Toeplitz operators and zeros of square-integrable random holomorphic sections
Alexander Drewitz Bingxiao Liu George Marinescu




 
Universität zu Köln,  Department Mathematik/Informatik,
    Weyertal 86-90,   50931 Köln, Germany
    adrewitz@uni-koeln.de
bingxiao.liu@uni-koeln.de
gmarines@math.uni-koeln.de
The authors are partially supported by the 
DFG Priority Program 2265 `Random Geometric Systems' (Project-ID 422743078).
G. M. is partially supported by DFG funded projects SFB/TRR 191
(Project-ID 281071066-TRR 191), 
and the ANR-DFG project QuaSiDy (Project-ID 490843120).





We use the theory of abstract Wiener spaces to construct a probabilistic model
for Berezin–Toeplitz quantization on a complete Hermitian complex manifold
endowed with a positive line bundle.
We associate to a function with compact support (a classical observable)
a family of square-integrable
Gaussian holomorphic sections. Our focus then is on the asymptotic distributions
of their zeros in the semi-classical limit, in particular, we prove
equidistribution results, large deviation estimates,
and central limit theorems of the random zeros on the support of the given function.
One of the key ingredients of our approach is the local asymptotic expansions of
Berezin–Toeplitz kernels with non-smooth symbols.

 




Toeplitz operators and zeros of square-integrable random holomorphic sections
    Alexander Drewitz, Bingxiao Liu and George Marinescu
    April 29, 2024
=============================================================================







§ INTRODUCTION




In this paper we prove several probabilistic
results on the action of a 
classical observable on random quantum states,
via Berezin–Toeplitz quantization. More precisely, we investigate 
the distribution of zeros, laws of large numbers,
large deviation estimates and central limit theorems. 
Given a symplectic manifold X the Berezin–Toeplitz
quantization is a family of Hilbert spaces 
_ħ, where ħ is the Planck constant considered
as a small parameter, together with linear maps T_ħ from
^∞(X) to
the space of bounded linear operators (_ħ).
From a physics point of view, the manifold X can be seen as the phase space
of a physical system, a function f∈^∞(X)
as a classical observable, and the operator T_f,ħ=T_ħ(f)
the corresponding quantum observable.
A fundamental principle states that quantum mechanics
contains the classical one as the limiting case ħ→0.


Here our quantum spaces will be
_1/p:=H^0_(2)(X,L^p⊗ E), p∈ (thus ħ=1/p), consisting
of square-integrable
holomorphic sections of the p-tensor powers of a positive holomorphic
line bundle (L,h_L)→ X twisted with an auxiliary Hermitian holomorphic 
line bundle (E,h_E).
The operators T_f,p will be Toeplitz operators with symbol 
f∈^∞(X), or more generally, also non-continuous symbols.
The Berezin–Toeplitz quantization and the underlying techniques
have many applications, ranging from symplectic topology <cit.>,
asymptotics of the analytic torsion forms <cit.>,
topological quantum field theory <cit.>, 
entanglement entropy <cit.>, to
non-commutative geometry <cit.>.

In this paper we focus on probabilistic aspects
of the Berezin–Toeplitz quantization.
For this purpose, we recall that in <cit.> we introduced a general method 
to randomize the quantum states in H^0_(2)(X,L^p⊗ E) 
by using Toeplitz operators and considering
random combinations of pure states.
For appropriate symbols f such that T_f,p is Hilbert-Schmidt
and injective, we consider random sections of the form

    S_f,p=∑_j=1^d_pη^p_jλ^p_jS^p_j

in
H^0_(2)(X,L^p⊗ E), 
where {η^p_j}_j=1^d_p denotes a sequence of independent and 
identically distributed (i.i.d.) standard
complex Gaussian variables, (λ^p_j)_j=1^d_p
is the point spectrum of T_f,p on H^0_(2)(X,L^p⊗ E), 
and {S^p_j}_j=1^d_p is an orthonormal basis 
of H^0_(2)(X,L^p⊗ E) of H^0_(2)(X,L^p⊗ E) such that
T_f,pS^p_j=λ^p_jS_j.
The rigorous definition of the probability
distribution on H^0_(2)(X,L^p⊗ E) in <cit.> proceeds by using
the machinery of constructing Gaussian measures on an abstract
Wiener space introduced  by Gross <cit.>. 
Our results concern the zero divisors of the random sections
S_f,p. First, we describe how
the classical observable f and its quantum counterpart T_f,p
influence statistical properties of S_f,p such as its expectation, variance, etc.
Subsequently, we consider the semiclassical limit 
p→∞ and obtain, as in many inverse problems, several
features of the geometric input and of the observable f. 
Note that on small length scales of order of the Planck scale 1/√(p), 
one loses track of the special features of the geometrical setting and
obtains a universal limiting behavior of random zeros 
<cit.>. 
In our setting, we will observe a universal limiting behavior which (to leading order) is independent of the specific choice of our function f, as long as we restrict ourselves to the support of f; cf. Corollary
<ref> and in Theorem <ref> for further details.



The distribution results from <cit.> apply to functions f
which vanish up to order two. Their derivations are based on the
asymptotics of the kernels of Toeplitz operators
on the support of their symbol f and the calculation
of the first several terms in the asymptotics.
In this article, we take a different approach and
prove a large deviation estimate from which the
distribution of the zeros follows, independent of the vanishing
order of the symbol. Moreover, the results hold
for a large class of non-smooth symbols f. For X compact we will provide semiclassical
estimates for the lowest eigenvalues of T_f,p,
which are intimately linked to
the distribution of zeros.







 §.§ Geometric setting and square-integrable random holomorphic sections

We now describe the geometric setting of our paper.
Let (X,J,Θ) be a connected complex Hermitian (paracompact) manifold of 
complex dimension n, where J denotes the canonical complex 
structure of X, and Θ denotes a J-compatible Hermitian 
form. Then we have an induced
Riemannian metric g^TX(·,·)=Θ(·,J·) on X,
and the corresponding Riemannian volume form dV:=Θ^n/n!. 
With 𝔽= or, let ℒ^∞(X,𝔽)
denote the space of (essentially) bounded measurable functions on X,
and let ℒ^∞_c(X,) denote the subspace of
ℒ^∞(X,) consisting of functions with compact (essential) support.


Let L, E be two holomorphic line bundles on X equipped with 
smooth Hermitian metrics h_L, h_E, and let R^L, R^E
denote their corresponding Chern curvatures. 
The first Chern form of (L,h_L) is defined as

    c_1(L,h_L):=√(-1)/2π R^L,

which is a real (1,1)-form on X representing
the first Chern class in both de Rham and Dolbeault cohomologies.

In the semiclassical setting, we assume (L,h_L) to be positive and
consider the high tensor powers of (L,h_L), that is, for p∈_≥ 1, 
the Hermitian line bundle
(L^p⊗ E,h_p):=(L^⊗ p⊗ E, h_L^⊗ p⊗ h_E)
on X. The space of square-integrable sections of L^p⊗ E
with respect to h_p and dV is denoted by 
ℒ^2(X,L^p⊗ E), endowed with the -norm
·^2_.
The quantum space in this paper will be the space
of square-integrable holomorphic sections of L^p⊗ E,

    H^0_(2)(X,L^p⊗ E):={s_p∈ H^0(X,L^p⊗ E) :  s_p^2_=∫_X |s_p(z)|^2_h_pdV(z)<∞}.

Then H^0_(2)(X,L^p⊗ E) together with the -inner product
becomes a (separable) complex Hilbert space. 
We set

    d_p:=_ H^0_(2)(X,L^p⊗ E) ∈∪{∞}

and denote by

    P_p:ℒ^2(X,L^p⊗ E)→
    H^0_(2)(X,L^p⊗ E)

the orthogonal (Bergman) projection.
One fundamental method to study the sequence of Hilbert spaces 
H^0_(2)(X,L^p⊗ E) is through their associated reproducing kernels
P_p(x,y), called Bergman kernels. 
The asymptotic expansion of Bergman kernels as p→∞
was obtained in <cit.> under the following
very general geometric conditions, which we will also
assume in this paper.


The Riemannian metric g^TX is complete on X, 
and there exist ε_0>0, 
C_0>0 such that

    √(-1)R^L≥ε_0Θ,  √(-1)(R^det+R^E)>-C_0Θ,  and   |∂Θ|_g^TX< C_0.



The first inequality in (<ref>) says that
(L,h_L) is uniformly positive on X, and Condition <ref>
also implies that there exist C>0, p_0∈, such that d_p≥ Cp^n 
for p≥ p_0.
Condition <ref> is a necessary assumption in Sections 
<ref> and  <ref> – <ref> 
which deal with semi-classical limits.

In <cit.>, we constructed a Gaussian random holomorphic section S_p in H^0_(2)(X,L^p⊗ E) via the formula

    S_p:=∑_j=1^d_pη^p_j S^p_j,

where {S^p_j}_j is an orthonormal basis of H^0_(2)(X,L^p⊗ E), and {η^p_j}_j is a family of i.i.d. standard complex Gaussian random variables. Moreover, we have uniqueness in the sense that the distribution of S_p does not depend on the choice of the orthonormal basis {S^p_j}_j. In <cit.>, we have studied equidistribution results, large deviation estimates and hole probabilities for the zeros of S_p in the semi-classical limit. We also refer to <cit.> for further discussions on  random zeros and random holomorphic sections in complex geometry.


However, when d_p=∞, the random section S_p turns out to be almost surely not  square-integrable on X. 
Then the Berezin–Toeplitz quantization came into our construction in <cit.> to define a Gaussian random -holomorphic sections.
Given f∈ℒ^∞_c(X,) a real bounded measurable function with compact (essential) support, the Toeplitz operators associated to f are defined for p∈,
 
    T_f,p:=P_p M_f ∈End(H^0_(2)(X,L^p⊗ E)),

where M_f denotes the pointwise multiplication by f. 
Moreover, T_f,p is a self-adjoint Hilbert-Schmidt operator. 
We also set T^2_f,p=T_f,p∘ T_f,p and denote by T^2_f,p(x,y)
the smooth integral kernel of T^2_f,p with respect to the metric
h_p and the volume form dV.


Let T_f,p denote the range of T_f,p in H^0_(2)(X,L^p⊗ E),
and let H^0_(2)(X,L^p⊗ E, f):= T_f,p
=( T_f,p)^⊥
be the closure of the range of T_f,p, which itself is also a Hilbert space. 
As explained in <cit.>, 
regarding T_f,p as an injective Hilbert-Schmidt operator on 
H^0_(2)(X,L^p⊗ E, f) and applying the theory of abstract Wiener space, 
we get a unique Gaussian probability measure _f,p on
H^0_(2)(X,L^p⊗ E, f) which provides a model
for the action of T_f,p on S_p defined in (<ref>).


The probabilistic Berezin–Toeplitz quantization associated to the
symbol f∈ℒ^∞_c(X,) is defined as the
sequence of Gaussian random -holomorphic sections
{S_f,p,p∈}, where each S_f,p
denotes the random variable taking values in 
H^0_(2)(X,L^p⊗ E, f)⊂ H^0_(2)(X,L^p⊗ E) 
with law _f,p. 
An equivalent definition is given by formula (<ref>).


In this paper, we study the asymptotic behavior of the sequence of random
(1,1)-currents {[Div(S_f,p)]}_p on X
defined by the integration currents on the zero divisors of S_f,p as
p→ +∞. In the case of compact Kähler manifolds 
and a smooth function f, such questions were also independently
studied by Ancona–Le Floch <cit.> motivated by understanding the
Kodaira embedding twisted by Toeplitz operators.
Moreover, provided a smooth function f, in <cit.> 
(for compact Kähler manifolds) and in <cit.>, 
the equidistribution results of [Div(S_f,p)] as
p→+∞ were proved on the subset of the support of f, 
where f only vanishes up to order 2. 
The present article aims to contribute to the above body of work by providing a more profound understanding of the following natural questions:

	
  (i). Do the above equidistribution and the large deviation results for [Div(S_f,p)] on the support of f still hold true when f has higher vanishing orders or a lower regularity?
	
  (ii). How are the random zeros distributed asymptotically outside the support of f? Can one quantify the difference between random zeros 1/p[Div(S_f,p)] and the expected limit c_1(L,h_L) on a subset where f is supported on the most part of it?

  (iii). Does the central limit type behavior of
[Div(S_f,p)] hold on the support of f? 
More concretely, following the work of Shiffman-Zelditch 
<cit.> for S_p on 
a compact Kähler manifold, one can ask for the analogues 
of their results to [Div(S_f,p)] but probably 
only on the support of f.

  (iv).  When X is compact, a problem related to the above is to describe the asymptotic behavior of the
spectra of  {T_f,p}_p.  In particular, when f is non-negative
and not fully supported on X, how does the lowest eigenvalues of
T_f,p decay to 0 as p→+∞?







 §.§ Asymptotic distribution of zeros of random -holomorphic sections

At first, we introduce some notions on the regularity and the support of functions on X. For f ∈ℒ^∞(X,𝔽), let 
‖ f‖_ℒ^∞=inf{C>0:  |f|≤ C a.e.} 
denote its essential supremum norm with respect 
to the measure dV.
The essential support of f on X (with respect 
to dV), denoted by f, is the smallest closed subset of X such that f vanishes almost everywhere
on its complement. When f is also continuous, then 
f coincides with the support 
supp f={x∈ X:f(x)≠0} of f. We will always call f the support of f. Note that we say f to be smooth (resp. 𝒞^k) an open subset U⊂ X if 
there is a smooth (resp. 𝒞^k) function f on U such that 
f|_U-f=0 almost everywhere in U (with respect to 
dV). 






With = or ,  let ℒ^∞_const(X,) denote the subspace of ℒ^∞(X,) consisting of functions which are constant outside a compact subset, i.e.,

    ℒ^∞_const(X,)={f∈ℒ^∞(X,) | there exists  c_f∈ such that  f-c_f∈ℒ^∞_c(X,)}.





For f ∈ℒ^∞(X,), and U an open subset of X, we say that f is of class 𝒞^k (k∈∪{∞}) almost everywhere on U if there exists a closed subset D⊂ U of Lebesgue measure 0 such that f|_U∖ D∈𝒞^k(U∖ D,). We also say that f is of class 𝒞^k (k∈∪{∞}) almost everywhere near U if it is so on an open neighbourhood of U.



(i) A smooth function f∈𝒞^k(X,) is always of class
𝒞^k almost everywhere near any given open subset of X.

(ii) Let U be an open subset of X such that
∂ U:=U∖ U has Lebesgue measure zero in X. 
Then the characteristic function 
	
    f(x)=1_U(x):=1    if  x∈ U,
     0    if  x∉U,

	is smooth almost everywhere near U.

We will identify the (1,1)-form R^L with the Hermitian matrix

    Ṙ^L∈End(T^(1,0)X)

such that for W,Y∈ T^(1,0)X,

    R^L(W, Y̅)=⟨Ṙ^LW,Y̅⟩_g^TX⊗.


We can now introduce the following quantity in order
to give a bound on the regularity of f, 
which is necessary in our methods to obtain 
the asymptotic results for random zeros.



For any relatively compact subset U⊂ X, set 

    κ(R^L,U):=sup{maxspec (Ṙ^L_x),  x∈ U}≥ε_0,

and  define

    m(U):=⌈ (6n+6)κ(R^L, U)/ε_0⌉∈.

In the prequantum case Θ=c_1(L,h_L), we can take 
κ(R^L,U)=ε_0=2π and m(U)=m(X):=6n+6,
which is independent of the subset U. 



We now introduce a quantity that measures
the relative position of an open set with respect to
the the support of a function f.



Let f ∈ℒ^∞_c(X,)
and let U be an open subset of X. Define

    r(f,U):=sup{r> 0 : geodesic ball 𝔹(x,r)⊂ U∖ f 
    },

where the geodesic ball is taken with respect to g^TX.
Since U is assumed to be open, if 
U∖ f≠∅, 
we have r(f,U)>0. When U⊂  f, 
we say that f has full support on U. In this case, 
there is no nontrivial geodesic ball in U∖ f, 
and we set r(f,U)=0.



The norm ·_U,-2 for the (1,1)-currents is defined
Definition <ref> setting α=2. For any two subsets 
U, U' of X, the notation U⋐ U' means that 
U is compact and contained in U'. 
Our first main result is a concentration estimate
which gives an upper bound on the deviation
of 1/p[Div(S_f,p)] from c_1(L,h_L)
on an open set in the semi-classical limit in terms of
the relative position with respect to the support of f.


Let (X,J,Θ) be a connected Hermitian complex manifold 
and let (L,h_L), (E,h_E) be two holomorphic line bundles
on X with smooth Hermitian metrics.
Furthermore, assume that Condition <ref> holds and fix 
a pair of nonempty open subsets (U,U') of X such that U⋐ U'. 
Then there exists a constant r(U,U')>0 such that if f∈ℒ^∞_c(X,) is of 𝒞^m(U')+1 almost everywhere on U' with 

    δ_0(f):=(r(f,U')/r(U,U'))^1/(2n+2)<1/2  ,

then U∩  f≠∅, and for any 
	δ > δ_0(f), there 
	exists a constant C=C(U',f,δ)>0 and p_0>0 such that for all p≥ p_0 
	we have
	
    (  1/p[(S_f,p)] 
    		-  c_1(L,h_L)_U,-2>δ/π )≤ 
    		e^-C p^n+1.

	As a consequence, we have
	
    (  lim sup_p→+∞1/p[(S_f,p)] 
    		-  c_1(L,h_L)_U,-2≤δ_0(f)/π ) =1.




Observe that when U⊂ f, 
we have r(f,U)=0, so we need the strict inequality 
δ > δ_0(f)

in the statement of Theorem <ref>.



In general, it is difficult to determine r(U,U')
precisely.
By the proof of
Proposition <ref> the constant r(U,U')>0
depends on the geometry of g^TX, the complex structure of X,
and several auxuliary constants.
However, we can still give a rough formula (<ref>) for r(U,U'),
which is certainly not sharp.


The question of quantum ergodicity (mass distribution) for a 
sequence of holomorphic sections as tensor power p tending
to infinity is a parallel problem to the asymptotic distributions
of their zeros as integration currents, whose central objects are
the following measures on X defined by the holomorphic sections.


The mass distribution of a section s_p∈ H^0(X,L^p⊗ E) 
is defined as the measure

    M_p(s_p):=1/p^n|s_p(z)|^2_h_pdV

on X, where dV=Θ^n/n!. 
If s_p is square-integrable, then M(s_p) is a finite measure.


 For Gaussian (or sub-Gaussian) holomorphic sections on compact Kähler manifolds or certain random polynomials on ^n, such problems were investigated in <cit.>. In Section <ref>, we also consider the mass distribution of our random -holomorphic sections S_f,p. In particular, we get a law of large numbers for ∫_X g(z)M_p(S_f,p)(z). Now we can state the result, as an analog of <cit.>, whose proof will be given in Section <ref>.



Let (X,J,Θ) be a connected Hermitian complex manifold
and let (L,h_L), (E,h_E) be two holomorphic line bundles
on X with smooth Hermitian metrics.
We assume that the Condition <ref> holds. 
Let U be a relative compact open subset of X, 
and fix a nontrivial 
f∈𝒞^m(U)+1_c(U,_≥ 0). 
Then for any g∈𝒞^0_c(U), we have ℙ-a.s. that

    lim_N→ +∞1/N∑_1≤ p≤ N∫_U 
    g(z)M_p(S_f,p)(z)=
    ∫_U g(z)f(z)^2dV^L(z),

where the volume form dV^L:=c_1(L,h_L)^n/n! 
in the limit is defined independently from Θ.







 §.§ Large deviation and equidistribution on the 
support of f



As a special case of Theorem <ref>, 
we can give a type of large deviation estimates for the zeros of 
S_f,p on the support of f.
Such kind of estimates are also referred as concentration 
of measure. As a consequence, we obtain the equidistribution
of random zeros on the support of f, 
see also <cit.> and 
<cit.>). 


The following theorem generalizes the previous results
in <cit.> by removing 
the conditions on the smoothness of f and the vanishing
orders of f on the given domain. Note that the quantity m(U) 
is defined by (<ref>).




Let (X,J,Θ) be a connected complex Hermitian manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics.
We assume that the condition <ref> holds. Fix 0≠ f∈ℒ^∞_c(X,). Let U be an open subset of X such that U⊂ess.supp  f and f is of class 𝒞^m(U)+1 almost everywhere on U. Then for any 
	δ >0, and V⋐ U, there 
	exists a constant C=C(f,δ, V)>0 such that for all sufficiently large p∈,
	we have
	
    (  1/p[(S_f,p)] 
    -  c_1(L,h_L)_V,-2>δ )≤ 
    e^-C p^n+1.

	As a consequence, we have -a.s. that
		
    lim_p→+∞1/p[(S_f,p)] 
    		-  c_1(L,h_L)_U,-2=0.







In <cit.> we proved 
large deviation estimates and equidistribution results for
⟨1/p[(S_f,p)]-  c_1(L,h_L),φ⟩
for a fixed φ∈Ω^n-1,n-1_c(X),
in the case of Gaussian random holomorphic sections 
S_p (defined in (<ref>)).
In <cit.>, we proved the almost sure 
convergence (<ref>) under extra finiteness condition. 
But actually, Theorem <ref> holds for the 
random holomorphic sections S_p, since
all the asymptotic expansions
for T^2_f,p(x,y) necessary to prove Theorem <ref>
have analogoues for P_p(x,y).



We now provide an interesting consequence of Theorem <ref>.
For any Borel subset U⊂ X 
we set
 
    Vol^L_2n(U):=∫_U c_1(L,h_L)^n/n!>0.

Analogously, if Y is a complex submanifold of X with complex codimension 1, we define the (2n-2)-dimensional volume with 
respect to c_1(L,h_L) of Y as 
 
    Vol^L_2n-2(Y):=∫_Y c_1(L,h_L)^n-1/(n-1)!|_Y.





For s_p∈ H^0(X,L^p⊗ E)∖{0}, the (2n-2)-dimensional volume of the divisor (s_p) (cf. (<ref>))
in an open subset U⊂ X as follows:

    ^L_2n-2((s_p)∩ U)=∑_Y⊂ 
    	Z(s_p)ord_Y(s_p)Vol^L_2n-2(Y∩ U) ·

If we use this volume to measure the size of the zeros of s_p in U, 
then Theorem <ref> leads to the following result.

We assume the same geometric conditions on X, L, E as in Theorem <ref>. Fix 0≠ f∈ℒ^∞_c(X,). Let U be an open subset of X such that U⊂ess.supp  f and f is of class 𝒞^m(U)+1 almost everywhere on U.
	If V is a nonempty relatively compact open subset of U such that 
	∂ V has zero measure in X, 
	then for any δ >0, there 
	exists a constant c=c_f,δ,V>0 such that for all sufficiently large p,
	we have
	
    (  |1/p^L_2n-2((S_f,p)∩ V) 
    		- n^L_2n(V) |>δ )≤ 
    		e^-c  p^n+1.

	In addition, there exists a 
	constant C_f,V>0 such that for p≫ 0,
	
    ((S_f,p)∩ V=∅)
    		≤ e^-C_f,V p^n+1 .



The right-hand side of (<ref>)
is called the hole probability for the random sections S_f,p.







 §.§ Expectation of random zeros and pluripotential theory on X


As a part of the equidistribution results for the zeros of S_f,p, we also need to study the convergence of the expectation of 1/p[(S_f,p)] as (1,1)-currents on X. In Theorem <ref>, we show that

    1/p[[(S_f,p)]]=c_1(L,h_L)+√(-1)/2π p∂∂logT_f,p^2(x,x)+1/pc_1(E,h_E).

Hence the main point is to study the current √(-1)/2π p∂∂logT_f,p^2(x,x). Then by the observation from (<ref>) that 1/p[[(S_f,p)]] is a positive current on X, we can apply the techniques from the pluripotential theory, especially the theory of quasi-plurisubharmonic (quasi-psh) functions, to study the asymptotic properties of logT_f,p^2(x,x) as p→+∞. We will recall some basics for plurisubharmonic functions in Section <ref>.

As a consequence, we have the following theorem in a great generality.

Let (X,J,Θ) be a connected complex Hermitian manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics.
We assume that the condition <ref> holds. For f∈ℒ^∞_c(X,), if there exists a small open ball B≠∅ such that f|_B∈𝒞^1(B) and f never vanishes on B. Let U be a connected open subset of X which is relatively compact and U∩ B≠∅, then there exist constants C>0,  C'≥ n depending only on X, U, L and f such that for all p≥ 1, any sequence of nonempty open subsets {A_p}_p≥ 1 of U, we have 
	
    1/Vol(A_p)∫_A_p T^2_f,p(x,x)dV(x)≥exp(-Cp/Vol(A_p)-C'logp),

	where Vol(A_p):=∫_A_pdV(x)>0.
	Moreover, with the same constant C>0 as above and for all p≫ 0, we have
	
    1/p𝔼[[(S_f,p)]]-c_1(L,h_L)_U,-2≤C/π.

	There exists a subsequence {p_j}_j=1^∞⊂ that is increasing to +∞ and a quasi-psh function f on U such that we have the convergence of (1,1)-currents of order 2 on U,
	
    lim_j→ +∞1/p_j𝔼[[(S_f,p_j)]]|_U=c_1(L,h_L)|_U+√(-1)/2π∂∂f≥ 0.

	In particular, f|_B∩ U≡ 0.


A remark on Theorem <ref> for a compact Hermitian manifold X is that when f is non-negative (hence T_f,p is injective) the constant C in (<ref>) and (<ref>) can be determined by the lowest eigenvalues of T_f,p. Then a consequence, as we will explain in Sections <ref> and <ref>, is that the above constant C is related to the size of ess.supp  f (or equivalently X∖ess.supp  f).



When we consider the case U⊂ess.supp  f, we get the convergence of 1/p𝔼[[(S_f,p)]] on U as p→+∞. The precise statement is given as follows.

