from the base locus $BsH^{0}(X,L_{p})$ of $H^{0}(X,L_{p}),$ and by (2.2) we
have that

(2.3) $\log{K_{p}}\in
L^{1}(X,\omega^{n})\,\,\,\text{and}\,\,\,\gamma_{p}-c_{1}(L_{p},h_{p})=\frac{1}{2}dd^{c}\log{K_{p}}.$

which amounts to say that $\gamma_{p}$ has the same cohomology class as
$c_{1}(L_{p},h_{p}).$ If $\Phi_{p}:X\dashrightarrow\mathbb{P}^{d_{p}-1}$ is
the Kodaira map defined by the basis $\\{S_{j}^{p}\\}_{j=1}^{d_{p}},$ that is

$\Phi_{p}(x)=[S_{1}^{p}(x):\ldots:S_{d_{p}}^{p}(x)]\,\,\,\,\text{for }x\in
U_{p}.$

Then we have $\gamma_{p}=\Phi_{p}^{*}(\omega_{FS}),$ which justifies their
name. Before finishing this section, let us give an important tool also known
as Poincaré-Lelong formula(see [MM1]): Given $s\in H^{0}(X,L_{p}),$ we have

(2.4) $[Z_{s}]=c_{1}(L_{p},h_{p})+dd^{c}\log{|s|_{h_{p}}}.$

##  3\. Variance Estimate

In this section, we will establish the proof of Theorem 1.1 using an inductive
approach based on the codimension $k$. We will start with the case of
codimension $1$, serving as the initial step in our induction process. To
begin, let us make some initial observations.

Let $s_{p}\in H^{0}(X,L_{p})$. We can write it as

$s_{p}=\sum_{j=1}^{d_{p}}{a_{j}^{p}S^{p}_{j}}=\langle a,\Gamma_{p}\rangle,$

where
$\Gamma_{p}=(S_{1}^{p},\ldots,S^{p}_{d_{p}}),\,a=(a_{1}^{p},\ldots,a_{d_{p}}^{p})\in\mathbb{C}^{d_{p}}$
and $\\{S^{p}_{j}\\}_{j=1}^{d_{p}}$ is an orthonormal basis of
$H^{0}(X,L_{p})$.

Let $x\in X$. $U\subseteq X$ be an open trivializing neighborhood of $x$ and
$e_{p}$ be a holomorphic frame of $L_{p}$ in $U$. Then locally
$S^{p}_{j}=f_{j}e_{p}$, where $f_{j}$ are holomorphic functions in $U$ and so,
by writing $f=(f_{1},\ldots,f_{d_{p}}),$

$s_{p}=\sum_{j=1}^{d_{p}}{a_{j}^{p}f_{j}e_{p}}=\langle a,f\rangle e_{p}.$

By Poincaré-Lelong formula (2.4), on the neighborhood $U$, we have

$[Z_{s_{p}}]=dd^{c}\log{|\langle a,f\rangle|}=dd^{c}\log|\langle
a,\Gamma_{p}\rangle|_{h_{p}}+c_{1}(L_{p},h_{p}).$

Now, for any $\phi\in\mathcal{D}^{n-1,n-1}(X)$, we define the following random
variable

(3.1) $W_{s_{p}}:=[Z_{s_{p}}]-c_{1}(L_{p},h_{p})=dd^{c}\log{|\langle
a,\Gamma_{p}\rangle|_{h_{p}}}.$

By a basic feature of the variance,

(3.2) $\mathrm{Var}\langle[Z_{s_{p}}],\phi\rangle=\mathrm{Var}\langle
W_{s_{p}},\phi\rangle.$

Therefore, in the light of (3.2) it is enough to estimate $\mathrm{Var}\langle
W_{s_{p}},\phi\rangle$. Employing certain methods from [SZ99] and [SZ08] in
our setting, we have the following theorem for codimension one.

######  Theorem 3.1.

Under the hypotheses of Theorem 1.1, if $s_{p}\in H^{0}(X,L_{p})$, then for
all $p\geq 1$ and any $\phi\in\mathcal{D}^{n-1,n-1}(X)$, the following
variance estimate holds true

$\langle\mathrm{Var}[\widehat{Z_{s_{p}}}],\phi\rangle=\frac{(C_{p})^{2/\alpha}}{A^{2}_{p}}(B_{\phi}\mathrm{Vol}(X))^{2},$

where $B_{\phi}$ is a constant depending on the form $\phi$.

###### Proof.

First we have, by the relation (3.1),

(3.3) $\mathbb{E}\langle
W_{s_{p}},\,\phi\rangle^{2}=\int_{H^{0}(X,L_{p})}\int_{X}\int_{X}{\log{|\langle
a,\Gamma_{p}(x)\rangle|_{h_{p}}}\log{|\langle
a,\Gamma_{p}(y)\rangle|_{h_{p}}}dd^{c}\phi(x)dd^{c}\phi(y)d\sigma_{p}(s)}.$

Writing

$|\Gamma_{p}(x)|_{h_{p}}:=\Big{(}\sum_{j=1}^{d_{p}}{|S^{p}_{j}(x)|^{2}_{h_{p}}}\Big{)}^{1/2}=\sqrt{K_{p}(x)}$

gives $\Gamma_{p}(x)=|\Gamma_{p}(x)|_{h_{p}}u_{p}(x)$ (so that
$|u_{p}|_{h_{p}}=1$) and we insert $\Gamma_{p}=|\Gamma_{p}|_{h_{p}}u_{p}$ into
the integrand in (3.3), which breaks it into four terms:

(3.4)
$\displaystyle\log{|\Gamma_{p}(x)|_{h_{p}}}\log{|\Gamma_{p}(y)|_{h_{p}}}+\log{|\Gamma_{p}(x)|_{h_{p}}}\log{|\langle
a,u_{p}(y)\rangle|_{h_{p}}}+\log{|\Gamma_{p}(y)|_{h_{p}}}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}$ (3.5) $\displaystyle+\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}\log{|\langle a,u_{p}(y)\rangle|_{h_{p}}}.$

Before continuing with the variance estimate, we will see that
$\mathbb{E}\langle W_{s_{p}},\,\phi\rangle$ is bounded. To do this, we
establish an auxiliary inequality to begin with. First by the relation (2.1),
we have

(3.6) $\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)}\leq
b\,\|\phi\|_{\mathscr{C}^{2}}\int_{X}{\log{\Big{(}|\Gamma_{p}(x)|_{h_{p}}\Big{)}}\,\omega^{n}(x)}<\infty$

since $\log{K_{p}}\in L^{1}(X,\omega^{n})$ for every $p\geq 1$. Now, it is
evident that

$\displaystyle\big{|}\mathbb{E}\langle W_{s_{p}},\phi\rangle|$
$\displaystyle=|\int_{H^{0}(X,L_{p})}\int_{X}{(\log{|\Gamma_{p}(x)|_{h_{p}}}+\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}})}dd^{c}\phi(x)d\sigma_{p}(s_{p})\big{|}$
$\displaystyle\leq\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})+\int_{H^{0}(X,L_{p})}\int_{X}{\big{|}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}\big{|}}}\big{|}dd^{c}\phi(x)\big{|}d\sigma_{p}(s_{p})$

The first integral has an upper bound by (3.6) and $\sigma_{p}$ being a
probability measure on $H^{0}(X,L_{p})$. The second double integral is also
bounded from above. To see this, by the identification
$H^{0}(X,L_{p})\simeq\mathbb{C}^{d_{p}}$ and the moment condition (1.6)
combined with Hölder’s inequality and the relation (2.1), we get

(3.7) $\int_{X}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}}d\sigma_{p}(a)dd^{c}\phi(x)\leq(C_{p})^{\frac{1}{\alpha}}\,b\,\|\phi\|_{\mathcal{C}^{2}}Vol(X),$

where

$\rho_{p}(x)=\Big{(}\frac{f_{1}(x)}{\sqrt{\sum_{j=1}^{d_{p}}{|f_{j}(x)|^{2}}}},\ldots,\frac{f_{d_{p}}(x)}{\sqrt{\sum_{j=1}^{d_{p}}{|f_{j}(x)|^{2}}}}\Big{)}.$

It follows from Fubini-Tonelli’s theorem that

(3.8) $\int_{H^{0}(X,L_{p})}\int_{X}{\big{|}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}\big{|}}\big{|}dd^{c}\phi(x)\big{|}d\sigma_{p}(s_{p})=\int_{X}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}}d\sigma_{p}(a)\big{|}dd^{c}\phi(x)\big{|}.$

Hence, we are done.

We now return to the variance estimate of $W_{s_{p}}$. In order to do this, we
also expand the second term $\langle\mathbb{E}[W_{s_{p}}],\,\phi\rangle^{2}$
of the variance by using the expression above for expected distribution :

$\mathbb{E}\langle
W_{s_{p}},\phi\rangle=\int_{H^{0}(X,L_{p})}\int_{X}{(\log{|\Gamma_{p}(x)|_{h_{p}}}+\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}})}dd^{c}\phi(x)d\sigma_{p}(s_{p}),$

which gives the following

$\big{(}\mathbb{E}\langle W_{s_{p}},\phi\rangle)^{2}=J_{1}+2J_{2}+J_{3},$

where

(3.9)
$J_{1}=\big{(}\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})}\big{)}^{2}$
(3.10)
$J_{2}=\big{(}\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})}\big{)}\big{(}\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})\big{)}$

and lastly,

(3.11) $J_{3}=\big{(}\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})}\big{)}^{2}$

Note that all of the integrals $J_{1},J_{2}$ and $J_{3}$ are finite since
$\mathbb{E}\langle W_{s_{p}},\phi\rangle$ is bounded.

