$\frac{\partial^{2}\varphi}{\partial^{2}x_{2}}(0)-\frac{\partial^{2}\varphi}{\partial^{2}y_{2}}(0)=2\frac{\partial^{2}\varphi}{\partial\overline{z}_{2}\partial
z_{2}}(0)\neq 0$. Let us say
$\frac{\partial^{2}\varphi}{\partial^{2}x_{2}}(0)\neq 0$. Since
$\frac{\partial\varphi}{\partial x_{2}}(0)=0$ we find $\frac{\partial
g}{\partial x_{2}}(0)=0$ and hence that $\frac{\partial\eta}{\partial
x_{2}}(0)\neq 0$ where $\eta$ is defined by
$\eta(x^{\prime},y):=\left(\frac{\partial\varphi}{\partial
x_{2}}\right)(g(x^{\prime},y),x^{\prime},y)$. By the implicit function theorem
there exists a smooth real valued function $h$ with $h(0)=0$ such that near
the origin $\eta^{-1}(0)$ can be written as the graph of $h$ that is
$x_{2}=h(x_{3},\ldots,x_{n+1},y)$. For
$x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}$ write
$x^{\prime\prime}=(x_{3},\ldots,x_{n+1})$. Using Taylor expansion we find

$g(h(x^{\prime\prime},y)+t,x^{\prime\prime},y)=g(h(x^{\prime\prime},y),x^{\prime\prime},y)+\frac{\partial
g}{\partial
x_{2}}(h(x^{\prime\prime},y),x^{\prime\prime},y)t+\frac{1}{2}\left(\frac{\partial^{2}g}{\partial^{2}x_{2}}\right)(h(x^{\prime},y),x^{\prime},y)t^{2}+O(t^{3}).$

By the construction of $h$ we have $\frac{\partial g}{\partial
x_{2}}(h(x^{\prime\prime},y),x^{\prime\prime},y)=0$ for all
$(x^{\prime\prime},y)$ near zero. Furthermore,
$\frac{\partial^{2}\varphi}{\partial^{2}x_{2}}(h(x^{\prime\prime},y),x^{\prime\prime},y)\neq
0$ for all $(x^{\prime\prime},y)$ near zero. Hence we find
$g(h(x^{\prime\prime},y)+t,x^{\prime\prime},y)=g(h(x^{\prime\prime},y),x^{\prime\prime},y)+(f(t,x^{\prime\prime},y)t)^{2}$
where $f$ is smooth function defined in a neighborhood of zero in
$\mathbb{R}^{2n+1}$ where $f$ is bounded and bounded from below by some
positive constant and has bounded derivatives. Now it is enough to show that
the function

(4.1.2)
$\displaystyle(t,x^{\prime\prime},y)\mapsto\left(\sqrt{(g(h(x^{\prime\prime},y),x^{\prime},y)+t^{2})^{2}+y_{1}^{2}}\right)^{-1}$

is integrable in a sufficiently small open neigborhood of zero in
$\mathbb{R}^{2n+1}$. Because then the integrability of $\iota^{*}\beta$ on
$bG\cap U$ for some open neighbourhood $U\subset\mathbb{C}^{n+1}$ around zero
follows from the transformation law.
We find

$\frac{1}{\sqrt{(g(h(x^{\prime\prime},y),x^{\prime\prime},y)+t^{2})^{2}+y_{1}^{2}}}\leq\frac{1}{\sqrt{(\min\\{g(h(x^{\prime\prime},y),x^{\prime\prime},y),0)\\}+t^{2})^{2}+y_{1}^{2}}}$

and

$\int_{-\varepsilon}^{\varepsilon}\left(\sqrt{(\min\\{g(h(x^{\prime\prime},y),x^{\prime\prime},y),0\\}+t^{2})^{2}+y_{1}^{2}}\right)^{-1}dt\leq\frac{1}{\sqrt{|y_{1}|}}(|\log(|y_{1}|)|+4)$

by Lemma 4.8 for sufficiently small $\varepsilon>0$ and all
$(x^{\prime\prime},y)$, $y_{1}\neq 0$, in a neighborhood of zero in
$\mathbb{R}^{2n}$. Note that the function
$(x^{\prime\prime},y)\mapsto\left(\sqrt{|y_{1}|}\right)^{-1}(|\log(|y_{1}|)|+4)$
is integrable in a small neighborhood of $0\in\mathbb{R}^{2n}$. Then, since
the function in (4.1.2) is non-negative, its integrability near zero follows
from the Fubini-Tonelli theorem. ∎

From Lemma 4.9 we can deduce Lemma 4.4.

###### Proof of Lemma 4.4.

Since $f$ is smooth up to $bG$ we can assume that $f$ is the restriction of a
smooth function denoted by $f_{1}$ defined on an open neighborhood of
$\overline{G}$. Since $Y$ is a complex manifold and $(\partial f)_{p}\neq 0$
we find holomorphic functions $f_{2},\ldots,f_{n+1}$,
$f_{2}(p)=\ldots=f_{n+1}(p)=0$, defined on an open neighborhood of $p$ in $Y$
such that the differential $dF_{p}$ of $F=(f_{1},\ldots,f_{n+1})$ in $p$ is
invertable. Hence $F$ is a diffeomorphism on its image in an open neighborhood
$U$ of $p$ in $Y$ such that $F(p)=0$, $F|_{U\cap G}$ is holomorhic and
$F|_{U\cap bG}$ is CR. By shrinking $U$ we can assume that
$F(U\cap\overline{G})$ can be written as $\\{\varphi<0\\}$ for some smooth
real valued function $\varphi$ defined on an open neighborhood around zero in
$\mathbb{C}^{n+1}$ satisfying the properties in Lemma 4.9. Furthermore, in the
coordinates given by $F$ we find that $f=z_{1}$ and $\partial f=dz_{1}$ on
$F(U\cap\overline{G})$. Then the claims (i)-(iv) follow from Lemma 4.9.
It remains to show that (v) holds. By Stokes formula we find

$\int_{G}\frac{\overline{f}}{|f|^{2}+\delta}\partial
f\wedge\overline{\partial}\psi=\int_{bG}\log(|f|^{2}+\delta)\wedge\overline{\partial}\psi-\int_{G}\log(|f|^{2}+\delta)\wedge\partial\overline{\partial}\psi$

for all $\delta>0$ and all $\psi\in\Omega_{c}^{n,n}(\overline{G})$ since the
forms in the integrals are smooth. We have $|f|/(|f|^{2}+\delta)\leq 1/|f|$
and $-\log(|f|^{2}+\delta)\leq-2\log(|f|)$ for all $\delta>0$. Furthermore, we
have $\lim_{\delta\to 0}\overline{f}/(|f|^{2}+\delta)=1/f$ and
$\lim_{\delta\to 0}\log(|f|^{2}+\delta)=2\log(|f|)$ pointwise almost
everywhere. Now by the already proven statements (i)-(iv) in Lemma 4.4 we can
choose $U\subset Y$ around $p$ small enough such that
$\iota^{*}(\frac{1}{f}\partial f\wedge\overline{\partial}\psi)$ and
$\iota^{*}(\log(|f|)\overline{\partial}\psi$ are integrable on $bG$ and
$\log(|f|)\wedge\partial\overline{\partial}\psi$ is integrable on $G$ for all
$\psi\in\Omega_{c}^{n,n}(\overline{G}\cap U)$. For
$\psi\in\Omega_{c}^{n,n}(\overline{G}\cap U)$ we then conclude from the
dominated convergence theorem that

$\displaystyle\int_{G}\frac{1}{f}\partial f\wedge\overline{\partial}\psi$
$\displaystyle=$ $\displaystyle\lim_{\delta\to
0}\int_{G}\frac{\overline{f}}{|f|^{2}+\delta}\partial
f\wedge\overline{\partial}\psi$ $\displaystyle=$ $\displaystyle\lim_{\delta\to
0}\int_{bG}\log(|f|^{2}+\delta)\wedge\overline{\partial}\psi-\int_{G}\log(|f|^{2}+\delta)\wedge\partial\overline{\partial}\psi$
$\displaystyle=$
$\displaystyle\int_{bG}\log(|f|^{2})\wedge\overline{\partial}\psi-\int_{G}\log(|f|^{2})\wedge\partial\overline{\partial}\psi.$

∎

###  4.2. Zero Sets of Holomorhic Functions on Domains with Boundary

Now we are ready to give a definition of
$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi$ where
$f\in\mathcal{O}\cap\mathscr{C}^{\infty}(\overline{G})$ is regular with
respect to $bG$ and $\psi\in\Omega_{c}^{n,n}(\overline{G})$ does not need to
vanish near the boundary.

######  Definition 4.10.

With the notations above let
$f\in\mathcal{O}(G)\cap\mathscr{C}^{\infty}(\overline{G})$, $f\not\equiv 0$,
be regular with respect to $bG$. Given $\psi\in\Omega_{c}^{n,n}(\overline{G})$
we define

(4.2.1)
$\displaystyle\text{\,\,\,\,\,\,\,}\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi:=-\int_{bG}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{bG}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{G}\log(|f|)\wedge\partial\overline{\partial}\psi.$

We note that the right-hand side of (4.2.1) is well defined by Corollary 4.5.
Furthermore, we see that for $\psi\in\Omega_{c}^{n,n}(G)$ in (4.2.1) we obtain
the usual definition of
$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi$. Recall that by
Stokes formula we find for any
$\alpha\in\mathscr{C}^{\infty}(\overline{G})=\Omega^{0,0}(\overline{G})$ and
$\beta\in\Omega_{c}^{n,n}(\overline{G})$ that

(4.2.2)
$\displaystyle\int_{G}\partial\overline{\partial}\alpha\wedge\beta=-\int_{bG}\iota^{*}(\partial\alpha\wedge\beta)-\int_{bG}\iota^{*}(\alpha\overline{\partial}\beta)+\int_{G}\alpha\partial\overline{\partial}\beta.$

Given $\psi\in\Omega_{c}^{n,n}(\overline{G})$ with
$\text{supp}(\psi)\cap\\{f=0\\}=\emptyset$ we find that $\log|f|$ is smooth in
a neighborhood of $\text{supp}(\psi)$. Then from (4.2.2) it follows that
$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi=0$.