Let (X,J,Θ) be a connected complex Hermitian manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics.
We assume that the condition <ref> holds. Fix 0≠ f∈ℒ^∞_c(X,). Let U be an open subset of X such that U⊂ess.supp  f, and we assume that f is of class 𝒞^1 almost everywhere near U. Then we have, as p→ +∞,
	
    1/plogT_f,p^2(x,x)_ℒ^1(U,)→ 0.

Then we have, as p→ +∞,
		
    1/p𝔼[[(S_f,p)]]- c_1(L,h_L)_U,-2→ 0.


Note that without any extra condition, we can not conclude (<ref>) directly from the almost sure convergence (<ref>), so that in the proof of Theorem <ref>, the use of certain compactness result for quasi-psh functions is necessary in our method. The proofs of both theorems above are given in Section <ref>.









 §.§ Central limit theorem for random zeros on the support


The following theorem extends <cit.> and <cit.> to our Toeplitz setting on possibly noncompact Hermitian manifolds. Moreover, as pointed out in <cit.>, such result also holds true for the Gaussian holomorphic sections {S_p}_p (defined by (<ref>)) on noncompact Hermitian manifolds.


Let (X,J,Θ) be a connected complex Hermitian manifold and
let (L,h_L), (E,h_E) be two holomorphic line bundles on X
with smooth Hermitian metrics.
We assume that Condition <ref> is satisfied. 
Fix f∈𝒞^∞_c(X,) which is not identically zero,
and let U be an open subset of X such that U⊂{f≠ 0}.
Let φ be a real (n-1,n-1)-form on X with 𝒞^3-coefficients such that φ⊂ U and ∂∂φ≢0, set
	
    Z_f,p(φ):=⟨[Div(S_f,p)],φ⟩,

then as p→∞, the distribution of the random 
variables
	
    Z_f,p(φ)-[Z_f,p(φ)]/√(Var[Z_f,p(φ)])

converges weakly to 𝒩_(0,1).


Note that the smoothness assumption 
f∈𝒞^∞_c(X,)
can be relaxed to f∈𝒞^m_c(X,)
for sufficiently large m. 
In the case of compact Kähler manifolds, 
Shiffman and Zelditch <cit.>
also computed explicitly the variance Var[Z_f,p(φ)]. 
The same method also applies to our case due to our results for the 
normalized Berezin–Toeplitz kernels 
(in particular Theorem <ref> below). 
The proof of Theorem <ref> will be given in Section <ref>.

For a real (n-1,n-1)-form  φ on X with 𝒞^3-coefficients, recall that L(φ)∈𝒞^1(X,) is defined by

    √(-1)∂∂φ = L(φ) c_1(L,h_L)^n/n!.

If U is as in Theorem <ref> and φ⊂ U, then in Theorem <ref>, we prove that
for p≫ 0, 

    Var[Z_f,p(φ)]=p^-n(ζ(n+2)/4π^2∫_U |L(φ)(z)|^2 dV^L(z)+𝒪(p^-1/2+ϵ)),

where

    ζ(n+2)=∑_k=1^∞1/k^n+2.
 
Note that ∫_U |L(φ)(z)|^2 dV^L(z) is a positive quantity independent of the metric Θ on X.

With the same assumptions in Theorem <ref>, 
we have 1/p[Z_f,p(φ)]→⟨ c_1(L,h_L),φ⟩ as p→+∞. Therefore, as a consequence of 
(<ref>), Theorem <ref>, and the definition of convergence in distribution as the pointwise convergence of the distribution functions towards the distribution function of the limiting random variable in all points of continuity,

we get the following universality result.

With the same assumptions in Theorem <ref>, set

    σ(U,h_L,φ):= ζ(n+2)/4π^2∫_U |L(φ)(z)|^2 dV^L(z),

then the distribution of the real random variable 

    p^n/2⟨[Div(S_f,p)]-p c_1(L,h_L),φ⟩

converges weakly to 𝒩_(0,σ(U,h_L,φ))
as p→ +∞. 

This shows the local universality character of the distribution of
zeros of random -holomorphic sections in the sense that the
limiting distribution of the random variable (<ref>)

is independent of the function f∈𝒞^∞_c(X,)
under the condition that U⊂{f≠ 0}.





 §.§ Berezin–Toeplitz kernel with non-smooth symbols


The Berezin–Toeplitz kernel T^2_f,p(x,y) is an essential element in all our proofs to the above results for the Gaussian -holomorphic sections S_f,p on X. For a smooth symbol f, the asymptotic expansions of T^2_f,p(x,y) (in the non-compact setting) were given by the seminal works of Ma–Marinescu <cit.> using the techniques of analytic localization. This method remain applicable for the non-smooth symbol f as given by Barron, Ma, Marinescu and Pinsonnault in <cit.>. Our results for T^2_f,p(x,y) presented in Section <ref> for a non-smooth symbol f can be regarded as the local versions of the results proved in <cit.>, and our proofs are still built on the techniques of analytic localization developed in <cit.>.


In this paper, our (local or global) regularity condition on the symbol f will be among {ℒ^∞,𝒞^0, 𝒞^1, 𝒞^m+1, 𝒞^∞}, depending on the different contexts, as is already alluded to by the theorems in the previous Sections. The smoothness on f generally is not needed (in Sections <ref> and <ref>) but we impose this condition in Section <ref> and Section <ref> for the sake of simplicity. The 𝒞^m+1-regularity (with m=m(U)) is necessary to have a proper near-diagonal asymptotics for the normalized Berezin–Toeplitz kernels N_f,p (cf. Definition <ref>), which plays the role of correlation function of the holomorphic Gaussian field S_f,p. Therefore, for the large deviation estimates and their consequences, such (local) regularity is assumed. Moreover, in most case, we only need the function f to be 𝒞^m+1 outside a negligible closed subset, so that we can include the interesting examples of f such as the cut-off functions, indicator functions, etc, in our framework. In the cases where only the uniform bounds or the leading terms of T^2_f,p(x,y) are needed, we can assume the function f to be only locally 𝒞^0, 𝒞^1 or just ℒ^∞.


We fix a real function 
f∈ℒ^∞_const(X,). Then T_f,p is a self-adjoint bounded operator, and we will mainly be concerned with the nonnegative operator T_f,p^2:=T_f,p∘ T_f,p.


	For p∈^*, the normalized Berezin–Toeplitz 
	kernel associated to the given f is defined by
	
    N_f,p(x,y):=|T_f,p^2(x,y)|_h_p,x⊗ h^*_p,y/√(T_f,p^2(x,x))·√(T_f,p^2(y,y)),    x,y∈ X,

	whenever T_f,p^2(x,x)≠ 0 and T_f,p^2(y,y)≠ 0.




For x∈ X, let inj_x^X>0 be the supremum of the radius 
r>0 such that the geodesic map T_xX∋ v↦exp_x(v)∈ 
X is a diffeomorphism when restricting to the open ball 
B^T_xX(0,r)⊂ T_xX. Fix a relatively compact open subset 
U⋐ X, set

    inj^X_U=inf_x∈ Uinj_x^X >0.



The following theorem extends 
<cit.> to this new setting of Berezin–Toeplitz operators. For x∈ X, the distance 
function Φ_x will be defined in (<ref>). The constant 
ε_0 was introduced in our assumption 
(<ref>).



Let (X,J,Θ) be a connected complex Hermitian manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics.
We assume that the condition <ref> holds. Fix f∈ℒ^∞_const(X,).
	Let U be a relatively compact open subset of X such that f 
	is smooth on an open neighbourhood of U, closure of 
	U in X, and f does not vanish in U. Then there 
	exist 0<δ_U<inj^X_U/4 such that the 
	following uniform estimates on the normalized Berezin–Toeplitz kernel hold 
	for x,y∈ U:
	For k≥ 1, there exists a constant M_k>0 (we may take M_k=12k) such that for 
	any fixed 
	b≥√(M_k/ε_0), we have for 
	all p ≫ 0 with b√(log 
		p/p)≤δ_U that
	
    N_f,p(x,y)=   (1+R_p,x(v'))exp(-p4 Φ_x(0,v')^2),  
               uniformly for dist(x,y)≤ b√(log 
    		p/p) , with y=exp_x(v'), v'∈ T_xX; 
       𝒪(p^-k),   uniformly for dist(x,y) ≥ 
    		b√(log p/p),

where 

    sup_x∈ U,  v'∈ T_x X, v'≤ b√(logp/p) |R_p,x(v')|→ 0

 as p→ +∞. More precisely, we have the following estimate, for any fixed ϵ∈ ]0,1/2], there exists C=C(f,ϵ,U)>0 such that for any x∈ U, v'∈ T_x X, v'≤ b√(logp/p),

    |R_p,x(v')|≤ C p^-1/2+ϵ.



The proof of above theorem is given in Section <ref>. Moreover, essentially by the same proof, we get a different version of Theorem <ref> under a lower regularity assumption on f as follows, a brief proof to it is also given in Section <ref>.


Recall that m(U)=⌈ (6n+6)κ(R^L, U)/ε_0⌉ is given in (<ref>), this definition actually follows from the choice M_n+1=12(n+1) and b=√(M_n+1/ε_0) in the proof of Theorem <ref>. Then we have the following results.




	Let (X,J,Θ) be a connected complex Hermitian manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics.
We assume that the condition <ref> holds. Fix f∈ℒ^∞_const(X,).
	Let U be a relatively compact open subset of X such that f 
	is of 𝒞^m+1 with m=m(U) on an open neighbourhood of U and f does not vanish in U. Then there 
	exist 0<δ_U<inj^X_U/4 such that the 
	following uniform estimates on the normalized Berezin–Toeplitz kernel hold 
	for x,y∈ U: set b=√(12n+12/ε_0), we have for 
	all p ≫ 0 with b√(log 
		p/p)≤δ_U that
	
    N_f,p(x,y)=   (1+o(1))exp(-p4 Φ_x(0,v')^2),  
               uniformly for dist(x,y)≤ b√(log 
    		p/p) , with y=exp_x(v'), v'∈ T_xX; 
       𝒪(p^-n-1),   uniformly for dist(x,y) ≥ 
    		b√(log p/p).




An advantage of the above results, Theorem <ref>
and Corollary <ref>, is that we only need the
local regularity of f on the part where we want N_f,p
to have the asymptotic formula as in (<ref>) or 
(<ref>), and away from U, there is no 
regularity assumption on f. 







 §.§ Lowest eigenvalue of Toeplitz operators on compact manifolds


Now we focus on a compact Hermitian manifold (X,Θ). 
In this case, H^0_(2)(X,L^p⊗ E)=H^0(X,L^p⊗ E) is finite dimensional, 
and the Gaussian holomorphic section S_p (see (<ref>)) 
can be regarded as the identity map on H^0(X,L^p⊗ E) 
after equipping H^0(X,L^p⊗ E) with the standard 
Gaussian probability measure associated to the -inner product. 
For a real bounded (measurable) function f on X, the random section 
S_f,p in Definition <ref> is equivalent to

    S_f,p=T_f,pS_p.


Even in this case, the problem about the asymptotic distribution of the random zeros 
[Div(S_f,p)] outside the support of f 
remains open. In Section <ref>, 
we present simulations of the zeros of T_f,pS_p
on the Riemann sphere ℂℙ^1, where S_p
is the SU(2)-polynomial. More precisely, if 
is a the unit disc of a standard local chart U_0≃,
which is a geodesic ball in X=ℂℙ^1 of 
g^TX_FS-radius 
r_FS=√(π)/4≃ 0.44311…, 
a simulation for p=20 is shown in Figure <ref>.
The left picture draws the 20000 roots of 1000 times of 
realizations of S_1_ , 20 
(that lie in that coordinate box), and the right picture is the 
density histogram according to the Fubini-Study distance of the 
zeros from the origin z=0, where the bell-shape curve represents
the density c_1(𝒪(1),h_FS)=
ω_FS.





From Figure <ref>, with p=20, near the origin
(inside the support of f=1_), we see the simulated zeros
approximate c_1(𝒪(1),h_FS) quite well,
but outside the support
(the part r_FS> √(π)/4), the simulated zeros behave
very differently from c_1(𝒪(1),h_FS).
More simulation results will be given in Section <ref>
to illustrate the convergence results on the support of f
from Sections <ref> and <ref>.

From (<ref>), the asymptotics of 1/plogT_f,p^2(x,x) is a crucial term to study 1/p[[(S_f,p)]]. By the asymptotic expansion of T_f,p^2(x,x) (see Theorem <ref>), we conclude that 

    logT_f,p^2(x,x)≤ Clogp.

When f is smooth and (L,h_L) is prequantum (that is Θ=c_1(L,h_L)), the lower bounds for logT_f,p^2(x,x) were obtained in <cit.>, <cit.>, <cit.> under the assumption that f vanishes only up to order 2. But a proper lower bound for logT_f,p^2(x,x) in general case is missing. 

 The above expected lower bounds relate clearly to the lowest nonzero eigenvalues of T^2_f,p. Let us focus on the non-negative function f. For a nontrivial f∈ℒ^∞(X,_≥ 0), T_f,p is injective and positive, set λ^p_min(f):=minSpec(T_f,p) >0, then on X, we get a lower bound for 1/plogT_f,p^2(x,x):

    1/plogT_f,p^2(x,x)≥2logλ^p_min(f) /p +logP_p(x,x)/p.

The easy case is that the essential infimum of f is strictly positive on X, so that we can conclude 1/plogT_f,p^2(x,x)→ 0 uniformly on X as p→ +∞. 

When a nontrivial f≥ 0 has a 𝒞^1-vanishing point in X, 
then λ^p_min(f) decays to 0 as p→ +∞
(see the analogous statements in Corollary <ref> and
Remark <ref>, see also <cit.>). 
Then a first step to study λ^p_min(f) is to understand how 
fast it decays to 0 when f vanishes at some points in X.

In Section <ref>,  for the Hermitian line bundle (L,h_L)=(𝒪(1),h_FS) on the Riemann sphere ℂℙ^1, 
we have computed explicitly the Toeplitz spectra for three types of functions
and obtain three different asymptotic behavior for λ^p_min. 
Let U_0≃ denote a standard complex chart for ℂℙ^1.

	
  * For k∈_≥ 1, set f_k(z):=|z|^2k/(1+|z|^2)^k
	on U_0≃, then f_k has only one vanishing point at z=0
	with vanishing order 2k. We have
	
    λ^p_min(f_k)=k!p^-k(1+k(k+3)/2p+𝒪(p^-2)).

	
  * For f(z):=e^-1/|z|^2 on U_0≃, 
	then f has only one vanishing point at z=0 with vanishing order ∞. We have
	
    λ^p_min(f)=e^-2√(p)(1+o(1)).


  * Let 𝔹⊂ℂℙ^1 be a geodesic ball, set 
f(z):=1_𝔹(z) the indicator function for 𝔹. 
Assume 𝔹≠ℂℙ^1, hence Vol(𝔹) <1. 
We have
	
    λ^p_min(1_𝔹)=Vol(𝔹)^p+1.


In Question <ref>, following the above examples, 
we summarize a question for the lowest Toeplitz eigenvalues
for the general compact Hermitian complex manifolds. 


Now we present some partial results on λ^p_min,
whose proofs will be given in Section <ref>. 
A complete answer still remains open.

Let (X,J,Θ) be a connected, compact Hermitian complex manifold 
and let (L,h_L), (E ,h_E) be holomorphic line bundles on X 
with smooth Hermitian metrics. Assume h_L to be positive.
Fix f∈ℒ^∞(X,_≥ 0) which is not identically zero.

	
  (i) For N∈_≥ 1, if there exists x_0∈ X
	such that f is 𝒞^2N+1 near x_0 and f 
	vanishes at x_0 with vanishing order 2N, then there exists
	C>0 such that for all p>0,
	
    minSpec(T_f,p)≤ Cp^-N.

	
  (ii) If there exists x_0∈ X such that f is smooth
	near x_0 and f vanishes at x_0 with vanishing order +∞, 
	then for any ℓ∈, there exists C_ℓ>0 such that for all p>0,
		
    minSpec(T_f,p)≤ C_ℓ p^-ℓ.





The following result provides a supportive evidence for the situation 
like (<ref>), which also refines the lower bound in (<ref>) 
in compact case.

Let (X,J,Θ) be a connected, compact Hermitian complex manifold and let 
(L,h_L), (E ,h_E) be holomorphic line bundles on X with smooth Hermitian metrics. 
Assume h_L to be positive. For f∈ℒ^∞(X,_≥ 0)
which is not identically zero and is continuous near a nonvanishing point, 
there exist constants C'>0, c'>0 depending only on X, L, E and 
f such that for all p≫ 0,

    minSpec(T_f,p)≥ C e^-cp

If f ≠ X, then for any A>0, there exists C'=C'(f,A)>0 such that for all p≫ 0,

    minSpec(T_f,p)≤ C' e^-A√(plogp)







 §.§ Organization of the paper

This paper is organized as follows:

In Section <ref>, we give the asymptotic expansions of the Berezin–Toeplitz kernels T^2_f,p(x,y) under the local regularity assumption on f.

In Section <ref>, we recall the definition of Gaussian -holomorphic sections given in <cit.> and the related results.


In Section <ref>, we prove Theorem <ref>, where the key intermediate result is the Proposition <ref>. In particular, we give the proof of Proposition <ref> in Section <ref>.

In Section <ref>, we study the asymptotic distribution of random zeros on the support of f. In particular, the proofs of Theorems <ref>, <ref>, <ref> and <ref> are given. The results on the lowest eigenvalues of T_f,p for a compact Hermitian manifold X are given in Section <ref>.

At last, in Section <ref>, we discuss the number variance
and the asymptotic normality of the zeros of S_f,p
on f for a real smooth function f with compact support.



 §.§ Acknowledgments

AD and BL thank NYU Shanghai for their hospitality.
We also thank Xiaonan Ma and Stéphane Nonnenmacher
for useful discussions.




§ TOEPLITZ OPERATORS AND ASYMPTOTICS OF TOEPLITZ KERNELS

Let (X,J,Θ) be a connected complex Hermitian (paracompact) manifold of 
complex dimension n, where J denotes the canonical complex 
structure of X, and Θ denotes a J-compatible Hermitian 
form. Then we have an induced
Riemannian metric g^TX(·,·)=Θ(·,J·) on X. We 
denote by R^det the curvature of the Chern connection 
∇^det on K^*_X:=(T^(1,0)X) with respect 
to induced Hermitian metric by g^TX. Let dist(·,·) denote the Riemannian distance 
of (X, g^TX). Let (L,h_L), (E,h_E) be two Hermitian holomorphic line bundle 
on X, and let ∇^L, ∇^E denote the corresponding Chern 
connections with the respective curvature forms R^L, R^E.
We always assume Condition <ref> to hold. 






 §.§ Bergman projections and the asymptotics of Bergman kernels
 
The Riemannian volume form on (X,g^TX) is denoted by 
dV=Θ^n/n!. For p∈ℕ_>0, set (L^p⊗ E, h_p):=(L^⊗ 
p⊗ E, h_L^⊗ p⊗ h_E). For s, s'∈𝒞^∞_c(X,L^p⊗ E), the -inner 
product is defined as follows,

    ⟨ s,s'⟩_(X,L^p⊗ E):=∫_X⟨ 
    	s(x),s'(x)⟩_h_p,xdV(x).

Let (X, L^p⊗ E) be the completion of 
𝒞^∞_c(X,L^p⊗ E) with respect 
to the above -inner product. Let H^0(X, L^p⊗ E) 
denote the space of global holomorphic sections of L^p⊗ E 
on X. We set

    H^0_(2)(X, L^p⊗ E):=(X,L^p⊗ E)∩ 
    	H^0(X,L^p⊗ E).

Then it is a separable Hilbert subspace of (X, L^p⊗ E). 
Set

    d_p:=_ H^0_(2)(X, L^p⊗ E) ∈∪{∞}.


Let P_p: (X,L^p⊗ E)→ 
H^0_(2)(X,L^p⊗ E) denote the obvious orthogonal 
projection, which is called the Bergman projection. It has a smooth 
Schwartz integral kernel, denoted by P_p(x,x')∈ (L^p⊗ 
E)_x⊗ (L^p⊗ E)^∗_x'. Following the work of 
Ma-Marinescu <cit.>, we have the following 
results on the asymptotics of Bergman kernels (under the assumption 
(<ref>)):

	
  * (Off-diagonal estimates) For any ℓ,m∈ℕ, 
	ε>0, a 
	compact subset K⊂ X, there exists 
	C_K,ℓ,m,ε>0 such that for all x,x'∈ K, 
	dist(x,x')≥ε, we have
	
    |P_p(x,x')|_𝒞^m≤ C_K,ℓ,m,ε 
    		p^-ℓ.

	Here the 𝒞^m-norm is induced by ∇^L, 
	∇^E and h_p, g^TX.
	
  * (On-diagonal expansion) There exist coefficients 
	b_r∈𝒞^∞(X,), r∈, such 
	that for any compact subset K⊂ X, any k,ℓ∈, 
	there exists C_k,ℓ,K>0 such that for p∈^∗,
	
    |1/p^nP_p(x,x)-∑_r=0^kb_r(x)p^-r|_𝒞^ℓ(K)≤ C_k,ℓ,K p^-k-1,

where 
	b_0(x)=(Ṙ^L/2π) (recall (<ref>) for the definition of Ṙ^L), and an explicit formula for
b_1 was given as in 
<cit.>.

Furthermore, the near-diagonal expansion for P_p(x,x') also holds 
uniformly on any given compact subset of X. To describe this 
expansion, we need to introduce some notation as follows.

Fix a point x_0∈ X. Let {f_j}_j=1^n be an orthonormal basis of 
(T_x_0^1,0X, g_x_0^TX(·,·)) such that

    Ṙ^L_x_0 f_j=μ_j(x_0)f_j,

where μ_j(x_0), j=1, …, n are the eigenvalues of 
Ṙ^L_x_0. We have

    μ_j(x_0)≥ε_0, b_0(x_0)=∏_j=1^nμ_j(x_0)/2π.

Set 
e_2j-1=1/√(2)(f_j+f_j), 
e_2j=√(-1)/√(2)(f_j-f_j), j=1, …, n. 
Then they form an orthonormal basis of the (real) tangent vector 
space (T_x_0X, g_x_0^TX). Now we introduce the complex 
coordinate for T_x_0X (with respect to the complex structure 
J_x_0). If v=∑_j=1^2n v_je_j∈ T_x_0X, we can write

    v=∑_j=1^n(v_2j-1+√(-1)v_2j)1/√(2)f_j + ∑_j=1^n(v_2j-1-√(-1)v_2j)1/√(2)f_j.

Set z=(z_1,…,z_n) with z_j=v_2j-1+√(-1)v_2j, 
j=1,…, n. We call z the complex coordinate of v∈ T_x_0X. 
Then by (<ref>),

    ∂/∂ z_j = 1/√(2)f_j ,  ∂/∂z_j = 1/√(2)f_j,

so that

    v=∑_j=1^n(z_j∂/∂ z_j 
    	+z_j∂/∂z_j).

Note that |∂/∂ z_j|^2_g_x_0^TX=|∂/∂z_j|^2_g_x_0^TX=1/2.
For v, v'∈ T_x_0X, let z, z' denote the corresponding 
complex coordinates. Define

    𝒫_x_0(v,v')=∏_j=1^nμ_j(x_0)/2πexp(-1/4∑_j=1^nμ_j(x_0)(|z_j|^2
    +|z'_j|^2-2z_jz'_j)).
	
Define a weighted distance function Φ^TX_x_0(v,v') as follows,

    Φ^TX_x_0(v,v')^2=∑_j=1^nμ_j(x_0)|z_j-z'_j|^2.

Then 

    |𝒫_x_0(v,v')|=∏_j=1^nμ_j(x_0)/2πexp(-1/4Φ^TX_x_0(v,v')^2).

For sufficiently small δ_0>0, we identify the small open 
ball
B^X(x_0,2δ_0) in X with the ball B^T_xX(0, 
2δ_0) in T_x_0X via the geodesic coordinate. Let 
κ(v) be the positive smooth function such that 

    dV(exp_x_0(v))=κ(v)dV_Eucl(v),

where dV_Eucl denotes the Euclidean volume form 
on T_x_0X with respect to g^TX_x_0. In particular, κ(0)=1.


There exists C_2>0 such that for v,v'∈ 
B^T_x_0X(0, 2δ_0), we 
have

    C_2dist(exp_x_0(v),exp_x_0(v'))≥Φ^TX_x_0(v,v')≥1/C_2dist(exp_x_0(v),exp_x_0(v')).

In particular,

    Φ^TX_x_0(0,v)≥ε_0^1/2dist(x_0,exp_x_0(v))=ε_0^1/2‖ v‖.

Moreover, if we consider a compact subset K⊂ X, the 
constants δ_0 and C_1 can be chosen uniformly for all 
x_0∈ K. Similarly, by (<ref>), on a compact subset K, we have

    Φ^TX_x_0(0,v)^2≤κ(R^L, K) 
    	‖ v‖^2,   x_0∈ K.




We trivialize the line bundle L on B^T_x_0X(0, 2δ_0) 
using the parallel transport with respect to ∇^L along 
the curve [0,1]∋ t↦ tv, v∈ B^T_x_0X(0, 
2δ_0). Under this trivialization, for v,v' ∈ B^T_x_0X(0, 
2δ_0), 

    P_p(exp_x_0(v),exp_x_0(v'))∈End(L_x_0^p⊗ E_x_0)=.



By <cit.>, for any compact subset K⊂ X, there exists a constant C'>0 so that for any ℓ, 
m, N∈ℕ, there 
exists δ>0 and constant C=C(K,ℓ,m,N)>0 such that for 
x∈ K, v,v'∈ T_xX, multi-indices α, α'∈ℕ^2n with |α|+|α'|≤ m
‖ v‖, ‖ v'‖≤ 2δ, we have

    |∂^|α|+|α'|/∂ 
    		v^α∂ (v')^α'(1/p^nP_p(exp_x(v),exp_x(v'))-∑_r=0^Nℱ_r(√(p)v,√(p)v')κ^-1/2(v)κ^-1/2(v')p^-r/2)|_𝒞^ℓ(K)
       ≤ 
    		Cp^-(N+1-m)/2(1+√(p)‖ v‖+√(p)‖ 
    		v'‖)^2(N+n+ℓ)+2+mexp(-C'√(p)‖ v-v'‖)+𝒪(p^-∞).