According to the four terms given in (3.4) and (3.5), we have
$\mathbb{E}\langle W_{s_{p}},\,\phi\rangle^{2}=B_{1}+2B_{2}+B_{3}$, where

(3.12)
$B_{1}:=\int_{H^{0}(X,L_{p})}\int_{X}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}\log{|\Gamma_{p}(y)|_{h_{p}}}dd^{c}\phi(x)dd^{c}\phi(y)d\sigma_{p}(s)},$

(3.13)
$B_{2}:=\int_{H^{0}(X,L_{p})}\int_{X}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}\log{|\langle
a,u_{p}(y)\rangle|_{h_{p}}}dd^{c}\phi(x)dd^{c}\phi(y)d\sigma_{p}(s)}.$

(The second term $\log{|\Gamma_{p}(x)|_{h_{p}}}\log{|\langle
a,u_{p}(y)\rangle|_{h_{p}}}$ and the third one
$\log{|\Gamma_{p}(y)|_{h_{p}}}\log{|\langle a^{p},u_{p}(x)\rangle|_{h_{p}}}$
in (3.4) are actually the integrands that yield the same result) and finally,

(3.14)
$B_{3}:=\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)}\int_{\mathbb{C}^{d_{p}}}{\log{|\langle
a,\rho_{p}(x)\rangle|}\,\log{|\langle a,\rho_{p}(y)\rangle|}d\sigma_{p}(a)}$

From (3.6), the moment assumption (1.6) and the Fubini-Tonelli’s theorem, we
see that $B_{1},B_{2}$ and $B_{3}$ are all finite, moreover, we have that
$B_{1}=J_{1}$ and $B_{2}=J_{2}$. Therefore, the only integrals that survive
are $J_{3}$ and $B_{3}$, which are not always equal to each other, so we
obtain

(3.15) $\mathrm{Var}\langle W_{s_{p}},\phi\rangle=B_{3}-J_{3},$

so it will suffice to estimate the term $B_{3}$ from above to complete the
variance estimation. To this end, by Tonelli’s theorem and Hölder’s inequality
with $\frac{1}{\alpha}+\frac{1}{\beta}=1$, where $\alpha\geq 2$ is the
constant satisfying the moment condition (1.6), we obtain

$\displaystyle B_{3}$
$\displaystyle\leq\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}\,\big{|}\log{|\langle
a,\rho_{p}(y)\rangle|}\big{|}d\sigma_{p}(a)}$
$\displaystyle\leq\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)}{\Big{\\{}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}^{\alpha}d\sigma_{p}(a)}\Big{\\}}^{\frac{1}{\alpha}}\,\Big{\\{}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(y)\rangle|}\big{|}^{\beta}d\sigma_{p}(a)}\Big{\\}}^{\frac{1}{\beta}}}$
$\displaystyle\leq\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)(C_{p})^{\frac{1}{\alpha}}}\Big{\\{}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(y)\rangle|}\big{|}^{\beta}}d\sigma_{p}(a)\Big{\\}}^{\frac{1}{\beta}}$

If we apply again Hölder’s inequality to the innermost integral in the last
line (since $\alpha\geq 2\geq\beta$ allows us to do so), we get

(3.16)
$B_{3}\leq\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)(C_{p})^{\frac{2}{\alpha}}}.$

Consequently, using the total variation inequality (2.1) of $dd^{c}$ twice in
(3.16) concludes the following

(3.17)
$B_{3}\leq(C_{p})^{2/\alpha}b^{2}\|\phi\|^{2}_{\mathscr{C}^{2}}\mathrm{Vol}(X)^{2},$

which, by normalization, gives the desired variance estimate. ∎

Before proceeding to the proof of our main theorems, we present a lemma
concerning cohomology classes of integration currents that will be
instrumental in the sequel. This lemma has been previously proven in [SZ08](as
part of proposition 2.2), and for the sake of reader, we include its proof
here.

######  Lemma 3.2.

Let $(X,\omega)$ be a compact Kähler manifold with the fixed Kähler form
$\omega.$ If $Z_{s_{p}^{1}},\ldots,Z_{s_{p}^{k}}$ are smooth and intersect
transversally, we have

(3.18)
$\left\langle\big{[}Z_{\Sigma_{p}^{k}}\big{]},\omega^{n-k}\right\rangle=\int_{X}c_{1}(L_{p},h_{p})^{k}\wedge\omega^{n-k}$

###### Proof.

For $k=1,$ this is just a consequence of Poincaré-Lelong formula and the fact
that $\omega$ is a closed form. Indeed,

$\left\langle\big{[}Z_{s_{p}}\big{]},\omega^{n-1}\right\rangle=\int_{X}c_{1}(L_{p},h_{p})\wedge\omega^{n-1}+\int_{X}dd^{c}\log|s_{p}|_{h_{p}}\wedge\omega^{n-1}=\int_{X}c_{1}(L_{p},h_{p})\wedge\omega^{n-1}.$

Let us now suppose that the assertion (3.18) is true for $k-1$ sections
$\Sigma_{p}^{k-1}=(s_{p}^{2},\ldots,s_{p}^{k}).$ Then by the induction
hypothesis and the base step of induction, it yields that

$\displaystyle\left\langle\big{[}Z_{s_{p}^{1}}\cap
Z_{\Sigma_{p}^{k-1}}\big{]},\omega^{n-k}\right\rangle$
$\displaystyle=\int_{Z_{s_{p}^{1}}}c_{1}(L_{p},h_{p})^{k-1}\wedge\omega^{n-k}$
$\displaystyle=\int_{X}c_{1}(L_{p},h_{p})\wedge
c_{1}(L_{p},h_{p})^{k-1}\wedge\omega^{n-k}=\int_{X}c_{1}(L_{p},h_{p})^{k}\wedge\omega^{n-k}$

Finally, based on the observation that
$\left\langle\big{[}Z_{\Sigma_{p}^{k}}\big{]},\omega^{n-k}\right\rangle=\left\langle\big{[}Z_{s_{p}^{1}}\cap
Z_{\Sigma_{p}^{k-1}}\big{]},\omega^{n-k}\right\rangle$, where
$\Sigma_{p}^{k}=\big{(}s_{p}^{1},\Sigma_{p}^{k-1}\big{)}$, we complete the
proof. ∎

Next we proceed with the proof of the variance estimate in higher
codimensions. We adapt some of the methods in [Shif] into our setting.

###### Proof of Theorem 1.1.

Theorem 1.1 provides the case $k=1$ of induction on the codimension $k$. Now
let $n\geq k\geq 2$ and we suppose that the variance estimate of Theorem (1.1)
is true for $k-1$ sections. We pick a system of $k$ independent random
holomorphic sections $\Sigma_{p}^{k}=(s^{1}_{p},\ldots,s^{k}_{p})\in
H^{0}(X,L_{p})^{k}$ and we write
$\Sigma_{p}^{k}=(\Sigma_{p}^{k-1},s^{k}_{p})$, where
$\Sigma_{p}^{k-1}=(s^{1}_{p},\ldots,s_{p}^{k-1})$ and
$Z_{\Sigma_{p}^{k}}=Z_{\Sigma_{p}^{k-1}}\wedge Z_{s^{k}_{p}}.$ Let
$\phi\in\mathcal{D}^{n-k,n-k}(X)$ be a test form. Since
$\mathbb{E}[Z_{\Sigma_{p}^{k}}]=\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\wedge\mathbb{E}[Z_{s^{k}_{p}}]$
in view of the independence of the random holomorphic sections
$s^{1}_{p},\ldots,s^{k}_{p}$, we first have

(3.19) $\displaystyle\mathrm{Var}\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle$
$\displaystyle=\mathbb{E}\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle^{2}-(\mathbb{E}\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle)^{2}$
(3.20)
$\displaystyle=\mathbb{E}\langle[Z_{\Sigma_{p}^{k-1}}]\wedge[Z_{s^{k}_{p}}],\phi\rangle^{2}-(\langle\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\rangle)^{2}.$

We shall use the following

$\langle\big{\\{}[Z_{\Sigma_{p}^{k-1}}]\wedge[Z_{s^{k}_{p}}]\big{\\}},\phi\rangle^{2}-\langle\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\rangle^{2}=I_{1}+I_{2}$

where

$\displaystyle I_{1}$ $\displaystyle=I_{1}(\Sigma_{p}^{k-1},s^{k}_{p})$
$\displaystyle=\langle[Z_{\Sigma_{p}^{k-1}}]\wedge[Z_{s^{k}_{p}}],\phi\rangle^{2}-\langle[Z_{\Sigma_{p}^{k-1}}]\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\rangle^{2}$

and

$I_{2}:=I_{2}(\Sigma_{p}^{k-1})=\langle[Z_{\Sigma_{p}^{k-1}}]\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\rangle^{2}-(\langle\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\rangle)^{2}$

for generic choice of $\Sigma_{p}^{k-1}$ and $s^{k}_{p}.$

We observe that

(3.21)
$\mathrm{Var}\langle[Z_{\Sigma_{p}^{k}}],\phi\rangle=\mathbb{E}[I_{1}]+\mathbb{E}[I_{2}].$

Now let $Y:=\big{\\{}x\in X:\Sigma_{p}^{k-1}(x)=0\big{\\}}$. Notice that
$\langle[Z_{\Sigma_{p}^{k-1}}]\wedge[Z_{s^{k}_{p}}],\phi\rangle=\langle[Z_{s^{k}_{p}}]\big{|}_{Y},\phi|_{Y}\rangle$.
Initially, we will find an estimation on $\mathbb{E}[I_{1}]$. To do so, the
first step is to integrate $I_{1}$ over $H^{0}(X,L_{p})$ then use the
observation above and apply Theorem 3.1:

$\displaystyle\int_{H^{0}(X,L_{p})}{I_{1}(\Sigma_{p}^{k-1},s^{k}_{p})d\sigma_{p}(s^{k}_{p})}$
$\displaystyle=\int_{H^{0}(X,L_{p})}{\langle[Z_{s^{k}_{p}}]\big{|}_{Y},\phi|_{Y}\rangle^{2}-(\langle\mathbb{E}\langle[Z_{s^{k}_{p}}]\big{|}_{Y},\phi|_{Y}\rangle})^{2}$
$\displaystyle=\mathrm{Var}\langle[Z_{s^{k}_{p}}]\,\big{|}_{Y},\phi|_{Y}\rangle$
$\displaystyle\leq(C_{p})^{\frac{2}{\alpha}}\Big{(}B_{\phi}\big{|}_{Y}\int_{Y}{\omega^{n-k+1}}\Big{)}^{2}$
$\displaystyle\leq(C_{p})^{\frac{2}{\alpha}}\Big{(}B_{\phi}\int_{Y}{\omega^{n-k+1}}\wedge
c_{1}(L_{p},h_{p})^{k-1}\Big{)}^{2}$
$\displaystyle\leq(C_{p})^{2/\alpha}\,\big{(}B_{\phi}\,Vol(X)(2A_{p})^{k-1}\big{)}^{2},$

where the last inequality is obtained by using the fact that
$\int_{X}{\omega^{n-k+1}\wedge
c_{1}(L_{p},h_{p})^{k-1}}\leq(2A_{p})^{k-1}Vol(X)$ for $p\geq p_{1},$ which is
a simple consequence of the diophantine approximation condition (1.1).