######  Remark 4.11.

From (v) in Lemma 4.4 it follows that any point $p\in bG$ with $f(p)=0$ has an
open neighborhood $U\subset Y$ such that

$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi=-\int_{bG}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{G}\frac{1}{2f}\partial f\wedge\overline{\partial}\psi$

holds for all $\psi\in\Omega_{c}^{n,n}(\overline{G}\cap U)$.

######  Lemma 4.12.

With the notations above let
$f\in\mathcal{O}(G)\cap\mathscr{C}^{\infty}(\overline{G})$, $f\not\equiv 0$,
be regular with respect to $bG$. Let $\psi\in\Omega_{c}^{n,n}(\overline{G})$
be a form which satisfies $(\psi\wedge\partial f)_{p}=0$ or
$(\psi\wedge\overline{\partial}f)_{p}=0$ at any point $p\in\overline{G}$. Then

$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi=0.$

###### Proof.

For $\delta>0$ we have that $\alpha_{\delta}:=\log(|f|^{2}+\delta)$ is smooth
on $\overline{G}$. We calculate

$\partial\overline{\partial}\alpha_{\delta}=\frac{\delta}{(|f|^{2}+\delta)^{2}}\partial
f\wedge\overline{\partial}f.$

Then for $\psi\in\Omega_{c}^{n,n}(\overline{G})$ with $\psi\wedge\partial f=0$
or $\psi\wedge\overline{\partial}f=0$ at any point we observe that
$\int_{G}\partial\overline{\partial}\alpha_{\delta}\wedge\psi=0$. From Stokes
formula (see (4.2.2)) we then find

$0=-\int_{bG}\frac{\overline{f}}{|f|^{2}+\delta}\partial
f\wedge\psi-\int_{bG}\log(|f|^{2}+\delta)\wedge\overline{\partial}\psi+\int_{G}\log(|f|^{2}+\delta)\partial\overline{\partial}\psi$

for all $\delta>0$. We have $|f|/(|f|^{2}+\delta)\leq 1/|f|$ and the pullback
of $z\mapsto 1/|f|$ to $bG$ is locally integrable by Corollary 4.5.
Furthermore, $-\log(|f|^{2}+\delta)\leq-2\log(|f|)$ and $z\mapsto-\log(|f|)$
is locally integrable on $G$ and its pullback to $bG$ is also locally
integrable by Corollary 4.5. It then follows from the dominated convergence
theorem that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\lim_{\delta\to
0}\left(-\int_{bG}\frac{\overline{f}}{|f|^{2}+\delta}\wedge\psi-\int_{bG}\log(|f|^{2}+\delta)\wedge\overline{\partial}\psi+\int_{G}\log(|f|^{2}+\delta)\partial\overline{\partial}\psi\right)$
$\displaystyle=$ $\displaystyle-\int_{bG}\frac{1}{f}\partial
f\wedge\psi-\int_{bG}\log(|f|^{2})\wedge\overline{\partial}\psi+\int_{G}\log(|f|^{2})\partial\overline{\partial}\psi$

Hence we obtain

$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi:=-\int_{bG}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{bG}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{G}\log(|f|)\wedge\partial\overline{\partial}\psi=0.$

∎

Before we can state the main theorem of this section we need the following.

######  Lemma 4.13.

Let $f\in\mathcal{O}(G)\cap\mathscr{C}^{\infty}(\overline{G})$ be a
holomorphic function on $G$ which is smooth up to the boundary and $p\in bG$ a
point with $f(p)=0$ such that $f$ is regular at $p$ with respect to $bG$.
There exists an open neighborhood $U\subset Y$ around $p$ and a smooth
submanifold $X$ of $Y$ through $p$ with real dimension $\dim_{\mathbb{R}}X=2n$
such that

  * (i)

$X$ is connected and orientable.

  * (ii)

$U\cap G\cap\\{f=0\\}$ is a complex submanifold of $G$ of complex dimension
$\dim_{\mathbb{C}}=n$,

  * (iii)

$U\cap G\cap\\{f=0\\}$ is an open relatively compact subset of $X$,

  * (iv)

$U\cap bG\cap\\{f=0\\}$ has measure zero in $X$.

###### Proof.

We can write $f=F|_{\overline{G}}$ where $F$ is a smooth function defined on
an open neighborhood of $\overline{G}$ in $Y$. Since $\partial f_{p}\neq 0$
and $f(p)=0$ we have that there exists an open set $V$ of $Y$ around $p$ such
that $X:=F^{-1}(0)\cap V$ is a real connected orientable submanifold of $Y$
with real dimension $\dim_{\mathbb{R}}X=2n$ and $p\in X$ which proves (i).
Since $G$ is open in $Y$ we find that $G\cap X$ is open in $X$. In addition we
can choose $V$ such that $\partial f$ does not vanish on $X\cap G$. Then,
since $X\cap G=\\{f=0\\}\cap V$ and $f$ is holomorphic on $G$, we have that
$X\cap G$ is a complex $n$-dimensional submanifold of $G$. Choosing $U$ around
$p$ small enough we have that (ii) and (iii) holds. In order to prove (iv) we
choose coordinates around $p$ as in the proof of Lemma 4.4 keeping in mind
that $f=F|_{\overline{G}}$. Hence we can consider the case that $V$ is an open
neighborhood of $\mathbb{C}^{n+1}$ around zero, $X=\\{z\in V\mid z_{1}=0\\}$,
$G\cap V=\\{z\in V\mid\varphi(z)<0\\}$, $bG\cap V=\\{z\in
V\mid\varphi(z)=0\\}$ for some smooth function $\varphi\colon V\to\mathbb{R}$
with $\varphi(0)=0$, $d\varphi_{p}\neq 0$ for all $p\in V$ and $\varphi$
satisfies at least one of the conditions (i) or (ii) in the assumptions of
Lemma 4.9. Given the case that $\frac{\partial\varphi}{\partial z_{j}}(0)\neq
0$ for some $j\geq 2$ we can choose $U$ around zero small enough such that
$U\cap bG\cap X$ is a submanifold of $X$ of real dimension $2n-1$. Hence
$U\cap bG\cap\\{f=0\\}$ has measure zero in $X$. Now consider the case that
$\frac{\partial\varphi}{\partial z_{j}}(0)=0$ for all $j\geq 2$. By the Levi
form assumptions on $\varphi$ we can assume without loss of generality that
$\frac{\partial^{2}\varphi}{\partial^{2}x_{2}}(0)-\frac{\partial^{2}\varphi}{\partial^{2}y_{2}}(0)=2\frac{\partial^{2}\varphi}{\partial\overline{z}_{2}\partial
z_{2}}(0)\neq 0$. Let us say
$\frac{\partial^{2}\varphi}{\partial^{2}x_{2}}(0)\neq 0$. Then choosing $U$
around zero small enough we can assume
$\frac{\partial^{2}\varphi}{\partial^{2}x_{2}}(z)\neq 0$ on $U$. Consider the
sets $A:=\\{z\in U\mid\frac{\partial\varphi}{\partial x_{2}}(z)\neq 0\\}$ and
$B:=U\setminus A$. Then $A\cap X\cap bG$ is a submanifold of $X$ of real
dimension $2n-1$. Hence $A\cap X\cap bG$ has measure zero in $X$. Furthermore,
$B\cap X$ is real submanifold of $U$ of real dimension $2n-1$. It follows that
$B\cap X\cap bG$ has measure zero in $X$. We conclude $bG\cap U\cap X=(A\cup
B)\cap X\cap bG$ has measure zero in $X$. ∎

Now we are ready to state the Lelong-Poincaré formula for domains with
boundary.

######  Theorem 4.14.

Let $Y$ be a complex manifold of dimension $\dim_{\mathbb{C}}Y=n+1$. Let
$G\subset Y$ be a domain with smooth boundary $bG$. Let
$f\in\mathcal{O}(G)\cap\mathscr{C}^{\infty}(\overline{G})$, $f\not\equiv 0$,
be regular with respect to $bG$. We have

(4.2.3)
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\frac{i}{\pi}\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi$

for all $\psi\in\Omega_{c}^{n,n}(\overline{G})$ where
$\left[\operatorname{Div}\\!\left(f\right)\right]$ denotes the zero divisor of
$f$ and the right-hand side is given by Definition 4.10.

######  Remark 4.15.

We note that the left-hand side in (4.2.3) is well-defined by Lemma 4.13.
Furthermore, it follows from (iv) in Lemma 4.13 that the value of
$\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)$ for
$\psi\in\Omega_{c}^{n,n}(\overline{G})$ stays the same when integration is
performed on $G$ instead of $\overline{G}$. This issue should be considered
with respect to Example 4.2 and Example 4.3.