The functions ℱ_r, r∈ are given as follows,

    ℱ_r(v,v')=𝒫_x(v,v')𝒥_r(v,v'),

where 𝒥_r(v,v') is a polynomial in v,v' of degree ≤ 3r, whose 
coefficients are smooth in x∈ X. In particular,

    𝒥_0=1.

The notation 𝒪(p^-∞) means that this term is 
bounded uniformly by C^''p^-k for any given k∈ 
and with some constant C^''>0 which is independent of the 
choices of
x∈ K,  
v, v'∈ T_xX and p involved in (<ref>).


One crucial step in the approach of Ma-Marinescu <cit.> for the Bergman kernel expansions is the localization the 
calculation of the Bergman kernel near one given point <cit.> due to the 
finite propagation speed of solutions of hyperbolic equations. For our 
convenience of explaining the proofs of Toeplitz kernels in next 
Section, we sketch this localization technique in the sequel.

Let _p denote the -operator on 
𝒞^∞_c(X, 
Λ^∙T^*(0,1)X⊗ L^p⊗ E), and let ^∗_p 
denote its formal adjoint with respect to the -inner product. 
We always take the maximal extensions of _p, 
^∗_p as differential operators on 
(X,Λ^∙T^*(0,1)X⊗ L^p⊗ 
E). Since (X,g^TX) is assumed to be complete, then the maximal 
extension of ^∗_p coincides with the Hilbert adjoint of 
the maximal extension of _p. We still use the same notation to 
denote the above operators.

The Kodaira Laplacian 
^E_p=_p^∗_p+^∗_p_p is 
a densely defined, positive operator. In our setting, it has a unique 
self-adjoint extension, denoted also by ^E_p and called Gaffney extension, whose domain is 
given by

    Dom(^E_p)={s∈Dom(_p)∩Dom(^∗_p) : _ps∈Dom(_p^∗), ^∗_ps∈Dom(_p)}.

Then we have

    H^0_(2)(X,L^p⊗ E)= (_p|_(X, L^p⊗ 
    E))=(^E_p|_(X, L^p⊗ 
    E)).


The 
assumptions in (<ref>) implies a spectral gap 
<cit.> so that there exists C>0, C'>0 such that

    Spec(^E_p)⊂{0}∪ [ Cp-C',+∞[ .

Moreover, the higher Dolbeault 
-cohomology groups vanish for p≫ 0.
We are mainly concerned with ^E_p|_(X, L^p⊗ 
E), which will be denoted simply by ^E_p in the sequel. 


Recall that the injectivity radius inj^X_U is defined in (<ref>). Fix δ∈ [0,inj^X_U/4[. Let h: → [0,1] be an even 
smooth function such that

    h(s)=
    	1    for   |s|≤δ/2,
    
    	0    for   |s|≥δ.

Set

    H(a)=(∫_-∞^+∞h(s)ds)^-1∫_-∞^+∞e^√(-1)sah(s)ds.

Then H(a) is an analytic even function that also lies in the 
Schwartz space 𝒮() and H(0)=1.

We always consider the integer p to be such that Cp-C'≥ 0 (the 
constants are from the spectral gap (<ref>)). Set

    ϕ_p(a)=1_[√(Cp-C'),+∞[ (|a|)H(a).

It is still an even function. Note that by the functional calculus, 
we have the operators H(D_p), ϕ_p(D_p) well-defined as 
bounded operators on (X,L^p⊗ E) with smooth integral 
kernels. In particular, if p>C'/C, we have

    H(D_p)-P_p=ϕ_p(D_p).


If Q,Q' are two differential operators of order m, m' with 
compact support respectively, then for any ℓ>0, there exists 
C_ℓ>0 such that for p≥ C'/C, we have for any 
s∈𝒞^∞_c(X, L^p⊗ E),

    ‖ Qϕ_p(D_p)Q's‖_≤ C_ℓ p^-ℓ‖ s‖_.

As a consequence, the Bergman projection P_p can be approximated 
by H(D_p) up to an reminder of 𝒪(p^-ℓ) as well 
as in the level of their integral kernels. In particular, 
H(D_p)(x,x') only depends on the restriction of D_p to 
B^X(x,δ), and we have

    H(D_p)(x,x')=0,  for dist(x,x')≥δ.

As a consequence, the off-diagonal estimate (<ref>) 
follows from (<ref>), (<ref>) and 
(<ref>). For the on-diagonal expansion 
(<ref>) and the near-diagonal expansion 
(<ref>), the computation localizes to the small ball 
B^X(x,δ). The details are referred to <cit.>.

Finally, we recall the off-diagonal estimates for P_p on a compact complex manifold.
[<cit.> and <cit.>]
Let (X,J,Θ) be a connected, compact Hermitian complex manifold and let (L,h_L), (E,h_E) be holomorphic line bundles on X with smooth Hermitian metrics. Assume h_L to be positive. 
Then there exist constants c>0 and p_0∈ such that for any m∈, there exists a constant C_m>0 such that for p≥ p_0, and for all x,y∈ X,

    |P_p(x,y)|_𝒞^m≤ C_m p^n+m/2e^-c√(p)dist(x,y).



Moreover, for any δ>0, A>0, there exist constants p_0∈, C_δ,A>0 such that for all x,y∈ X with dist(x,y)≥δ, p≥ p_0, we have

    |P_p(x,y)|_h_p,x⊗ h^*_p,y≤ C_δ,A e^-A√(plogp).



The first part of the above theorem was proved by 
Ma–Marinescu <cit.>, and their result 
holds for general (noncompact) complete manifolds with 
bounded geometry. Under the off-diagonal assumption 
dist(x,y)≥δ, for the case where 
m=0, Θ=c_1(L,h_L) and E= trivial line bundle, 
(<ref>) is called Agmon estimate, 
and also follows from the parametix of the Szegő kernels on
pseudo-convex domains (such as <cit.>), 
see also <cit.>. 
The sharper off-diagonal estimate (<ref>) 
was proved by Christ <cit.> by studying the 
near-diagonal estimate of the Green kernels for 
Kodaira Laplacians acting on (0,1)-forms.




 §.§ Berezin–Toeplitz quantization

We have defined the following spaces of bounded measurable functions 
in the Introduction: 

    ℒ^∞_c(X,)
    ⊂ℒ^∞_const(X,)
    ⊂ℒ^∞(X,).



	For any f∈ℒ^∞(X,), set
	
    T_f,p: H^0_(2)(X, L^p⊗ E) → 
    H^0_(2)(X,L^p⊗ E),  
    		T_f,p:=P_pM_f,

where M_f denotes the pointwise multiplication by f. 
The family of bounded operators {T_f,p}_p∈^∗
is called Toeplitz operator associated 
to the symbol f, and we call T_f,p the
Toeplitz operator of level p. Equivalently, T_f,p can be seen
as an operator on ℒ^2-spaces,
T_f,p:ℒ^2(X, L^p⊗ E) 
→ℒ^2(X,L^p⊗ E), 
T_f,p:=P_pM_fP_p.


The map which associates to a function f the operator T_f,p on 
(X, L^p⊗ E) is the Berezin–Toeplitz 
quantization of level p. The map 
ℒ^∞(X,)∋ f↦{T_f,p}_p∈^∗ is called the Berezin–Toeplitz 
quantization <cit.>. 
Clearly, we have

    ‖ T_f,p‖_op≤‖ f 
    	‖_ℒ^∞,

where ‖ T_f,p‖_op denotes the operator norm 
of T_f,p.

For f∈ℒ^∞(X,) and p∈^∗, T_f,p always has a smooth integral kernel given by

    T_f,p(x,x')=∫_X 
    P_p(x,x^'')f(x^'')P_p(x^'',x')
    dV(x^'').

Note also that the Hilbert adjoint of T_f,p is 
T_f,p. If f has compact support, 
an easy modification of the arguments in <cit.> shows that all T_f,p, p∈^*, are 
Hilbert-Schmidt. 

In <cit.>, the asymptotic expansion of 
T_f,p(x,x') as p→ +∞ has been studied in 
detail with the assumption that f is smooth on X. However, 
without the global smoothness of f, the same arguments
presented in <cit.> can still be utilized to obtain 
the analogues of <cit.>. Note that in <cit.> 
Toeplitz operators with 𝒞^k symbol
were considered (see also <cit.>), and the asymptotics of their kernels 
were established using the arguments from <cit.>. In particular, the first part of the following 
theorem for compact X was already given in <cit.>.




	Assume the geometric setting as in Condition <ref>.
	Given f∈ℒ^∞_const(X,). Then for 
	any compact subset K⊂ X and m∈, there exists 
C_K,m>0 such that for x∈ K and for all p≥ 1, we have 
the on-diagonal 
estimate

    |T_f,p(x,x)|_𝒞^m(K)≤ 
    	C_K,m p^n+m/2.

	For any m,ℓ∈, ε >0, 
and a compact subset K⊂ X, there exists 
C_K,m,ℓ,ε>0 such that for x,x'∈ K with 
dist(x,x')≥ε and for all p≥ 1, we have the off-diagonal 
estimate

    |T_f,p(x,x')|_𝒞^m≤ 
    	C_K,m,ℓ,ε p^-ℓ.

Let U⊂ X be an open 
	subset such that that f is smooth on U. There 
	exists a family of polynomials {Q_r,x_0(f)∈[v,v'], 
	v,v'∈ T_x_0X}_x_0∈ U with the same parity as r, 
	and smooth in x_0∈ U such that for any compact subset 
	K⊂ U, there exist δ_K>0, C'>0 such that for any 
	ℓ,m,N∈, v,v'∈ T_x_0X, ‖ v‖, ‖ v'‖≤δ_K, multi-indice α, α'∈ℕ^2n with |α|+|α'|≤ m, there exist 
C_K,ℓ,m,N>0, 
	M_K,ℓ,m,N∈ such that
	
    |∂^|α|+|α'|/∂ 
    			v^α∂ (v')^α'(1/p^nT_f,p(exp_x_0(v),exp_x_0(v'))
                   -∑_r=0^N(Q_r,x_0(f)𝒫_x_0)(√(p)v,√(p)v')κ^-1/2(v)κ^-1/2(v')p^-r/2)|_𝒞^ℓ(K)
           ≤ 
    		C_K,ℓ,m,Np^-(N+1-m)/2(1+√(p)‖ v‖+√(p)‖ 
    		v'‖)^M_K,ℓ,m,Nexp(-C'√(p)‖ 
    		v-v'‖)+𝒪(p^-∞).

In particular, we have Q_0,x_0(f)=f(x_0). 
For x_0∈ U, if f vanishes at x_0 with vanishing order N∈, then we have 
Q_r,x_0(f)≡ 0 for r≤ N-1. If 
	x_0∉ f, then we have 
	Q_r,x_0(f)≡ 0 for r=0,1,⋯. 


By considering the operator ϕ_p(D_p) defined 
in previous Section (cf. (<ref>), (<ref>)), 
we get by (<ref>), 
(<ref>) and (<ref>), 

    T_f,p=H(D_p)M_fH(D_p)+𝒪(p^-∞).

For any compact subset K⊂ 
X, for x,x'∈ K, we have

    T_f,p(x,x')   =(H(D_p)M_fH(D_p))(x,x')+𝒪_K(p^-∞)
       =∫_X 
    H(D_p)(x,x^'')f(x^'')H(D_p)(x^'',x')
    dV(x^'')+𝒪_K(p^-∞).

Recalling the definitions (<ref>), 
(<ref>) of the functions h and H, we have

by (<ref>),

    (H(D_p)M_fH(D_p))(x,x')=0,  for dist(x,x')≥ 2δ.

Then, using the same arguments from the proof of (<ref>) 
from the estimate (<ref>), we get (<ref>).

Now let us consider a compact subset K⊂ X and
take take 0<δ< inj^X_K/4 
in the definition of function h in (<ref>). 
Then for x_0∈ K and x,x'∈ 
B^X(x_0,δ) the asymptotic expansion of 
T_f,p(x,x') is the same as (H(D_p)M_fH(D_p))(x,x'), 
and as in <cit.>, the computations of the expansion only depend 
on the values of f on B^X(x_0, 2δ)⊂ U, then the 
lack of global smoothness of f will not make any difference on this 
computations near x_0. More precisely, we consider the point 
x_0∈ K, we trivialize the line bundles L, E on the small ball 
B^X(x_0, 3δ) along the radial geodesics from the center 
x_0 with respect to their Chern connections, so that locally the 
line bundles L, E are identified with the trivial line bundles 
given by L_x_0, E_x_0 respectively. Under this 
trivialization, for v,v'∈ T_x_0X, ‖ v ‖,‖ v'‖ 
<3δ, we have

    P_p(exp_x_0(v),exp_x_0(v'))∈End(L_x_0^p⊗ E_x_0)=.

We regard P_p,x_0(v,v'):= P_p(exp_x_0(v),exp_x_0(v')), (v,v'∈ T_x_0X, ‖ v ‖,‖ v'‖ 
<3δ), as a smooth section function over TX×_K TX. We 
refer to <cit.> for more details.

Let ρ:→ 
[0,1] be a smooth even function such that

    ρ(s)=
    		1      if   |s|<2, 
    
    		0      if   |s|>4.

Then for v,v'∈ T_x_0X with ‖ v‖, ‖ v'‖ < 
δ, we have

    T_f,p(exp_x_0(v), exp_x_0(v'))
       =∫_v”∈ T_x_0 
    	X   
    	P_p,x_0(v,v^'')ρ(‖ 
    	v^''‖/δ)f(exp_x_0(v^''))P_p,x_0(v^'', v')κ_x_0(v^'')dv^''+𝒪_K(p^-∞).

where κ_x_0 is given as the function κ in 
(<ref>) but we put the subscript x_0
 to indicate the base point x_0∈ K. Note that in 
 (<ref>), we do not need f to be smooth near x_0.
 
To conclude (<ref>), we can proceed as in the proof of <cit.>. Indeed, the derivatives on variable x_0 at a given 
point x_0 can be 
replaced by the derivatives on v and v' in (<ref>), 
which eventually makes derivatives on v or v' of 
P_p,x_0(v,v'), then the factor p^n+m/2 follows from 
(<ref>). 


Now we assume that f is smooth on U and K is a compact subset 
of U. In the above arguments to obtain  (<ref>), we 
take 0<δ< min{inj^X_K, dist(K, 
X∖ U)}/4. 
Following exactly the same arguments in <cit.>, the identity (<ref>) gives the expansion 
(<ref>) with Q_0,x_0(f)=f(x_0) for the case m=0. 

For general m∈, we note that the expansions of 
∂^|α|+|α'|/∂ v^α∂ (v')^α'T_f,p(exp_x_0(v),exp_x_0(v')) is given by computing the following integration

    p^n∫_v”∈ T_x_0 
    	X ∂^|α|/∂ v^αP_p,x_0(v,v^'')κ_x_0^1/2(v^'')ρ(‖ 
    	v^''‖/δ)f(exp_x_0(v^''))∂^|α'|/∂ (v')^α'P_p,x_0(v^'', v')κ_x_0^1/2(v^'')dv^'',

where we apply (<ref>) for 
∂^|α|/∂ 
v^αP_p,x_0(v,v^'')κ_x_0^1/2(v^'') and ∂^|α'|/∂ (v')^α'P_p,x_0(v^'', v')κ_x_0^1/2(v^''). Then (<ref>) for general m≥ 0 follows from the analogous arguments as in <cit.> (also cf. <cit.>) and a 
simple observation on <cit.>: for 
polynomials F, G∈[v,v'] there exists a polynomial 
𝒦[F,G]∈[v,v'] such that 

    ∫_v^''∈^2n 
    	F(v,v^'')𝒫(v,v^'')G(v^'',v')𝒫(v^'',v')dv^''
    	=𝒦[F,G](v,v')𝒫(v,v'),

 and for multi-indice α, α'∈ℕ^2n, we have

    ∫_v^''∈^2n∂^|α|/∂ 
    	v^α(F(v,v^'')𝒫(v,v^''))∂^|α'|/∂ (v')^α'(G(v^'',v')𝒫(v^'',v'))dv^''           
    
    	=∂^|α|+|α'|/∂ v^α∂ (v')^α'(𝒦[F,G](v,v')𝒫(v,v')).

This way, we complete the proof of (<ref>).


The formula for general 
Q_r,x_0(f)(v,v') is given by <cit.> inductively when f is 𝒞^r near x_0, more precisely, we have

    Q_r,x_0(f)=∑_ℓ_1+ℓ_2+|α|=r𝒦[𝒥_ℓ_1, x_0, ∂^α f_x_0/∂ v^α(0)v^α/α!𝒥_ℓ_2, x_0],

where 𝒥_ℓ, x_0 is the polynomial in (<ref>) of degree ≤ 3ℓ.
Thus we conclude directly that Q_r,x_0(f) is a polynomial whose coefficients are given in terms of the derivatives of f at x_0 up to order r, by induction on r, we get Q_ℓ,x_0(f)≡ 0 for ℓ≤ r-1 if f vanishes at x_0 up to order r. If 
x_0∉ f, we can modify f to an 
equivalent function which is identically 0 near x_0 so that 
they define the same Toeplitz operator, in particular, from again
<cit.>, we conclude Q_r,x_0(f)≡ 0.
The proof is completed.



	In fact, as in <cit.> (cf. Theorem <ref>), from 
	(<ref>), we can improve the estimate 
	(<ref>) to the following result: for any compact 
	subset K⊂ X, any m∈ and δ'>0, there exist constants 
	C,c>0 such that for x,x'∈ K, dist(x,x')≤δ' and p≥ 1, we have
	
    |T_f,p(x,x')|_𝒞^m≤ 
    		Cp^n+m/2e^-c√(p) dist(x,x')+𝒪(p^-∞).



As in <cit.>, instead of assuming f to be smooth on U, we can consider an assumption of lower regularity. We have the following result, which is essentially a local version of <cit.>.

Assume the geometric setting as given in Condition <ref>.
	Given f∈ℒ^∞_const(X,). For N∈. Let U⊂ X be an open 
	subset such that that f is 𝒞^N+1 on U. For any compact subset 
	K⊂ U, there exists δ_K>0 such that for any 
	m∈{0,1,⋯,N}, v,v'∈ T_x_0X, ‖ v‖, ‖ v'‖≤δ_K, there exist 
C_K,m,N>0, 
	M_K,m,N∈ such that
	
    |1/p^nT_f,p(exp_x_0(v),exp_x_0(v'))
                   -∑_r=0^m(Q_r,x_0(f)𝒫_x_0)(√(p)v,√(p)v')κ^-1/2(v)κ^-1/2(v')p^-r/2|_𝒞^0(K)
           ≤ 
    		C_K,m,Np^-(m+1)/2(1+√(p)‖ v‖+√(p)‖ 
    		v'‖)^M_K,m,Nexp(-C'√(p)‖ 
    		v-v'‖)+𝒪(p^-∞),

	where the polynomials Q_r,x_0(f) (r=0,⋯, N) are the same as given in Theorem <ref> but their coefficients are of class 𝒞^N+1-r in variable x_0∈ U.


We proceed as in the proof of Theorem <ref>, notice that (<ref>) still holds uniformly for x_0∈ K, this way, we only need the local computation to conclude (<ref>), and the local computation is the same as in <cit.>.



As a consequence of Theorem <ref>, analogous to 
<cit.>, <cit.> and employing the same arguments in their proof, we also have the following result:

Assume the geometric setting as given in Condition <ref>.
	For f∈ℒ^∞_c(X,), if there 
	exists a point x_0∈ X such that |f(x_0)|=‖ 
	f‖_ℒ^∞ and f is 𝒞^1 near x_0, 
	then 
	
    lim_p→ +∞‖ 
    		T_f,p‖_op=‖ 
    		f‖_ℒ^∞.

	More precisely, there exists a constant C>0 such that for p≫ 0,
		
    ‖ 
    		T_f,p‖_op≥‖ 
    		f‖_ℒ^∞-C/√(p).



 	In fact, the result (<ref>) still holds true if we only assume f 
 	to be continuous at its (essential) maximum point x_0, which was proved in 
 	<cit.> for the case of compact manifolds.


The following result is an extension of <cit.>,
where it is proved for compact case, and the same proof applies
in our setting since the (essential) support of f is assumed to be compact.
[cf. <cit.>]
	For f∈ℒ^∞_c(X,), then T_f,p is a trace class. We have the 
	following expansion as p→ +∞: for any N∈,
	
    Tr[T_f,p]=∑_r=0^N t_r,fp^n-r+𝒪(p^n-N-1),

	where t_r,f=∫_X f(x)b_r(x) dV(x), the functions b_r (r=0, 1, …) are given in (<ref>).


Let us discuss a bit more on the asymptotics of the spectrum of T_f,p.
We restrict us to the case 
f∈ℒ^∞_c(X,_≥ 0). 
Then T_f,p is a self-adjoint compact operator on 
H^0_(2)(X,L^p⊗ E). When d_p=∞, 
the residual spectrum of T_f,p contains only 0,
and each nonzero eigenvalue in the point 
spectrum of T_f,p always has finite multiplicity.
 When f is given as the indicator function of a (Borel) 
 subset of X, the asymptotic statistics of the eigenvalues 
 of T_f,p were studied by 
 <cit.> 
 in various settings for compact or noncompact X. 
 Moreover, for a continuous function f≥ 0 with compact support, 
 the spectral densitiy measure 
μ_f,p of T_f,p is defined as the sum of the Dirac masses
at all the eigenvalues (counted with multiplicities) of T_f,p,
which is locally finite. 
A result of <cit.> shows that as p→ +∞,
we have the weak convergence of measures on (0, |f|_𝒞^0],

    p^-nμ_f,p→ f_*(1/n!c_1(L,h_L)^n).

This extends the results for compact Kähler manifolds or domains in ^n,
such as <cit.>.

In Section <ref>, we will give more results
for the asymptotics of the lowest eigenvalues of T_f,p
on a compact Hermitian manifold.










 §.§ Compositions of Toeplitz operators

Now we consider the composition T_f,p∘ T_g,p (or simply T_f,pT_g,p) of Toeplitz 
operators for two 
functions f,g∈ℒ^∞_const(X,). Based at 
the previous Section, the 
following theorem is an easy extension of some results presented by 
Ma and Marinescu in <cit.> and <cit.>.


Assume the geometric setting as in Condition <ref>.
	Given f, g∈ℒ^∞_const(X,). Then for any compact subset K⊂ X, there exists 
C_K,m>0 such that for x∈ K and for all p≥ 1, we have 
the on-diagonal 
estimate

    |(T_f,pT_g,p)(x,x)|_𝒞^m≤ 
    	C_K,m p^n+m/2.

	
For any m,ℓ∈, ε >0, 
and a compact subset K⊂ X, there exists 
C_K,m,ℓ,ε>0 such that for x,x'∈ K with 
dist(x,x')≥ε and for all p≥ 1, we have the off-diagonal 
estimate

    |(T_f,pT_g,p)(x,x')|_𝒞^m≤ 
    	C_K,m,ℓ,ε p^-ℓ.


	
	Let U⊂ X be an open 
	subset such that that both f and g are smooth on U. There 
	exists a family of polynomials {Q_r,x_0(f,g)∈[v,v'], 
	v,v'∈ T_x_0X}_x_0∈ U with the same parity as r, 
	and smooth in x_0∈ U such that for any compact subset 
	K⊂ U, there exist δ_K>0, C'>0 such that for any 
	ℓ,N,m∈, v,v'∈ T_x_0X, ‖ v‖, ‖ v'‖≤δ_K and multi-indices α, α'∈ℕ^2n with |α|+|α'|≤ m, there exist C=C_K,ℓ,m,N>0, 
	M_K,ℓ,m,N∈ such that
	
    |∂^|α|+|α'|/∂ 
    			v^α∂ (v')^α'(1/p^n(T_f,pT_g,p)(exp_x_0(v),exp_x_0(v'))                                        
    
    			-∑_r=0^N(Q_r,x_0(f,g)𝒫_x_0)(√(p)v,√(p)v')κ^-1/2(v)κ^-1/2(v')p^-r/2)|_𝒞^ℓ(K)   
    ≤ 
    		Cp^-(N+1-m)/2(1+√(p)‖ v‖+√(p)‖ 
    		v'‖)^M_K,ℓ,m,Nexp(-C'√(p)‖ 
    		v-v'‖)+𝒪(p^-∞).

	In particular, we have Q_0,x_0(f,g)=f(x_0)g(x_0). If 
	x_0∈ U and
	x_0∉ f ∩ g, then we have 
	Q_r,x_0(f,g)≡ 0 for r=0,1,⋯.



	The proof of this theorem follows from the analogous arguments in 
	the proof of <cit.> and of Theorem 
	<ref> in previous Section. In particular, for 
	x_0∈ X, for v,v'∈ T_x_0X with ‖ v‖, 
	‖ v'‖≤δ (δ>0 is chosen properly), then 
	we have (cf. <cit.>)
	
    (T_f,pT_g,p)(exp_x_0(v),exp_x_0(v'))
           =∫_v^''∈ T_x_0X 
    				T_f,p(exp_x_0(v),exp_x_0(v^''))ρ(‖ v^''‖/δ)T_f,p(exp_x_0(v^''),exp_x_0(v^'))κ_x_0(v^'')dv^''
                       +𝒪(p^-∞).

	Then combining (<ref>) with (<ref>) 
	and (<ref>), we conclude the estimate 
	(<ref>). Then similarly, by (<ref>), we 
	get (<ref>). When f, g are smooth near x_0, the 
	expansion (<ref>) follows from the combination of the expansion (<ref>), (<ref>) and (<ref>) by the arguments given in <cit.>. Note that an explicit formula for Q_r,x_0(f,g) is 
given in <cit.>(see also (<ref>)). The proof is complete.