Now integrating over $H^{0}(X,L_{p})^{k-1}$ and taking the last inequality
just above into account it yields that

$\displaystyle\mathbb{E}[I_{1}]$
$\displaystyle=\int_{H^{0}(X,L_{p})^{k-1}}\int_{H^{0}(X,L_{p})}{I_{1}(\Sigma_{p}^{k-1},s^{k}_{p})d\sigma_{p}(s^{k}_{p})d\sigma^{k-1}_{p}(\Sigma_{p}^{k-1})}\leq(C_{p})^{2/\alpha}\,\big{(}B_{\phi}\,Vol(X)(2A_{p})^{k-1}\big{)}^{2},$

since $\sigma^{k-1}_{p}$ is a product probability measure on
$H^{0}(X,L_{p})^{k-1}$. This finishes the estimation of $\mathbb{E}[I_{1}]$.

In order to get the upper bound for $\mathbb{E}[I_{2}]$, first observe that

(3.22)
$\mathbb{E}[I_{2}]=\mathbb{E}\big{\langle}\big{(}[Z_{\Sigma_{p}^{k-1}}]-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\big{)}\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\big{\rangle}^{2}.$

Also we see that

$\displaystyle\big{\langle}\big{(}[Z_{\Sigma_{p}^{k-1}}]-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\big{)}\wedge\mathbb{E}[Z_{s^{k}_{p}}],\phi\big{\rangle}^{2}$
$\displaystyle=\Big{\\{}\int_{H^{0}(X,L_{p})}{\big{\langle}\big{(}[Z_{\Sigma_{p}^{k-1}}]-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}]\big{)}\wedge[Z_{s^{k}_{p}}],\phi\big{\rangle}d\sigma_{p}(s^{k}_{p})}\Big{\\}}^{2}$
$\displaystyle\leq\int_{H^{0}(X,L_{p})}{\Big{\\{}\big{\langle}([Z_{\Sigma_{p}^{k-1}}]-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}])\wedge[Z_{s^{k}_{p}}],\phi\big{\rangle}\Big{\\}}^{2}d\sigma_{p}(s^{k}_{p})},$

where, in the second line, we have used Cauchy-Schwarz inequality. Analogous
to the case for $\mathbb{E}[I_{1}]$, this time we consider the zero set of
single $s^{k}_{p}$, namely, the set $Y:=\big{\\{}x\in
X:s^{k}_{p}(x)=0\big{\\}}$. We first get

(3.23)
$\mathbb{E}[I_{2}]\leq\int_{H^{0}(X,L_{p})^{k-1}}\int_{H^{0}(X,L_{p})}\Big{\\{}\big{\langle}([Z_{\Sigma_{p}^{k-1}}]-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}])\wedge[Z_{s^{k}_{p}}],\phi\big{\rangle}\Big{\\}}^{2}d\sigma_{p}(s^{k}_{p})d\sigma^{k-1}_{p}(\Sigma_{p}^{k-1}).$

As has been argued for $\mathbb{E}[I_{1}]$ above, since
$\langle(Z_{\Sigma_{p}^{k-1}}-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}])\wedge
Z_{s^{k}_{p}},\phi\rangle=\langle(Z_{\Sigma_{p}^{k-1}}-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}])\big{|}_{Y},\phi\big{|}_{Y}\rangle$,
by invoking Fubini-Tonelli’s theorem, (3.23) becomes

(3.24)
$\mathbb{E}[I_{2}]\leq\int_{H^{0}(X,L_{p})}\int_{H^{0}(X,L_{p})^{k-1}}{\Big{\\{}\big{\langle}([Z_{\Sigma_{p}^{k-1}}]-\mathbb{E}[Z_{\Sigma_{p}^{k-1}}])\big{|}_{Y},\phi\big{|}_{Y}\big{\rangle}\Big{\\}}^{2}d\sigma^{k-1}_{p}(\Sigma_{p}^{k-1})d\sigma_{p}(s^{k}_{p})}.$

The inner integral is, by definition, the variance of
$Z_{\Sigma_{p}^{k-1}}\big{|}_{Y}$, so (3.24) takes the following form:

(3.25)
$\mathbb{E}[I_{2}]\leq\int_{H^{0}(X,L_{p})}{\mathrm{Var}\big{\langle}[Z_{\Sigma_{p}^{k-1}}]\big{|}_{Y},\phi|_{Y}\big{\rangle}d\sigma_{p}(s^{k}_{p})}.$

By using the induction hypothesis we have

$\displaystyle\mathrm{Var}\big{\langle}[Z_{\Sigma_{p}^{k-1}}]\big{|}_{Y},\phi|_{Y}\big{\rangle}$
$\displaystyle\leq(C_{p})^{2/\alpha}\,\Big{(}B_{\phi}\big{|}_{Y}(2A_{p})^{k-2}\int_{Y}\omega^{n-1}\Big{)}^{2}$
$\displaystyle\leq(C_{p})^{2/\alpha}\,\Big{(}B_{\phi}\
(2A_{p})^{k-2}\int_{X}\omega^{n-1}\wedge c_{1}(L_{p},h_{p})\Big{)}^{2}$
$\displaystyle\leq(C_{p})^{2/\alpha}\,\big{(}B_{\phi}\,Vol(X)(2A_{p})^{k-1}\big{)}^{2},$

where the last inequality is obtained by using the fact that
$\int_{X}{\omega^{n-1}\wedge c_{1}(L_{p},h_{p})}\leq 2A_{p}Vol(X)$ for $p\geq
p_{2},$ which is a simple consequence of the diophantine relation (1.1). Now
integrating over $H^{0}(X,L_{p})$ and taking the last inequality just above
into account implies that

$\displaystyle\mathbb{E}[I_{2}]\leq(C_{p})^{2/\alpha}\,\big{(}B_{\phi}\,Vol(X)(2A_{p})^{k-1}\big{)}^{2}$

since $\sigma^{1}_{p}$ is a product probability measure on $H^{0}(X,L_{p}).$
Lastly, using the relation (3.21) and subsequently applying normalization ends
the proof of Theorem 1.1.

∎

##  4\. Equidistribution of Zeros of Random Sections

In this section, we provide the proof for Theorem 1.1. We begin by first
establishing the asymptotic behavior of the expected zero distribution.
Subsequently, by utilizing the variance estimate from the previous section in
conjunction with the expected distribution, we prove that, subject to one
summability condition, the zero sets have the self-averaging property.

Let $\Sigma_{p}^{k}=(s_{p}^{1},s_{p}^{2},\cdots,s_{p}^{k})$ be a $k-$system of
i.i.d. random holomorphic sections $s_{p}^{j}\in H^{0}(X,L_{p})$ for
$j=1,2,\cdots,k\,$ where $1\leq k\leq\textup{dim}_{\mathbb{C}}X$. Our aim is
to investigate the distribution of the simultaneous zero locus, namely

$Z_{\Sigma_{p}^{k}}:=\\{x\in X:s_{p}^{1}(x)=\cdots=s_{p}^{k}(x)=0\\}$

Consider the probability spaces

$(\mathcal{H}_{p}^{k},\sigma_{p}^{k})=\Big{(}\prod_{j=1}^{k}H^{0}(X,L_{p}),\,\prod_{j=1}^{k}\sigma_{p}\Big{)}\,\,\,\,\text{and}\,\,\,\,(\mathcal{H}_{\infty}^{k},\sigma_{\infty}^{k})=\Big{(}\prod_{p=1}^{\infty}\mathcal{H}_{p}^{k},\,\prod_{p=1}^{\infty}\sigma_{p}^{k}\Big{)}.$

These spaces consist of random, i.i.d. $k$-systems of holomorphic sections of
$L_{p}$ and sequences of $k$-systems of holomorphic sections of $L_{p}$ with
increasing $p,$ respectively. Now since the base locus
$Bs(H^{0}(X,L_{p}))=\emptyset$ for $p\geq p_{0}$ (by (1.3)), and our
probability measure does not charge the pluripolar subsets, Bertini’s Theorem
(see Appendix) implies that with probability one the zero sets $Z_{s_{p}^{j}}$
are smooth and intersect transversally. In particular, for
$\sigma_{p}^{k}-$almost every system $\Sigma_{p}^{k}\in\mathcal{H}_{p}^{k}$,
the zero set $Z_{\Sigma_{p}^{k}}$ is a complex submanifold of codimension $k$
and is achieved as a complete intersection of the zero loci of individual
sections. Moreover, almost surely the current of integration over
$Z_{\Sigma_{p}^{k}}$ is given by

$[Z_{\Sigma_{p}^{k}}]=[Z_{s_{p}^{1}}]\wedge\cdots\wedge[Z_{s_{p}^{k}}]$

###  4.1. Expected Distribution of Zeros

In this part, we embark on the proof of the first assertion in Theorem 1.1. To
achieve this, we will employ an inductive argument. First, we will establish
expected distribution for the case of codimension one, and subsequently extend
our proof to cover higher codimensions.