###### Proof of Theorem 4.14.

It is enough to show that for any point $p\in\overline{G}$ there is an open
neighborhood $U\subset Y$ around $p$ such that the statement holds for all
$\psi\in\Omega_{c}^{n,n}(U\cap\overline{G})$. For $p\in G$ we can take $U=G$
and the statement follows from the usual Lelong-Poincaré formula. Given $p\in
bG$ with $f(p)\neq 0$ we can choose $U$ around $p$ such that $f$ does not
vanish on $U\cap\overline{G}$. Then on the one hand one has
$(\left[\operatorname{Div}\\!\left(f\right)\right],\psi)=0$ for all
$\psi\in\Omega_{c}^{n,n}(U\cap\overline{G})$. On the other hand we have that
$\log|f|$ is smooth on $U\cap\overline{G}$. Hence
$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi=0$ for all
$\psi\in\Omega_{c}^{n,n}(U\cap\overline{G})$ by Definition 4.10 and (4.2.2)
which shows that (4.2.3) holds in that case. Now take $p\in bG$ with $f(p)=0$.
Choosing local coordinates as in the proof of Lemma 4.4 we can consider an
open neighborhood of $V\subset\mathbb{C}^{n+1}$ around $p=0$, $f(z)=z_{1}$,
$G\cap V=\\{z\in V\mid\varphi(z)<0\\}$, $bG\cap V=\\{z\in
V\mid\varphi(z)=0\\}$ for some smooth function $\varphi\colon V\to\mathbb{R}$
with $\varphi(0)=0$, $d\varphi_{0}\neq 0$ and there exists
$v\in\mathbb{C}^{n+1}$ with $\partial\varphi_{0}(v)=0$ and
$(\partial\overline{\partial}\varphi)_{0}(v,\overline{v})\neq 0$. We will
first show that we can assume $\frac{\partial\varphi}{\partial z_{1}}(z)\neq
0$ on $V$. If $\frac{\partial\varphi}{\partial z_{1}}(0)\neq 0$ this follows
immediately by shrinking $V$. If $\frac{\partial\varphi}{\partial z_{1}}(0)=0$
we have that $\frac{\partial\varphi}{\partial z_{j}}(0)\neq 0$ for some $j\geq
2$. Then replace the coordinate $z_{j}$ by $z_{j}+z_{1}$. In the new
coordinates we have $f(z)=z_{1}$, $\frac{\partial\varphi}{\partial
z_{1}}(0)\neq 0$ and the other assumptions on $\varphi$ are still valid on
some small open neighborhood around zero. Then by shrinking $V$ it follows
that $\frac{\partial\varphi}{\partial z_{1}}(z)\neq 0$ on $V$. Let $B$ be the
open ball in $\mathbb{C}^{n}$ centered at zero of radius $\varepsilon>0$. Let
$D$ be the open disc in $\mathbb{C}$ centered at zero of radius
$\varepsilon>0$. Given $\varepsilon>0$ small enough we have $U:=D\times
B\subset\subset V$ is an open neighborhood around zero relatively compact in
$V$. For $z=(z_{1},\ldots,z_{n+1})$ write $z^{\prime}=(z_{2},\ldots,z_{n+1})$
and for $z^{\prime}\in B$ put $D(z^{\prime})=\\{z\in
D\mid\varphi(z,z^{\prime})<0\\}$. By the assumptions on $\varphi$ we have for
any $z^{\prime}\in B$ that $D(z^{\prime})$ is an open set with smooth boundary
in $D$ (that is the boundary of $D(z^{\prime})$ relative to $D$ is smooth). We
define $A_{1}:=\\{z^{\prime}\in B\mid\varphi(0,z^{\prime})<0\\}$,
$A_{2}:=\\{z^{\prime}\in B\mid\varphi(0,z^{\prime})=0\\}$ and
$A_{3}:=\\{z^{\prime}\in B\mid\varphi(0,z^{\prime})>0\\}$. Then $A_{j}$,
$j=1,2,3$, are pairwise disjoint with $B=A_{1}\cup A_{2}\cup A_{3}$.
Furthermore, $A_{1}$ and $A_{3}$ are open and from (iv) in Lemma 4.13 we have
that $A_{2}$ has measure zero in $\mathbb{C}^{n}$. Now take
$\psi\in\Omega_{c}^{n,n}(U\cap\overline{G})$. We can write $\psi=\sum_{1\leq
j,k\leq n+1}\psi_{j,k}\widehat{dz_{j}}\wedge\widehat{d\overline{z}_{k}}$ for
some $\psi_{j,k}\in\mathscr{C}_{c}^{\infty}(U\cap\overline{G})$ where
$\widehat{dz_{j}}=dz_{1}\wedge\ldots\wedge dz_{j-1}\wedge
dz_{j+1}\wedge\ldots\wedge dz_{n+1}$ and
$\widehat{d\overline{z}_{k}}=\overline{\widehat{dz_{k}}}$. Since
$\\{f=0\\}\cap U\cap G=\\{0\\}\times A_{1}$ it follows that

$\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\int_{A_{1}}\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}.$

Put
$\psi^{\prime}=\psi-\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}$.
Since $\partial f=dz_{1}$ we obtain from Lemma 4.12 that
$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi^{\prime}=0$. It
remains to show that
$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}=-i\pi\int_{A_{1}}\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}$.
Choosing $\varepsilon>0$ in the definition of $U$ small enough we find by
Remark 4.11 that

$\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}=-\int_{bG}\iota^{*}(\frac{1}{2z_{1}}\psi_{1,1}dz_{1}\wedge\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}})-\int_{G}\frac{1}{2z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}\wedge\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}.$

Given $z^{\prime}\in A_{3}$ we have that $0\notin D(z^{\prime})$ and hence
$z_{1}\mapsto 1/z_{1}$ is smooth on $D(z^{\prime})$. We obtain from Stokes
formula in $\mathbb{C}$ that

(4.2.4)
$\displaystyle\int_{bD(z^{\prime})}\frac{\psi_{1,1}}{z_{1}}dz_{1}=-\int_{D(z^{\prime})}\frac{1}{z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}.$

Given $z^{\prime}\in A_{1}$ we have that $0\in D(z^{\prime})$. Then it follows
from the Cauchy-Pompeiu formula (see Remark 4.16 below) that

(4.2.5) $\displaystyle 2\pi
i\psi_{1,1}(0,z^{\prime})=\int_{bD(z^{\prime})}\frac{\psi_{1,1}}{z_{1}}dz_{1}+\int_{D(z^{\prime})}\frac{1}{z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}.$

Since $B$ is the union of the pairwise disjoint sets $A_{j}$, $j=1,2,3$, and
$A_{2}$ has measure zero we find by the Fubini theorem that

$\int_{bG}\iota^{*}(\frac{1}{z_{1}}\psi_{1,1}dz_{1}\wedge\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}})=\int_{A_{1}\cup
A_{3}}\left(\int_{bD(z^{\prime})}\frac{1}{z_{1}}\psi_{1,1}dz_{1}\right)\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}$

and

$\int_{G}\frac{1}{z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}\wedge\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}=\int_{A_{1}\cup
A_{3}}\left(\int_{D(z^{\prime})}\frac{1}{z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}\right)\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}.$

From (4.2.4) and (4.2.5) it follows that

$\int_{A_{3}}\left(-\int_{bD(z^{\prime})}\frac{\psi_{1,1}}{z_{1}}dz_{1}-\int_{D(z^{\prime})}\frac{1}{z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}.\right)\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}=0.$

and

$\int_{A_{1}}\left(-\int_{bD(z^{\prime})}\frac{\psi_{1,1}}{z_{1}}dz_{1}-\int_{D(z^{\prime})}\frac{1}{z_{1}}\frac{\partial\psi_{1,1}}{\partial\overline{z}_{1}}dz_{1}\wedge
d\overline{z}_{1}.\right)\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}=-2\pi
i\int_{A_{1}}\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}.$

Hence we can conclude

$\frac{i}{\pi}\int_{G}\partial\overline{\partial}\log(|f|)\wedge\psi=\int_{A_{1}}\psi_{1,1}\widehat{dz_{1}}\wedge\widehat{d\overline{z}_{1}}=\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)$

and the claim follows. ∎

######  Remark 4.16.

Let $U\subset\mathbb{C}$ be a domain. Let $D$ be an open subset of $U$ such
that its boundary relative to $U$ denoted by $b^{U}D$ is smooth. Let
$\overline{D}^{U}$ be the closure of $D$ relative to $U$. For any
$\psi\in\mathscr{C}_{c}^{\infty}(\overline{D}^{U})$ and any $w\in D$ one has

$\psi(w)=\frac{1}{2\pi
i}\left(\int_{b^{U}D}\psi\frac{dz}{z-w}+\int_{D}\frac{\partial\psi}{\partial\overline{z}}\frac{dz\wedge
d\overline{z}}{z-w}\right).$

This formula is a version of the Cauchy-Pompeiu formula for non bounded
domains in $\mathbb{C}$. For the proof take a small disc $B$ around $w$
relatively compact in $D$. The statement then follows from Stokes theorem
applied with respect to the domain $D\setminus\overline{B}$ and the Cauchy-
Pompeiu formula for the disc $B$.

###  4.3. Zero Sets of CR Functions

We are now going to prove two results (see Theorem 4.19 and Theorem 4.21)
concerning the zero sets of CR functions on strictly pseudoconvex CR
manifolds. For the proofs we need the following.

######  Proposition 4.17.

Let $(X,T^{1,0}X)$ be as in Theorem 1.1 and let $p\in X$ be a point. There
exists an open neighborhood $V$ around $p$, a domain
$Y\subset\mathbb{C}^{n+1}$ around zero, a smooth function $\varphi\colon
Y\to\mathbb{R}$ with non-vanishing differential $d\varphi$ and a CR embedding
$F\colon V\to Y$ with the following properties:

  * (i)

$F(V)=\\{\varphi=0\\}$, $F(p)=0$ and $\\{\varphi=0\\}$ is connected,

  * (ii)

$G:=\\{z\in Y\mid\varphi(z)<0\\}$ is a domain in $Y$ with smooth boundary
(relative to $Y$),

  * (iii)

$\sum_{0\leq j,k\leq n}\frac{\partial^{2}\varphi}{\partial
z_{j}\partial\overline{z}_{k}}(z)w_{j}\overline{w}_{k}>0$ for all
$z\in\\{\varphi=0\\}$ and all $w=(w_{0},\ldots,w_{n})$ with
$\sum_{j=0}^{n}w_{j}\frac{\partial\varphi}{\partial z_{j}}(z)=0$.

###### Proof.

By the assumptions on $X$ we have that $X$ is CR embeddable into the complex
euclidean space. Hence we can find an open neighborhood $V\subset X$ around
$p$, an open set $Y\subset\mathbb{C}^{n+1}$ around zero, a smooth function
$\varphi\colon Y\to\mathbb{R}$ with non-vanishing differential $d\varphi$ and
a CR embedding $F\colon V\to Y$ with $F(V)=\\{\varphi=0\\}$ and $F(p)=0$.
Shrinking $V$ and $Y$ we can assume that $\\{\varphi=0\\}$ is connected. Since
$X$ is strictly pseudoconvex, by possibly changing the sign of $\varphi$ we
can assume that (iii) is satisfied. Shrinking $Y$ again we obtain that $G$ is
a domain in $Y$. ∎

######  Remark 4.18.

In the following when we apply Proposition 4.17 we consider the closure and
boundary of $G$ always relative to $Y$, that is,
$\overline{G}:=\overline{G}^{Y}$ and $bG:=b^{Y}G$.