The polynomial Q_r,x_0(f,g)(v,v') in v,v' has degree ≤ 3r, and its coefficients are given by polynomials in terms of the derivatives of f, g at x_0 up to order r.


We use the notation in <cit.> (see also (<ref>)), we have

    Q_r,x_0(f,g)=∑_ℓ_1+ℓ_2=r𝒦[Q_ℓ_1,x_0(f),Q_ℓ_2,x_0(g)].

By (<ref>) and (<ref>), we have Q_ℓ_1,x_0(f) ≤ 3ℓ_1, Q_ℓ_2,x_0(g) ≤ 3ℓ_2, hence the degree of Q_r,x_0(f,g) (in v,v') ≤ 3r. The rest part for the coefficients is also deduced from (<ref>). The proof is completed.




	Analogous to Remark <ref>, we can improve the estimate 
	(<ref>) to the following result: for any compact 
	subset K⊂ X, any m∈ and δ'>0, there exist constants 
	C,c>0 such that for x,x'∈ K, dist(x,x')≤δ' and p≥ 1, we have
	
    |(T_f,pT_g,p)(x,x')|_𝒞^m≤ 
    		Cp^n+m/2e^-c√(p) dist(x,x')+𝒪(p^-∞).




As an analog of Theorem <ref>, combining the localized computation (<ref>) with the arguments in <cit.>, we conclude the following near-diagonal expansions for the kernel of T_f,pT_g,p.

	Assume the geometric setting as given in Condition <ref>. Given N∈ and f, g∈ℒ^∞_const(X,).
	Let U⊂ X be an open 
	subset such that that both f|_U,  g|_U∈𝒞^N+1(U). Then for any compact subset 
	K⊂ U, there exists δ_K>0 such that for any 
	m∈{0,1,⋯,N}, v,v'∈ T_x_0X, ‖ v‖, ‖ v'‖≤δ_K, there exist C_K,m,N>0, 
	M_K,m,N∈ such that
	
    |1/p^n(T_f,pT_g,p)(exp_x(v),exp_x(v'))                                        
    
    			-∑_r=0^m(Q_r,x(f,g)𝒫_x)(√(p)v,√(p)v')κ^-1/2(v)κ^-1/2(v')p^-r/2|_𝒞^0(K)   
    ≤ 
    		C_K,m,Np^-(m+1)/2(1+√(p)‖ v‖+√(p)‖ 
    		v'‖)^M_K,m,Nexp(-C'√(p)‖ 
    		v-v'‖)+𝒪(p^-∞),

	where the coefficients Q_r,x(f,g) are the polynomials given in Theorem <ref>. 








 §.§ Normalized Berezin–Toeplitz kernels;
proof of Theorem <ref>

In this Section, we give the proof of Theorem <ref>.
Due to the expansion presented in Theorem <ref>, the 
	proof of this theorem is an easy modification of the proofs of 
	<cit.>. We include the details as 
	follows. 


	Note that since f is smooth on an open neighbourhood of 
U, so that we can apply the asymptotic expansion 
(<ref>) for all points x,y∈ U. In particular, we have 
the uniform expansion on U,

    T_f,p^2(x,x)=p^nf(x)^2b_0(x)+𝒪(p^n-1),

where b_0(x) is as given in (<ref>). Since 
we assume f to be nonvanishing on U, then there exists 
c_U>0 such that for all sufficiently large p>0, we have

    T_f,p^2(x,x)≥ c_Up^n.



	
	
	We start by proving the second estimate of the 
theorem. One way to see this estimate is from (<ref>), 
where the constant M_k is determined by the constant c that appears in 
the exponential term. In the sequel, we prove it
by the arguments in <cit.>.


Note that U is relatively compact in X, so 
U is compact and all the results of Theorem 
<ref> are 
applicable for the points in U. Let δ_K>0 be the sufficiently small quantity 
stated in the last part of Theorem 
<ref> with K=U. Then by 
(<ref>), if 
x,y∈ U is such that dist(x,y)≥δ_K, we have

    |T_f,p^2(x,y)|_h_p,x⊗ h^*_p,y≤ C_U,0,k,δ_K  p^-k.


For the given k, we will determine a constant M_k later on.
We fix a large enough p_0∈ℕ such that

    b √(logp_0/p_0)≤δ_K/2.

For p>p_0, if x,y∈ U is such that 
b√(logp/p)≤dist(x,y)< δ_K, then 
we take advantage of the expansion in (<ref>) with N=2k, x_0=x, 
v=0, 
y=exp_x(v'), and v'∈ T_xΣ, in order to obtain

    |1/p^nT^2_f,p(x,y)-∑_r=0^2k(Q_r,x(f,f)𝒫_x)(0,√(p)v')κ^-1/2(v')p^-r/2|   
    ≤ 
    		Cp^-(k+1/2)(1+√(p)‖ 
    		v'‖)^M_K,0,0,2kexp(-C'√(p)‖ 
    		v'‖)+𝒪(p^-∞).


Now for k ≥ 1, by Lemma <ref>, we have 

    ℓ(r):=max{_v'Q_r,x(f,f)(0,v'), x∈ U}≤ 3r.




Note that ‖ v'‖ =dist(x,y). By 
(<ref>), (<ref>), (<ref>),
 and the properties of 
Q_r,x(f,f) in Theorem <ref>, together with the fact 
that δ_K>‖ v'‖≥ 
b√(logp/p), for r=0,1,⋯, 2k, we get that

    |(Q_r,x(f,f)𝒫_x)(0,√(p)v')κ^-1/2(v')|≤ C_K,r 
    	p^ℓ(r)/2exp (-ε_0/4 
    	b^2logp ),

where the constant C_K,r>0 does not depend on x∈ U. If

    ε_0/4b^2≥ 
    	3k,

then we have for r=0,…, 2k,

    |(Q_r,x(f,f)𝒫_x)(0,√(p)v')κ^-1/2(v')p^n-r/2|≤ 
    	C_K,rp^n-k.

We may take M_k= 12k in 
our constraint for b.
Finally, combining (<ref>)–(<ref>), we get the 
desired estimate for any p>p_0.


We next prove the first part of our theorem. For 
this purpose, we only need to consider sufficiently large p such that 
b√(logp/p)≤δ_K/2, or p≥ p_0.

Recall that κ(R^L, U)>0 is defined in (<ref>). We set

    m(U,b):=⌈b^2/2κ(R^L, U)⌉.

By (<ref>), for ‖ v'‖ =|z'|≤ 
b√(logp/p), we have

    exp(p/4Φ^TX_x(0,v')^2 ) ≤ p^m(U,b)/2.



In the expansion (<ref>), we take x_0=x, y=exp_x(v'), 
N=m(U,b), so dist(x,y)=‖ v'‖ =|z'|≤ 
b√(logp/p), where z'∈ is the complex coordinate 
for v'. We infer

    |1/p^nT^2_f,p(x,y)-∑_r=0^m(U,b)(Q_r,x(f,f)𝒫_x)(0,√(p)v')κ^-1/2(v')p^-r/2|
    		≤ 
    		Cp^-(m(U,b)+1)/2.

Since ‖ v'‖≤ b√(logp/p), by (<ref>) we infer that
|Q_r,x(f,f)(0,√(p)v')|≤ C_f,U|logp|^ℓ(r)/2 for some constant C_f,U>0. Note that |logp|^ℓ(r)/2 p^-r/2=𝒪(p^-1/2+ϵ) for r≥ 1 as p grows.


Note that f is a real function which does not vanish near U. The expansion (<ref>) in combination with (<ref>) then supplies us with

    exp(p/4Φ^TX_x(0,v')^2 )T^2_f,p(x,y)/√(T_f,p^2(x,x))√(T_f,p^2(y,y))   =b_0(x)f(x)^2κ^-1/2(v')+𝒪(p^-1/2+ϵ)/√(f(x)^2b_0(x)+𝒪(p^-1))√(f(y)^2b_0(y)+𝒪(p^-1))
       =1+𝒪(‖ v'‖ +p^-1/2+ϵ)
       =1+o(1),  as p→ +∞.

In (<ref>), the small term o(1) in the last line represents our function R_p,x(v'), then (<ref>) follows clearly.

Since in the asymptotic expansion (<ref>) we 
have trivialized the line bundle near x using the Chern 
connections, we have

    |T_f,p^2(x,y)|_h_p,x⊗ 
    	h^∗_p,y=|T_f,p^2(x,y)|.

Combining (<ref>) and (<ref>), we get the first part of (<ref>). This completes the proof of Theorem 
<ref>.	


Now, when f has a lower regularity such as 𝒞^m+1 with m=m(U,b), the above arguments still hold with the bounded choices of k. We give the proof of Corollary <ref> as follows.




The inequality (<ref>) does not require a specific regularity on f due to (<ref>). Then by Theorem <ref>, when f|_U is of 𝒞^m(U)+1(U), the arguments (<ref>) – (<ref>) holds true with k=n+1 and b=√(12n+12/ε_0), so that the second part of (<ref>) holds. The first part of (<ref>) follows from the same arguments as in last step of the proof of Theorem <ref>, i.e., (<ref>) – (<ref>). This way, we complete our proof.




Note that the upper bound (<ref>) is not optimal for R_p,x(v'), since we have R_p,x(0)=0 and ∇ R_p,x(0)=0 (here ∇ denotes the coordinate derivatives in v').
The following estimate is an analog of <cit.> in our Berezin–Toeplitz setting.

With the same assumptions in Theorem <ref> (in particular, f is smooth near U), the term R_p,x(v') satisfies the following estimate: there exists C_1=C_1(f,ϵ, U)>0 such that for all sufficiently large p, x∈ U, v'∈ T_x X with v'≤ b√(logp),

    |R_p,x(v'/√(p))|≤ C_1v'^2 p^-1/2+ϵ


For given k, ℓ∈, there exist a sufficiently large b>0 such that there exist a constant C_2>0 such that for all x,y∈ U, dist(x,y)≥ b√(logp/p), we have

    |∇^ℓ_x,yN_f,p(x,y)|≤ C_2 p^-k.



Since f is smooth on a neighborhood of U, then the expansion (<ref>) holds on U with 𝒞^ℓ-norm, this way, (<ref>) follows by repeating the same arguments in the first part of the proof of Theorem <ref>. 

By (<ref>), we conclude that for all x∈ U, v'∈ T_xX, v'≤δ_K (where K=U), we have

    |∇^2 R_p,x(v')|≤ Cp^1/2(1+(√(p)v')^M),

for some large integer M∈.

Set H_p,x(v')=R_p,x(v'/√(p)). Then for v'≤ b√(logp),

    |∇^2 H_p,x(v')|≤ C_ϵ,U p^-1/2+ϵ.

Then 

    |∇ H_p,x(v')|≤sup_t∈ [0,1] |(∇^2 H_p,x)(tv')|·v'≤  C_ϵ,U p^-1/2+ϵv'.

Then (<ref>) follows after a similar computation as in (<ref>).













§ GAUSSIAN ℒ^2-HOLOMORPHIC SECTIONS VIA TOEPLITZ 
OPERATORS

In this section we recall the Gaussian -holomorphic 
section of a Hermitian line bundle on a complex manifold defined 
through a given Hilbert-Schmidt Toeplitz operator, this kind of 
random holomorphic sections were constructed in <cit.>, taking
advantage of the theory of abstract Wiener spaces.




 §.§ Gaussian ℒ^2-holomorphic 
sections


We now recall the construction of Gaussian ℒ^2-holomorphic 
sections given in <cit.>. Let (F,h_F) be a holomoprhic line bundle on X with smooth Hermitian metric h_F. Furthermore, by H^0_(2)(X,F) we denote the space of ℒ^2-holomorphic sections of F on X, which is a separable Hilbert space with the Hilbert metric given by the ℒ^2-inner product induced by h_F and dV. Set

    d_F:=_ H^0_(2)(X,F)∈∪{∞}.

In this Section, we always assume that d_F>0 and by P: (X,F)→ H^0_(2)(X,F) we denote the Bergman projector.

Let ℒ^∞_c(X,)
be the space of measurable essentially bounded real functions on X
with compact essential support.
For f∈ℒ^∞_c(X,)
the Toeplitz operator 
T_f=PM_f: H^0_(2)(X,F)→ H^0_(2)(X,F)
is Hilbert-Schmidt and self-adjoint.
We will consider in the sequel only non-trivial functions 
f≠0∈ℒ^∞_c(X,).



When d_F<∞, for f≥ 0 with compact support, 
T_f is a linear isomorphism of H^0_(2)(X,F)
with strictly positive eigenvalues. When d_F=∞, 
the residual spectrum of T_f contains only 0, 
and each nonzero eigenvalue in the point 
spectrum of T_f always has finite multiplicity. 
In particular, the nonzero eigenvalues of T^2_f form
a decreasing sequence of strictly positive real numbers, 

    λ^2_1≥λ^2_2≥…≥λ^2_m≥…→ 0,

where λ_1,λ_2,… denote the 
nonzero values in the point spectrum of T_f, 
repeated according to their multiplicities. If furthermore f≥ 0, 
the eigenvalues λ_1,λ_2,… are strictly positive.

Next, we introduce
	
    H^0_(2)(X,L,f):=( T_f)^⊥=T_fH^0_(2)(X,L)⊂ 
    		H^0_(2)(X,L),


which is a Hilbert space, and the sections in H^0_(2)(X,L,f) 
are the -holomorphic 
sections of L detected by f. We consider the (self-adjoint)
Hilbert-Schmidt operator
	
    T_f^♯:=T_f|_H^0_(2)(X,L,f): 
    		H^0_(2)(X,L,f)→ 
    		H^0_(2)(X,L,f).

Note that for a notrivial f≥ 0 with compact support, 
T_f is injective and H^0_(2)(X,L,f)=H^0_(2)(X,L).
We furthermore set d_F(f):=_ H^0_(2)(X,L,f)∈∪{∞},
and it is immediate that d_F(f)≤ d_F. 

The following lemma now is elementary.

For f∈ℒ^∞_c(X,)∖{0}
we have

    d_F(f)≥1/|f|^2_ℒ^∞(X)∫_X T^2_f(x,x)dV(x).



Suppose that T_f≠ 0 (that is, d_F(f)≥ 1). In this case, we can choose
 an orthonormal basis {S_j}^d_F(f)_j=1  of 
H^0_(2)(X,F,f) with respect to the -metric 
 such that

    T_fS_j=λ_jS_j.



On H^0_(2)(X,L,f) with d_F(f)≥ 1, the operator T_f^♯ is one-to-one and Hilbert-Schmidt, and
·_f:=T^♯_f· defines a Hermitian measurable norm on 
H^0_(2)(X,F,f) due to <cit.>. We furthermore denote by ℬ_f(X,F) the 
completion of H^0_(2)(X,F,f) with respect to 
·_f and set

    ℓ^2_f()={(a_j∈)_j≥ 1 : ∑_j≥ 
    	1λ_j^2|a_j|^2<∞}.

Endowed with the induced norm this is clearly a separable Hilbert space, and using the basis as in (<ref>), we have

    ℬ_f(X,F)≃ℓ^2_f().




	Assume 0≠ f∈ℒ^∞_c(X,) and T_f≠ 0. 
	Then the operator T_f extends uniquely to an isomorphism of Hilbert spaces 
	
    T_f: (ℬ_f(X,F),·_f)→ (H^0_(2)(X,F,f),·_(X,F)).



Given 0≠ f∈ℒ^∞_c(X,_≥ 0), if d_F(f)<∞, 
we set

    (ℬ_f(X,F),·_f)= 
    (H^0_(2)(X,F,f),·_f),  and  T_f:=T^♯_f.

This unifies the notation for both cases d_F<∞ and 
d_F=∞.


Let ℬ_f(X,F)^* be the topological dual space of ℬ_f(X,F). 
If α∈ℬ_f(X,F)^*, then it is uniquely determined by the  continuous linear functional on α|_H^0_(2)(X,F,f) on H^0_(2)(X,F,f). This 
way, we regard ℬ_f(X,F)^* as a (dense) subspace of 
H^0_(2)(X,F,f)^*, where H^0_(2)(X,F,f)^* can be identified with H^0_(2)(X,F,f) via the -inner product. By a slight abuse of notation we denote by 
𝒮_f the Borel σ-algebra of ℬ_f(X,F). Then each 
α∈ℬ_f(X,F)^* is a Borel-measurable function from 
ℬ_f(X,F) to .
For V⊂ℬ_f(X,F)^*⊂ H^0_(2)(X,F,f) an arbitrary finite 
dimensional subspace we introduce the notation

    ϕ_V: ℬ_f(X,F)   → V, ϕ_V(b)=∑_j=1^_ V (b,v_j)v_j,

where {v_j} is an orthonormal basis of (V, ·_(X,F)).
Gross <cit.> proved the following result. 
[L. Gross <cit.>, see also <cit.>]
	Let ·_f denote the measurable norm on H^0_(2)(X,F,f) as introduced after (<ref>). 
	There exists a unique probability measure 𝒫_f on 
	(ℬ_f(X,F),𝒮_f) such that for  V⊂ℬ_f(X,F)^* any finite dimensional 
	subspace,
	
    𝒫_f(ϕ^-1_V(U))=μ_V, ·_(X,F)(U),

	for all Borel subset  U of V, where μ_V, ·_(X,F) denotes the standard 
	Gaussian measure on V with respect to the Hermitian metric 
	associated with ·_(X,F). The triple 
	(ℬ_f(X,F),𝒮_f,𝒫_f) is called an abstract 
	Wiener space.









Let _f be the
Gaussian probability measure on H^0_(2)(X,F,f) given 
by the pushforward of the probability measure 𝒫_f from Theorem <ref> through the 
isomorphism in (<ref>). This way, we randomize the sections 
in H^0_(2)(X,F,f). A Gaussian (random) -holomorphic section S_f of F associated to a nonzero f∈ℒ^∞_c(X,) is a random variable valued in H^0_(2)(X,F,f) with law _f, i.e., S_f∼_f. 





	When 0<d_F(f)<∞, then ℬ_f(X,F)=H^0_(2)(X,F,f), 
	and 𝒫_f=_st is exactly the standard Gaussian 
	probability measure on H^0_(2)(X,F,f) with respect to the 
	-inner product; so _f is the pushforward of _st
	via the isomorphism T_f. For 0<d_F(f)<∞, let S denote the random holomorphic section valued in H^0_(2)(X,F,f) with law _st, that is
	
    S=∑_j=1^d_F(f)η_j S_j,

	where {S_j}_j is an orthonormal basis of H^0_(2)(X,F,f) as in (<ref>) and {η_j} is a sequence of i.i.d. standard complex Gaussian random variables. In this setting, the random section S_f associated to f is given equivalently by 
	
    S_f=T^♯_f S=T_f S.





The following lemma is an easy modification of <cit.>

Assume d_F(f)∈_≥ 1∪{∞}, 
0≠ f∈ℒ^∞_c(X,). 
For any nonzero S∈ H^0_(2)(X,F), the random variable on 
(H^0_(2)(X,F,f), _f) defined via 

    H^0_(2)(X,F,f)∋ 
    s↦⟨ s, S⟩_(X,F)∈

is a centered complex Gaussian variable with 
variance T_fS^2_(X,F). 




In the case d_F(f)=∞, we consider the orthonormal basis consisting of eigensections of T_f as in (<ref>). Setting

    η_j=1/λ_j⟨S_f,S_j⟩_(X,F)

the {η_j}_j=1^∞ form a sequence of i.i.d. standard complex Gaussian random variables. Then our random section S_f associated to f is given equivalently by the formula
	
    S_f=∑_j=1^∞η_j λ_j S_j.

The well-definedness of the sum in (<ref>) is already discussed in <cit.>. Also note that (<ref>) then is consistent with (<ref>) and (<ref>) which treated the  case d_F(f)<∞.



 §.§ Currents and the Poincaré-Lelong formula

The zero-set of a holomorphic section is a complex analytic set which 
is in general singular. The analytic tool used to deal with 
singularities in complex geometry is the theory of currents, introduced
by de Rham <cit.> (see <cit.> and especially 
<cit.> for complete expositions).  

Let X be a complex manifold of dimension n and let E
be a complex vector bundle on X. The space
of smooth sections of E is denoted by ^∞(X,E)
and is endowed with the ^∞-topology of uniform
convergence of all derivatives on compact sets.
The space
of smooth sections of E with compact support 
is denoted by ^∞_ c(X,E)
and is endowed with the topology of inductive limit
of spaces of smooth sections with support on a given compact set.
In particular, we denote by 
Ω^n-1,n-1_c(X) the space
of smooth (n-1,n-1)-forms with compact support.

The space of (1,1)-currents on X 
is the topological dual of the space Ω^n-1,n-1_c(X) 
(called test forms in this context). In the sequel, we 
let ⟨ T, φ⟩ be the pairing between a (1,1)-current T 
and a test form φ∈Ω^n-1,n-1_c(X).
A (1,1)-current is called of order k∈_0 if it is
continuous in the ^k-topology, equivalently, it
extends as a linear continuous functional to the space
of (n-1,n-1)-forms of class ^k with compact support.


For an open subset U⊂ X, if T is a (1,1)-current of order 0 (for example, a poisitive (1,1)-current), we define the following norm of T on U for α∈,

    T_U,-α:=sup|⟨ T,φ⟩|,

where φ runs over all test forms in Ω^n-1,n-1_c(U) with |φ|_𝒞^α≤ 1.


For any analytic hypersurface V⊂ X, we define
the current of integration [V] on V by 

    φ↦∫_Vφ:=∫_V_regφ,
      φ∈Ω^n-1,n-1_c(X),
 
where 
V_reg is the regular set of V (a complex submanifold 
of codimension 1). By a theorem 
of Lelong (<cit.> <cit.>) the current 
of integration on V is a closed positive (1,1)-current (hence a stronly positive (1,1)-current due to <cit.>).
It is clear that [V] is a current of order 0 on X.

Let F be a holomorphic line bundle on X.
For a holomorphic section s∈ H^0(X,F)∖{0} the divisor 
(s) of s is defined as the formal sum

    (s)=∑_V ⊂ Z(s)ord_V(s) V,

where V runs over all the irreducible analytic hypersurfaces 
contained in Z(s), and ord_V(s) denotes the 
vanishing order of s along V.
Let Z(s) denote the 
set of zeros of s, which is 
a purely 1-codimensional analytic subset of X. The current
of integration (with multiplicities) on the divisor (s)
is defined by 

    [(s)]=∑_V ⊂ Z(s)ord_V(s) [V],  ⟨[(s)],φ⟩=
    ∫_(s)φ:=
    ∑_V ⊂ Z(s)ord_V(s)∫_Vφ .

Assume that F is endowed with a smooth Hermitian
metric h_F. By the Poincaré-Lelong formula <cit.> we have

    [(s)]=√(-1)/2π∂∂̅log|s|^2_h_F+ c_1(F,h_F),
    	  for s∈ H^0(X,F).

This important formula is crucial for our purposes. It
links the zero-divisor to the curvature and to the
logarithm of the pointwise norm of a section, 
which is analytically easier to tackle and allows the introduction
of the Bergman kernel or Berezin-Toeplitz kernel into the picture.   




 §.§ Expectation of zeros of Gaussian -holomorphic sections

We use the same notation as in Section <ref>. In this Section we always assume d_F(f)∈_≥ 1∪{∞} with 0≠ f∈ℒ^∞_c(X,). Let T_f^2(x,y) denote the smooth kernel of the operator T_f^2. We consider the orthonormal basis of H^0_(2)(X,F,f) consisting of eigensections of T_f as in (<ref>), then we have the nonnegative function on X,

    T_f^2(x,x)=∑_j=1^d_F(f)λ_j^2 |S_j(x)|^2_h_F.



Similar to the proof of <cit.>, we get that the (1,1)-current ∂∂logT_f^2(x,x) is well-defined on X. If we proceed as in <cit.>, in particular, as in the proof of <cit.>, we conclude the following result.

[Definition of positive current γ_f(X,F)]
Assume d_F(f)∈_≥ 1∪{∞} with 0≠ f∈ℒ^∞_c(X,). The following current is a closed positive (1,1)-current on X,

    γ_f(F,h_F):=c_1(F,h_F)+√(-1)/2π∂∂logT_f^2(x,x).




The base locus of H^0_(2)(X,F,f) is the proper analytic set

    _f(X,F):={x∈ X :  s(x)=0 for all 
    	s∈ H^0_(2)(X,F,f)}.

Then _f(X,F)={x∈ X : T^2_f(x,x)=0}. Hence γ_f(F,h_F)
is a smooth form if _f(X,F)=∅. Moreoever, when f≥ 0, we also have _f(X,F)={x∈ X : P(x,x)=0}=: (X,F). 


Assume d_F(f)∈_≥ 1∪{∞} with 0≠ f∈ℒ^∞_c(X,). Let S_f be the random -holomorphic section of F associated to f defined in Definition <ref>. Then [Div(S_f)] is random variable valued in the space of (1,1)-currents on X (equipped with weak topology). Moreover, the expectation [[Div(S_f)]] exists as a closed positive (1,1)-current on X and we have

    [[Div(S_f)]]=γ_f(F,h_F).



For m∈_>0, consider the random (1,1)-current μ_m(S_f) as follows, for any test form φ∈Ω^n-1,n-1_c(X),

    ⟨μ_m(S_f),φ⟩:=⟨√(-1)/2π∂∂̅log(|S_f|^2_h_F+1/m)+ c_1(F,h_F), φ⟩.

It is clear that ⟨μ_m(S_f),φ⟩ is a random variable valued in  (i.e., a measurable function on the underlying probability space). Moreover, by (<ref>) and dominated convergence, we have

    lim_m→∞⟨μ_m(S_f),φ⟩=⟨ [Div(S_f)],φ⟩.

Therefore, ⟨ [Div(S_f)],φ⟩ is a random variable, and then [Div(S_f)] is a well-defined random variable valued in the space of (1,1)-currents with respect to the weak topology. 

Analogous to <cit.>, using the computations as in <cit.>(cf. <cit.>), we can conclude that for φ∈Ω^n-1,n-1_c(X),

    [⟨ [Div(S_f)],φ⟩]=⟨γ_f(F,h_F),φ⟩∈,

then we get (<ref>). This way, we complete our proof.