######  Theorem 4.1.

Let $(L_{p},h_{p})_{p\geq 1}$, $(X,\omega)$ and $\sigma_{p}$ be as defined
above. Assume that they satisfy the conditions (1.1), (1.3) and (1.6) . If $\
\lim_{p\rightarrow\infty}\frac{C_{p}^{1/\alpha}}{A_{p}}=0,$ then

(4.1) $\frac{1}{A_{p}}\,\mathbb{E}[Z_{s_{p}}]\longrightarrow\omega$

in the weak* topology of currents as $p\rightarrow\infty.$

###### Proof.

Let $s_{p}\in H^{0}(X,L_{p}),$ then

$s_{p}=\sum_{j=1}^{d_{p}}{a_{j}^{p}S^{p}_{j}}=\langle a,\Gamma_{p}\rangle,$

where $\Gamma_{p}=(S_{1}^{p},\ldots,S^{p}_{d_{p}}),\
\text{and}\,a=(a_{1},\ldots,a_{d_{p}})\in\mathbb{C}^{d_{p}}$. Let $x\in X$,
$U\subseteq X$ be an open neighborhood of $x$ and $e_{p}$ be a holomorphic
frame of $L_{p}$ in $U$. Then locally $S^{p}_{j}=f_{j}e_{p}$, where $f_{j}$
are holomorphic functions in $U$ and so, by writing
$f=(f_{1},\ldots,f_{d_{p}}),$ we have

$s_{p}=\sum_{j=1}^{d_{p}}{a_{j}^{p}f_{j}e_{p}}=\langle a,f\rangle e_{p}.$

By Poincaré-Lelong formula (2.4), on the neighborhood $U$, we have

(4.2) $[Z_{s_{p}}]=dd^{c}\log{|\langle a,f\rangle|}=dd^{c}\log|\langle
a,\Gamma_{p}\rangle|_{h_{p}}+c_{1}(L_{p},h_{p}).$

Let us now fix $\phi\in\mathcal{D}^{n-1,n-1}(X),$ without lost of generality
we may assume that $supp(\phi)\subset U$ as the general case follows by
covering $supp(\phi)$ by $U_{\alpha}$’s and using the compatibility
conditions. Using the definition of expectation and incorporating (4.2) along
with the observation that $c_{1}(L_{p},h_{p})$ is independent of $s_{p}$ we
have

(4.3)
$\frac{1}{A_{p}}\langle\mathbb{E}[Z_{s_{p}}],\phi\rangle=\frac{1}{A_{p}}\langle
c_{1}(L_{p},h_{p}),\phi\rangle+\frac{1}{A_{p}}\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\langle
a,\Gamma_{p}(x)\rangle|_{h_{p}}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})$

Let us denote the second term above by $I_{p}$, then by exploiting the fact
that $\Gamma_{p}(x)=|\Gamma_{p}(x)|_{h_{p}}u_{p}(x)$ (so that
$|u_{p}|_{h_{p}}=1$) one has

(4.4) $\displaystyle I_{p}$
$\displaystyle\leq\frac{1}{A_{p}}\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)d\sigma_{p}(s_{p})+\frac{1}{A_{p}}\int_{H^{0}(X,L_{p})}\int_{X}{\big{|}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}\big{|}}}\big{|}dd^{c}\phi(x)\big{|}\big{|}d\sigma_{p}(s_{p})$

Utilizng (2.1) and the fact that $\frac{1}{A_{p}}\log K_{p}(x)\rightarrow 0$
as $p\rightarrow\infty,$ in $L^{1}(X,\omega)$ we obtain the following

(4.5)
$\frac{1}{A_{p}}\int_{H^{0}(X,L_{p})}\int_{X}\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)d\sigma_{p}(s)\leq\frac{B_{\phi}}{2A_{p}}\int_{X}\log
K_{p}(x)\omega^{n}(x)\rightarrow 0$

as $p\rightarrow\infty.$ Additionaly, using the identification
$H^{0}(X,L_{p})\simeq\mathbb{C}^{d_{p}}$, the moment condition (1.6) along
with Hölder’s inequality and the relation (2.1), we get

(4.6) $\int_{X}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}}d\sigma_{p}(a)dd^{c}\phi(x)\leq(C_{p})^{\frac{1}{\alpha}}B_{\phi}\,Vol(X),$

where

$\rho_{p}(x)=\Big{(}\frac{f_{1}(x)}{\sqrt{\sum_{j=1}^{d_{p}}{|f_{j}(x)|^{2}}}},\ldots,\frac{f_{d_{p}}(x)}{\sqrt{\sum_{j=1}^{d_{p}}{|f_{j}(x)|^{2}}}}\Big{)}.$

It follows from Fubini-Tonelli’s theorem that

$\int_{H^{0}(X,L_{p})}\int_{X}{\big{|}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}\big{|}}\big{|}dd^{c}\phi(x)\big{|}d\sigma_{p}(s_{p})=\int_{X}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}}d\sigma_{p}(a)\big{|}dd^{c}\phi(x)\big{|}.$

Consequently, employing the given hypothesis, we deduce that

(4.7) $\frac{1}{A_{p}}\int_{H^{0}(X,L_{p})}\int_{X}{\big{|}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}\big{|}}\big{|}dd^{c}\phi(x)\big{|}d\sigma_{p}(s_{p})\leq\frac{C_{p}^{1/{\alpha}}}{A_{p}}\,B_{\phi}\,Vol(X)\rightarrow
0,$

as $p\rightarrow\infty$. In turn, (4.5) and (4.7) imply that $I_{p}\rightarrow
0,$ as $p\rightarrow\infty.$ Finally, by using (1.1) we obtain that
$\frac{1}{A_{p}}\langle
c_{1}(L_{p},h_{p}),\phi\rangle\rightarrow\langle\omega,\phi\rangle,$ thus
concluding the proof.

∎

######  Theorem 4.2.

Let $(L_{p},h_{p})_{p\geq 1}$, $(X,\omega)$ and $\sigma_{p}$ be as defined
above. Assume that they satisfy the conditions (1.1), (1.3) and (1.6). Then
for $1\leq k\leq\textup{dim}_{\mathbb{C}}X$

$\mathbb{E}\big{[}Z_{\Sigma_{p}^{k}}\big{]}=\mathbb{E}\big{[}Z_{s_{p}^{1}}\big{]}\wedge\cdots\wedge\mathbb{E}\big{[}Z_{s_{p}^{k}}\big{]}$

Moreover, if $\ \lim_{p\rightarrow\infty}\frac{C_{p}^{1/\alpha}}{A_{p}}=0,$
then

$\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

###### Proof.

Let $\phi\in\mathcal{D}^{n-k,n-k}(X),$ then

(4.8)
$\big{|}\big{\langle}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\big{\rangle}\big{|}=\Big{|}\frac{1}{A_{p}^{k}}\int_{Z_{\Sigma_{p}^{k}}}\phi\,\Big{|}\leq\frac{1}{A_{p}^{k}}\sup\left\|\phi\right\|\mathrm{Vol}(Z_{\Sigma_{p}^{k}})$

Using Lemma 3.2 above, and the prequantization condition (1.1)

(4.9)
$\mathrm{Vol}(Z_{\Sigma_{p}^{k}})=\big{\langle}\big{[}Z_{\Sigma_{p}^{k}}\big{]},\omega^{n-k}\big{\rangle}=\int_{X}c_{1}(L_{p},h_{p})^{k}\wedge\omega^{n-k}\leq(2A_{p})^{k}\,\mathrm{Vol}(X)$

for all $p\geq p_{1}.$ Thus, combining both of the inequalities above, we
obtain that for $p\geq p_{1},$

(4.10)
$\big{|}\big{\langle}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\big{\rangle}\big{|}\leq
2^{k}\ \mathrm{Vol}(X)\sup\left\|\phi\right\|<\infty$

which means that
$\big{\langle}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\big{\rangle}$ is
bounded for almost all $\Sigma_{p}^{k}.$ Consequently,
$\mathbb{E}[\widehat{Z_{\Sigma_{p}^{k}}}]$ is a well-defined current of
bidegree $(k,k)$.

Let us now prove the first assertion. To accomplish this, we induct on the
codimension $k.$ The base step, $k=1$ is obvious. Suppose that the claim holds
for $k-1$ sections, say $\Sigma_{p}^{k-1}=(s_{p}^{2},\ldots,s_{p}^{k}).$ Fix
$s_{p}^{1}\in H^{0}(X,L_{p})$ so that $Y:=Z_{s_{p}^{1}}$ is a complex
submanifold. For almost all $s_{p}^{j},$ we write
$(s_{p}^{j})^{\prime}=s_{p}^{j}|_{Y}$ and
$H^{0}(X,L_{p})^{\prime}:=H^{0}(X,L_{p})\big{|}_{Y},$ which in turn for almost
all $\Sigma_{p}^{k}=(s_{p}^{1},\ldots,s_{p}^{k})\in\mathcal{H}_{p}^{k}$ give
rise to the notation $(\Sigma_{p}^{k})^{\prime}=\Sigma_{p}^{k}\big{|}_{Y},$
and $(\mathcal{H}_{p}^{k})^{\prime}=\mathcal{H}_{p}^{k}\big{|}_{Y},$ where
$\Sigma_{p}^{k}\big{|}_{Y}=(s_{p}^{1}|_{Y},\ldots,s_{p}^{k}|_{Y}),$ and
$\mathcal{H}_{p}^{k}=H^{0}(X,L_{p})^{k}$ respectively. We endow
$H^{0}(X,L_{p})^{\prime}$ with the push-forward probability measure
$\sigma_{p}^{\prime}:=\theta_{*}\sigma_{p},$ where
$\theta:H^{0}(X,L_{p})\rightarrow H^{0}(X,L_{p})^{\prime}$ is the restriction
map. The induced measure on $(\mathcal{H}_{p}^{k})^{\prime}$ will be denoted
by $(\sigma_{p}^{\prime})^{k}.$ By the independence and induction hypothesis
applied to $Y=Z_{s_{p}^{1}}$ and $(\mathcal{H}_{p}^{k-1})^{\prime}$, and
observation that $Z_{\Sigma_{p}^{k}}=Z_{(\Sigma_{p}^{k-1})^{\prime}}$ we have,
for $\phi\in\mathcal{D}^{n-k,n-k}(X),$

$\displaystyle\int_{\mathcal{H}_{p}^{k-1}}\big{\langle}\big{[}Z_{\Sigma_{p}^{k}}\big{]},\phi\big{\rangle}d\sigma_{p}^{k-1}(\Sigma_{p}^{k-1})$
$\displaystyle=\int_{(\mathcal{H}_{p}^{k-1})^{\prime}}\left\langle\big{[}Z_{(\Sigma_{p}^{k-1})^{\prime}}\big{]},\phi\big{|}_{Y}\right\rangle
d(\sigma_{p}^{\prime})^{k-1}((\Sigma_{p}^{k-1})^{\prime})$
$\displaystyle=\left\langle\mathbb{E}[Z_{(s_{p}^{2})^{\prime}}]\wedge\cdots\wedge\mathbb{E}[Z_{(s_{p}^{k})^{\prime}}],\phi\big{|}_{Y}\right\rangle$
$\displaystyle=\int_{Z_{s_{p}^{1}}}\mathbb{E}[Z_{s_{p}^{2}}]\wedge\cdots\wedge\mathbb{E}[Z_{s_{p}^{k}}]\wedge\phi$

Finally, integrating over all $s_{p}^{1}$ in the last expression, leads us to
the desired result.