######  Theorem 4.19.

Let $(X,T^{1,0}X)$ be as in Theorem 1.1 and let $f\in
H^{0}_{b}(X)\cap\mathscr{C}^{\infty}(X)$ be a smooth CR function such that
$df_{p}\neq 0$ for any $p\in X$ with $f(p)=0$. Then $1/f$ is integrable on
$X$.

######  Remark 4.20.

We note that the assumptions on $f$ do not imply in general that $0$ is a
regular value of the map $f\colon X\to\mathbb{C}\simeq\mathbb{R}^{2}$. (See
Example 4.3).

###### Proof of Theorem 4.19.

We show that any point $p\in X$ has an open neighborhood such that $1/f$ is
integrable there. Let $p\in X$ be a point. If $f(p)\neq 0$ there is nothing to
show. So we assume $f(p)=0$. By Proposition 4.17 we find an open neighborhood
around $p$ which can be CR isomorphically identified with a set
$bG=\\{\varphi=0\\}\subset Y\subset\mathbb{C}^{n+1}$ where $Y,G,\varphi$
satisfy the conditions in Proposition 4.17. By [Hoermander_2000, Theorem
2.6.13] after shrinking $Y$ we find a function
$F\in\mathcal{O}(G)\cap\mathscr{C}^{\infty}(\overline{G})$ with $F|_{bG}=f$.
In addition, since $df_{p}\neq 0$, we can achieve by shrinking $Y$ again that
$df$ does not vanish on $bG$. Since $bG$ is strictly pseudoconvex we have that
$F$ is regular with respect to $bG$. Then it follows from Lemma 4.4 that $1/f$
is integrable in a small neighborhood around $p$. ∎

In the situation of Theorem 4.19 assuming that $0$ is a regular value of the
map $f\colon X\to\mathbb{C}$ we have that $\\{f=0\\}:=f^{-1}(0)$ is a smooth
compact oriented submanifold in $X$ of real codimension two. Given
$\psi\in\Omega^{2n-1}(X)$ we then define

(4.3.1)
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right):=\int_{\\{f=0\\}}\ell^{*}\psi$

where $\ell\colon\\{f=0\\}\to X$ denotes the inclusion map. We have the
following.

######  Theorem 4.21.

Let $(X,T^{1,0}X)$ be as in Theorem 1.1 and let $f\in
H^{0}_{b}(X)\cap\mathscr{C}^{\infty}(X)$ be a smooth CR function such that $0$
is a regular value of $f\colon X\to\mathbb{C}$. Then for any
$\psi\in\Omega^{2n-1}(X)$ we have

$\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\frac{1}{2\pi
i}\int_{X}\frac{df}{f}\wedge d\psi.$

###### Proof.

The theorem follows immediately from Theorem 4.23 below. ∎

It turns out that the result in Theorem 4.21 holds for any oriented smooth
real manifold and any smooth complex valued function which has zero as a
regular value what we are going to explain now.

Let $M$ be an oriented smooth real manifold of dimension $n\geq 2$. Given a
smooth function $f\in\mathscr{C}^{\infty}(M)$ such that zero is a regular
value of the map $f\colon M\to\mathbb{C}$ we have that $\\{f=0\\}:=\\{x\in
M\mid f(x)=0\\}$ is an oriented smooth submanifold of dimension $n-2$. We then
define

(4.3.2)
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\int_{\\{f=0\\}}\iota^{*}\psi$

for $\psi\in\Omega^{n-2}_{c}(M)$ where $\iota\colon\\{f=0\\}\to M$ denotes the
inclusion map. We would like to show that
$\left[\operatorname{Div}\\!\left(f\right)\right]$ is given by $(2\pi
i)^{-1}d(df/f)$ in the sense of distributions (see Theorem 4.23). Therefore we
need the following.

######  Lemma 4.22.

Let $M$ be an oriented smooth real manifold of dimension $n\geq 2$. Given a
function $f\in\mathscr{C}^{\infty}(M)$ such that zero is a regular value of
the map $f\colon M\to\mathbb{C}$ we have that $1/f$ is locally integrable.

###### Proof.

We need to show that any point $p\in M$ has an open neighborhood where $1/f$
is integrable. Given $f(p)\neq 0$ there is nothing to show. So let us assume
$f(p)=0$. By the assumptions on $f$ we can choose local coordinates
$(x_{1},\ldots,x_{n})$ in an open neighborhood $U$ around $p$ such that
$x_{1}=\operatorname{Re}f$, $x_{2}=\operatorname{Im}f$. In this coordinates we
find $1/f=\frac{1}{x_{1}+ix_{2}}$. We note that
$(x_{1},x_{2})\mapsto|x_{1}+ix_{2}|^{-1}=|(x_{1},x_{2})|^{-1}$ is locally
integrable on $\mathbb{R}^{2}$. Then the claim follows from the Fubini-Tonelli
theorem. ∎

######  Theorem 4.23.

Let $M$ be an oriented smooth real manifold of dimension $n\geq 2$. For any
function $f\in\mathscr{C}^{\infty}(M)$ such that zero is a regular value of
the map $f\colon M\to\mathbb{C}$ we have
$\left[\operatorname{Div}\\!\left(f\right)\right]=(2\pi i)^{-1}d(df/f)$ in the
sense of distributions, that is,

(4.3.3)
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\frac{1}{2\pi
i}\int_{M}\frac{df}{f}\wedge d\psi$

for all $\psi\in\Omega^{n-2}_{c}(M)$.

We note that the right-hand side of (4.3.3) is well-defined by Lemma 4.22.

######  Remark 4.24.

Theorem 4.23 can be obtained from the results due to Harvey–Semmes [HS92]
under much more general assumptions on the function $f$. Here, we will give an
elementary proof for the simple case that zero is a regular value of the map
$f\colon X\to\mathbb{C}$.

###### Proof of Theorem 4.23.

It is enough to show that any point $p\in M$ has an open neighborhood where
the statement holds. Let $p\in M$ be a point with $f(p)\neq 0$. Consider the
open neighborhood $U:=M\setminus\\{f=0\\}$ around $p$. We have that $df/f$ is
smooth on $U$ with $d(df/f)=0$. Given $\psi\in\Omega^{n-2}_{c}(U)$ we find by
Stokes theorem that

$\int_{M}\frac{df}{f}\wedge d\psi=\int_{U}\frac{df}{f}\wedge
d\psi=\int_{U}d\left(\frac{df}{f}\right)\wedge\psi=0.$

On the other hand we have
$\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=0$. Hence
(4.3.3) holds on $U$.
Now let $p\in M$ be a point with $f(p)=0$. By the assumptions on $f$ we find
local coordinates $x_{1},x_{2},\ldots,x_{n}$ on an open neighborhood $U\subset
M$ around $p$ such that $\operatorname{Re}f=x_{1}$, $\operatorname{Im}f=x_{2}$
and so that $p$ is mapped to the origin in $\mathbb{R}^{n}$. In these
coordinates, after shrinking $U$, we can identify $U$ with an open
neighborhood in $\mathbb{R}^{n}$ of the form $U=D\times B$, where
$D\subset\mathbb{R}^{2}$, $B\subset\mathbb{R}^{n-2}$ are balls centered in
zero, such that $p=0\in\mathbb{R}^{n}$ and $\\{f=0\\}\cap U=\\{x\in U\mid
x_{1}=x_{2}=0\\}=\\{0\\}\times B$. We use the notation
$x=(x_{1},x_{2},x^{\prime})\in\mathbb{R}^{2}\times\mathbb{R}^{n-2}$ with
$x^{\prime}=(x_{3},\ldots,x_{n})$. Furthermore, put
$z:=x_{1}+ix_{2}\in\mathbb{C}$ with

$dz=dx_{1}+idx_{2},\,\,d\overline{z}=dx_{1}-idx_{2},\,\,\frac{\partial}{\partial
z}=\frac{1}{2}\left(\frac{\partial}{\partial x_{1}}-i\frac{\partial}{\partial
x_{2}}\right),\,\,\frac{\partial}{\partial\overline{z}}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{1}}+i\frac{\partial}{\partial x_{2}}\right).$

Using this notation we find $df/f=dz/z$ and

$d\nu=\partial\nu+\overline{\partial}\nu+d^{\prime}\nu$

for all $\nu\in\mathscr{C}^{\infty}(U)$ where
$\partial\nu=\frac{\partial\nu}{\partial z}dz$,
$\overline{\partial}\nu=\frac{\partial\nu}{\partial\overline{z}}d\overline{z}$,
$d^{\prime}\nu=\sum_{j=3}^{n}\frac{\partial\nu}{\partial x_{j}}dx_{j}$. Note
that we can declare $\partial,\overline{\partial},d^{\prime}$ to act on forms
in $\Omega^{\bullet}(U)$ in the standard way.
Let $\psi\in\Omega^{n-2}_{c}(U)$ be arbitrary. Since
$2dx_{1}=dz+d\overline{z}$, $2idx_{2}=dz-d\overline{z}$ we can write
$\psi=\alpha_{0}+\psi^{\prime}$ with

$\psi^{\prime}=dz\wedge\alpha_{1}+d\overline{z}\wedge\alpha_{2}+dz\wedge
d\overline{z}\wedge\alpha_{3}$

such that $\alpha_{0}\in\Omega^{n-2}_{c}(U)$,
$\alpha_{1},\alpha_{2}\in\Omega^{n-3}_{c}(U)$,
$\alpha_{3}\in\Omega^{n-4}_{c}(U)$ and $\iota_{\frac{\partial}{\partial
x_{j}}}\alpha_{k}=0$ for $j=1,2$ and $k=0,\ldots,3$. It follows that
$\alpha_{0}=\nu dx_{3}\wedge\ldots\wedge dx_{n}$ for some smooth function
$\nu\in C^{\infty}_{c}(U)$ and hence we obtain $\iota^{*}\psi=\nu\circ\iota
dx_{3}\wedge\ldots\wedge dx_{n}$. We conclude that