§ ASYMPTOTICS OF RANDOM ZEROS OF -HOLOMORPHIC SECTIONS


Let us recall the main assumption in the semi-classical limit – Condition <ref> – as follows: we 
always assume (X,g^TX) to be a complete Riemannian manifold, and 
that there exist ε_0>0, 
C_0>0 
such that

    √(-1)R^L≥ε_0Θ, √(-1)(R^det+R^E)>-C_0Θ,   |∂Θ|_g^TX< C_0.

If X is compact and (L,h_L) is positive, then the above 
conditions always hold true. Due to our assumptions on (X,Θ) and L, E, for p≫ 0, we have d_p:= _ H^0_(2)(X,L^p⊗ E)≥ 1. We may assume that d_p≥ 1 for all p≥ 1.


For p≥ 1, let S_f,p denote the random -holomorphic section of L^p⊗ E associated to the nontrivial f∈ℒ^∞_c(X,) defined in Definition <ref>.


The goal of this section is to study the asymptotic behavior of the zeros of random section S_f,p as p→∞, where the asymptotic behavior of T_f,p(x,x') is a crucial ingredient. As we saw in Section <ref>, in order to analyze T_f,p(x,x') in the semi-classical limit, we always need to assume certain regularities on f.

We start with the following lemma, which enable us to apply the results from Section <ref> for the line bundles L^p⊗ E. Recall that the function b_0(x)≠ 0 is given in (<ref>). 


For f∈ℒ^∞_c(X,), if there exists an open subset U≠∅ such that f|_U∈𝒞^1(U) and f never vanishes on U, then for all sufficiently large p, we have T_f,p≠ 0, hence d_p(f)≥ 1.

Moreover, fix a proper open subset V of U, we have

    d_p(f)≥p^n/|f|^2_ℒ^∞(X)∫_V f^2(x)b_0(x)dV(x)+𝒪(p^n-1).



If T_f,p=0, then for all x∈ X, T_f,p(x,x)≡ 0. Since f|_U∈𝒞^1(U), we can apply Theorem <ref> to f on U so that fix x∈ U, we have

    T_f,p(x,x)=p^n f(x)b_0(x)+𝒪_x(p^n-1).

By our assumption f(x)≠ 0, then for all sufficiently large p, T_f,p(x,x)≠ 0. Therefore, we have T_f,p≠ 0.

Note that

    ∫_X T_f,p^2(x,x)dV(x)≥∫_V T_f,p^2(x,x)dV(x).

Similarly, we can apply Theorem <ref> to f on U, then (<ref>) follows from Lemma <ref> and (<ref>).







 §.§ Bounded measurable functions and their supports

This Section is a preparation for the proof of Theorem <ref>. Recall that the quantity r(f,U) for f∈ℒ^∞_c(X,) is defined in Definition <ref>. Now we give some examples of function f such that r(f,U) has a good geometric sense.


	
  i) For t∈, set sin^+(tπ):=max{0,sin(tπ)}. Now fix N∈_≥ 1 and R≫ 0, let 𝔹(0,R)⊂ be the open ball in  with radius R, and define for z=x+√(-1)y∈,
	
    f_N,R(z):=1_𝔹(0,R)(z) sin^+(Nxπ)sin^+(Nyπ).

	Then f_N,R is smooth almost everywhere on , and we have for R≫ 0,
	
    r(f_N,R, 𝔹(0,R))=√(2)/2N.

	In Figure <ref>, it shows the support in black blocks of the function f_8,2(z) (that is, N=8, R=2) on .
	

  ii) Let {B_j}_j=1^N be a finite collection of disjoint geodesic open ball of respective radius r_j>0 in a compact Hermtian manifold (X,Θ), set V:=X∖∪_j B_j, then f:=1_V is a function which is smooth almost everywhere on X, and r(f,X)=max{ r_j :  j=1,…,N}.

  iii) Let (Σ,ω) be a compact Riemann surface with a Kähler form ω. Let K be the 1-skelton of a (smooth) triangluation of Σ. Then for a sufficient small ε>0, let K(ε) be the closed subset defined as the ε-neighbourhood of K in Σ (which is still a proper subset). This is a thick graph embbeded in Σ. Then we can use the quantity r(1_K(ε),Σ) to quantify the approximation of the thick graph K(ε) to Σ. Taking a sequence of strict refinements of K and a suitable decreasing sequence of ε, then the sequence of the quantities r(1_K(ε),Σ) will converge to 0.




Let g^T^2n denote the standard Riemannian
metric on ^n≃^2n given by the 
standard Euclidean inner product on ^2n. 
The following proposition is elementary.


For every compact subset K⊂ X there exist 
constants t_K>0, c_K≥ 1 such that for every point
x∈ K, the open geodesic ball 𝔹(x,t_K) in 
(X,g^TX) is always included in a connected, 
open holomorphic local chart (V_x⊂ X,ϕ_x) 
centered at x such that ϕ_x is a biholomophism 
between the open subset V_x and ϕ_x(V_x)⊂^n 
with the following properties:


  * ϕ_x(x)=(0,…,0), and g^TX_x=
ϕ_x^∗(g^T^2n_0);

  * ϕ(V_x) is a convex domain of ^n;

  * let dist^ϕ_x(·,·) 
be the distance function on V_x× V_x 
associated to the Riemannian metric 
ϕ_x^∗(g^T^2n), then for any y,y'∈ V_x,

    1/c_Kdist^ϕ_x(y,y')≤dist(y,y')≤ c_K dist^ϕ_x(y,y').


As a consequence, for a given x∈ K, and for 
f ∈ℒ^∞(X,_≥ 0), we have

    r(f,V_x)/c_K≤ r_^n(f∘ϕ_x^-1,
    ϕ_x(V_x))≤ c_K r(f,V_x),

where r_^n(f∘ϕ_x^-1,ϕ_x(V_x)) 
is the number defined as in (<ref>)
with respect to the standard Riemannian metric g^T^2n on ^n.



The choices of (t_K, c_K) for a given K in 
Proposition <ref> are not unique, 
in fact we can always make t_K as small as we want. 
Moreover, if we take a sufficiently small t_K>0, 
then the constant c_K will be very close to 1.






 §.§ A concentration estimate; proof of Theorem <ref>

In this section, we prove Theorem <ref>
which provides a general concentration estimate for the difference between
the zeros of the random section S_f,p and the expected 
limit c_1(L,h_L) on a given domain, provided that the function f 
is supported on the large part of this domain. 
Recall that the norm ·_U,-2 for the (1,1)-currents
is given Definition <ref> with α=2, and that 
the quantity m(U)∈ defined in (<ref>),

then Theorem <ref> is a consequence 
of the following proposition. 


With the same assumption in Theorem <ref> for X and for the line bundles L,E. Fix a pair of nonempty open subsets (U,U') of X with U⊂⊂ U', then there exists a constant r(U,U')>0 such that f∈ℒ^∞_c(X,) which is of 𝒞^m(U')+1 almost everywhere on U' with 

    (r(f,U')/r(U,U'))^1/(2n+2)<1/2  ,

then U∩   f≠∅, and for any 
	δ > (r(f,U')/r(U,U'))^1/(2n+2), we have a constant C=C_U',f,δ>0 such that for all sufficiently large p

    (
    		∫_U|log|S_f,p(x)|_h_p| 
    		 dV(x) >
    		δ p )≤ e^-Cp^n+1.



Now we explain how to get Theorem <ref>
from Proposition <ref>, and the proof of Proposition 
<ref> is defered to Section <ref>.

Combining the first part of Proposition <ref> with Lemma <ref>, we get that for all sufficiently large p, T_f,p≠ 0, that is d_p(f)≥ 1. Then the results from Section <ref> apply. In particular, the random section S_f,p is not the zero variable.


The Poincaré-Lelong formula (<ref>) shows that 
	
    √(-1)/π∂∂log|S_f,p|_h_p=[(S_f,p)]-pc_1(L,h_L)-c_1(E,h_E)

as an identity of (1,1)-currents on X.	Let U be the open subset as assumed in Theorem <ref>. Now fix φ∈Ω^n-1,n-1_c(X) with supp φ⊂ U (that is, φ∈Ω^n-1,n-1_c(U)). Note that supp φ⊂ U⊂ U'.
Then

    ⟨1/p[(S_f,p)],φ⟩
    -∫_Xc_1(L,h_L)∧φ=√(-1)/pπ∫_Xlog|S_f,p|_h_p ∂∂φ +1/p⟨ c_1(E,h_E),φ⟩.

Since φ has a compact support in U, so has 
∂∂φ. 

Then
	
    |√(-1)/pπ∫_Xlog|S_f,p|_h_p ∂∂φ|≤|φ|_𝒞^2(U)/pπ∫_U|log|S_f,p(x)|_h_p| dV(x).

Applying Proposition <ref>
to get a constant r(U,U')>0, we fix a f as in the theorem 
with 
    (r(f,U')/r(U,U'))^1/(2n+2) <1/2,
 
then U∩   f≠∅. 
When we take 
    δ > (r(f,U')/r(U,U'))^1/(2n+2),
 
we can always fix a sufficiently small ε>0 such that

    δ-2ε > (r(f,U')/r(U,U'))^1/(2n+2).
 
Since the term 1/p c_1(E,h_E) converges to 0 as 
p→∞, then there exists an integer p_0∈
(depending on (E,h^E)) such that for any 
φ∈Ω^n-1,n-1_c(U) and all p≥ p_0,

    |1/p⟨ c_1(E,h_E),φ⟩|
    ≤ε |φ|_𝒞^2(U)/π ·
 
Applying Proposition <ref> to right-hand side of 
(<ref>) with δ-ε, we get

    (1/p∫_U|log|S_f,p(x)|_h_p| 
    		 dV(x)>
    		δ-2ε)≤ e^-Cp^n+1.

For p≥ p_0, except the event from (<ref>) of probability 
≤ e^-Cp^n+1, we have for all φ∈Ω^n-1,n-1_c(U),

    |⟨1/p[(S_f,p)] 
    		-  c_1(L,h_L),φ⟩|
       ≤|φ|_𝒞^2(U)/pπ∫_U|log|S_f,p(x)|_h_p| dV(x)+ |1/p⟨ c_1(E,h_E),φ⟩|
       ≤1/π(|φ|_𝒞^2(U)(δ-2ε)+ε |φ|_𝒞^2(U)) ≤ |φ|_𝒞^2(U)δ-ε/π,

Equivalently, except the event in (<ref>) of probability 
≤ e^-Cp^n+1, we have 

    1/p[(S_f,p)] 
    		-  c_1(L,h_L)_U,-2≤δ-ε/π.

This way, we get (<ref>). 
To prove (<ref>), applying Borel-Cantelli lemma to 
(<ref>), we get

    (lim sup_p→ +∞ 1/p[(S_f,p)] 
    		-  c_1(L,h_L)_U,-2≤δ/π )=1.

By taking a sequence of δ's that decreases to δ_0(f),
we infer (<ref>) from (<ref>). 







 §.§ Local sup-norms of random -holomorphic sections


Before proving Proposition <ref>, we need to investigate 
the probabilities for ℳ^U_p(S_f,p)
taking both atypically large and small values, respectively. 
We follow the same strategy as in <cit.>.

Let U⊂ X be a relatively compact open subset. For s_p∈ H^0(X, L^p⊗ E), we set

    ℳ^U_p(s_p)=sup_x∈ 
    	U|s_p(x)|_h_p<+∞.



 Assume Condition <ref> to hold. Fix 0≠ f∈ℒ^∞_c(X,_≥ 0). Let U⊂ X be a relatively compact open subset, then for any δ>0, there exists a constant C_U,δ>0 such 
	that for p∈_>1, 
	
    (ℳ^U_p(S_f,p)≥ 
    		e^δ p)≤ e^-δ 
    		p^n+1+C_U,δp^nlogp .



	We fix δ>0 and let r>0 be sufficiently small so that we can 
	choose a finite set of points {x_j}_j=1^ℓ⊂ U such 
	that the geodesic open balls B^X(x_j,r), j=1,…, ℓ, 
	form an open covering of U. Since r is 
	sufficiently small, then we can assume that each larger ball
	B^X(x_j,2r) lies in a complex chart (hence viewed as an 
	open subset of ^n), and that for each j, we can fix a 
	local holomorphic frame e_L,j (resp. e_E,j) of L (resp. E) on a neighborhood of B^X(x_j, 2r) 
	with sup_x∈ B^X(x_j,2r) |e_L,j(x)|_h_L=1 and sup_x∈ B^X(x_j,2r) |e_E,j(x)|_h_E≤ 1.
	Set
	
    ν=min{inf_x∈ 
    		B^X(x_j,2r)|e_L,j(x)|_h_L : j=1 , …, ℓ}.

	It is clear that 0<ν≤ 1. By fixing r small enough, we 
	can and do assume that 
	 
    -logν≤δ/4 ·
 	
	Then there 
	exists a constant C>0 such that for each j=1,…, ℓ, if 
	f is a holomorphic function on a neighborhood of 
	B^X(x_j,2r), then
	
    sup_x∈ B^X(x_j,r)|f(x)|≤ 
    		Cf_ℒ^2(B^X(x_j,2r)),

	where the volume form dV(x) on X is used in the norm 
	·_ℒ^2(B^X(x_j,2r)).
	Note that the choices of x_j, r, ℓ, and the constants 
	ν, C
	are independent of the tensor power p. Set 
		U=∪_j B^X(x_j,2r)⊃ U.
	For p∈, s_p∈ H^0(X,L^p⊗ E), on each B^X(x_j,2r), we write
	
    s_p|_B^X(x_j,2r)=f_je_L,j^⊗ p⊗ e_E,j,

	where f_j is a holomorphic function on the chart in ^n 
	corresponds to B^X(x_j,2r). Then we have
	
    ℳ^U_p(s_p)=sup_x∈ U|s_p(x)|_h_p   ≤max_jsup_x∈ 
    			B^X(x_j,r)|f_j(x)|
       ≤ Cmax_j{f_j_ℒ^2(B^X(x_j,2r))}
       ≤C'/ν^pmax_j{s_p_ℒ^2(B^X(x_j,2r),L^p⊗ E)}
       ≤C'/ν^ps_p_ℒ^2(U,L^p⊗ E),

where the constant C' independent of p is determined by 

    C/C'=min{inf_x∈ 
    B^X(x_j,2r)|e_E,j(x)|_h_E : j=1 , 
    …, ℓ}=: ν_E.

The next step is to estimate the quantity 
[S_f,p^2p^n_ℒ^2(U,L^p)] 
for p≥ 2. Applying Hölder's inequality with 
1/p^n+p^n-1/p^n=1, we get 
	
    [S_f,p^2p^n_ℒ^2(U,L^p⊗ E)]
    ≤(U)^p^n-1[∫_U
    |S_f,p(x)|^2p^n_h_p(x)dV].

For p≥ 2, let {S^p_m}_m=1^d_p(f) be an orthonormal basis of H^0_(2)(X,L^p⊗ E,f)⊂ H^0_(2)(X,L^p⊗ E) with respect to -inner product and consisting of eigensections of T_f,p with nonzero eigenvalues {λ^p_m≠ 0}_m. As explained in (<ref>), we can construct a sequence of i.i.d. standard complex Gaussian variable {η^p_m}_m such that we can write

    S_f,p=∑_m=1^d_p(f)η^p_m λ^p_m S^p_m.

As in (<ref>), on a neighborhood of B^X(x_j, 2r), write

    S^p_m=f^p_me_L,j^⊗ p⊗ e_E,j.

	If x∈ B^X(x_j,2r), set
	
    F_j(x)=∑_m=1^d_p(f)η^p_mλ^p_m f^p_m(x).

	Then F_j(x) is a complex Gaussian random variable with 
	(total) variance ∑_m=1^d_p(f)|λ^p_m f^p_m(x)|^2. 
	By our 
	assumption on the local frame e_L,j, we get
	
    ∑_m=1^d_p(f)|λ^p_m f^p_m(x)|^2≤1/ν^2pν_E^2T^2_f,p(x,x).

	Then we 
	have
	
    [|F_j(x)|^2p^n]=p^n!
    (∑_m=1^d_p(f)|λ^p_m f^p_m(x)|^2)^p^n.

As a consequence, we get that for x∈U,

    [|S_f,p(x)|_h_p^2p^n]≤[|F_j(x)|^2p^n]
    ≤p^n!/ν^2p^n+1ν_E^2p^n(T^2_f,p(x,x))^p^n.

Since we are in the context of σ-finite measures and 
	the integrands are non-negative, Tonelli's Theorem applies, so that
	
    [∫_U|S_f,p(x)|^2p^n_h_pdV(x)]≤p^n!/ν^2p^n+1ν_E^2p^n∫_U (T^2_f,p(x,x))^p^ndV(x).

Moreover, by the estimate for T^2_f,p(x,x) given in (<ref>) on a compact 
subset, there exists a 
	constant C_U,f>0 (independent of p) such that for 
	p∈, x∈U,
	
    T^2_f,p(x,x)≤ C_U,fp^n.

		Combining (<ref>) with the above inequalities, we infer that
	
    [S_f,p^2p^n_ℒ^2(U,L^p⊗ E)]≤(C_U(U)/ν_E^2)^p^n·p^n!/ν^2p^n+1(p^n)^p^n.

	By applying (<ref>) to S_f,p, we get
	
    [ℳ^U_p(S_f,p)^2p^n]   ≤(C/ν^pν_E)^2p^n[S_f,p^2p^n_ℒ^2(U,L^p⊗ E)]
    		≤(Cp^n)^2p^n/ν^4p^n+1,

	where C>0 is a constant independent of p.
Then (<ref>) follows from Chebyshev's inequality and the inequality
	1/ν≤ e^δ/4 from (<ref>).




Now we consider the probabilities of small values of 
ℳ^U_p(S_f,p), and we will adapt the 
ideas in <cit.>, <cit.>: viewing S_f,p as a line bundle valued holomorphic Gaussian field on X and studying its correlation function.



Assume Condition <ref> to hold.
	 Fix f∈ℒ^∞_c(X,_≥ 0) and assume that on a small open ball B of X, f|_B is 𝒞^m(B)+1(B) (the quantity m(B) is given in (<ref>)) and nonvanishing. 	Let U be a relatively compact open subset in X such that U∩ B≠∅, then there exist constants C_U>0, C_U'>0 such 
	that for all δ>0 and p∈, 
	
    (ℳ^U_p(S_f,p)≤ 
    		e^-δ p)≤ e^-C_Uδ p^n+1+C_U' 
    		p^nlogp .



From the local uniformity of the Berezin–Toeplitz kernel expansions,
as explained in Section <ref>, it follows that for every compact subset 
K⊂ B
there exists p(K) such that for all p≥ p(K) and all x∈ K
we have  T^2_f,p(x,x)>0. 

Now for any x∈ B we fix some λ_x∈ L_x, μ_x∈ E_x with 
 |λ_x|_h_L=|μ_x|_h_E=1, and set
 
    ξ_x=⟨λ_x^⊗ p⊗μ_x, 
     S_f,p(x)⟩_h_p/√(T^2_f,p(x,x))  ,

 whenever T^2_f,p(x,x)≠ 0. Then ξ_x is a complex Gaussian random variable. Moreover, for 
any two points x,y∈ B, we have 
 
    |𝔼[ξ_xξ_y]|
    =N_f,p(x,y),

 where N_f,p(x,y) is defined by (<ref>).
	

Since U∩ B≠∅, so we can find a small open ball C:=B(x_0,r)⊂ U∩ B, then f is 𝒞^m(B)+1 and never vanishes near C. Then the asymptotic equations in (<ref>) from Corollary <ref> holds true for all x,y∈ C and for all p≫ 0. In particular, we have

    (ℳ^U_p(S_f,p)≤ 
    		e^-δ p)≤(ℳ^C_p(S_f,p)≤ 
    		e^-δ p).

Next step, by (<ref>), we may proceed using the similar arguments in <cit.> or <cit.>, we can 
	prove a more general version of (<ref>) as follows: for a sequence of positive numbers 
{λ_p}_p∈,

    (ℳ^U_p(S_f,p)≤λ_p)≤ e^Cp^nlogλ_p +C'p^nlog 
    	p,    p≫ 0.

Then, for any 
δ>0, choosing λ_p=e^-δ p in (<ref>), we recover 
(<ref>). This completes our 
proof.

Combining Propositions <ref> and <ref>, we arrive at the following.

Let (X,J,Θ) be a connected Hermitian complex manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics. Assume Condition <ref> to hold. Fix f∈ℒ^∞_c(X,_≥ 0) and assume that on a small open ball B of X, f|_B is 𝒞^m(B)+1(B) and nonvanishing. Let U be a relatively compact open subset in X such that U∩ B≠∅, then there exist constants C=C_U,f,δ>0 such 
	that for all δ>0 and p∈,
	
    (|logℳ^U_p(S_f,p)|≥δ p) ≤ e^-Cp^n+1.









 §.§ Proof of Proposition <ref>

Now we give the proof of Proposition <ref>, which follows by combining from the arguments in <cit.>
with Corollary <ref>. Since the kernels T^2_f,p(x,y) behave in a more complicated way, depending on the values of f, than the Bergman kernels, some steps are required necessary modifications from that in <cit.>. In the same time, we also include several explicit function estimates in order to work out a rough formula for the quantity r(U,U') in the statement of Proposition <ref>.

For t>0, set

    log^+t=max{logt,0}, log^-t:=log^+(1/t)=max{-logt,0}.

Then

    |logt|=log^+t+log^-t.



Let U be the open subset as assumed in the proposition.
Then for any 
nonzero holomorphic section s_p∈ H^0(X,L^p⊗ E), we have that |log|s_p|_h_p| is integrable on 
U with respect to dV. We now start with showing that for any δ >0,

    (
    	∫_Ulog^+|S_f,p(x)|_h_p dV(x)≥δ p)≤ e^-C_U,f,δp^n+1.

For this purpose, observe that on U we have

    log^+|S_f,p|_h_p≤|logℳ^U_p(S_f,p)|,

which then supplies us with

    (
    		∫_Ulog^+|S_f,p(x)|_h_p dV(x)≥δ p)
    	≤(
    		|logℳ^U_p(S_f,p)|≥δ/(U) p),

where (U) denotes the volume of U with respect to 
dV. 

Note that we assume that f is of class 𝒞^m+1 with m=m(U') almost everywhere near U' and with 

    U∩   f≠∅  ,

the set f, by its definition, can not be null measure in any given open subset, then there exists a small open ball 𝔹⊂  f ∩ U such that f is of class 𝒞^m(B)+1 (since m(B)≤ m(U')) and never vanishing on 𝔹. Therefore, Corollary <ref> applies to this open subset 𝔹 of U and the function f. Then combining (<ref>) with Corollary <ref>, we obtain (<ref>) with any given δ>0.







The next step is to study 
∫_Ulog^-|S_f,p(x)|_h_p dV(x). At first, let us construct a particular 
family of holomorphic local charts to cover U. 
Since U is relatively compact in U', 
we apply Proposition <ref> for 
K=U to choose the constants (t_K, c_K) 
such that the t_K-neighbourhood of U is still included 
in U' and the line bundles L, E can be locally 
trivialized on any small (geodesic) balls of radius 2t_K on U.

For each point x∈U, let (V_x, ϕ_x)
be the local chart as in Proposition <ref>. Set 

    A_x:=ϕ_x^-1({1/2t_K/c_K<|z|<3/4t_K/c_K})⊂ U'.

Then the family {A_x}_x∈U 
is an open covering of U, 
hence there exists a finite number of points 
{x_j}_j=1^N⊂U 
such that {A_x_j}_j=1^N 
is already an open covering of U. 
We also set

    A_x⊂ B_x:=ϕ_x^-1({|z|<
    t_K/c_K}) ⊂ V_x.

By (<ref>), we have

    r_^n(f∘ϕ_x^-1,ϕ_x(B_x))
    ≤ c_K r(f,V_x)≤ c_K r(f,U').

For t_K/c_K>δ^2n+2 > c_K r(f,U'), 
any open coordinate ball (of ^n) in {|z|<t_K/c_K}
of radius δ^2n+2 intersects the 
essential support of f∘ϕ_x^-1.


Next we work on each A_x_j. 
In order to simplify the notation, after rescaling 
the coordinates of the local chart (V_x_j,ϕ_x_j), 
we may and we will consider the following model case: 
in the local coordinate z=(z_1,…, z_n), 
set A=𝔹(2,3):={2<|z|<3}, and 
B=𝔹(4):={|z|<4}. 
Assume that on the coordinate open ball 
𝔹(5)⊂ U', the holomorphic line bundles 
L and E can be trivialized by holomorphic local frames. 
In the following, we work on this coordinates (z_1,…,z_n)
and let r_^n(f,V) denote the number given as in 
(<ref>) with respect 
to the standard Riemannian metric g^T^2n on ^n.

For f∈ℒ^∞_c(X,_≥ 0)
which is of 𝒞^m(U')+1 almost everywhere near B 
with r_^n(f,B)≤1/2, 
we always have A ∩   f ≠∅.




Let e_L, e_E the be respective holomorphic local frames of the line bundles L, E 
restricting to the open ball B=𝔹(4). Set α_L(z)= 
log|e_L(z)|^2_h_L, α_E(z)= 
log|e_E(z)|^2_h_E. On this local chart, we can write

    S_f,p|_B=F_p(z)e^⊗ p_L⊗ e_E,

where F_p is a random holomorphic function on B. Then on B

    log|S_f,p|_h_p=log|F_p|+p/2α_L+1/2α_E.




Fixing ε>0, we take

    K_1:=2ε+1/2sup_z∈𝔹(4)|α_L(z)|.

Then by (<ref>) and (<ref>), 
we get that, except the event 
{|logℳ^A_p(S_f,p)|
≥ε p} of probability less than
e^-C_A,f,ε  p^n+1 , we have for all r∈ [1,3], 
		
    ∫_{|z|=r}log^+|F_p|(z) dσ_r(z)≤
    	K_1 p,

		where dσ_r denote the invariant probability measure on the sphere {|z|=r}.