We will now prove, using an inductive approach based on codimensions, that for
$\phi\in\mathcal{D}^{n-k,n-k}(X),$

(4.11)
$\Big{\langle}\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\Big{\rangle}=\left\langle\frac{1}{A_{p}^{k}}c_{1}(L_{p},h_{p})^{k},\phi\right\rangle+R_{X,p}(\phi)$

Here, $\big{|}R_{X,p}(\phi)\big{|}\leq
2^{n-1}\frac{\mathrm{Vol}(X)}{(2A_{p})^{\textup{codim}(X)}}B_{\phi}\big{[}(n+1)\frac{\log
A_{p}}{2A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}.$ We refer to $R_{X,p}$
as a remainder current. Similar to the previous part, by writing
$Y=Z_{s_{p}^{1}}$ and using the induction assumption on $Y,$ one finds, for
$\phi\in\mathcal{D}^{n-k,n-k}(X),$

$\displaystyle\int_{\mathcal{H}_{p}^{k-1}}\langle\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\rangle
d\sigma_{p}^{k-1}(\Sigma_{p}^{k-1})$
$\displaystyle=\langle\mathbb{E}\big{[}\widehat{Z_{(\Sigma_{p}^{k-1})^{\prime}}}\big{]},\phi\big{|}_{Y}\rangle$
$\displaystyle=\int_{Z_{s_{p}^{1}}}\frac{1}{A_{p}^{k-1}}\,c_{1}(L_{p},h_{p})^{k-1}\wedge\phi+R_{Y,p}(\phi)$
$\displaystyle=\Big{\langle}\big{[}Z_{s_{p}^{1}}\big{]},\frac{1}{A_{p}^{k-1}}\,c_{1}(L_{p},h_{p})^{k-1}\wedge\phi\Big{\rangle}+R_{Y,p}(\phi)$

where

(4.12) $|R_{Y,p}(\phi)|\leq 2^{n-1}B_{\phi}\big{|}_{Y}\big{[}(n+1)\frac{\log
A_{p}}{2A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}\int_{Y}\frac{\omega^{n-1}}{2A_{p}}$

Using Lemma 3.2 and the fact that
$\int_{X}\frac{c_{1}(L_{p},h_{p})}{2A_{p}}\wedge\omega^{n-1}\leq Vol(X)$ for
sufficiently large $p,$ we have

(4.13) $\displaystyle|R_{Y,p}(\phi)|$ $\displaystyle\leq
2^{n-1}B_{\phi}\big{[}(n+1)\frac{\log
A_{p}}{2A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}\int_{X}\frac{c_{1}(L_{p},h_{p})}{2A_{p}}\wedge\omega^{n-1}$
(4.14) $\displaystyle\leq
2^{n-1}\frac{Vol(X)}{(2A_{p})^{\textup{codim}(X)}}B_{\phi}\big{[}(n+1)\frac{\log
A_{p}}{2A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}.$

By taking the average over $s_{p}^{1}\in H^{0}(X,L_{p})$ and using the
information about codimension one, specifically

(4.15)
$\mathbb{E}[\widehat{Z_{s_{p}^{1}}}]=\frac{1}{A_{p}}c_{1}(L_{p},h_{p})+\frac{1}{2A_{p}}dd^{c}\log
K_{p}(x)+\frac{1}{A_{p}}dd^{c}\log\big{|}\langle
a,u_{p}(x)\rangle\big{|}_{h_{p}}$

one finds

$\displaystyle\left\langle\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\right\rangle$
$\displaystyle=\int_{\mathcal{H}_{p}^{1}}\int_{\mathcal{H}_{p}^{k-1}}\langle\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\rangle
d\sigma_{p}^{k-1}(\Sigma_{p}^{k-1})d\sigma_{p}^{1}(s_{p}^{1})$
$\displaystyle=\frac{1}{A_{p}}\int_{\mathcal{H}_{p}^{1}}\
\big{\langle}\big{[}Z_{s_{p}^{1}}\big{]},\frac{1}{A_{p}^{k-1}}\,c_{1}(L_{p},h_{p})^{k-1}\wedge\phi\big{\rangle}d\sigma_{p}^{1}(s_{p}^{1})+\int_{\mathcal{H}_{p}^{1}}R_{Y,p}(\phi)d\sigma_{p}^{1}(s_{p}^{1})$
$\displaystyle=\big{\langle}\mathbb{E}[\widehat{Z_{s_{p}^{1}}}],\frac{1}{A_{p}^{k-1}}\,c_{1}(L_{p},h_{p})^{k-1}\wedge\phi\big{\rangle}+\int_{\mathcal{H}_{p}^{1}}R_{Y,p}(\phi)d\sigma_{p}^{1}(s_{p}^{1})$
$\displaystyle=\langle\frac{1}{A_{p}^{k}}c_{1}(L_{p},h_{p})^{k},\phi\rangle+\int_{\mathcal{H}_{p}^{1}}\int_{X}\frac{1}{2A_{p}}\log
K_{p}(x)\frac{c_{1}(L_{p},h_{p})^{k-1}}{A_{p}^{k-1}}dd^{c}\phi(x)d\sigma_{p}(s_{p}^{1})$
$\displaystyle+\int_{\mathcal{H}_{p}^{1}}\int_{X}\frac{1}{A_{p}}\log\big{|}\langle
a,u_{p}(x)\rangle\big{|}_{h_{p}}\frac{c_{1}(L_{p},h_{p})^{k-1}}{A_{p}^{k-1}}dd^{c}\phi(x)d\sigma_{p}(s_{p}^{1})+\int_{\mathcal{H}_{p}^{1}}R_{Y,p}(\phi)d\sigma_{p}^{1}(s_{p}^{1})$

Choosing $p\in\mathbb{N}$ sufficiently large so that $A_{p}\geq M_{0}$ and the
assumption (1.3) is satisfied, we obtain that

$A_{p}^{n-1}\leq K_{p}(x)\leq A_{p}^{n+1}$

for all such $p.$ In turn, we have $\frac{1}{2A_{p}}\log
K_{p}(x)\leq(n+1)\frac{\log A_{p}}{2A_{p}}.$ Now, using the moment codition
(1.6), relation (1.1), total variation inequality (2.1), the previous estimate
for $\big{|}R_{Y,p}(\phi)\big{|},$ and the induction hypothesis

(4.16)
$\displaystyle\left\langle\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\right\rangle=\langle\frac{1}{A_{p}^{k}}c_{1}(L_{p},h_{p})^{k},\phi\rangle+2^{k-1}Vol(X)B_{\phi}\big{[}(n+1)\frac{\log
A_{p}}{A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}+R_{Y,p}(\phi)$

Thus we have

(4.17)
$\left\langle\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\right\rangle=\left\langle\frac{1}{A_{p}^{k}}c_{1}(L_{p},h_{p})^{k},\phi\right\rangle+R_{X,p}(\phi)$

where

$\big{|}R_{X,p}(\phi)\big{|}\leq\big{|}R_{Y,p}(\phi)\big{|}+2^{k-1}Vol(X)B_{\phi}\big{[}(n+1)\frac{\log
A_{p}}{A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}\leq
2^{n-1}Vol(X)B_{\phi}\big{[}(n+1)\frac{\log
A_{p}}{A_{p}}+\frac{C_{p}^{1/\alpha}}{A_{p}}\big{]}$

Finally, passing to the limit as $p\rightarrow\infty$ we obtain the result. ∎

###  4.2. Almost Sure Distribution of Zeros

In this subsection, we will delve into the proof of the second assertion from
Theorem 1.2, which deals with the almost sure behaviour of random zeros.

######  Theorem 4.3.

Let $(X,\omega)$ be a compact Kähler manifold of
$\textup{dim}_{\mathbb{C}}X=n$ and let $(L_{p},h_{p})_{p\geq 1},$be a sequence
of Hermitian holomorphic line bundles on $X$ with $\mathscr{C}^{2}$ metrics
$h_{p}$. Assume that the conditions (1.1) and (1.3) hold. If $\
\sum_{p=1}^{\infty}\frac{C_{p}^{2/\alpha}}{A_{p}^{2}}<\infty,$ then for
$\sigma_{\infty}^{k}-$ almost every sequence
$\mathbf{\Sigma_{k}}=\left\\{\Sigma_{p}^{k}\right\\}_{p\geq
1}\in\mathcal{H}_{\infty},$

$\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

###### Proof.

Fix $\phi\in\mathcal{D}^{n-k,n-k}(X),$ and pick
$\mathbf{\Sigma_{k}}=\left\\{\Sigma_{p}^{k}\right\\}_{p\geq
1}\in\mathcal{H}_{\infty}.$ Let us examine the non-negative current valued
random variables

(4.18)
$\mathcal{X}_{p}(\mathbf{\Sigma_{k}}):=\big{\langle}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}-\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\big{\rangle}^{2}\geq
0$

By appealing to the equivalent characterization of variance, notice that

(4.19)
$\int_{\mathcal{H}_{\infty}}\mathcal{X}_{p}(\mathbf{\Sigma_{k}})d\sigma_{\infty}^{k}(\mathbf{\Sigma_{k}})=\big{\langle}\mathrm{Var}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\,\phi\big{\rangle}$