(4.3.4)
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\int_{\\{f=0\\}\cap
U}\iota^{*}\psi=\int_{\\{0\\}\times
B}\iota^{*}\psi=\int_{B}\nu(0,x^{\prime})dx_{3}\ldots dx_{n}.$

We will now show that (4.3.3) holds on $U$ in two steps: First we prove
$\int_{U}\frac{df}{f}\wedge d\psi^{\prime}=0$. Second we calculate
$\int_{U}\frac{df}{f}\wedge d\alpha_{0}$.
We have $dz\wedge d\psi^{\prime}=dz\wedge d\overline{z}\wedge
d^{\prime}\alpha_{2}$. For $z\neq 0$ fixed put $U_{z}=\\{z\\}\times B$ and
denote by $\ell\colon U_{z}\to U$ the inclusion map. We obtain

$\displaystyle\int_{U_{z}}\ell^{*}(d^{\prime}\alpha_{2})=\int_{U_{z}}\ell^{*}(d\alpha_{2})=\int_{U_{z}}d\ell^{*}\alpha_{2}=0.$

since $\ell^{*}\alpha_{2}$ has compact support in $U_{z}$. Since
$z\mapsto\frac{1}{z}$ is locally integrable on $\mathbb{C}$ we conclude from
the Fubini-Tonelli theorem that

(4.3.5) $\displaystyle\int_{U}\frac{df}{f}\wedge
d\psi^{\prime}=\int_{U}\frac{dz}{z}\wedge d\overline{z}\wedge
d^{\prime}\alpha_{2}=\int_{D}\left(\int_{U_{z}}\ell^{*}(d^{\prime}\alpha_{2})\right)\frac{dz\wedge
d\overline{z}}{z}=0.$

We have $dz\wedge d\alpha_{0}=\frac{\partial\nu}{\partial\overline{z}}dz\wedge
d\overline{z}\wedge dx_{3}\wedge\ldots\wedge dx_{n}$. Again by the Fubini-
Tonelli theorem we find

$\int_{U}\frac{df}{f}\wedge
d\alpha_{0}=\int_{U}\frac{\partial\nu}{\partial\overline{z}}\frac{dz}{z}\wedge
d\overline{z}\wedge dx_{3}\wedge\ldots\wedge
dx_{n}=\int_{B}\left(\int_{D}\frac{\partial\nu}{\partial\overline{z}}\frac{dz\wedge
d\overline{z}}{z}\right)dx_{3}\ldots dx_{n}.$

Note that for any $x^{\prime}\in B$ we have that $z\mapsto\nu(z,x^{\prime})$
has compact support in $D$. Then, using the Cauchy-Pompeiu formula, we obtain

(4.3.6) $\displaystyle\int_{U}\frac{df}{f}\wedge d\alpha_{0}=2\pi
i\int_{B}\nu(0,x^{\prime})dx_{3}\ldots dx_{n}.$

Since $\psi=\alpha_{0}+\psi^{\prime}$ we conclude from (4.3.4), (4.3.5) and
(4.3.6) that

$\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\int_{B}\nu(0,x^{\prime})dx_{3}\ldots
dx_{n}=\frac{1}{2\pi i}\int_{U}\frac{df}{f}\wedge d\psi.$

∎

##  5\. Equidistribution on CR manifolds

In this section we are going to prove Theorem 1.6 and Theorem 1.7 stated in
the introduction. We will first understand $\mathcal{C}_{f}$ as a distribution
valued random variable defined on a sequence of probability spaces
$(A_{k})_{k>0}$ contained in the space of smooth CR functions (see Definition
5.1 and Lemma 5.2). A direct calculation leads to the proof of Theorem 1.6
(see Theorem 5.5 and Corollary 5.6). In order to prove Theorem 1.7 (see
Corollary 5.15) we need variance estimates for $\mathcal{C}_{f}$ (see Theorem
5.13). It turns out that those estimates can be obtained from the
understanding of a sequence of bilinear forms $(\theta_{k})_{k>0}$ (see
Definition 5.10) when $k$ becomes large (see Lemma 5.11 and Lemma 5.12).

Let $(X,T^{1,0}X)$ be a compact orientable strictly pseudoconvex CR manifold
such that the Kohn-Laplacian has closed range. Denote by $\xi$ a contact form
on $X$ such that the respective Levi form is positive definite and let
$\mathcal{T}$ be the respective Reeb vector field uniquely determined by
$\iota_{\mathcal{T}}\xi\equiv 1$ and $\iota_{\mathcal{T}}d\xi\equiv 0$. Let
$P\in L^{1}_{\mathrm{cl}}(X)$ be a first order formally self-adjoint classical
pseudodifferential operator with $\sigma_{P}(\xi)>0$ where $\sigma_{P}$
denotes the principal symbol of $P$. Denote by $T_{P}=\Pi P\Pi$ the
corresponding Toeplitz operator. Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$
be the positive eigenvalues of $T_{P}$ counting multiplicity and let
$f_{1},f_{2}\ldots\in H_{b}^{0}(X)\cap\mathscr{C}^{\infty}(X)$ be a respective
orthonormal system of eigenvectors. In particular, we have
$T_{P}f_{j}=\lambda_{j}f_{j}$, $(f_{j},f_{j})=1$ and $(f_{j},f_{\ell})=0$ for
all $j,\ell\in\mathbb{N}$, $j\neq\ell$. Let
$\chi\in\mathscr{C}^{\infty}((\delta_{1},\delta_{2}))$, $\chi\not\equiv 0$, be
a cut-off function for some $0<\delta_{1}<\delta_{2}<1$ and put
$\eta(t):=|\chi(t)|^{2}$, $t\in\mathbb{R}$, with
$\tau_{j}:=\int_{\mathbb{R}}t^{n+j}\eta(t)dt$ for $j\in\mathbb{N}_{0}$. Put
$N_{k}=|\\{j\in\mathbb{N}\mid\lambda_{j}\leq k\delta_{2}\\}|$. Let
$\kappa\colon\mathbb{R}\to\mathbb{C}$ be a function such that
$|\kappa(k)|^{2}/(1+|k|^{n})$ is bounded in $k$. For $k>0$ put
$A_{k}=\text{span}\left(\\{1\\}\cup\\{f_{j}\mid\lambda_{j}\leq\delta_{2}k\\}\right)$.
On $A_{k}$ we consider the probability measure $\mu_{k}$ induced by the
standard complex Gaussian measure $\mu^{G}$ on $\mathbb{C}^{N_{k}+1}$ and the
map

$\mathbb{C}^{N_{k}+1}\ni a\mapsto
a_{0}\kappa(k)+\sum_{j=1}^{N_{k}}a_{j}\chi_{k}(\lambda_{j})f_{j}\in A_{k}$

where $\chi_{k}(t)=\chi(k^{-1}t)$.

######  Definition 5.1.

Let $f\in H^{0}_{b}(X)\cap\mathscr{C}^{\infty}(X)$ such that $df_{p}\neq 0$
for all $p\in X$ with $f(p)=0$. We define

$\mathcal{C}_{f}\colon\Omega^{2n}(X)\to\mathbb{C},\,\,\,\,\mathcal{C}_{f}(\psi)=\frac{1}{2\pi
i}\int_{X}\frac{df}{f}\wedge\psi.$

We note that under the assumptions on $f$ in Definition 5.1 we have that
$\mathcal{C}_{f}$ is well defined by Theorem 4.19. Furthermore, given $f\in
H^{0}_{b}(X)\cap\mathscr{C}^{\infty}(X)$ such that zero is a regular value of
the map $f\colon X\to\mathbb{C}$ it follows that $df_{p}\neq 0$ for all $p\in
X$ with $f(p)=0$ (cf. Remark 4.20).

Since we wish to understand $\mathcal{C}_{f}$ as a random variable on $A_{k}$,
we have to make sure that it is defined on a set $\tilde{A}_{k}\subset A_{k}$
with $\mu_{k}(\tilde{A_{k}})=1$. We have the following.

######  Lemma 5.2.

Put $\tilde{A}_{k}:=\\{f\in A_{k}\mid\text{0 is a regular value of }f\colon
X\to\mathbb{C}\\}$. For any $k>0$ with

$|\kappa(k)|^{2}+\sum_{j=1}^{N_{k}}|\chi_{k}(\lambda_{j})|^{2}|f_{j}(x)|^{2}>0,\,\,\,\forall
x\in X$

we have that $\tilde{A}_{k}$ is $\mu_{k}$-measurable with
$\mu_{k}(A_{k}\setminus\tilde{A}_{k})=0$.

Lemma 5.2 is a direct consequence of the following.

######  Lemma 5.3.

Let $M$ be a compact manifold and let
$g_{0},g_{1},\ldots,g_{N}\in\mathscr{C}^{\infty}(M)$ such that
$|g_{0}(x)|^{2}+\ldots+|g_{N}(x)|^{2}>0$ for all $x\in M$. Define
$\tau:\mathbb{C}^{N}\to\mathscr{C}^{\infty}(M)$,
$\tau(a)=\sum_{j=0}^{N}a_{j}g_{j}$ and put

$Q:=\\{a\in\mathbb{C}^{N+1}\mid\text{0 is a regular vlaue of the map
}\tau(a)\colon X\to\mathbb{C}\\}.$

We have that $Q$ is open such that $\mathbb{C}^{N+1}\setminus Q$ has Lebesgue
measure zero.