For 0<r<4, consider the Poisson kernel for the ball 𝔹(r),

    𝒫_r(ξ,z):=r^2n-2r^2-|ξ|^2/|ξ-z|^2n,

where |z|=r, ξ∈𝔹(r). Since log|F_p| is a subharmonic function on B, by the sub-mean inequality in terms of Poisson kernel, we conclude that for any point ξ∈𝔹(1), 1<r<3,

    ∫_{|z|=r}𝒫_r(ξ, z)|log|F_p|(z)|dσ_r(z)≤ 2∫_{|z|=r}𝒫_r(ξ, z)log^+|F_p|(z)dσ_r(z)-log|F_p|(ξ).

As a consequence, we have, for r∈[2,3],

    ∫_{|z|=r}|log|F_p|(z)|dσ_r(z)
       ≤ 12(9/2)^2n-1∫_{|z|=r}log^+|F_p|(z)dσ(z)-2(3/2)^2n-1log|F_p|(ξ).


Note that 𝔹(1)⊂ B, following from the condition r_^n(f,B)≤1/2, Corollary <ref> applies to 𝔹(1). We conclude that except for an event of probability less than e^-C_𝔹(1),f,1p^n+1, we can always find a point ξ_0∈𝔹(1) with the property:

    log|F_p|(ξ_0)>-p.

Then combining it with (<ref>) and (<ref>), we get that except for an event Ω_1 of probability less than e^-C_B,f,K_2p^n+1, we have that for all r∈[2,3],

    ∫_{|z|=r}|log|F_p|(z)|dσ_r(z)≤ K_2 p,

where K_2>0 is a sufficiently large constant given by

    K_2=12(9/2)^2n-1 K_1+2(3/2)^2n-1.







Given δ∈(0,1/2] (the case of δ > 1/2
will become a consequence of this one), we fix a decomposition of the 
unit sphere {|z|=1} into a disjoint union of sets I_1, I_2, ⋯, 
I_q with (Euclidean) diameter ≃δ^2n+2. 
The number q is a large integer depending on δ. 
The following inequality was established in <cit.>: there exists a constant C_n>0 independent of δ∈  
(0,1/2] such that for any r∈(1,3], any collection of points {ξ_k}_k=1^q ⊂𝔹(4) with dist(ξ_k, (r-δ)I_k)< δ^2n+2 and any subharmonic function u on 𝔹(4),

    ∫_{|z|=r}u(z)dσ_r(z)≥∑_j=1^q σ_1(I_k)u(ξ_k)-C_nδ∫_{|z|=r}|u(z)|dσ_r(z).

The main idea to prove (<ref>) in <cit.>
is to use the Poisson kernel as in (<ref>) for each
term u(ξ_k), and C_n is a constant related to the estimate
on the derivatives of 𝒫_r, and in our setting, 
we can take C_n=36^n(4n+4).


From this point we proceed as in <cit.> but with some necessary modifications, and we will also produce a rough formula for the constant r(U,U') needed in our proposition. Fix δ∈  
(0,1/2], and let f∈ℒ^∞_c(X,_≥ 0) which is of 𝒞^m(U')+1 almost everywhere near B such that r_^n(f,B)≤2/3δ^2n+2.

Set Q:=⌈9/4δ^-2n-2⌉ and set r_j=3/2+2/3δ^2n+2j, for 0≤ j≤ Q. Then for 1≤ k ≤ q, 0≤ j≤ Q, set

    V_kj:={z∈ B : dist^^n(z,r_jI_k)< 1/2δ^2n+2}.

Then {V_kj}_k,j form a covering for A, moreover, each V_kj always contains a coordinate open ball of raidus δ^2n+2, so that Corollary <ref> applies: there exists C_kj>0 such that for all p≫ 0
	
    (|logℳ^V_kj_p(S_f,p)|> δ p) ≤ e^-C_kjp^n+1.

	Set the event Ω_δ:=∪_k,j{|logℳ^V_kj_p(S_f,p)|≥δ p}, then we have that for p≫ 0,
		
    (Ω_δ) ≤ e^-C_B,f,δp^n+1.




When S_f,p lies outside Ω_δ, we can always find ξ_kj∈ V_kj with 

    log |S_f,p|_h_p(ξ_kj)≥ -δ p.

Fix r∈ ]2,3[, there exists j∈{0,1,…,Q} such that 

    |r-δ-r_j|≤2/5δ^2n+2.
 
Then dist^^n(rI_k, ξ_kj)<2δ, 
so that we have

    ∫_{|z|=r}α_L(z)dσ_r(z)≥∑_j=1^q σ_1(I_k)α_L(ξ_kj)-2δsup_z∈ B |dα_L(z)|.

A similar inequality holds also for α_E.
Moreover, we have dist^^n((r-δ)I_k,
ξ_kj)<δ^2n+2, then we can apply (<ref>) 
for log |F_p|.
We combine this with (<ref>), so 
when S_f,p lies outside Ω_1∪Ω_δ, we have

    -∫_{|z|=r}log|S_f,p|_h_p(z)dσ_r(z)
           ≤ 
    		-∑_k=1^qσ_r(I_k) 
    		log|S_f,p|_h_p(ξ_jk)+C_nδ∫_{|z|=r}|log|F_p||(z) dσ_r(z)
               +pδsup_z∈ 
    		B|dα_L(z)| + δsup_z∈ 
    		B|dα_E(z)|
           ≤δ p+C_n K_2 δ p+pδsup_z∈ 
    		B|dα_L(z)| + δsup_z∈ 
    		B|dα_E(z)|.

We set

    K_3:=1+C_n K_2+ sup_z∈ 
    		B|dα_L(z)| + ε >0.

Let b_0>0 be such that for all z∈ A, dV(z)≤ b_0 dλ(z), where dλ(z) denotes the standard Lebesgue measure on ^n. Then taking the integral with repsect to r∈[2,3] and rescaling dσ_r to the standard spherical measure for the sphere of radius r, we conclude, by taking advantage of (<ref>), that

    (
    	-∫_Alog|S_f,p|_h_p dV≥ 
    	b_0 K_3 π^n/n!(9^n-4^n)δ p)≤ e^-C_B,f,δp^n+1, ∀ p≫ 0.

Combining the above estimate with (<ref>), we get that for p≫ 0,

    (
    	∫_A|log|S_f,p|_h_p(z)| dV(z)> 
    	b_0 K_3 π^n/n!(9^n-4^n)δ p)≤ e^-C'_B,f,δp^n+1.



Now we reformulate (<ref>) for A_x_j. By the choice of (t_K,c_K) from Proposition <ref> for K=U, we can identify A_x_j with A and B_x_j with B via rescaling the coordinate by 4c_K/t_K. Then we can take b_0=(t_K/4)^2n. Therefore, we get, for p≫ 0,

    (
    	∫_U|log|S_f,p|_h_p(z)| dV(z)> N
    	b_0 K_3 π^n/n!(9^n-4^n)δ p)≤ e^-C^''_B,f,δp^n+1.


The condition for (<ref>) to hold is that 
r_^n(f,B)≤2/3δ^2n+2
(with δ≤1/2). A sufficient condition for this is

    r(f,U')≤min{t_K/6c_K^2δ^2n+2,
    t_K/8c_K^2}.

Finally, we set

    r(U,U'):=t_K/6c_K^2min{3· 4^n, (1/N
    	b_0 K_3 π^n/n!(9^n-4^n))^2n+2}>0,

which fulfills our purpose as stated in Proposition <ref>,
in particular, it implies that U∩   f≠∅.
This completes the proof.







 §.§ Expected mass of random -holomorphic sections





 Recall that the volume form that is used to define the
 -inner products is dV:=Θ^n/n!. 
Similarly, since c_1(L,h_L) is uniformly positive on X
we define a different volume form on X,

    dV^L:=c_1(L,h_L)^n/n!,
      dV^L(z)=b_0(z) dV(z),

where the positive function 
b_0(z)=(Ṙ_z^L/2π) on X 
is given by (<ref>).

Recall that for a holomorphic section 
s_p∈ H^0(X,L^p⊗ E) we introduced 
the measure M_p(s_p) on X
in Definition <ref>. 
The following proposition gives preliminary results on 
the expectation of mass distribution 
M_p(S_f,p) of S_f,p.
The proof follows from the asymptotic expansion
of T^2_f,p given in Section <ref>.

Let (X,J,Θ) be a connected Hermitian complex manifold
and let (L,h_L), (E,h_E) be two holomorphic line bundles
on X with smooth Hermitian metrics. 
Assume Condition <ref> to hold. 
Fix a nontrivial 
f∈𝒞^1_c(X,_≥ 0), or 
f=1_A for a relative compact domain A⊂ X 
with a continuous boundary, and let 
{S_f,p∈ H^0_(2)(X,L^p⊗ E)}_p 
be the associated random -holomorphic sections defined 
via {T_f,p}_p. Then for any z∈ X, we have

    [|S_f,p(z)|^2_h_p]=T^2_f,p(z,z).

Hence we have the locally uniform weak-star convergence of measures on X,

    [M_p(S_f,p)]→ f^2dV^L.




The identity (<ref>) follows from the series
formula for S_f,p given in (<ref>)
(see also (<ref>)). 
Then for any function g∈𝒞^0_c(X), we have

    [∫_X g(z)M_p(S_f,p)(z)]=
    1/p^n∫_X g(z)T^2_f,p(z,z)dV(z).

When f∈𝒞^1_c(X,_≥ 0), by Theorem <ref>, we have 

    T^2_f,p(z,z)=p^n f(z)^2 b_0(z)+𝒪(p^n-1/2).

We conclude (<ref>) from (<ref>).

Now we consider the case f=1_A. Fix a function g∈𝒞^0_c(X) and let K be a compact subset with supp  g⊂ K. By (<ref>), we have for x∈ K

    1/p^nT^2_f,p(x,x)≤ C_K,

with some constant C_K≥ 1.
For any ε>0, let δ>0 be a small number such that 

    Vol({z∈ X  : dist(z,∂ A)≤δ})≤ε/3 C_K.

By the uniform expansion of T^2_f,p(x,x) Theorem <ref>
for the points away from ∂ A, we get that there exits
p_0∈ such that for all p≥ p_0

    |1/p^n∫_z∈ X, dist(z,∂ A)≥δ g(z)T^2_f,p(z,z)dV(z)-
    ∫_A(δ)g(z)dV^L(z)|≤g_ℒ^∞ε/3,

where A(δ):={z∈ A : dist(z,∂ A)≥δ}.

Assembling the above estimates together we have for p≥ p_0,

    | 1/p^n∫_X g(z)T^2_f,p(z,z)dV(z)- ∫_A g(z)dV^L(z)| ≤g_ℒ^∞ε.

Then we conclude that

    1/p^n∫_X g(z)T^2_f,p(z,z)dV(z)→∫_A g(z)dV^L(z),   p→∞.

The proof is complete.





We need the following lemma to prove Proposition <ref>.

With the same assumptions as in Proposition <ref>
for the geometric data, let U be a relative compact open subset 
of X, and fix a nontrivial 
f∈𝒞^m(U)+1_c(U,_≥ 0). 
Then for any g∈𝒞^0_c(U), set Y_p^g
=∫_X g(z)M_p(S_f,p)(z), we have

    Var[Y_p^g]:=[|Y_p^g|^2]-|[Y_p^g]|^2
    =∫_X |g(z)|^2 f(z)^4dV^L(z)+o(1).



Note that we have

    [|Y_p^g|^2]   =1/p^2n∫_X× X g(z)g(w)[|S_f,p(z)|^2_h_p|S_f,p(w)|^2_h_p]dV(z)dV(w)
       =1/p^2n∫_X× X g(z)g(w)T^2_f,p(z,z)T^2_f,p(w,w)[|S_f,p(z)|^2_h_p/T^2_f,p(z,z)|S_f,p(w)|^2_h_p/T^2_f,p(w,w)]dV(z)dV(w).

By <cit.>, we have

    [|S_f,p(z)|^2_h_p/T^2_f,p(z,z)|S_f,p(w)|^2_h_p/T^2_f,p(w,w)]=1+N_f,p(z,w)^2.

Since f is assumed to be 𝒞^m(U)+1, then N_f,p(z,w)^2 satisfies the asymptotics (<ref>), as a consequence, we have (cf. (<ref>)–(<ref>))

    [|Y_p^g|^2]=|[Y_p^g]|^2+∫_X |g(z)|^2|f(z)|^4dV^L(z)+o(1).

This proof is complete.




By Proposition <ref>, we have

    [Y^g_p]=∫_X g(z)f(z)^2dV^L(z)+o(1).

Then as N→+∞, we have

    1/N∑_1≤ p ≤ N[Y^g_p]→∫_X g(z)f(z)^2dV^L(z).

Note that {Y^g_p}_p is a sequence of independent variables, 
and by Lemma <ref>, we have Var[Y^g_p]=
𝒪(1), so that by the strong law of large numbers for pairwise independent random variables
(see <cit.>), we have

    1/N∑_1≤ p≤ N∫_X g(z)M_p(S_f,p)(z)-1/N∑_1≤ p ≤ N[Y^g_p]→ 0, almost  surely.

Combining the above convergence with (<ref>)
we conclude the proof.



When X is compact, Bayraktar <cit.>
considered the sub-Gaussian holomorphic sections and proved
the quantum ergodicity for their mass distribution
by using the Hanson-Wright inequality.
This result is stronger than Proposition <ref>.















§ EXPECTATION AND EQUIDISTRIBUTION ON THE SUPPORT
OF THE SYMBOL


We continue our discussion of zeros of S_f,p in Section <ref>,
especially, we prove that the random zeros will be asymptotically uniformly distributed
with respect to c_1(L,h_L) on parts inside the support of f where it is
𝒞^m(U)+1 with f ⊂ U⋐ X.
These results are applications of Theorem <ref> specializing to the case r(f,U')=0.



 §.§ Equidistribution of random zeros on the support; 
proof of Theorem <ref>


In <cit.>, the zeros of random sections S_f,p
were studied for the case 0≠ f∈𝒞^∞_c(X,), 
and the equidistribution results were proven for their zeros as p→∞
on the support of f with the assumption that f vanishes up to order 2.
Here, we remove the condition on the vanishing orders of f for the equidistribution results.


 By our assumption U⊂ess.supp  f, we get r(f,U)=0, in 
 Theorem <ref>, we have δ_0(f)=0, thus (<ref>) 
 holds for all δ>0. Similarly, taking δ_0(f)=0 in (<ref>), 
 we get (<ref>).



Note that the estimate (<ref>) is a direct consequence of 
(<ref>) by taking 
δ=n^L_2n(U'). 
Hence, it is sufficient to prove (<ref>). For this purpose, let 
1_U' denote the indicator function of U' on X.
Let δ>0 be arbitrary, and take ψ_1, ψ_2∈𝒞^∞_0(X,) with supports in U such that
0≤ψ_1≤1_U'≤ψ_2≤ 1, and

    ∫_Xψ_1c_1(L,h_L)^n/n!≥^L_2n(U')-δ,   ∫_Xψ_2c_1(L,h_L)^n/n!≤^L_2n(U')+δ .

Note that the existence of such functions is guaranteed by the 
assumption that ∂ U' has measure 0 with respect to 
dV, hence also to 1/n!c_1(L,h_L)^n.
For j∈{1,2}, set φ_j=1/(n-1)!ψ_jc_1(L,h_L)^n-1. 
By applying Theorem <ref> to φ_j separately, we 
get exactly (<ref>). 




Another interesting object that is closely related to this equidistribution result
is to study the asymptotic behavior of 1/p𝔼[[(S_f,p)]]
and compare it with c_1(L,h_L), the expected limit. By Lemma <ref>
and Theorem <ref>, we have

    1/p𝔼[[(S_f,p)]]-c_1(L,h_L)=
    1/p c_1(E,h_E)+√(-1)/2π p∂∂logT_f,p^2(x,x).




Now we show that, under a global finiteness on the geometry of the manifold, 
the convergence of 1/p𝔼[[(S_f,p)]] to c_1(L,h_L)
on f can be obtained as a consequence of the equidistribution 
result (<ref>). Certainly, this is a weaker version of the convergence 
given in Theorem <ref> via the method of pluripotential theory on X.

We assume the same geometric conditions on X, L, E as in Theorem <ref>. 
Furthermore, we assume one of the following conditions to hold:


  (i) Θ is Kähler and ∫_X  c_1(L,h_L) ∧Θ^n-1<∞, or

  (ii) ∫_X c_1(L,h_L)^n<∞.

Fix 0≠ f∈ℒ^∞_c(X,). 
Let U be an open subset of X such that U⊂ess.supp  f, 
and we assume that f is of class 𝒞^m(U)+1 almost everywhere on U. 
Then we have
	
    1/p𝔼[[(S_f,p)]]|_U → c_1(L,h_L)|_U;
       √(-1)/2π p∂∂logT_f,p^2(x,x)|_U→ 0,

where the limits in (<ref>) are taken with respect 
to the convergence of (1,1)-currents on U.


Applying Theorem <ref>, we get that for any φ∈Ω^n-1,n-1_c(U), we have
	
    (  lim_p→∞⟨1/p[(S_f,p)],
    		φ⟩=⟨ c_1(L,h_L),φ⟩ )=1.
	

Now we assume that first case: Θ is Kähler and 
∫_X  c_1(L,h_L) ∧Θ^n-1<∞. For all φ∈Ω^n-1,n-1(X), s_p∈ H^0(X,L^p), due to the 
	positivity of the current [(s_p)], we have
	
    |⟨[(s_p)],φ⟩|≤ 
    		|φ|_𝒞^0(X)⟨[(s_p)],
    		Θ^n-1⟩=p|φ|_𝒞^0(X)∫_Xc_1(L,h_L)∧Θ^n-1.

 and the last identity follows from the Poincaré-Lelong formula. We 
	may set C=∫_Xc_1(L,h_L)∧Θ^n-1<∞, then
	
    1/p|⟨[(s_p)],φ⟩|≤ 
    		C|φ|_𝒞^0(X).

Then applying the dominated convergence theorem to the limit in (<ref>), we infer that 
	
    lim_p→∞[⟨1/p[(S_f,p)],
    φ⟩]=⟨ c_1(L,h_L),φ⟩.

	By the definition of [[(S_f,p)]],
	we can rewrite the above limit as
		
    lim_p→∞⟨1/p[[(S_f,p)]],
    		φ⟩=⟨ c_1(L,h_L),φ⟩.

	This way, we get the first line of (<ref>).

	By Theorem <ref>, we have
	
    1/p[[(S_f,p)]]-c_1(L,h_L)=1/pc_1(E,h_E)+
    	√(-1)/2π p∂∂logT_f,p^2(x,x).

	Then the second line of (<ref>) follows from the first one.
	
For the second case ∫_X c_1(L,h_L)^n<∞, we just replace Θ by 
c_1(L,h_L) in the above arguments, and certainly we have to change the 
𝒞^0-norm for test forms to the one induced by c_1(L,h_L), 
then we conclude the same conclusion (<ref>). This way, we complete our proof.



It is an interesting question to ask that if the finiteness assumptions listed in 
Proposition <ref> together with (<ref>) 
would imply that d_p<+∞, it seems so for the case of Riemannian surfaces. 
But the authors have no awareness of such results in general.








 §.§ Proofs of Theorems <ref> and <ref>

Now we give a general result on 𝔼[[(S_f,p)]]
that is obtained from the pluripotential theory on X. 

Let us recall some basic facts about the plurisubharmonic functions. 
Let X be a connected complex manifold of complex dimension n≥ 1, 
and we fix a smooth Kähler form ω on X. 
Recall that a quasi-plurisubharmonic (quasi-psh for short) function 
on an open domain of X is locally the difference of a plurisubharmonic function and a smooth one. 
For any open subset U⊂ X, set PSH(U,ω) the space of quasi-psh functions 
on U with respect to ω, that consists of the upper semi-coutinuous functions 
φ: U→ [-∞,+∞[ with the propoerty 
dd^cφ+ω≥ 0 as (1,1)-current on U, 
where dd^c=√(-1)/π∂∂. 

The following two propositions are well-known facts for the functions in 
PSH(U,ω), which follow from the corresponding properties of 
plurisubharmonic functions on the domains of ^n. For their proofs, we refer to 
<cit.>.

Fix an open domain U in X, and a compact subset K⊂ U.
 Let ℱ⊂PSH(U,ω) be a family 
 of quasi-psh functions on U such that for each 
 φ∈ℱ, max_z∈ Kφ(z)=0, then ℱ|_K 
 is a bounded subset in ℒ^q(K,) (∞ > q≥ 1), 
 and it is also a relatively compact subset of ℒ^1(K,).



Let U be a relatively compact open domain U in X.
Let φ_ℓ (ℓ∈) be a sequence of quasi-psh functions 
on U with dd^cφ≥ -ω on 
U, which have a uniform upper bound on U. 
Then, only one the following situations holds

	
  * either φ_ℓ converge uniformly to -∞ 
	on any compact subset of U, or
	
  * there exists a subsequence φ_ℓ_j converges 
	in ℒ^q-norm (∞ > q≥ 1) on U to a quasi-psh 
	function φ on U with dd^cφ≥ -ω.  



The above propositions allow us to establish some estimates on the 
integrals of Berezin–Toeplitz kernel function T^2_f,p(x,x), 
and then we can deduce the properties on 𝔼[[(S_f,p)]]. 
At first, we give the proof of Theorem <ref>.


With our assumption on f, by Lemma <ref>, we get T_f,p≠ 0 
for all sufficiently large p, then we have the corresponding random 
-holomorphic section S_f,p which is not identically zero. 
By Theorem <ref>, we have the identity of (1,1)-currents on X for p≫ 0:

    0≤1/p[[Div(S_f,p)]]=
     √(-1)/2π p∂∂log T^2_f,p(x,x) + 
     c_1(L,h_L)+1/pc_1(E,h_E).


Let C_E>0 be a constant such that c_1(E,h_E)≤ C_E c_1(L,h_L) on 
U. Then for all p≥⌈ C_E ⌉, we have

    c_1(L,h_L)+1/pc_1(E,h_E)≤ 2 c_1(L,h_L), on U.
 

Set 

    u_p(x):=1/plog T^2_f,p(x,x),

 and we work on the connected compact subset K=U. 
 Then by (<ref>) and (<ref>),  
 u_p is a quasi-psh function with respect to the Kähler form 
 4 c_1(L,h_L) on K, that is, dd^cu_p+4 c_1(L,h_L)≥ 0 on K.
Set now

    m_p:=max_x∈ K u_p(x).

Then applying (<ref>) on K and Theorem <ref> 
for the open subset U∩ B⊂ K, there exist constants C' ≥ n, 
p_0≥ 2 such that for all p≥ p_0,

    0≤ m_p≤C'logp/p

Therefore {u_p-m_p}_p≫ 0 is a family of quasi-psh functions on K 
(actually on an open neighbourhood of K) such that each one has the maximum 
0 in K. By Proposition <ref>, we conclude that  {u_p-m_p}_p≫ 0 
is a bounded set in ℒ^q(K,) with any q≥ 1. In particular, 
there exists a constant C>0 such that for all p≫ 0, we have

    u_p-m_p_ℒ^1(K,):=∫_K |u_p(x)-m_p|dV(x)≤ C.

As a consequence, for all p≫ 0, we get

    ∫_K |logT^2_f,p(x,x)|dV(x)≤ Cp+C'Vol(K)logp,

Then for any nonempty open subset A⊂ K, we have

    ∫_A |logT^2_f,p(x,x)|dV(x)≤ Cp+C'Vol(A)logp,

We conclude that for all p≫ 0,

    -Cp-C'Vol(A)logp≤∫_A
    logT^2_f,p(x,x)dV(x)≤ C'Vol(A)logp,

Then considering the probability space modelled as 
(A,1/Vol(A)dV|_A) and applying Jensen's 
inequality with log(t), we get

    1/Vol(A)∫_A logT^2_f,p(x,x)dV(x) ≤log(1/Vol(A)∫_A T^2_f,p(x,x)dV(x) )

Combining it with (<ref>), we get

    log(1/Vol(A)∫_A T^2_f,p(x,x)dV(x) )
    ≥ -Cp/Vol(A)-C'logp.

Therefore, we get exactly (<ref>). 

The estimate (<ref>) follows from combining (<ref>) 
and (<ref>) for any test function in Ω^n-1,n-1_c(U). 
Finally, since the quasi-psh functions {u_p}_p≫ 0 form a bounded set of 
ℒ^1(K,) hence relatively compact by Proposition <ref>, 
then there exists a subsequence {u_p_j}_j which converges in 
ℒ^1(K,) to a quasi-psh function f. 
Then applying (<ref>), we get (<ref>).
In particular, since u_p converges uniformly to 0 on B, 
we also have f|_B∩ U≡ 0. This way, we finish our proof.



When f^2≥ c>0 with some constant c>0, 
the lower bound in (<ref>)
is trivial due to the asymptotic expansions of Bergman kernel function P_p(x,x). 
When f vanishes identically on a nonempty open subset V of U 
(which is given as in Theorem <ref>), 
by Theorem <ref>, we have

    T^2_f,p(x,x)=𝒪_V(p^-∞), 
     uniformly for all  x∈V.

In this case, the lower bound in (<ref>) is nontrivial and provides
a better understanding of the decay of T^2_f,p(x,x) on U.

Following the properties introduced in the proof of Theorem <ref>, 
we can generalize Proposition <ref> to obtain the convergence of 
1/p𝔼[[(S_f,p)]] on the support of f. 


We use the same notation as in the proof of Theorem <ref>. 
Note that U is compact in X. Since f is of class 𝒞^1 
almost everywhere near U and it has nonvanishing points,
then the family {u_p}_p defined in (<ref>) is a bounded set in
ℒ^2(K,) following Proposition <ref>. 
Let C_2>0 be the constant such that for all p≥ p_0,

    u_p^2_ℒ^2(U,):=
    ∫_U|u_p(x)|^2dV(x)≤ C_2.