Using Theorem 1.1 along with the summability condition given by the
hypothesis, we obtain

(4.20)
$\sum_{p=1}^{\infty}\int_{\mathcal{H}_{\infty}}\mathcal{X}_{p}(\mathbf{\Sigma_{k}})d\sigma_{\infty}^{k}(\mathbf{\Sigma_{k}})=\sum_{p=1}^{\infty}\big{\langle}\mathrm{Var}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\,\phi\big{\rangle}<\infty$

By the relation (4.19) above and invoking Beppo-Levi Therom from the standard
measure theory, we get

(4.21)
$\int_{\mathcal{H}_{\infty}}\sum_{p=1}^{\infty}\mathcal{X}_{p}(\mathbf{\Sigma_{k}})d\sigma_{\infty}^{k}(\mathbf{\Sigma_{k}})=\sum_{p=1}^{\infty}\big{\langle}\mathrm{Var}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\,\phi\big{\rangle}<\infty$

This implies that, for $\sigma_{\infty}^{k}-$ almost every sequence of
$k-$systems $\mathbf{\Sigma_{k}}\in\mathcal{H}_{\infty}$, the series
$\sum_{p=1}^{\infty}\mathcal{X}_{p}(\mathbf{\Sigma_{k}})$ converges, leading
to the conclusion that $\mathcal{X}_{p}\rightarrow 0$ $\sigma_{\infty}^{k}-$
almost surely. By definition (4.18) of random variables $\mathcal{X}_{p}$ this
also indicates that

(4.22)
$\big{\langle}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}-\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\big{\rangle}\rightarrow
0$

$\sigma_{\infty}^{k}-$almost surely. Combining this last information with
Theorem (4.2), we conclude that for $\sigma_{\infty}^{k}-$ almost every
sequence,

(4.23) $\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty$. ∎

##  5\. Some Special Cases

In [BCM], certain types of measures which satisfy the assumption (1.6) have
been investigated as special cases. We will now provide some insights
regarding a few of these measures in connection with Theorem 1.1 and 1.2. The
first two measures to be considered here will be the Gaussian and the Fubini-
Study measures, both of which are unitary invariant measures that come with
certain advantages in estimations.

###  5.1. Gaussian and Fubini-Study

In what follows, $\lambda_{n}$ represents the Lebesgue measure on
$\mathbb{C}^{n}$ (identified with $\mathbb{R}^{2n}$). We will present the
variance estimate simultaneously for both Gaussian and Fubini-Study cases,
with detailed explanations provided for the Gaussian case as the computations
are exactly the same. It turns out that, in these cases, the constants $C_{p}$
reduce to the ones independent of $p$ and Theorem 1.1 remains valid for every
$\alpha\geq 1$. The standard Gaussian measure is precisely defined as follows,
for $a=(a_{1},\ldots,a_{n})\in\mathbb{C}^{n},$

(5.1) $d\sigma_{n}(a)=\frac{1}{\pi^{n}}e^{-||a||^{2}}d\lambda_{n}(a),$

and the Fubini-Study measure $\mathbb{C}\mathbb{P}^{n}\supset\mathbb{C}^{n}$
is defined as as:

(5.2)
$d\sigma_{n}(a)=\frac{n!}{\pi^{n}}\frac{1}{(1+||a||^{2})^{n+1}}d\lambda_{n}(a).$

As for these two measures, we record two facts (Lemma 4.8, Lemma 4.10) from
[[BCM]]: Given that $\sigma_{n}$ is the Gaussian measure, for every integer
$n\geq 1$ and every $\alpha\geq 1$, we have

(5.3) $\int_{\mathbb{C}^{n}}{|\log{|\langle
a,v\rangle|}|}^{\alpha}d\sigma_{n}(a)=2\int_{0}^{\infty}{r|\log{r}|^{\alpha}e^{-r^{2}}dr},\,\,\,\,\forall
v\in\mathbb{C}^{n},\,\,\,||v||=1;$

if $\sigma_{n}$ is the Fubini-Study, then for every integer $n\geq 1$ and
every $\alpha\geq 1$

(5.4) $\int_{\mathbb{C}^{n}}{|\log{|\langle
a,v\rangle|}|}^{\alpha}d\sigma_{n}(a)=2\int_{0}^{\infty}{\frac{r|\log{r}|^{\alpha}}{(1+r^{2})^{2}}dr},\,\,\,\,\forall
v\in\mathbb{C}^{n},\,\,\,||v||=1.$

It is evident that they are indeed independent of the dimension $n$.

Let’s first show that

(5.5) $\mathbb{E}[W_{s_{p}}]=dd^{c}\log{|\Gamma_{p}|_{h_{p}}}.$

Since

(5.6) $\mathbb{E}\langle
W_{s_{p}},\phi\rangle=\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\Gamma_{p}(x)|_{h_{p}}}dd^{c}\phi(x)d\sigma_{d_{p}}(s)+\int_{H^{0}(X,L_{p})}\int_{X}\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}}dd^{c}\phi(x)d\sigma_{d_{p}}(s_{p}),$

we need to show that the second double integral is zero. By (4.1),

$\int_{H^{0}(X,L_{p})}\int_{X}{\log{|\langle
a,u_{p}(x)\rangle|_{h_{p}}}}dd^{c}\phi(x)d\sigma_{d_{p}}(s_{p})=\int_{X}\int_{\mathbb{C}^{d_{p}}}{\log{|\langle
a,\rho_{p}(x)\rangle|}}d\sigma_{d_{p}}(a)dd^{c}\phi(x).$

Hence, in the form of currents, this gives

$\langle dd^{c}\big{\\{}\int_{\mathbb{C}^{d_{p}}}{\log{|\langle
a,\rho_{p}(x)\rangle|}d\sigma_{d_{p}}(a)}\big{\\}},\phi\rangle=0$

since the integral acted by $dd^{c}$ is a constant independent of $x$ due to
the relation (5.3), which gives (5.5).

Now, the first term is $\mathbb{E}\langle W_{s_{p}},\phi\rangle^{2}$ owing to
the relation (5.5), which cancels out the second term of the variance. Also,
similar to the above reasoning, we see that the integrals of the second and
third terms become zero by using the fact (5.3). Therefore we estimate only
the fourth term, which results in the variance of $W_{s_{p}}$, that is

(5.7) $\mathrm{Var}\langle
W_{s_{p}},\phi\rangle=\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)}\int_{\mathbb{C}^{d_{p}}}{\log{|\langle
a,\rho_{p}(x)\rangle|}\,\log{|\langle
a,\rho_{p}(y)\rangle|}d\sigma_{d_{p}}(a)}.$

By Cauchy-Schwarz inequality, the relations (5.3) and (2.1) , we get

$\displaystyle\mathrm{Var}\langle W_{s_{p}},\phi\rangle$
$\displaystyle\leq\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}\,\big{|}\log{|\langle
a,\rho_{p}(y)\rangle|}\big{|}d\sigma_{d_{p}}(a)}$
$\displaystyle\leq\int_{X}\int_{X}{dd^{c}\phi(y)dd^{c}\phi(x)}{\Big{\\{}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(x)\rangle|}\big{|}^{2}d\sigma_{d_{p}}(a)}\Big{\\}}^{\frac{1}{2}}\,\Big{\\{}\int_{\mathbb{C}^{d_{p}}}{\big{|}\log{|\langle
a,\rho_{p}(y)\rangle|}\big{|}^{2}d\sigma_{d_{p}}(a)}\Big{\\}}^{\frac{1}{2}}}$
$\displaystyle\leq\Lambda\|\phi\|^{2}_{\mathscr{C}^{2}}\,b^{2}\,\mathrm{Vol}^{2}(X),$

where $\Lambda:=2\int_{0}^{\infty}{r|\log{r}|^{2}e^{-r^{2}}}dr$, which we have
obtained the Gaussian (Fubini-Study) version of Theorem 3.1. Thus, by carrying
out the proof of this theorem in the same way, what we have is the next
theorem:

######  Theorem 5.1.

Under the same assumptions of Theorem 1.1, let $\sigma^{k}_{d_{p}}$ be the
product Gaussian (Fubini-Study) measure on $H^{0}(X,L_{p})^{k}$ given by (5.1)
(by (5.2)). Then for any $\phi\in\mathcal{D}^{n-k,n-k}(X)$, one gets

$\mathrm{Var}\big{\langle}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\,\phi\big{\rangle}\leq\frac{1}{A^{2}_{p}}\Lambda_{k}\,B^{2}_{\phi}\,\mathrm{Vol}^{2}(X),$

where $\Lambda_{k}=2^{k-1}\int_{0}^{\infty}{r|\log{r}|^{\alpha}e^{-r^{2}}dr}$
($\Lambda_{k}=2^{k-1}\int_{0}^{\infty}{\frac{r|\log{r}|^{\alpha}}{(1+r^{2})^{2}}dr}$).

We infer from Theorem 4.2 and (5.5) the following theorem

######  Theorem 5.2.

With the same assumptions of Theorem 1.2 , let $\sigma_{p}$ be the Gaussian
(Fubini-Study) measure on $H^{0}(X,L_{p})\simeq\mathbb{C}^{d_{p}}$ given by
(5.1) (by 5.2). Then, for $1\leq k\leq\textup{dim}_{\mathbb{C}}X$

(5.8)
$\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$ In addition, if
$\sum_{p=1}^{\infty}\frac{1}{A^{2}_{p}}<\infty,$ then for
$\sigma^{k}_{\infty}-$almost every sequence
$\\{\Sigma_{p}^{k}\\}\in\mathcal{H}^{k}_{\infty}$ we have

(5.9) $\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

When we consider the prequantum line bundle setting, where
$(L_{p},h_{p})=(L^{p},h^{p})$ and $c_{1}(L,h)=\omega\,$ in Theorem 5.1 and
Theorem 5.2, we recover the results of Shiffman-Zelditch ([SZ99], [SZ08]).