###### Proof.

Given an open set $U\subset M$ we define $Q(U)\subset\mathbb{C}^{N+1}$ by
saying $a\in Q(U)$ if and only if there exists an open neighborhood $V$ around
$\overline{U}$ such that 0 is a regular value of the map $\tau(a)|_{V}\colon
V\to\mathbb{C}$. The compactness of $M$ implies that $Q(U)$ is open for any
open set $U\subset M$. Since $Q=Q(M)$ we conclude that $Q$ is open. Let $p\in
M$ be a point. Since $|g_{0}(x)|^{2}+\ldots+|g_{N}(x)|^{2}>0$ there exist an
open neighborhood $U\subset M$ around $p$, $j\leq j\leq N$ and a constant
$c>0$ such that $|g_{j}(x)|\geq c$ for all $x\in U$. Without loss of
generality assume $j=0$. We define
$\tau^{\prime}:\mathbb{C}^{N}\to\mathscr{C}^{\infty}(M)$,
$\tau^{\prime}(a)=\sum_{j=1}^{N}a_{j}g_{j}$. Given
$a^{\prime}\in\mathbb{C}^{N}$ we define $W(a^{\prime},U)\subset\mathbb{C}$ by
saying $\lambda\in W(a^{\prime},U)$ if and only if there exists an open
neighborhood $V$ around $\overline{U}$ such that $-\lambda$ is a regular value
of the map $\tau^{\prime}(a)/g_{0}|_{V}\colon V\to\mathbb{C}$. Let $x\in M$ be
a point with $g_{0}(x)\neq 0$ and
$a=(a_{0},a^{\prime})\in\mathbb{C}\times\mathbb{C}^{N}$ with $\tau(a)(x)=0$.
It follows that $\tau^{\prime}(a^{\prime})(x)=-a_{0}g_{0}(x)$ and we find

$\left(d\frac{\tau^{\prime}(a^{\prime})}{g_{0}}\right)_{x}=\frac{(d\tau^{\prime}(a^{\prime}))_{x}}{g_{0}(x)}-\frac{\tau^{\prime}(a^{\prime})}{g_{0}(x)}\frac{(dg_{0})_{x}}{g_{0}(x)}=\frac{(d\tau(a))_{x}}{g_{0}(x)}.$

From this observation it follows that for any
$a=(a_{0},a^{\prime})\in\mathbb{C}\times\mathbb{C}^{N}$ we have $a\in Q(U)$ if
and only if $a_{0}\in W(a^{\prime},U)$. By the Theorem of Sard we have that
$\mathbb{C}\setminus W(a^{\prime},U)$ has Lebesgue measure zero for any
$a^{\prime}\in\mathbb{C}^{N}$. Furthermore, since $Q(U)$ is open we have that
$\mathbb{C}^{N+1}\setminus Q(U)$ is Lebesgue measurable. Since
$\mathbb{C}^{N+1}\setminus Q(U)=\\{(a_{0},a^{\prime})\in\mathbb{C}^{N+1}\mid
a_{0}\in\mathbb{C}\setminus W(a^{\prime},U)\\}$ it follows from the Fubini
theorem that $\mathbb{C}^{N+1}\setminus Q(U)$ has Lebesgue measure zero.
Using the construction above, by the compactness of $M$ we can find finitely
many open sets $U_{1},\ldots,U_{m}\subset M$ with $\bigcup_{j=1}^{m}U_{j}=M$
such that $\mathbb{C}^{N+1}\setminus Q(U_{j})$ has Lebesgue measure zero for
all $1\leq j\leq m$. Since $\mathbb{C}^{N+1}\setminus
Q=\bigcup_{j=1}^{m}\mathbb{C}^{N+1}\setminus Q(U_{j})$ we conclude that
$\mathbb{C}^{N+1}\setminus Q$ has Lebesgue measure zero. ∎

###  5.1. Proof of Theorem 1.6

We are now going to prove Theorem 1.6. We need to introduce some useful
notation before. For $d\in\mathbb{N}$ we denote by $\langle\cdot,\cdot\rangle$
the standard inner product on $\mathbb{C}^{d+1}$ that is $\langle
a,b\rangle=\sum_{j=0}^{d}a_{j}\overline{b_{j}}$ for
$a=(a_{0},\ldots,a_{d}),b=(b_{0},\ldots,b_{d})\in\mathbb{C}^{d+1}$.
Furthermore, given $a\in\mathbb{C}^{d+1}$ we put $|a|=\sqrt{\langle
a,a\rangle}$. For $k>0$ consider the map

(5.1.1) $\displaystyle F_{k}\colon
X\to\mathbb{C}^{N_{k}+1},\,\,\,\,F_{k}=(\kappa(k),\chi_{k}(\lambda_{1})f_{1},\ldots,\chi_{k}(\lambda_{N_{k}})f_{N_{k}}).$

Recall that $\eta_{k}=|\chi_{k}|^{2}$. We put
$S_{k}(x,y):=\eta_{k}(T_{P})(x,y)=\sum_{j=1}^{N_{k}}\eta_{k}(\lambda_{j})f_{j}(x)\overline{f_{j}(y)}$
and $B_{k}(x):=S_{k}(x,x)$ for $x,y\in X$ and $k>0$. It turns out that
$|F_{k}|^{2}=|\kappa(k)|^{2}+B_{k}$ and $\langle
F_{k}(x),F_{k}(y)\rangle=|\kappa(k)|^{2}+S_{k}(x,y)$ for all $x,y\in X$ and
$k>0$. We further consider $dF_{k}$ component wise, that is,

$dF_{k}=(0,\chi_{k}(\lambda_{1})df_{1},\ldots,\chi_{k}(\lambda_{N_{k}})df_{N_{k}}).$

Then $\langle a,\overline{dF_{k}}\rangle$ is a one form on $X$ for any
$a\in\mathbb{C}^{N_{k}+1}$. We have for example $\langle
F_{k}(x),(dF_{k})_{y}\rangle=d_{y}S_{k}(x,y)$ and for $f=\langle
a,\overline{F_{k}}\rangle$ we obtain $df=\langle a,\overline{dF_{k}}\rangle$.

######  Lemma 5.4.

There exists $k_{0}>0$ such that for any $k\geq k_{0}$ we find a constant
$C_{k}>0$ with

(5.1.2) $\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\frac{|\langle
a,\overline{dF_{k}}\rangle(V)|}{|\langle
a,\overline{F_{k}(x)}\rangle|}d\mu^{G}(a)\leq C_{k}|V|.$

for all $x\in X$ and $V\in T_{x}X$.

###### Proof.

By Theorem 2.10 we can choose $k_{0}>0$ such that there exists a constant
$C>0$ with $|F_{k}|^{2}\geq Ck^{n+1}$ for all $k\geq k_{0}$. Furthermore, we
have that $\int_{\mathbb{C}^{N_{k}+1}}|a_{0}|^{-1}\|a\|d\mu^{G}(a)<\infty$.
Since $F_{k}(x)/\|F_{k}(x)\|$, $x\in X$, has unit length we obtain from the
unitary invariance of $d\mu^{G}$ that

$\int_{\mathbb{C}^{N_{k}+1}}\frac{|a|}{|a_{0}|}d\mu^{G}(a)=\int_{\mathbb{C}^{N_{k}+1}}\frac{|a|}{|\langle
a,\overline{F_{k}(x)}/|F_{k}(x)|\rangle|}d\mu^{G}(a).$

Given $k\geq k_{0}$, $x\in X$ and $V\in T_{x}X$ we find

(5.1.3) $\displaystyle\left|\frac{\langle
a,\overline{dF_{k}}\rangle(V)}{\langle a,\overline{F_{k}(x)}\rangle}\right|$
$\displaystyle\leq$ $\displaystyle\frac{|a||VF_{k}/|F_{k}(x)||}{|\langle
a,\overline{F_{k}(x)}/|F_{k}(x)|\rangle|}$

It follows that

$\int_{\mathbb{C}^{N_{k}+1}}\left|\frac{\langle
a,\overline{dF_{k}}\rangle(V)}{\langle
a,\overline{F_{k}(x)}\rangle}\right|d\mu^{G}(a)\leq\int_{\mathbb{C}^{N_{k}+1}}\frac{|a|}{|a_{0}|}d\mu^{G}(a)\frac{|VF_{k}|}{|F_{k}(z)|}<\infty.$

Since $|VF_{k}|^{2}=\left((d_{x}\otimes
d_{y}S_{k}(z,w))\right)|_{x=y}(V\otimes V)$ we conclude from Lemma 2.11 that
there is a constant $\tilde{C}_{k}>0$ with $|VF_{k}|\leq\tilde{C}_{k}|V|$ for
all $x\in X$ and $V\in T_{x}X$. Hence we can choose a constant $C_{k}>0$
independent of $x$ and $V$ such that (5.1.2) holds. ∎

Theorem 1.6 follows from the next theorem by putting $\kappa(k)=0$ and
replacing $\psi$ by $d\psi$ (see Corollary 5.6).

######  Theorem 5.5.

Under the same assumptions as in Theorem 1.1 and with the notations above
there exists $k_{0}>0$ such that for each $k\geq k_{0}$ we have that
$\mathcal{C}_{f}$ is a well-defined current for all $f$ outside a subset in
$A_{k}$ of $\mu_{k}$-measure zero, and given $\psi\in\Omega^{2n}(X)$ the
expectation value

$\mathbb{E}_{k}(\mathcal{C}_{f})(\psi):=\int_{A_{k}}\mathcal{C}_{f}(\psi)d\mu_{k}(f)$

exists with
$\mathbb{E}_{k}(\mathcal{C}_{f})(\psi)=\int_{X}\beta_{k}\wedge\psi$ where
$\beta_{k}$ is the smooth one form given by

(5.1.4)
$\displaystyle\beta_{k}(x)=\frac{d_{x}\eta_{k}(T_{P})(x,y)|_{x=y}}{2\pi
i(|\kappa(k)|^{2}+\eta_{k}(T_{P})(x,x))}$

for all $x\in X$ and $k\geq k_{0}$ where $\eta=|\chi|^{2}$.