As a consequence, for any A⊂U measurable, we have

    u_p_ℒ^1(A,)≤(∫_A |u_p(x)|^2dV(x))^1/2Vol(A)^1/2≤√(C_2)Vol(A)^1/2.

Combining this with the fact that {u_p}_p is a bounded in 
ℒ^2(K,), 
we get that the family {u_p}_p is uniformly integrable on U 
(see <cit.>).

By Theorem <ref>, 
the assumption on f 
implies that u_p converges to 0 almost everywhere on U with 
respect to the Lebesgue measure (or dV).
Finally by the Lebesgue-Vitali theorem (see <cit.>), 
we get that u_p converges to 0 in ℒ^1-norm on U, and 
(<ref>) follows.
Combining (<ref>) with (<ref>), 
we conclude the convergence (<ref>). The proof is complete.



(i) If we use the bound of ℒ^2-norm for u_p
such as (<ref>) in the proof of Theorem <ref>, 
then we replace the exponent -CpVol(A_p)^-1-C'logp 
in the right-hand side of (<ref>)
by -C_2p/√(Vol(A_p)).

(ii) In Theorem <ref>, if we assume a stronger condition on f, 
for example, if the function |log f^2| is integrable on U with respect to 
dV, 
then one can ask if the following convergence of (1,1)-currents on U as p→ +∞ holds,

    𝔼[[(S_f,p)]]|_U- p c_1(L,h_L)|_U→√(-1)/2π∂∂log f^2 +c_1(E,h_E)|_U  ?








 §.§ Random zeros and lowest Toeplitz eigenvalues on
compact manifolds

Now we apply the results from previous Sections to the case
of compact complex manifolds. 
In this Section, we always assume (X,Θ) to be a 
connected compact Hermitian manifold and (L,h_L) 
to be a positive holomorphic line bundle on X. 
A random section S_f,p as in (<ref>)
can be equivalently defined by 
(<ref>).



As in the Definition <ref> we can set by
the compactness of X,

    κ(R^L,X):=sup{maxspec (Ṙ^L_x),  x∈ X}≥ε_0,

and then set

    m(X):=⌈ (6n+6)κ(R^L, X)/ε_0⌉∈.

At first, we apply Theorems <ref> and 
<ref> to the case f=X.


Let (X,J,Θ) be a connected, compact complex Hermitian manifold and 
let (L,h_L), (E,h_E) be holomorphic line bundles on X with smooth 
Hermitian metrics. Assume h_L to be positive. Fix a real bounded function 
f which is of class 𝒞^m(X)+1 almost everywhere on X and 
with f=X (or equivalently f=X). Then we have -a.s. that

    lim_p→∞1/p[(S_f,p)]- c_1(L,h_L)_X,-2
    = 0,

If f is a real bounded function which is 𝒞^1 almost everywhere 
on X and with f=X, then as p→ +∞, we have

    1/plogT_f,p^2(x,x)_ℒ^1(X,)→ 0,

and

    1/p𝔼[[(S_f,p)]]
    - c_1(L,h_L)_X,-2→ 0.




The convergence of 1/plogT_f,p^2(x,x) 
to the zero function in (<ref>) can fail if we take the 
pointwise limit of 1/plogT_f,p^2(x,x). 
An example is given in Section <ref>, where the smallest eigenvalue 
minSpec(T_f,p) is exponentially small in p.






A consequence of Theorem <ref> is an improvement of the lower bound (<ref>).

Let (X,J,Θ) be a connected, compact complex Hermitian manifold and let (L,h_L), (E,h_E) be holomorphic line bundles on X with smooth Hermitian metrics. Assume h_L to be positive. Fix a real bounded function f (it can take negative values) which is of class 𝒞^1 almost everywhere on X and with f=X (or equivalently f=X). Then there exists a constant c>0 and a decreasing sequence of strictly positive numbers {r_p}_p≥ 1 with limit lim_p→∞r_p=0 which depend only on X, L and f such that for all p≥ 1, any sequence of nonempty open subsets {A_p}_p≥ 1 of X, we have
	
    1/Vol(A_p)∫_A_p T^2_f,p(x,x)dV(x)≥exp(-r_p p/Vol(A_p)).



By (<ref>) (which only require f to be 𝒞^1 almost everywhere on X), we fix a decreasing sequence r_p>0, p∈, such that

    1/plogT_f,p^2(x,x)_ℒ^1(X,)≤ r_p → 0.

Similarly to (<ref>), we get for any A⊂ X,

    -r_p p≤∫_A logT^2_f,p(x,x)dV(x)≤ r_p p,

When Vol(A)≠ 0, by Jensen's inequality with log(t), we get

    log(1/Vol(A)∫_A T^2_f,p(x,x)dV(x) ) ≥1/Vol(A)∫_A logT^2_f,p(x,x)dV(x) ≥ -r_p p/Vol(A)

This way, we conclude (<ref>).


Theorems <ref> and <ref> seem to indicate
is that the asymptotic behavior of the random zeros of 
1/p[Div(T_f,pS_p)] 
depends mainly on the support of the function f
rather than the precise values of f. 
Following this thread, we formulate 
some interesting questions. 
For the sake of simplicity, we will restrict ourselves to 
to consider only non-negative functions f, unless explicitly stated otherwise.

At first, we consider the following equivalence relation on the
non-negative functions on X: for 
f_1, f_2∈ℒ^∞(X,_≥ 0) 
(resp. 𝒞^k(X,_≥ 0)), 
we say they are comparable if there exists a constant 
C=C(f_1, f_2)≥ 1 such that on whole X,

    1/C f_1≤ f_2 ≤ C f_1.

Let ℰℒ^∞(X,_≥ 0) 
(resp. ℰ𝒞^k(X,_≥ 0)) be the set of all equivalent classes of comparable functions in ℒ^∞(X,_≥ 0) 
(resp. 𝒞^k(X,_≥ 0)), then it has a semigroup
structure given by the pointwise addition of functions 
with the zero element given by the zero function.

If f_1, f_2∈𝒞^0(X,_≥ 0) are comparable,
then they have exactly the same set-theoretic support,
that is {x∈ X :  f_1(x)>0}={x∈ X :  f_2(x)>0}, 
therefore supp  f_1=supp  f_2. 


On can then enquire about the relation between the asymptotic  
behaviors of the random zero sets [Div(T_f_1,pS_p)]
and [Div(T_f_2,pS_p)] in the case
when f_1, f_2 are comparable functions on X, 
which might not be fully supported on X.

Let (X,J,Θ) be a connected, compact complex Hermitian manifold and let (L,h_L), (E,h_E) be holomorphic line bundles on X with smooth Hermitian metrics. Assume h_L to be positive. Fix the sequence of the standard Gaussian random holomorphic sections {S_p}_p. If two nontrivial functions f_1, f_2∈ℒ^∞(X,_≥ 0) are comparable, is it true that -a.s.,

    lim_p→ +∞1/p[Div(T_f_1,pS_p)]-1/p[Div(T_f_2,pS_p)]_X,-2=0,

 and that

    lim_p→ +∞[1/p[Div(T_f_1,pS_p)]-1/p[Div(T_f_2,pS_p)]]_X,-2=0

 hold true?


By Theorem <ref>, 
when f_1,f_2 ∈𝒞^m(X)+1
 have full support on X, then the above two equations hold true. 
So the difficult case is when f_js are not fully supported. 
Note that if the results hold in general, they allow us to classify
the asymptotic behavior of the random zeros associated the function
f via their equivalence class in ℰℒ^∞(X,_≥ 0).



A potential approach to a positive answer to Question <ref>
is via establishing the following pointwise estimates for a pair 
of comparable functions (f_1, f_2) for all x∈ X,

    1/C^2T^2_f_1,p(x,x)≤ T^2_f_2,p(x,x) ≤C^2T^2_f_1,p(x,x).

Note that the condition (<ref>) only implies 

    1/CT_f_1,p(x,x)≤ T_f_2,p(x,x) ≤CT_f_1,p(x,x),

which, in general, could not infer (<ref>). Moreover, near the nonvanishing smooth point x of f_j, inequality (<ref>) always holds true for sufficiently large p after taking a bit large constant C (by the asymptotic expansion (<ref>)).

Another interesting question closely related to (<ref>)
or (<ref>) is to understand the asymptotic spectra of 
T_f,p or T^2_f,p as p→ +∞;
this is already partially addressed in Section <ref> 
and the last part Section <ref>. 
Then in the sequel, we will try to understand more how the lowest
eigenvalues of T_f,p decay to 0 as p→+∞
when f has vanishing points.



We start with a lemma for comparison of eigenvalues of Hermitian matrices.
[Weyl's inequality, see <cit.>]
For m∈_≥ 1. Let A, B be two Hermitian  m× m-matrices, and let λ_1 ≤λ_2 ≤…≤λ_m be the increasingly ordered eigenvalues of a Hermitian matrix, then we have the inequalities for the eigenvalues, for j,ℓ∈{1,…,m},

    λ_j+ℓ-m(A+B)≤λ_j(A)+λ_ℓ(B)≤λ_j+ℓ-1(A+B),

whenever the subscript makes sense.

As a consequence, we have the following comparison results for the eigenvalues of Toeplitz operators.

Let (X,J,Θ) be a connected, compact complex Hermitian manifold and let (L,h_L), (E,h_E) be holomorphic line bundles on X with smooth Hermitian metrics. Assume h_L to be positive. Let f_1, f_2∈ℒ^∞(X,_≥ 0) be such that each of them is not identically zero and f_1≤ f_2. For j=1,2, let 0<λ^p_min(f_j)=λ^p_1(f_j)≤λ^p_2(f_j)≤…≤λ^p_d_p(f_j)= λ^p_max(f_j) denote the ordered eigenvalues of T_f_j,p respectively. Then we have for all p∈, j=1,…,d_p,

    λ^p_j(f_1)≤λ^p_j(f_2).


As a consequence, if f_1, f_2∈ℒ^∞(X,_≥ 0) are comparable as in (<ref>), then we have for all p∈, j=1,…,d_p,

    1/Cλ^p_j(f_1)≤λ^p_j(f_2) ≤Cλ^p_j(f_1).



The comparison result (<ref>) is a direct consequence of the first part of this lemma, which is actually an application of <cit.>, we give a proof in the sequel.

Set h:=f_2-f_1≥ 0, then T_h,p= T_f_2, p - T_f_1,p is a positive semidefinite Hermitian operator. Take A=T_f_1,p, B=T_h,p and ℓ=1 in (<ref>), we conclude 

    λ^p_j(f_1)+λ^p_1(h)≤λ^p_j(f_2).

Since λ^p_1(h)≥ 0, we get (<ref>). The lemma is proved.









Among all the eigenvalues of T^2_f,p, we now concentrate on the lowest nonzero one. These lowest eigenvalues are also interesting in the sense of their correspondence to the lowest energy for the ground state in the Berezin–Toeplitz quantization
as in the work of Deleporte <cit.>. He obtained the asymptotic expansions for λ^p_min as well as for the corresponding eigensections when f≥ 0 is a smooth function with only nondegenerate vanishing points of order 2. Then the general results about the lowest eigenvalues are expected, so that we put it as the following conjecture/question:


Let (X,J,Θ) be a connected, compact complex Hermitian manifold and let (L,h_L), (E,h_E) be holomorphic line bundles on X with smooth Hermitian metrics. Assume h_L to be positive. For f∈𝒞^∞(X,) (it can take negative values), set κ(f):=sup_x∈ Xord_x(f)∈∪{+∞} (where ord_x(f) denotes the vanishing order of f at x), show that as p→ +∞

    minSpec^*(T^2_f,p)=≃ p^-κ(f)   ,  if  κ(f)<+∞; 
    ≃ e^-c_f √(p)   ,   if  κ(f)=+∞ and f=X 
    ≃ e^-c_f p   ,   if  κ(f)=+∞ and  X∖f≠∅
 
where c_f>0 is some constant depending on f, and Spec^*(T^2_f,p):=Spec(T^2_f,p)∖{0}, the sign ≃ means up to a multiplication of some nonzero constant. Moreover, a related interesting question is to describe the corresponding eigensections on X. Note that in Section <ref>, we give the examples on ℂℙ^1 where the lowest eigenvalues of T^2_f,p are the cases listed in (<ref>). Following the results on the off-diagonal decay of P_p(x,y) given in <cit.>, it is possible that we need to assume the analyticity on Θ and h_L, h_E to get the three nice cases in (<ref>).








The results in Proposition <ref> and Theorem <ref> shows a partial answer to the above question for a non-negative function f. In the sequel, we give their proofs as consequence of Theorems <ref>, <ref>, <ref> and the pluripotential theory on X.





In both case, we set λ^p_min(f)=minSpec(T_f,p), since f≥ 0 is nontrivial, we always have λ^p_min(f)>0. In the same time, we have the global point-wise estimate on X for all p,

    T_f,p(x,x)≥λ^p_min(f) P_p(x,x),

that is

    λ^p_min(f)≤T_f,p(x,x)/P_p(x,x).


Assume the case (i). Now we apply Theorem <ref> for the point x_0 together with the information on the Q_f,x_0(f) given in the last of Theorem <ref>, we get for all p≫ 0

    |1/p^nT_f,p(x_0,x_0)-(Q_2N,x_0(f)𝒫_x_0)(0,0)p^-N|≤ Cp^-N-1/2.

As a consequence, we have T_f,p(x_0,x_0)≤ Cp^n-N. 
In the same time, we have P_p(x_0,x_0)=b_0(x_0)p^n+𝒪(p^n-1). Taking these estimates in (<ref>), we obtain (<ref>).

For case (ii), since f is assumed to vanish with vanishing order +∞, by Theorem <ref> and the same arguments for (<ref>), we get that for any ℓ∈, we have a constant C_ℓ such that T_f,p(x_0,x_0)≤ C_ℓ p^n-ℓ. This way, we get (<ref>). The proof is completed.



We also have a uniform lower bound for T_f,p(x,x) by an explicit calculation on (Q_2N,x_0(f)𝒫_x_0)(0,0) in (<ref>): in fact, we have

    (Q_2N,x_0(f)𝒫_x_0)(0,0)=b_0(x_0)∑_α∈^2n,|α|=2N, C_α∂^2Nf/∂ v^α(x_0)>0,

where C_α is a constant depending only on α which is >0 if each component of α is even, and is 0 otherwise.
As a consequence, if f∈𝒞^2N+1(X,_≥ 0) and f vanishes on X with the vanishing order at most 2N, then there exists C>0 such that for all p>0, x∈ X,
	
    T_f,p(x,x)≥ Cp^n-N.

See also <cit.>. 
Using instead the expansion (<ref>) and Theorem <ref>, one can prove the analogous results of (<ref>) for T^2_f,p(x,x) with a general function f∈𝒞^ℓ(X,) which might take negative values, see also <cit.>.




Let B denote a very small (nonempty) open ball in X, and we consider the case f=1_B, the indicator function for B. In this case, d_p=𝒪(p^n), and we know T_f,p is injective and positive for all p≫ 0.

Now we set λ^p_min=λ^p_min(f):= minSpec(T_f,p) >0. For each p≫ 0, there exists a section s_p^min∈ H^0(X,L^p⊗ E) with s_p^min_ℒ^2(X,L^p⊗ E)=1 and

    λ^p_min=⟨ T_1_B,p s_p^min, s_p^min⟩_ℒ^2(X,L^p⊗ E)=∫_B |s_p^min(x)|^2_h_pdV(x).


Now we consider the family of integrable real functions v_p(x):=1/plog |s_p^min(x)|^2_h_p on X. As a consequence of Poincaré-Lelong formula, {v_p}_p≫ 0⊂PSH(X, 4c_1(L,h_L)), and this family of quasi-psh functions is bounded uniformly from above because of the uniform asymptotic expansion of P_p(x,x) on X. 

Now we prove that {v_p}_p≫ 0 is a bounded subset of ℒ^1(X,). If it is not true, we can assume that there exists a subsequence v_p_j such that

    v_p_j_ℒ^1(X,)→ +∞,  as j→ +∞.

Then we apply Proposition <ref> to this subsequence and X, we conclude that v_p_j has to converge uniformly to -∞ on X, but this contracts to our condition s_p^min_ℒ^2(X,L^p⊗ E)=1. Therefore, there exists a constant C>0 such that for all p≫ 0,

    v_p_ℒ^1(X,)≤ C.

Hence 

    ∫_B log |s_p^min(x)|^2_h_pdV(x) ≥ -Cp.


Similar to (<ref>), we get

    1/Vol(B)∫_B log|s_p^min(x)|^2_h_pdV(x) ≤log(1/Vol(B)∫_B |s_p^min(x)|^2_h_pdV(x) )

Then combining the above inequality with (<ref>), we conclude

    λ^p_min≥Vol(B) e^-Cp/Vol(B).

This shows that (<ref>) holds with f=1_B. In general, by our assumption on f from the statement, we can always find a small nonempty open ball B and a constant c>0 such that f≥ c1_B, then (<ref>) follows from Lemma <ref> for minimal eigenvalues and (<ref>). 

Now we prove (<ref>), we use the technique of choosing coherent sections and the estimate (<ref>). Assume now X∖ f ≠∅, then we fix x_0∉ f and a sufficiently small δ>0 such that the small geodesic ball B(x_0,2δ)⊂ X∖ f. We also assume that the line bundle L and E can be trivialized over B(x_0,2δ). Let e_L,x_0, e_E,x_0 denote the respective unit (smooth) frames of L_x_0, E_x_0, a coherent section S^p_x_0∈ H^0(X,L^p⊗ E) associated to e_L,x_0, e_E,x_0 is the unique section S^p_x_0 such that for any s_p∈ H^0(X,L^p⊗ E), we have

    s_p(x_0)=⟨ s, S^p_x_0⟩_(X,L^p⊗ E)e_L,x_0^⊗ p⊗ e_E,x_0.

In fact, we have S^p_x_0(x)=P_p(x,x_0)e_L,x_0^⊗ p⊗ e_E,x_0. Then we have

    S^p_x_0^2_(X,L^p⊗ E)=P_p(x_0,x_0)∼b_0(x_0)p^n,

and 

    |S^p_x_0(y)|_h_p,y=|P_p(y,x_0)|_h_p,y⊗ h_p,x_0^*.


As a consequence, we get

    ⟨ T_f,p S^p_x_0,S^p_x_0⟩_ℒ^2   =
    ∫_X f(y)|S^p_x_0(y)|^2_h_p,ydV(y)
       ≤f_ℒ^∞∫_dist(y,x_0)≥δ
    |P_p(y,x_0)|^2_h_p,y⊗ h_p,x_0^*dV(y)
       ≤ C_δ,AVol(X) 
    f_ℒ^∞ e^-A√(plogp),

where the last line follows from the estimate (<ref>).

Finally, we get

    λ^p_min≤⟨ T_f,p S^p_x_0,S^p_x_0⟩_ℒ^2/S^p_x_0^2_(X,L^p⊗ E)≤ Ce^-A√(plogp)/p^n,

then (<ref>) follows. The proof is complete.



As in <cit.>, when Θ, h_L, h_E are analytic, we have a sharper version of (<ref>) as follows

    sup_dist(x,y)≥δ|P_p(x,y)|_h_p,x⊗ h^*_p,y≤ C_δ e^-c_δ p.

Then in this case, when f is not fully supported on X, we have

    minSpec(T_f,p)≤ C' e^-c'p,

 which fits exactly the third case in (<ref>) under the analyticity condition.


As a consequence, we get the uniform point-wise lower bound for log T^2_f,p(x,x) on a compact manifold X.

With the same geometric assumption in Theorem <ref> or Lemma <ref>, for f∈ℒ^∞(X,_≥ 0) which is not identically zero and is continuous near a nonvanishing point, there exists a constant C_f>0 such that for all p≫ 0, x∈ X,

    log T^2_f,p(x,x)≥ -C_f p.





For any nontrivial f∈𝒞^0(X,_≥ 0), we set

    c(f):=lim inf_p→ +∞1/plogλ_p,min(f)∈_≤ 0.

By Theorem <ref>, the limit c(f) always exists (that is, finite) and is nonpositive. For f≡ 0, we set c(0):=-∞.


The following result is an easy consequence of Lemma <ref>.

When f>0 is a continuous function on X and never vanishes, we have c(f)=0. If f_1, f_2∈𝒞^0(X,_≥ 0) are comparable in the sense of (<ref>), then we have

    c(f_1)=c(f_2).

This way, we define a function c: ℰ𝒞^0(X,_≥ 0)→ [-∞,0].  






 §.§ Examples and simulations on Riemann sphere




In this Section, we present the examples of the Toeplitz operators on ℂℙ^1, in particular, we give the explicit computations on their spectra. In the last part, we present the random zeros of the sections given by certain Toeplitz operators acting on the SU(2)-random polynomials on .

Let us start with our basic settings. We consider one standard chart 
U_0≃ for ℂℙ^1. In this chart, the Fubini-Study metric is 
given by

    Θ=ω_FS=√(-1)/2πdz∧ 
    	dz̅/(1+|z|^2)^2.

It is also the volume form on  that we will use.

For p∈ℕ_>0, let 𝒪(p) denote the 
holomorphic line bundle on ℂℙ^1 of degree k. Then 
𝒪(p)=𝒪(1)^⊗ p, we equip 
𝒪(1) with the standard Fubini-Study Hermitian metric 
h_FS and equip 𝒪(p) with the induced 
Hermitian metric. Then 
Θ=ω_FS=c_1(𝒪(1),h_FS).

On this chart, the global holomorphic sections of 
𝒪(p) are given by all the polynomials in z with degree 
≤ p, i.e.,

    H^0(ℂℙ^1,𝒪(p))=Span_{1,z,⋯, z^p}.

The canonical orthonormal basis of 
H^0(ℂℙ^1,𝒪(p)) with respect to the 
-inner product is given by

    S^p_j(z)=√((p+1)pj) z^j,   j=0,1,⋯, p.


Note that SU(2) acts on ℂℙ^1 holomorphically 
and transitively.  On the chart  the action is given by

    g· z = a+b z/c+d z∈,   
    g=[ a b; c d ]∈SU(2).



The action of SU(2) always lifts to 𝒪(p)
and preserves the metrics 
Θ and h_FS. In particular, SU(2) acts 
on H^0(ℂℙ^1,𝒪(p)) isometrically with respect 
to -inner product. 

Note that for the explicit expressions of eigenvalues of T_f,p in Example <ref> for f_k and in Section <ref> for 1_r (r=1,1/4) were already computed in <cit.>.




 §.§ Lowest eigenvalues of Toeplitz operators for fully supported functions

In this part, we compute the spectra of Toeplitz operators associated to some special functions, as examples for the first two cases listed in (<ref>).



For k∈_≥ 1, on U_0≃, we take the function 

    f_k(z):=|z|^2k/(1+|z|^2)^k

Then f_k is a smooth nonnegative function on ℂℙ^1 
with the only vanishing point at z=0, and the vanishing order is 2k.

At first, we have for 0≤ j≠ℓ≤ p, then

    ⟨ T_f_k,p S^p_j, S^p_ℓ⟩_(ℂℙ^1,𝒪(p))=0

Then we have

    Spec  T_f_k,p={⟨ T_f_k,p S^p_j, S^p_j⟩_(ℂℙ^1,𝒪(p)) >0  :  j=0,1,…,p}.

By elementary techniques on the integrals, we get

    ⟨ T_f_k,p S^p_j, S^p_j⟩_(ℂℙ^1,𝒪(p)) = (j+k)!/j!(p+1)!/(k+p+1)!.

It is clear that this quantity increases as j grows. Then we have the following smallest (j=0) and biggest (j=p) eigenvalues of T_f_k,p:

    λ^p_min=k!(p+1)!/(k+p+1)!, λ^p_max=1-k/p+k+1.

In particular, we have, as p→ +∞,

    λ^p_min=k!p^-k(1+k(k+3)/2p+𝒪(p^-2)).

The corresponding eigensection is S^p_0.

The asymptotic expansion in (<ref>) gives an example for the first case listed in (<ref>).






On U_0≃, we take the function 

    f(z):=e^-1/|z|^2.

Then f_k is a smooth nonnegative function on ℂℙ^1 with the only vanishing point at z=0, and the vanishing order is +∞, but we have f=ℂℙ^1.

Similar to (<ref>), for 0≤ j≠ℓ≤ p, we have

    ⟨ T_f,p S^p_j, S^p_ℓ⟩_(ℂℙ^1,𝒪(p))=0

Then we have

    Spec  T_f,p={⟨ T_f,p S^p_j, S^p_j⟩_(ℂℙ^1,𝒪(p)) >0  :  j=0,1,…,p}.

By elementary techniques on the integrals, we get

    λ^p_j:=⟨ T_f,p S^p_j, S^p_j⟩_(ℂℙ^1,𝒪(p)) = (p+1)!/j!HyperU(p+2,j+2,1),

where HyperU(a,b,z) denotes the Tricomi confluent hypergeometric function, that is defined as

    HyperU(a,b,z):=1/Γ(a)∫_0^+∞ e^-ztt^a-1(1+t)^b-a-1dt.

Then we can rewrite

    λ^p_j=∫_0^+∞ e^-t(t/1+t)^p+1(1+t)^j/j!dt.

A direct computation shows that 

    λ^p_min:=λ^p_0< λ^p_1<⋯<λ^p_p=:λ^p_max.

For the biggest (j=p) eigenvalues of T_f,p, we have the following explicit formula

    λ^p_max   =1+∑_j=1^p (-1)^j(p-j)!/p!+(-1)^p+1/p!∫_0^+∞e^-t/1+tdt
       =1-1/p+𝒪(p^-2),    as  p→ +∞.

For the smallest eigenvalues, we have

    λ^p_min=∫_0^+∞e^-t(t/1+t)^p+1dt=(p+1)∫_0^+∞e^-tt^p/(1+t)^p+2dt.