###  5.2. Area Measure of Spheres

Let $\mathcal{A}_{n}$ be the surface area measure on the unit sphere
$S^{2n-1}$ in $\mathbb{C}^{n},$ given by
$\mathcal{A}_{n}=\frac{2\pi^{n}}{(n-1)!}.$ Let us consider the following
probability measure on $S^{2n-1}$

(5.10) $\sigma_{n}=\frac{1}{\mathcal{A}_{n}(S^{2n-1})}\mathcal{A}_{n}$

Given that $\sigma_{n}$ is the normalized area measure on the unit sphere , in
accordance with Lemma 4.11 from[BCM], for every $\alpha\geq 1$, there exists a
constant $C_{\alpha}>0$ such that for every integer $n\geq 2,$ we have:

(5.11) $\int_{\mathbb{C}^{n}}{|\log{|\langle
a,v\rangle|}|}^{\alpha}d\sigma_{n}(a)\leq C_{\alpha}\,(\log
n)^{\alpha},\,\,\,\,\forall v\in\mathbb{C}^{n},\,\,\,||v||=1;$

One should remark that, in this specific case even though the measure is
unitary invariant , the aforementioned upper bound is not a universal
constant.

Now, due to the fact that $C_{\alpha}\,(\log d_{p})^{\alpha}\leq
C_{\alpha}\,((n+2)\log A_{p})^{\alpha}$ for sufficiently large $p$, utilizing
Theorem 1.1 with $C_{p}=C_{\alpha}\,((n+2)\log A_{p})^{\alpha}$ leads to the
following variance estimate

######  Theorem 5.3.

Under the same assumptions of Theorem 1.1, let $\sigma_{p}:=\sigma_{d_{p}}$ be
the normalized area measure on the unit sphere of
$H^{0}(X,L_{p})\simeq\mathbb{C}^{d_{p}}$ given by (5.10). Then for any
$\phi\in\mathcal{D}^{n-k,n-k}(X)$ and sufficiently large $p$, one has

$\left\langle\mathrm{Var}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\right\rangle\leq\Big{(}\frac{\log{A_{p}}}{A_{p}}\Big{)}^{2}\Lambda_{k,n,\alpha}B_{\phi}^{2}\,$

where $\Lambda_{k,n,\alpha}=(2^{k-1}(n+2)\mathrm{Vol}(X)\
C_{\alpha}^{1/\alpha})^{2}$ is a positive constant.

Consequently, we have

######  Theorem 5.4.

Under the same assumptions of Theorem 1.2 , let $\sigma_{p}$ be the normalized
area measure on the unit sphere of $H^{0}(X,L_{p})\simeq\mathbb{C}^{d_{p}}$
given by (5.10). Then, for $1\leq k\leq\textup{dim}_{\mathbb{C}}X$

(5.12)
$\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$ In addition, if
$\sum_{p=1}^{\infty}\big{(}\frac{\log A_{p}}{A_{p}}\big{)}^{2}<\infty,$ then
for $\sigma_{\infty}^{k}-$almost every sequence $\\{\Sigma_{p}^{k}\\}_{p\geq
1}\in\mathcal{H}_{k}^{\infty}$ we have

(5.13) $\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

###  5.3. Random Holomorphic Sections with i.i.d. coefficients

In this context, we examine the probability space
$(H^{0}(X,L_{p}),\sigma_{p})$ where $\sigma_{p}$ is the product probability
measure induced by the probability distribution law $\mathbb{P}$ governing the
i.i.d. random coefficients $a_{j}^{p}$ in the representation (1.5). This
distribution possesses a bounded density $\psi:\mathbb{C}\rightarrow[0,M],$
and satisfies the property that there exist constants
$\epsilon>0\,\text{and}\,\delta>1,$ such that

(5.14)
$\mathbb{P}\big{(}\\{z\in\mathbb{C}:\log|z|>R\\}\big{)}\leq\frac{\epsilon}{R^{\delta}},\,\,\text{for
all }R\geq 1.$

This particular density type has been investigated in [Bay16] and [BCM], and
it encompasses distributions such as the real or complex Gaussian
distributions. Given such a measure $\sigma_{p}$ on $H^{0}(X,L_{p})$,
according to Lemma 4.15 of [BCM] we have the following for any
$1\leq\alpha\leq\delta:$

(5.15) $\int_{\mathbb{C}^{d_{p}}}{|\log{|\langle
a,v\rangle|}|}^{\alpha}d\sigma_{p}(a)\leq
B\,d_{p}^{\alpha/\delta},\,\,\,\,\forall
v\in\mathbb{C}^{d_{p}},\,\,\,||v||=1;$

where $B=B(M,\epsilon,\delta,\alpha)>0.$ In our present setting, for $p$
sufficiently large, $d_{p}\leq M_{0}Vol(X)A_{p}^{n}.$ Employing Theorem 1.1
with $C_{p}=DA_{p}^{n\frac{\alpha}{\delta}},$ where
$D=B\,(M_{0}Vol(X))^{\alpha/\delta}$ we obtain

######  Theorem 5.5.

Under the same assumptions of Theorem 1.1, if $\sigma_{p}$ is the probability
measure on $H^{0}(X,L_{p})\simeq\mathbb{C}^{d_{p}}$ defined as above. Then for
any $\phi\in\mathcal{D}^{n-k,n-k}(X)$ and sufficiently large $p$, one has

$\displaystyle\left\langle\mathrm{Var}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\right\rangle\leq\Big{(}\frac{1}{A_{p}^{1-n/\delta}}\Big{)}^{2}\big{(}2^{k-1}D^{1/\alpha}\,\mathrm{Vol}(X)B_{\phi}\big{)}^{2}$

where $D=(M_{0}\mathrm{Vol}(X))^{\alpha/\delta}B$ is a positive constant.

As a consequence, we have the following equidistribution result

######  Theorem 5.6.

Let $(L_{p},h_{p})_{p\geq 1}$, $(X,\omega)$ be as in Theorem 1.2 . Assume that
$\sigma_{p}$ is the probability measure on $H^{0}(X,L_{p})$ defined as above.
If $\delta>n$ then for $1\leq k\leq\textup{dim}_{\mathbb{C}}X$

(5.16)
$\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$ In addition, if
$\sum_{p=1}^{\infty}\frac{1}{A_{p}^{2-2n/\delta}}<\infty,$ where $\delta>2n$,
then almost surely

(5.17) $\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

###  5.4. Locally moderate measures

Consider a complex manifold $X$ and a positive measure $\sigma$ on $X$. In
accordance with [DNS], we define $\sigma$ as a locally moderate measure if,
for any open set $U\subset X$, any compact set $K\subset U$, and any compact
family $\mathscr{F}$ of plurisubharmonic functions on $U$, there exist
positive constants $M$ and $\beta$ such that

(5.18) $\int_{K}e^{-\beta\varphi}d\sigma\leq M,\ \ \text{for all
}\varphi\in\mathscr{F}.$

It is evident that $\sigma$ does not charge pluripolar sets. Furthermore, it
is noteworthy to mention that significant examples of such measures arise from
the Monge-Ampère measures associated with Hölder continuous plurisubharmonic
functions, for more details in this direction check [DNS]. According to ([BCM]
Lemma 4.16), if $\sigma_{p}$ is a locally moderate probability measure with
compact support in $\mathbb{C}^{d_{p}}\cong H^{0}(X,L_{p}),$ then for every
$\alpha\geq 1$

(5.19) $\int_{\mathbb{C}^{d_{p}}}{|\log{|\langle
a,v\rangle|}|}^{\alpha}d\sigma_{p}(a)\leq\Lambda_{p}R_{p}^{2\beta_{p}},\,\,\,\,\forall
v\in\mathbb{C}^{d_{p}},\,\,\,||v||=1;$

where $\Lambda_{p},\beta_{p}>0$ are positive constants and $R_{p}\geq 1$ such
that $\|a\|\leq R_{p}$ for all $a\in\mathrm{supp}\ \sigma_{p}.$ Continuing in
the same manner as the previous examples, we deduce the following results.

######  Theorem 5.7.

Under the same assumptions of Theorem 1.1, if $\sigma_{p}$ is a locally
moderate probability measure with compact support in $\mathbb{C}^{d_{p}}\cong
H^{0}(X,L_{p})$. Then for any $\phi\in\mathcal{D}^{n-k,n-k}(X)$ and
sufficiently large $p$, one has

$\displaystyle\left\langle\mathrm{Var}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]},\phi\right\rangle\leq\frac{(\Lambda_{p}R_{p}^{2\beta_{p}})^{2/\alpha}}{A_{p}^{2}}\,(2^{k-1}\mathrm{Vol}(X)B_{\phi}\,)^{2}$

where $\Lambda_{p},\beta_{p}>0$ are positive constants and $R_{p}\geq 1$ .

######  Theorem 5.8.

Let $(L_{p},h_{p})_{p\geq 1}$, $(X,\omega)$ be as in Theorem 1.2 . Assume that
$\sigma_{p}$ is the locally moderate probability measure on $H^{0}(X,L_{p})$
defined as above.

  * (i)

If $\
\lim_{p\rightarrow\infty}\frac{(\Lambda_{p}R_{p}^{2\beta_{p}})^{1/\alpha}}{A_{p}}=0$
then for $1\leq k\leq\textup{dim}_{\mathbb{C}}X$

$\mathbb{E}\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

  * (ii)

If $\
\sum_{p=1}^{\infty}\frac{(\Lambda_{p}R_{p}^{2\beta_{p}})^{2/\alpha}}{A_{p}^{2}}<\infty,$
then for $\sigma^{k}_{\infty}$-almost all $\\{\Sigma_{p}^{k}\\}_{p\geq
1}\in\mathcal{H}_{k}^{\infty}$

$\big{[}\widehat{Z_{\Sigma_{p}^{k}}}\big{]}\longrightarrow\omega^{k}$

in the weak* topology of currents as $p\rightarrow\infty.$

In [BCM], various significant probability measures are thoroughly examined,
including small ball probability measures, among others. For a more in-depth
exploration of such measures, refer to [Section 4, [BCM]].

##  6\. appendix

In this section, by adapting the proof of Proposition 3.2 in [CMM], we present
a general version of Bertini’s theorem applicable to any probability measure
$\sigma_{d}$ defined on $\mathbb{C}^{d}$ that assigns zero mass to pluripolar
subsets of $\mathbb{C}^{d}$. As a result, we demonstrate that the intersection
current is almost surely well-defined with respect to the product measure
induced by $\sigma_{d}$.