###### Proof.

Choose $k_{0}>0$ such that $|F_{k}(x)|>0$ holds for all $k\geq k_{0}$ and
$x\in X$. From Lemma 5.2 it follows for $k\geq k_{0}$ that $\mathcal{C}_{f}$
is a well-defined distribution for all $f$ outside a subset in $A_{k}$ of
$\mu_{k}$-measure zero. For $x\in X$ let $U^{x}_{k}$ be a unitary
transformation of $\mathbb{C}^{N_{k}+1}$ with
$U^{x}_{k}\overline{F_{k}(x)}/|F_{k}(z)|=(1,0,\ldots,0)$ and given
$b\in\mathbb{C}^{N_{k}+1}$ let us denote by $(U^{x}_{k}b)_{j}$ the $j$-th
component of the vector $U^{x}_{k}b$. We have $(U^{x}_{k}b)_{0}=\langle
F_{k}(x),\overline{b}\rangle/|F_{k}(x)|$ since $U^{x}_{k}$ is a unitary
transformation and basic properties of the inner product in
$\mathbb{C}^{N_{k}+1}$. We find
$\int_{\mathbb{C}^{N_{k}+1}}a_{j}/a_{0}d\mu^{G}(a)=0$ for all $j\geq 1$ . With
$\int_{\mathbb{C}^{N_{k}+1}}d\mu^{G}(a)=1$ it follows that

$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\frac{\langle
a,\overline{(dF_{k})_{x}}\rangle(V)}{\langle
a,\overline{F_{k}(x)}\rangle}d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\frac{\langle
a,U^{z}_{k}\overline{VF_{k}}\rangle}{a_{0}}d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\overline{(U^{x}_{k}\overline{(VF_{k})(z)}/|F_{k}(x)|)_{0}}$
$\displaystyle=$ $\displaystyle\frac{\langle
VF_{k},F_{k}(x)\rangle}{|F_{k}(x)|^{2}}$

for all $k\geq k_{0}$, $x\in X$ and $V\in T_{x}X$. We have
$|F_{k}(x)|^{2}=|\kappa(k)|^{2}+\eta_{k}(T_{P})(x,x)$ and $\langle
VF_{k},F_{k}(x)\rangle=(d_{x}\eta_{k}(T_{P})(x,y)|_{x=y})(V)$ for any $V\in
T_{x}X$. Hence given $\psi\in\Omega^{2n}(X)$ we conclude

$\displaystyle\int_{X}2\pi i\beta_{k}\wedge\psi$ $\displaystyle=$
$\displaystyle\int_{X}\left(\int_{\mathbb{C}^{N_{k}+1}}\frac{\langle
a,\overline{dF_{k}}\rangle}{\langle
a,\overline{F_{k}}\rangle}d\mu^{G}(a)\right)\wedge\psi$ $\displaystyle=$
$\displaystyle\int_{X}\int_{\mathbb{C}^{N_{k}+1}}\frac{\langle
a,\overline{dF_{k}}\rangle}{\langle a,\overline{F_{k}}\rangle}\wedge\psi
d\mu^{G}(a).$

By Lemma 5.2 we have the $\mathcal{C}_{f}$ is well defined for all $f$ in a
subset $\tilde{A}_{k}$ of $A_{k}$ with $\mu_{k}(\tilde{A}_{k})=1$. Using Lemma
5.4 we can apply the Fubini-Tonelli theorem and get

$\displaystyle\int_{X}\beta_{k}\wedge\psi$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\int_{X}\frac{1}{2\pi i}\frac{\langle
a,\overline{dF_{k}}\rangle}{\langle a,\overline{F_{k}}\rangle}\wedge\psi
d\mu^{G}(a)$ $\displaystyle=$ $\displaystyle\int_{A_{k}}\int_{X}\frac{1}{2\pi
i}\frac{df}{f}\wedge\psi d\mu_{k}(f)$ $\displaystyle=$
$\displaystyle\int_{\tilde{A}_{k}}\mathcal{C}_{f}(\psi)d\mu_{k}(f)=\mathbb{E}_{k}(\mathcal{C}_{f})$

for all $\psi\in\Omega^{2n}(X)$. ∎

######  Corollary 5.6.

Under the same assumptions as in Theorem 1.1 and with the notations above
there exist $k_{0}>0$ such that for each $k\geq k_{0}$ we have that
$\left[\operatorname{Div}\\!\left(f\right)\right]$ is a well-defined current
for all $f$ outside a subset in $A_{k}$ of $\mu_{k}$-measure zero and the
expectation value
$\mathbb{E}_{k}(\left[\operatorname{Div}\\!\left(f\right)\right])$ defined by

$\left(\mathbb{E}_{k}(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\int_{A_{k}}\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)d\mu_{k}(f),\,\,\,\,\psi\in\Omega^{2n-1}(X)$

exists with
$\mathbb{E}_{k}(\left[\operatorname{Div}\\!\left(f\right)\right])=d\beta_{k}$
where $\beta_{k}$ is given by (5.1.4).

###### Proof.

From Theorem 4.21 we obtain
$\left[\operatorname{Div}\\!\left(f\right)\right]=d\mathcal{C}_{f}$ in the
sense of distributions for any $f\in H^{0}_{b}(X)\cap C^{\infty}(X)$ such that
zero is a regular value of the map $f\colon X\to\mathbb{C}$. Hence, using
Lemma 5.2 and integration by parts, we obtain from Theorem 5.5 with
$\kappa(k)=0$ that

$\left(\mathbb{E}_{k}(\left[\operatorname{Div}\\!\left(f\right)\right]),\psi\right)=\mathbb{E}_{k}(\mathcal{C}_{f})(d\psi)=\int_{X}\beta_{k}\wedge
d\psi=\int_{X}d\beta_{k}\wedge\psi$

holds for all $\psi\in\Omega^{2n-1}(X)$ when $k$ is large enough. ∎

######  Lemma 5.7.

Under the same assumptions as in Theorem 1.1 and with $\beta_{k}$ given by
(5.1.4) we have

$k^{-1}\beta_{k}=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\frac{\xi}{\sigma_{P}(\xi)}+O(k^{-1})$

in $\mathscr{C}^{\infty}$-topology for $k\to+\infty$ where

(5.1.5)
$\displaystyle\operatorname{mv}(|\chi|^{2})=\frac{\int_{0}^{+\infty}t^{n+1}|\chi(t)|^{2}dt}{\int_{0}^{+\infty}t^{n}|\chi(t)|^{2}dt}.$

###### Proof.

The statement follows immediately from Theorem 2.10 and Lemma 2.11. ∎

######  Corollary 5.8.

In the situation of Theorem 5.5 there exists $k_{0},C>0$ such that

$\left|k^{-1}\mathbb{E}_{k}(\mathcal{C}_{f})(\psi)-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}\frac{\xi}{\sigma_{P}(\xi)}\wedge\psi\right|\leq
k^{-1}C\|\psi\|_{\mathscr{C}^{0}(X,\Lambda_{\mathbb{C}}^{2n}T^{*}X)}$

holds for all $\psi\in\Omega^{2n}(X)$ and $k\geq k_{0}$ where
$\operatorname{mv}(|\chi|^{2})$ is given by (5.1.5). As a consequence one has

$\lim_{k\to\infty}k^{-1}\mathbb{E}_{k}(\mathcal{C}_{f})=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\frac{\xi}{\sigma_{P}(\xi)}.$

###### Proof.

The result is a direct consequence of Theorem 5.5 and Lemma 5.7. ∎

###  5.2. Proof of Theorem 1.7

We will now prove Theorem 1.7. We start with the following lemma.

######  Lemma 5.9.

There is a constant $C>0$ such that for all $w\in\mathbb{C}\setminus\\{0\\}$
we have

$\int_{\mathbb{C}}\frac{d\mu^{G}(a)}{|a||a-w|}\leq\frac{C}{|w|}.$

###### Proof.

We have

$\int_{|a|\leq|w|/2}\frac{d\mu^{G}(a)}{|a||a-w|}\leq\frac{1}{|w|}\int_{|a|\leq|w|/2}\frac{d\mu^{G}(a)}{|a||a/w-1|}\leq\frac{2}{|w|}\int_{\mathbb{C}}\frac{d\mu^{G}(a)}{|a|}$

and

$\int_{|a|\geq|w|/2}\frac{d\mu^{G}(a)}{|a||a-w|}\leq\frac{2}{|w|}\int_{|a|\geq|w|/2}\frac{d\mu^{G}(a)}{|a-w|}\leq\frac{2}{|w|}\int_{\mathbb{C}}\frac{d\mu^{G}(a)}{|a-w|}.$

We have that the function $w\mapsto\int_{\mathbb{C}}\frac{d\mu^{G}(a)}{|a-w|}$
is bounded. The claim follows. ∎

It turns out that it in order to understand the variance of $\mathcal{C}_{f}$,
it is useful to consider the following objects.

######  Definition 5.10.

With the notation and assumptions above for any $k>0$ such that
$|F_{k}|^{2}>0$ put

$(Z_{k})_{x}:=\overline{(dF_{k})_{x}}-\frac{\langle
F_{k}(x),(dF_{k})_{x}\rangle}{|F_{k}(x)|^{2}}\overline{F_{k}(x)}$

and $\theta_{k}(a,x,y)\colon T_{x}X\times T_{y}X\to\mathbb{C}$,

$\displaystyle\theta_{k}(a,x,y)(V,W)$ $\displaystyle:=$
$\displaystyle\frac{\langle a,Z_{k}(V)\rangle\overline{\langle
a,Z_{k}(W)\rangle}}{\langle a,\overline{F_{k}(x)}\rangle\overline{\langle
a,\overline{F_{k}(y)}\rangle}}$

for $x,y\in X$ and $a\in\mathbb{C}^{N_{k}+1}$ such that $\langle
a,\overline{F_{k}(x)}\rangle\overline{\langle
a,\overline{F_{k}(y)}\rangle}\neq 0$.