Now let us give the asymptotic behavior of λ^p_min as p→ +∞. At first, we have

    (p+1)∫_0^+∞e^-tt^p/(1+t)^p+2dt    ≤ (p+1)∫_0^√(p)/2t^p/(1+t)^p+2dt 
            + (p+1)∫_√(p)/2^+∞e^-t(t/1+t)^pdt/(1+t)^2
       ≤(√(p)/2/1+√(p)/2)^p+1+p+1/√(p)/2+1e^-2√(p)+𝒪(1),

where the second term follows from finding the maximum of e^-t(t/1+t)^p.  Using the fundamental inequality for t>0,

    (t/1+t)^1+t≤ e^-1≤(t/1+t)^t,

we get (for all sufficiently large p)

    (p+1)∫_0^+∞e^-tt^p/(1+t)^p+2dt ≤ 2√(p)e^-2√(p)+𝒪(1).

For the lower bound, we get

    (p+1)∫_0^+∞e^-tt^p/(1+t)^p+2dt    ≥ (p+1)e^-√(p)∫_0^√(p)t^p/(1+t)^p+2dt 
       ≥ e^-√(p)(√(p)/1+√(p))^p+1
       ≥ e^-2√(p)-1/√(p),

where the last estimate follows from (<ref>).

Finally, we combining (<ref>) and (<ref>), we get an asymptotic formula for λ^p_min:

    λ^p_min=e^-2√(p)(1+o(1)),

or equivalently

    lim_p→∞logλ^p_min/√(p)=-2.

This gives an example for the second case listed in (<ref>).







 §.§ Spectrum of Toeplitz operators for indicator functions


For r>0, set the function 1_r on  as

    1_r(z)=1_𝔻(0,r)(z),

where 1_𝔻(0,r) denotes the indicator function 
for the open disc 𝔻(0,r) of radius r. In the sequel, for 
U⊂ or ℂℙ^1, let Vol(U) denote the 
Fubini-Study volume of U. In particular, we have

    Vol(𝔻(0,r))=r^2/1+r^2.

This disc 𝔻(0,r) is a geodesic ball in ℂℙ^1 
with radius 1/√(π)arctanr (with respect to 
Fubini-Study metric).

Now we consider the Toeplitz operators T_1_r,p acting on 
H^0(ℂℙ^1,𝒪(p)). 

	The spectrum of T_1_r,p is given by {λ^p_j(r), 
	j=0,1,⋯, p} where
	
    λ^p_j(r)=(1+r^2)^-p-1∑_i=0^p-jp+1p-i-jr^2i+2j+2.

	Moreover, we have
	
    λ^p_max(r):=max{λ^p_j(r), 
    	j=0,1,⋯, p}=λ^p_0(r)=1-1/(1+r^2)^p+1,
       λ^p_min(r):=min{λ^p_j(r), 
    	j=0,1,⋯, 
    	p}=λ^p_p(r)=(r^2/1+r^2)^p+1=Vol(𝔻(0,r))^p+1.

	For j=0,1,⋯, p, the λ^p_j(r)-eigenspace is 
	spanned by S^p_j (hence is 1-dimensional).


By definition, we have

    λ^p_j(r)   =⟨ T_1_r,p S^p_j, S^p_j⟩_ℒ^2
       =(p+1)pj∫_0^r^2t^j/(1+t)^p+2dt
       =(p+1)pj∫_0^1 t^j/(1+r^2 t)^p+2dt
       =r^2j+2p+1/j+1pj_2F_1(p+2,j+1;j+2;-r^2),

where _2F_1(a,b;c;z) denotes the hypergeometric function _2F_1. Using the fact that p and j are non-negative integers, the value of _2F_1(p+2,j+1;j+2;-r^2) can be worked out explicitly so that we get exactly (<ref>). The rest part is clear.


Using the SU(2)-symmetry, we get the following corollary.

	Let 𝔹 be a nontrivial geodesic ball in ℂℙ^1 with 
	Vol(𝔹)∈ ]0,1], then we have
	
    minSpec(T_1_𝔹,p)=Vol(𝔹)^p+1.

	In particular, if Vol(𝔹)<1, then 
	minSpec(T_1_𝔹,p) decays 
	exponentially to 0 as p→ +∞.


Finally, as a consequence of Lemma <ref>, we get

	For a f∈ℒ^∞(ℂℙ^1,_≥ 0), if there 
	exists two geodesic ball 𝔹⊂𝔹' and two 
	constants C≥ c>0 such that
	
    C1_𝔹'≥ f ≥ c1_𝔹,

	Then we have for p≥ 1,
	
    CVol(𝔹')^p+1≥minSpec(T_f,p) ≥ 
    		cVol(𝔹)^p+1.

	In particular, when Vol(𝔹')<1, 
	minSpec(T_f,p) decays 
	exponentially to 0 as p→ +∞.







 §.§ Simulations of random zeros on the support
of the symbol

In this part, we present some simulation results for the zeros of S_f,p on ℂℙ^1, where the function f is given as in Sections <ref> and <ref>. 

We now explain our computation model. Fix a concrete choice of f as above, we already know the precise spectrum of T_f,p: for j=0,…, p, let λ^p_j(f) be the eigenvalues of T_f,p with the eigensection S^p_j. Then we write on U_0≃,

    S_f,p(z):=∑_j=1^p η_j λ^p_j(f) √((p+1)pj) z^j,

where {η_j} is a sequence of i.i.d. standard complex Gaussian random variables, such random variables can be simulated properly by the mathematical computating softwares. Our random section S_f,p now becomes a random polynomial on  of degree p. In the special case where f≡ 1, S_1,p=S_p is exactly the SU(2)-polynomial.

In the sequel simulations, we always compare our random zeros with the expected distribution ω_FS on U_0≃. In order to visualize the comparison, we will classify all the random zeros for each simulation according to their Fubini-Study norms (defined below as r_FS), then we draw the corresponding density histograms. Such density histograms are viewed as the approximations to the probability density function with respect to the Fubini-Study norms of zeros of the polynomial S_f,p(z); in the same time, the corresponding probability density function of ω_FS in the Euclidea norm r=|z| is given by the function

    _≥ 0∋ r↦2r/(1+r^2)^2.

We consider the polar coordinate on U_0≃ with respect to ω_FS, so that for any z∈, r=|z|, it has the ω_FS-Riemannian distance r_FS=1/√(π)arctanr from z=0, in particular, r_FS=√(π)/2 corresponds to the North Pole {∞}∈ℂℙ^1. So that the diameter of (ℂℙ^1, ω_FS) is √(π)/2≃ 0.88622…, and the equator is given by r=|z|=1 or the circle with r_FS=√(π)/4≃ 0.44311…. 
For z∈≃ U_0, r_FS=1/√(π)arctan|z| is called the Fubini-Study norm of z. By (<ref>), in terms of the Fubini-Study norm r_FS∈ [0,√(π)/2], the density function of ω_FS on ℂℙ^1 is given by the function

    ψ(r_FS)=√(π)sin(2 √(π) r_FS).

Then by plotting the graph of ψ along with the density histograms of the Fubini-Study norms of zeros from the simulations, we can visualize directly the differences between ω_FS and the simulated random zeros.

Through all the simulations, we also keep the total number of random zeros to be a fixed large number such as 20000 for each comparison. This means, if we consider the degree p (such as p=20,50,100,200, which is a factor of 20000), we will repeat RN:=20000/p times of realizing S_f,p(z) and computing its zeros (RN is the short for repeating number), this way, we will get in total 20000 points as random zeros, then we draw the density histogram with respect to their Fubini-Study norms described as above.

We use MATLAB (version 23.2, R2023b) to perform the aforementioned simulations. Note that for large p (such as 100, 200 or bigger), the combinatorial numbers in the coefficients of (<ref>) can be extremly big, and the software can only compute properly their numerical approximations due to the precision limit on a laptop. Such simulations are not suitable for a precise quantitative analysis but they are good enough for our purpose of visualizing the random zeros and comparing them with the expected one. The following Figure <ref> shows one simulation for the zeros of SU(2)-polynomial. The picture in the left-hand side plots the roots that lie inside the square of side length 12 among all 20000 roots obtained via RN=1000 realizations of SU(2)-polynomials of degree p=20, and the one in the right-hand side compares the density histogram of the Fubini-Study norms of all these 20000 roots with the expected density function ψ(r_FS) (plotted as the red curve). From this comparison, we can see how well they fit to each other when p=20.


Now we consider two examples: f_3 from Section <ref> and 1_1 from Section <ref>.  
[The function f_3(z)=|z|^6/(1+|z|^2)^3]
Following Example <ref>, let us consider the function f_3(z)=|z|^6/(1+|z|^2)^3. The only vanishing point for f_3 is z=0 (corresponding to r=0), and the vanishing order is 6. Since f_3=ℂℙ^1, by Theorem <ref>, the random zeros 1/p[(S_f_3,p)] will converge to ω_FS as p→ +∞. We shall expect a nice approximation as shown in the right-hand side of Figure <ref> for sufficently large p.
 




We did 4 simulations for different degrees: p=20,  50,  100,  200. The results are displayed as 4 pictures in Figure <ref>. It is a straighforward observation that the simulated random zeros approximate better and better to the expected one represented by ψ(r_FS) as the degree p grows. This is exactly the main point proved in Theorem <ref>. In the same time, we see also that, different from the case of SU(2)-polynomial in Figure <ref>, the result with degree p=20 in Figure <ref> does not fit nicely with ψ(r_FS). The main deviation happens around z=0 (r_FS=0), which is exactly the unique vanishing point of the function f_3(z). Roughly speaking, this vanishing point is a sort of global minimizer of log T_f,p^2(x,x) on ℂℙ^1, such that for small p, one should observe more zeros around this minimizer.



Analogous to this example, one can also consider the function f(z)=e^-1/|z|^2 discussed in Example <ref>, then the point z=0 will still behave abnormally as in Figure <ref>, and since the vanishing order is +∞, we will need a very large p≫ 200 to observe a nice approximation of the simulation result to ψ(r_FS).




[The function 1_1(z)=1_𝔻(0,1)(z)]
A simulation result for the indicator function 1_1 was shown in Figure <ref>. As in previous example, we now increase the degrees: in Figure <ref>, we show how the simulated zeros change as the degree p goes from 20 to 200. As in Theorems <ref> and <ref>, the random zeros approximate to ω_FS on the support of 1_1(z) (the part {|z|≤ 1}={r_FS≤ 0.44311…}). But outside the support, the simulated zeros do not converge to c_1(L,h_L), and the observation is that the density of random zeros has a rapid drop outside but near the support, and has a concentration near the farthest point from the support. It is an interesting question to investigate such phenomenon in general.







If we increase the radius r, that is, enlarging the support of the function 1_r to define the Toeplitz operator, we will observe that the simulation of random zeros at degree p=20 behaves better than in Figure <ref> when we compare it with the density function ψ. In Figure <ref>, we show a simulation result for r=4 and p=20, RN=1000, and such result is supported by Theorem <ref> under the condition that the support of 1_r nearly fills in the whole part of ℂℙ^1.











§ SMOOTH STATISTICS FOR RANDOM ZEROS OF -HOLOMORPHIC SECTIONS

 In this section, we always consider the same geometric setting in Sections <ref> and <ref>, in particular, we assume the Condition <ref> to hold.




We also concentrate on the case f∈𝒞^∞_c(X,) and the associated random sections S_f,p, p∈. The goal of this section is to explain some partial extensions of the seminal results on the smooth statistics of random zeros obtained by Shiffman–Zelditch <cit.> on compact Kähler manifolds to the zeros of S_f,p inside the support of f. In particular, we focus on the number variance of ⟨ [(S_f,p)],φ⟩ with a given test form φ supported in f and the corresponding central limit theorem (see also <cit.>). Note that, except allowing X to be noncompact , another difference in our geometric setting from <cit.> is that we do not assume the connection between the Hermitian form Θ with c_1(L,h_L). The proofs essentially follow from the arguments
presented in <cit.>, 
and we will point out the necessary modifications for our setting.





 §.§ Number variance on the support of the symbol

For f∈𝒞^∞_c(X,)
which is not identically zero, we have studied in
Section <ref> the random (1,1)-currents 
[Div(S_f,p)] (p≫ 0) on f, 
especially, the expectations 
[[Div(S_f,p)]]. 
Now we are going to study the variance of [Div(S_f,p)].


Following <cit.>, we now introduce the variance current of [Div(S_f,p)]. Let π_1, π_2: X× X→ X denote the projections to the first and second factors. Then if S and T are two currents on X with respective degree r and q, then we define a current of degree r+q on X× X as follows

    S⊠ T:= π_1^* S∧π_2^* T.
 
In particular, [Div(S_f,p)]⊠ [Div(S_f,p)] defines a random (2,2)-current on X× X.

In the same time, we introduce the following notation: for a current T on X× X, we write

    ∂ T=∂_1 T+∂_2 T,

where ∂_1, ∂_2 denote the corresponding ∂-operators on the first and second factors of X× X. Let (z_1,…, z_n) be a local complex coordinate on the first factor of X× X, and let (w_1,…, w_n) be a local complex coordinate on the second factor of X× X, then we can write locally

    ∂_1=∑ dz_j ∂/∂ z_j, ∂_2=∑ dw_j ∂/∂ w_j.

Similarly, we also write ∂ T=∂_1 T+∂_2 T.



The variance current of [Div(S_f,p)], denoted as 𝐕𝐚𝐫[S_f,p], is a (2,2)-current on X× X defined by

    𝐕𝐚𝐫[S_f,p]:=[[Div(S_f,p)]⊠ [Div(S_f,p)]]-[[Div(S_f,p)]]⊠[[Div(S_f,p)]]



In order to simplify the notation, it is enough to consider only the real test forms Ω_c^n-1,n-1(X,). For φ∈Ω_c^n-1,n-1(X,), we have

    Var[⟨ [Div(S_f,p)],φ⟩]=⟨𝐕𝐚𝐫[S_f,p], φ⊠φ⟩.


Shiffman and Zelditch <cit.> established the framework to compute such variance current on a compact Kähler manifold, in particular, they obtained a pluri-bipotential for it. Their method can be easily adapted to our setting. Let us start with recalling the main ingredients from their results.

For t∈ [0,1], we set the function

    G(t):=-1/4π^2∫_0^t^2log(1-s)/sds=1/4π^2∑_j=1^∞t^2j/j^2.

This is an analytic function with radius of convergence 1. Moreover, for t∼ 0, we have G(t)=𝒪(t^2).

Set W_p={z∈ X :  T^2_f,p(z,z)=0}⊂ X.
Recall that the function N_f,p(z,w) on X× X
is defined in (<ref>). 
This is a smooth function on X× X∖ (W_p× X ∪ X× W_p) with values in [0,1]. In particular, for z∈ X∖ W_p, N_f,p(z,z)=1.
[cf. <cit.>]
For (z,w)∈ X× X∖ (W_p× X ∪ X× W_p), define

    Q_f,p(z,w):=G(N_f,p(z,w))=-1/4π^2∫_0^N_f,p(z,w)^2log(1-s)/sds.

Then Q_f,p(z,w) is a continuous function on (z,w)∈ X× X∖ (W_p× X ∪ X× W_p).


Since the near-diagonal behavior of N_f,p(z,w) depend on if there points z,w lie in the support of f or not, which are different from the case for Bergman kernel (such as in <cit.> or <cit.>). Following the computations in <cit.> and we use our results proved in Theorem <ref> and Lemma <ref>, we have the following results for Q_f,p(z,w).
[cf. <cit.>]
Let U be an open subset of X such that U⊂{f≠ 0} (hence U is compact).


  (i) Then there exists an integer p_0∈ such that for all p≥ p_0, T^2_f,p(z,z) never vanishes on U. Moreover, for all p≥ p_0, the function Q_f,p(z,w) is smooth on the region U× U∖Δ_U (Δ_U denotes the diagonal) and it is 𝒞^1 on U× U.

  (ii) Fix b≫ 0 and ϵ>0, then for all sufficiently large p and for x∈ U, v∈ T_x X with v≤ b√(logp), we have

    Q_f,p(x,exp_x(v'/√(p)))=G(exp(-Φ_x(0,v')^2/4))+𝒪(p^-1/2+ϵ),

where Φ_x(0,v') is defined in (<ref>).

  (iii) For given k, ℓ∈, there exist a sufficiently large b>0 such that there exist a constant C>0 such that for all z,w∈ U, dist(z,w)≥ b√(logp/p), we have

    |∇^ℓ_z,wQ_f,p(z,w)|≤ C p^-k.




For a real (n-1,n-1)-form  φ on X with 𝒞^3-coefficients, recall that L(φ)∈𝒞^1(X,) is defined by

    √(-1)∂∂φ =
     L(φ) c_1(L,h_L)^n/n! ·

Recall also that we have two volume forms 
dV=Θ^n/n! and dV^L:=c_1(L,h_L)^n/n! 
(see (<ref>)). Moreover, we have

    dV^L(z)=b_0(z) dV(z),

where the positive function 
b_0(z)=(Ṙ_z^L/2π) on X is given in (<ref>).

Let (X,J,Θ) be a connected complex Hermitian manifold and let (L,h_L), (E,h_E) be two holomorphic line bundles on X with smooth Hermitian metrics.
We assume Condition <ref> (see also (<ref>)) to hold. Fix f∈𝒞^∞_c(X,) which is not identically zero, and let U be an open subset of X such that U⊂{f≠ 0}. Then for sufficiently large p, we have the identity of (2,2)-currents on U× U,

    𝐕𝐚𝐫[S_f,p]|_U× U=-∂_1∂_1∂_2∂_2 Q_f,p(z,w)|_U× U=(√(-1)∂∂)_z  (√(-1)∂∂)_w Q_f,p(z,w)|_U× U.


Let φ be a real (n-1,n-1)-form on X with 𝒞^3-coefficients and φ⊂ U, then we have the formula for p≫ 0,

    Var[⟨ [Div(S_f,p)],φ⟩]=p^-n(ζ(n+2)/4π^2∫_U |L(φ)(z)|^2 dV^L(z)+𝒪(p^-1/2+ϵ)),

where

    ζ(n+2)=∑_k=1^∞1/k^n+2.
 


We sketch the proof based at the proofs of <cit.> and <cit.>. At first, fix a test form φ∈Ω^n-1,n-1_c(U,), a routine calculation (see also <cit.>) shows that

    Var[⟨[Div(S_f,p)], φ⟩]
       = -1/π^2 ∫_U× U (∂∂φ(z))∧
    (∂∂φ(w)) 
               ×[ log|
    T^2_f,p(z,z)^-1/2S_f,p(z)|_h_p·log|
    T^2_f,p(w,w)^-1/2S_f,p(w)|_h_p].

Then by Proposition <ref> (i) and <cit.>, on U× U and for all p≫ 0, we have 

    [ log|
    T^2_f,p(z,z)^-1/2S_f,p(z)|_h_p·log|
    T^2_f,p(w,w)^-1/2S_f,p(w)|_h_p]=γ^2/4+π^2G(N_f,p(z,w)),

where γ is the Euler's constant. Then we can rewrite (<ref>) as

    Var[⟨[Div(S_f,p)], φ⟩]    = - ∫_U× U (∂∂φ(z))∧
    (∂∂φ(w))G(N_f,p(z,w))
       =⟨ (√(-1)∂∂)_z  (√(-1)∂∂)_w Q_f,p(z,w),φ⊠φ⟩.

This way, we get (<ref>). In fact, (<ref>) still holds if φ is with 𝒞^3-coefficients, now we show (<ref>) by using Proposition <ref>-(ii) and (iii).

The first step is to rewrite the integral in the form

    ∫_U× U⋯=∫_z∈ U∫_{z}× U⋯.

As in <cit.>, we set for z∈ U,

    ℐ_p(z)   =∫_{z}× UQ_f,p(z,w)
    (√(-1)∂∂φ(w))
       =∫_{z}× U Q_f,p(z,w)L(φ)(w)
    b_0(w)dV(w).

Let b>0 be a fixed number which is sufficiently large as in
Proposition <ref> (iii) with k=n+1. 
Then we have for p≫ 0,

    ℐ_p(z)=∫_v∈ T_z X, v≤ 
    b√(logp) Q_f,p(z,exp_z(v/√(p)))
    (L(φ)b_0)(exp_z(v/√(p)))
    dV(exp_z(v/√(p)))   
    
    +𝒪(p^-n-1).


As in (<ref>), let 
dV_Eucl_,z(v) denote the Euclidean 
volume form on the real vector space (T_z X, g^TX_z). 
Then for v∈ T_z X, v≤ b√(logp), we have

    dV(exp_z(v/√(p)))=
    1/p^n(dV_Eucl_,z(v)+
    𝒪(√(p^-1logp))).

Since L(φ) is 𝒞^1 on U, 
then by Proposition <ref>-(ii) and 
<cit.>, 
we get for a fix small ϵ>0,

    ℐ_p(z)=(L(φ)b_0)(z)/p^n∫_v∈ T_z X, v≤ b√(logp)G(exp(-Φ_x(0,v')^2/4))
    dV_Eucl_,z(v)+𝒪(p^-n-1/2+ϵ).


By (<ref>), we can fix b to be large enough such that 

    ∫_v∈ T_z X, v≥ b√(logp)G(exp(-Φ_x(0,v')^2/4))
    dV_Eucl_,z(v) =𝒪(p^-n-1).

In the same time, by the formula (<ref>)
for Φ_x(0,v'), then for k∈_≥ 1, we have

    ∫_v∈ T_z X(exp(-kΦ_x(0,v')^2/2))
    dV_Eucl_,z(v) = 1/b_0(z) k^n.

Finally, combining the Taylor series (<ref>)
with (<ref>) – (<ref>), we get for z∈ U,

    ℐ_p(z)=p^-n(ζ(n+2)/4π^2
    L(φ)(z)+𝒪(p^-1/2+ϵ)),

after taking the integration with respect to z∈ U, we conclude exactly (<ref>). This way, we complete our proof.








 §.§ Asymptotic normality of random zeros;
proof of Theorem <ref>


The asymptotic normality of the zeros of random holomorphic functions 
or sections has been introduced and proved by Sodin–Tsirelson 
<cit.> for certain random holomorphic functions on 
 or 𝔻 and by Shiffman–Zelditch 
<cit.> for the random holomorphic sections of 
line bundles on a compact Kähler manifold. One key ingredient in 
their approaches is the normalized Bergman kernel which is
the covariance function of the corresponding Gaussian holomorphic fields on  or X, analogous to the construction in the proof of Proposition <ref>. Then the 
problem is reduced to the seminal result proved by Sodin and Tsirelson 
in <cit.> for the non-linear functionals of Gaussian process.

Let us recall the main result of <cit.>. Let (T,μ) be a measure space with a finite positive measure μ (with μ(T)>0). We also fix a sequence of measurable functions A_k: T→, k∈ such that on T,

    ∑_k |A_k(t)|^2≡ 1.


We consider a complex-valued Gaussian process on T defined as

    W(t):=∑_k η_k A_k(t).

Then {η_k} is a sequence of i.i.d. standard complex Gaussian random variables. Then for each t∈ T, W(t)∼𝒩_(0,1). The covariance function for W is ρ_W: T× T→ given by

    ρ_W(s,t):=[W(s)W(t)]=∑_k A_k(s)A_k(t).



Let {W_p}_p∈ be a sequence of independent Gaussian processes on T described as above, and let ρ_p(s,t) (p∈) denote the corresponding covariance functions. We also fix a nontrivial real function F∈ℒ^2(_+, e^-r^2/2r dr), and a bounded measurable function ψ: T→, set

    Z_p:=∫_T F(|W_p(t)|)ψ(t)dμ(t).


Sodin and Tsirelson proved the following results.

[<cit.>]
With the above constructions, and suppose that

	
  (i) 
    lim inf_p→ +∞∫_T∫_T |ρ_p(s,t)|^2αψ(s)ψ(t)dμ(s)dμ(t)/sup_s∈ T∫_T |ρ_p(s,t)|dμ(t)>0,

	for α=1 if f is monotonically increasing, or for all α∈ otherwise;
	
  (ii) 
    lim_p→+∞sup_s∈ T∫_T |ρ_p(s,t)|dμ(t)=0.


Then the distributions of the random variables

    Z_p-[Z_p]/√(Var[Z_p])

converge weakly to  the (real) standard Gaussian distribution 𝒩_(0,1) as p→ +∞.




Now we are ready to present the proof of Theorem <ref>.

The proof is an easy modification of <cit.>, together with the results in Theorem <ref> due to the assumption U⊂{f≠ 0}.

We take F(r)=logr, (T,μ)=(U,dV^L|_U), 
ψ(z)=1/π L(φ)(z) which satisfies the 
conditions in Theorem <ref>. 
Let σ:U→ L be a continuous section 
such that |σ(z)|_h_L≡ 1 on U. 
For each p, fix an orthonormal basis 
{S^p_j}_j=1^d_p(f) consisting of the eigensections of T_f,p
for nonzero eigenvalues {λ^p_j}_j. 
Then on U, we write

    S^p_j(z)=a^p_j(z)σ^⊗ p(z).

Then we can set A^p_j(z)=λ^p_j a^p_j(z)/√(T^2_f,p(z,z)),
which forms a sequence of measurable functions on U satisfying (<ref>). Then by (<ref>), we have the identity on U

    S_f,p(z)/√(T^2_f,p(z,z))=W_p(z) σ^⊗ p(z),

where W_p is the Gaussian process on U constructed
as in (<ref>). The covariance function 
ρ_p(z,w) for W_p is given by

    |ρ_p(z,w)|=N_f,p(z,w).

Let Z_p be the random variable defined
as in (<ref>) from W_p. 
Then (<ref>) implies that

    Z_f,p(φ)=Z_p + C_p,

where C_p is a deterministic constant. Thus the asymptotic 
normality of Z_f,p(φ) is equivalent to that of 
Z_p, which follows by checking the Conditions (i) and (ii) 
in Theorem <ref> for N_f,p(z,w). 
Finally, we apply Theorem <ref> and proceed
as in the last part of <cit.>. This completes the proof.














	Declarations



Conflict of interest. On behalf of all authors, the corresponding author states that there is
no conflict of interest.

Data sharing. Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

siam