######  Definition 6.1.

The analytic subsets $A_{1},\ldots,A_{k},\ k\leq n$ of a compact complex
manifold $X$ of dimension $n$ are said to be in general poisition if
$\mathrm{codim}A_{i_{1}}\cap\ldots\cap A_{i_{m}}\geq m$ for every $1\leq m\leq
k$ and $1\leq i_{1}<\ldots<i_{m}\leq k.$

######  Proposition 6.2.

Let $L\to X$ be a holomorphic line bundle over a compact complex manifold $X$
with $\dim_{\mathbb{C}}{X}=n$. Suppose the following:

  1. (i)

$V$ is a subspace of $H^{0}(X,L)$ with a basis $\\{S_{1},\ldots,S_{d}\\}$, and
the base locus $\text{Bs}(V)=\\{x\in X:S_{1}(x)=\ldots=S_{d}(x)=0\\}\subseteq
X$ such that $\dim{\text{Bs}(V)}\leq n-k$.

  2. (ii)

$Z(t)=\\{x\in X:\sum_{j=1}^{d}t_{j}S_{j}(x)=0\\}$, where
$t=(t_{1},\ldots,t_{d})\in\mathbb{C}^{d}$.

If $\sigma_{d}^{k}$ is the product measure on $(\mathbb{C}^{d})^{k}$ induced
by the probability measure $\sigma_{d}$ on $\mathbb{C}^{d}$ coming from the
identification $V\simeq\mathbb{C}^{d}$, then analytic sets
$Z(t^{1}),\ldots,Z(t^{k})$ are in general position for $\sigma_{d}^{k}-$
almost every $(t^{1},\ldots,t^{k})\in(\mathbb{C}^{d})^{k}$.

###### Proof.

The proof will be based on induction on $k$. Let

(6.1)
$H_{k}:=\\{(t^{1},\ldots,t^{k})\in(\mathbb{C}^{d})^{k}:\dim{Z(t^{1})\cap\ldots\cap
Z(t^{k})\cap Bs(V)}\leq n-k\\}.$

We start with the case $k=1$. If $Z(t^{1})\cap Bs(V)=\emptyset$, then
$Z(t^{1})$ is a complex submanifold of dimension $n-1$ whatever
$t^{1}\in\mathbb{C}^{d}$ is chosen, that is
$\\{t^{1}\in\mathbb{C}^{d}:\dim{Z(t^{1})}\leq n-1\\}=\mathbb{C}^{d}$. If
$Z(t^{1})\cap Bs(V)\neq\emptyset$, corresponding to the set
$H_{1}=\\{t^{1}\in\mathbb{C}^{d}:\dim{Z(t^{1})\cap Bs(V)}\leq n-1\\}$, we
consider the decomposition of the base locus

(6.2) $Bs(V)=\bigcup_{l=1}^{N_{0}}{E_{l}\cup Y},$

where $E_{l}$ are the irreducible components of $Bs(V)$ with $\dim{E_{l}}=n-1$
and $\dim{Y}\leq n-2$. The set $\\{t^{1}\in\mathbb{C}^{d}:E_{l}\subset
Z(t^{1})\\}$ has to be a proper algebraic subvariety of $\mathbb{C}^{d}$,
because, if $E_{l}\subset Z(t^{1})$ for all $t^{1}\in\mathbb{C}^{d}$, then
since $E_{l}\subset Bs(V)$, we must have that
$\dim{Z(t^{1})}\geq\dim{Z(t^{1})\cap Bs(V)}\geq n-1$, which is a contradiction
since any single analytic variety is always in general position. Therefore,
for any point $t^{1}\in\mathbb{C}^{d}\backslash H_{1}$, we get
$\dim{Z(t^{1})\cap Bs(V)}\geq n-1$. Since $Z(t^{1})\cap Bs(V)$ is an analytic
subvariety of $Bs(V)$, there exists $l\in\\{1,\ldots,N_{0}\\}$ such that
$E_{l}\subset Z(t^{1})\cap Bs(V)$, so we can write

$\mathbb{C}^{d}\backslash
H_{1}=\bigcup_{l=1}^{N_{0}}{\\{t^{1}\in\mathbb{C}^{d}:E_{l}\subset
Z(t^{1})\\}}.$

Since $\sigma_{d}$ puts no mass on pluripolar sets, it follows that
$\sigma_{d}(\mathbb{C}^{d}\backslash H_{1})=0$, which completes the initial
step for induction. Suppose that $\sigma^{k}_{d}(H_{k})=1$ for all $H_{k}$
defined as in (6.1). Let

(6.3)
$H_{k+1}=\\{(t^{1},\ldots,t^{k+1})\in(\mathbb{C}^{d})^{k+1}:\dim{Z(t^{1})\cap\ldots\cap
Z(t^{k+1})\cap Bs(V)}\leq n-k-1\\}.$

We need to show that $\sigma^{k+1}_{d}(H_{k+1})=1$. For this purpose, we show
that the $\sigma^{k+1}_{d}$-measure of the complement set $H^{c}_{k+1}$ is
zero. First, let us fix $t=(t^{1},\ldots,t^{k})\in H_{k}$. Define
$Z(t):=Z(t^{1})\cap\ldots\cap Z(t^{k})$ and

(6.4) $G(t):=\\{t^{k+1}\in\mathbb{C}^{d}:\dim{Z(t)\cap Bs(V)\cap
Z(t^{k+1})}\geq n-k-1\\}.$

It will suffice to prove that $\sigma_{d}(G(t))=0$. These sets $G(t)$ are
called the slices of the set $H^{c}_{k+1}$. Let

(6.5) $Z(t)\cap Bs(V)=\bigcup_{k=1}^{N_{0}}{E_{l}\cup Y},$

where $E_{l}$ are, as in the case $k=1$, the irreducible components of
$Z(t)\cap Bs(V)$ with $\dim{E_{l}}=n-k$ and $\dim{Y}=n-k-1$. If $t^{k+1}\in
G(t)$, then $Z(t)\cap Z(t_{k+1})\cap Bs(V)$ is an analytic subset of $Z(t)\cap
Bs(V)$ with $\dim{Z(t)\cap Z(t_{k+1})\cap Bs(V)}=n-k$, and this gives that
there is some $l\in\\{1,\ldots N_{0}\\}$ such that

(6.6) $E_{l}\subset Z(t)\cap Z(t_{k+1})\cap Bs(V).$

Hence we have

(6.7)
$G(t)=\bigcup_{l=1}^{N_{0}}{A_{l}(t)},\,\,A_{l}(t):=\\{t^{k+1}\in\mathbb{C}^{d}:E_{l}\subset
Z(t^{k+1})\\}.$

Now we see that not all sections become zero on $E_{l}$. Indeed, if it were
not so, by arguing as in the case $k=1$ above, $E_{l}\subset Bs(V)$ would
imply that $\dim{Z(t)\cap Bs(V)}\geq n-k$, contradicting $t\in H_{k}$. We may
then assume that $S_{d}\not\equiv 0$ on $E_{l}$. As before, by examining, this
time, the slices of $A_{l}(t)$, for any
$(t^{k+1}_{1},\ldots,t^{k+1}_{d-1})\in\mathbb{C}^{d-1}$, there exist at most
one $h\in\mathbb{C}$ such that $(t^{k+1}_{1},\ldots,t^{k+1}_{d-1},h)\in
A_{l}(t)$, otherwise, if there exist two different elements
$h,h^{\prime}\in\mathbb{C}$ with this property, we have

$\displaystyle t^{k+1}_{1}S_{1}+\ldots+t^{k+1}_{d-1}S_{d-1}+hS_{d}=0$
$\displaystyle
t^{k+1}_{1}S_{1}+\ldots+t^{k+1}_{d-1}S_{d-1}+h^{\prime}S_{d}=0,$

which immediately gives that $S_{d}\equiv 0$ on $E_{l}$, which is a
contradiction. Thus, $\sigma_{d}(A_{l}(t))=0$. This implies that
$\sigma_{d}(G(t))=0$, which finishes the proof. ∎

In our setting, because of the relation (1.3), there exists
$p_{0}\in\mathbb{N}$ such that $Bs(H^{0}(X,L_{p}))=\emptyset$ for all
$p>p_{0}$. Therefore, by taking $V=H^{0}(X,L_{p})\cong\mathbb{C}^{d_{p}}$ with
the measure $\sigma_{p}:=\sigma_{d_{p}},$ fixing an orthonormal basis
$\\{S^{p}_{1},\ldots,S^{p}_{d_{p}}\\}$ and using the arguments from
Proposition 3.3 in [CLMM] based on the results of Demailly (Corollary 2.11 and
Proposition 2.12 in [Dem93]), as a result of Proposition 6.2, we arrive at the
following proposition

######  Proposition 6.3.

There exists $p_{0}\in\mathbb{N}$ such that for all $p>p_{0}$,

  * (i)

The analytic subvarietes $Z_{s^{1}_{p}},\ldots,Z_{s^{k}_{p}}$ are all in
general position for $\sigma^{k}_{p}$-almost all
$(s^{1}_{p},\ldots,s^{k}_{p})\in H^{0}(X,L_{p})^{k}$.

  * (ii)

For $\sigma^{k}_{p}$-almost every
$\Sigma^{k}_{p}=(s^{1}_{p},\ldots,s^{k}_{p})\in H^{0}(X,L_{p})^{k}$, the
analytic subvariety $Z_{s^{1}_{p}}\cap\ldots\cap Z_{s^{k}_{p}}$ is of pure
dimension $n-k$ for any $k\in\\{1,\ldots,n\\}$.

  * (iii)

The intersection current
$[Z_{\Sigma^{k}_{p}}]:=[Z_{s^{1}_{p}}]\wedge\ldots\wedge[Z_{s^{1}_{p}}]$ is
well-defined and is equal to the current of integration with multiplicities
over the common zero set $Z_{\Sigma^{k}_{p}}$.

Acknowledgement: We thank Turgay Bayraktar for his suggestions regarding this
paper.

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