By the assumption $|F_{k}|^{2}>0$, for $x,y\in X$ fixed, it follows that
$\theta_{k}(a,x,y)$ is well defined for $\mu^{G}$-almost every
$a\in\mathbb{C}^{N_{k}+1}$. Recall the definition of $h^{F_{k}}$ (see
Definition 3.1) that is

$h^{F_{k}}(x,y)=\frac{|\langle
F_{k}(x),F_{k}(y)\rangle|^{2}}{|F_{k}(x)|^{2}|F_{k}(y)|^{2}},\,\,\,x,y\in X$

for all $k>0$ with $|F_{k}|^{2}>0$. In order to obtain estimates for the
variance of $\mathcal{C}_{f}$ we need to understand the behavior of
$\theta_{k}(a,x,y)$ and its integrals with respect to $a,x,y$ when $k$ becomes
large. In the following two lemmata we consider $\theta_{k}$ for sufficiently
large $k>0$.

######  Lemma 5.11.

With the notation and assumptions above there exist $k_{0}>0$ and constants
$c_{k}>0$ such that

(5.2.1)
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}|\theta_{k}(a,x,y)(V,W)|d\mu^{G}(a)\leq\frac{c_{k}|V||W|}{\sqrt{1-h^{F_{k}}(z,w)}}$

for all $k\geq k_{0}$, $x,y\in X$ with $x\neq y$ and all $V\in T_{x}X$, $W\in
T_{y}X$. Furthermore we can choose $k_{0}>0$ such that $\gamma_{k}(x,y)\colon
T_{x}X\times T_{y}X\to\mathbb{C}$,

(5.2.2)
$\displaystyle\gamma_{k}(x,y)(V,W):=\sqrt{1-h^{F_{k}}(x,y)}\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}(a,x,y)(V,W)d\mu^{G}(a)$

is well defined for all $k\geq k_{0}$ and $x,y\in X$, $x\neq y$, and that
there exist a constant $C>0$ with

(5.2.3) $\displaystyle|\gamma_{k}(x,y)(V,W)|$
$\displaystyle\leq\left(\left|\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{Z_{k}(W)}{|F_{k}(y)|}\right\rangle\right|+\frac{1}{1-h^{F_{k}}(x,y)}\left|\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}\right\rangle\right|\left|\left\langle\frac{Z_{k}(W)}{|F_{k}(y)|},\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle\right|\right).$

for all $k\geq k_{0}$, $x,y\in X$, $x\neq y$ and $V\in T_{x}X$, $W\in T_{y}X$.

###### Proof.

Recalling that $|F_{k}|^{2}=|\kappa(k)|^{2}+B_{k}$ by Theorem 2.10 we can
choose $k_{1}>0$ and a constant $C_{0}>0$ such that
$C_{0}^{-1}k^{n+1}\leq|F_{k}|^{2}\leq C_{0}k^{n+1}$ for all $k\geq k_{1}$. In
addition, we choose $k_{1}$ large enough such that for all $k\geq k_{1}$ and
$x,y\in X$ we have $h^{F_{k}}(x,y)=1$ if and only if $x=y$ (see Lemma 3.8).
Furthermore, since

$|VF_{k}|^{2}=(d_{x}\otimes d_{y}\eta_{k}(T_{P})(x,y))|_{x=y}V\otimes V$

for $V\in T_{x}X$ we find by Lemma 2.11 that there is a constant $C_{1}>0$
such that

(5.2.4) $\displaystyle|VF_{k}|\leq C_{1}k|F_{k}(x)||V|$

for all $k\geq k_{1}$, $x\in X$ and $V\in T_{x}X$. For $x,y\in X$, $V\in
T_{x}X$ and $W\in T_{y}X$ put

$I_{k}(x,y)(V,W):=\int_{\mathbb{C}^{N_{k}}+1}|\theta_{k}(a,x,y)(V,W)|d\mu^{G}(a).$

For $x,y\in X$ we can choose a unitary transformation $U^{x,y}_{k}$ of
$\mathbb{C}^{N_{k}+1}$ such
$U^{x,y}_{k}\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}=(1,0,\ldots,0)$ and
$U^{x,y}_{k}\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}=(\alpha,\beta,0\ldots,0)$
with $\alpha,\beta\in\mathbb{C}$, $|\alpha|^{2}+|\beta|^{2}=1$. One observes
that $\alpha|F_{k}(x)||F_{k}(y)|=\langle F_{k}(x),F_{k}(y)\rangle$ and hence
$|\alpha|^{2}=h^{F_{k}}(x,y)$. Then in addition we can choose $U^{x,y}_{k}$
such that $\beta=\sqrt{1-h^{F_{k}}(x,y)}$. Since $\|Z_{k}(V)\|\leq 2|VF_{k}|$
for all $V\in T_{x}X$ we find with (5.2.4) for $x\neq y$ that

$\displaystyle I_{k}(x,y)(V,W)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}}+1}\left|\frac{\langle
a,U_{k}^{x,y}Z_{k}(V)\rangle\overline{\langle
a,U^{x,y}_{k}Z_{k}(W)\rangle}}{|F_{k}(x)||F_{k}(y)||a_{0}||\alpha a_{0}+\beta
a_{1}|}\right|d\mu^{G}(a)$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{C}^{N_{k}}+1}\frac{|a|^{2}|Z_{k}(V)||Z_{k}(W)|}{|F_{k}(x)||F_{k}(y)||a_{0}||\alpha
a_{0}+\beta a_{1}|}d\mu^{G}(a)$ $\displaystyle\leq$
$\displaystyle(2kC_{1})^{2}|V||W|\left(\int_{\mathbb{C}^{2}}\frac{|a_{0}|}{|\alpha
a_{0}+\beta
a_{1}|}d\mu^{G}(a)+\int_{\mathbb{C}^{N_{k}}+1}\frac{\sum_{j=1}^{N_{k}+1}|a_{j}|^{2}}{|a_{0}||\alpha
a_{0}+\beta a_{1}|}d\mu^{G}(a)\right)$

By some unitary transformation argument we find that
$\int_{\mathbb{C}^{2}}\frac{|a_{0}|}{|\alpha a_{0}+\beta a_{1}|}d\mu^{G}(a)$
is uniformly bounded in $\alpha,\beta$ when $|\alpha|^{2}+|\beta|^{2}=1$.
Using Lemma 5.9 we find a constant $c>0$ independent of $\beta$ and
$a_{1}\in\mathbb{C}$ such that

(5.2.5) $\displaystyle\int_{\mathbb{C}}\frac{1}{|a_{0}||\alpha a_{0}+\beta
a_{1}|}d\mu^{G}(a_{0})\leq\frac{c}{\beta|a_{1}|}.$

With

$\int_{\mathbb{C}^{N_{k}}+1}\frac{\sum_{j=2}^{N_{k}+1}|a_{j}|^{2}}{|a_{0}||\alpha
a_{0}+\beta
a_{1}|}d\mu^{G}(a)\leq\frac{c}{\beta}\left(\int_{\mathbb{C}}\frac{1}{|a_{1}|}d\mu^{G}(a_{1})\right)\left(\int_{\mathbb{C}^{N_{k}-1}}|a^{\prime}|^{2}d\mu^{G}(a^{\prime})\right)$

we conclude that for any $k\geq k_{1}$ there is a constant $c_{k}>0$ depending
on $k$ such that $I_{k}(x,y)(V,W)\leq c_{k}(1+\frac{1}{\beta})|V||W|$ holds
for all $x,y\in X$, $x\neq y$, and $V\in T_{x}X$, $W\in T_{y}X$. Since
$\beta=\sqrt{1-h^{F_{k}}(z,w)}\leq 1$ we obtain (5.2.1).

Now we can prove the second part of the statement. For $x,y\in X$, $x\neq y$,
define $II_{k}(x,y)\colon T_{x}X\times T_{y}Y\to\mathbb{C}$,

$II_{k}(z,w)(V,W):=\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}(a,z,w)(V,W)d\mu^{G}(a).$

First, from (5.2.1) we obtain by the choice of $k_{1}$ at the beginning of the
proof that $II_{k}(x,y)$ is well defined for all $x,y\in X$, $x\neq y$, and
all $k\geq k_{1}$. In the following we will calculate $II_{k}(x,y)$.

Fix $V\in T_{x}X$, $W\in T_{y}X$ where $x\neq y$. Put
$\kappa_{j}=\left(U^{x,y}_{k}Z_{k}(V)\right)_{j}$ and
$\lambda_{j}=\left(U^{x,y}_{k}Z_{k}(W)\right)_{j}$ where we denote by
$(b)_{j}$ the $j$-th component of a vector $b\in\mathbb{C}^{N_{k}+1}$. We find

$\kappa_{0}=\left\langle
U^{x,y}_{k}Z_{k}(V),U^{x,y}_{k}\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle=\left\langle
Z_{k}(V),\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle=0$

and

$\beta\kappa_{1}=\alpha\kappa_{0}+\beta\kappa_{1}=\left\langle
U^{x,y}_{k}Z_{k}(V),U^{x,y}_{k}\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle=\left\langle
Z_{k}(V),\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle.$

Furthermore, we find

$\lambda_{0}=\left\langle
U^{x,y}_{k}Z_{k}(W),U^{x,y}_{k}\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle=\left\langle
Z_{k}(W),\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle$

and

$\alpha\lambda_{0}+\beta\lambda_{1}=\left\langle
U^{x,y}_{k}Z_{k}(W),U^{x,y}_{k}\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle=\left\langle
U^{x,y}_{k}Z_{k}(W),U^{x,y}_{k}\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle=0.$

Summing up we have

$\displaystyle\kappa_{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\kappa_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\left\langle
Z_{k}(V),\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle,$
$\displaystyle\lambda_{0}$ $\displaystyle=$ $\displaystyle\left\langle
Z_{k}(W),\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle,$
$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{\beta}\left\langle
Z_{k}(W),\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle.$

Furthermore, we observe that

$\sum_{j=0}^{N_{k}}\kappa_{j}\overline{\lambda_{j}}=\langle
U^{z,w}_{k}Z_{k}(V),U^{z,w}_{k}Z_{k}(W)\rangle=\langle
Z_{k}(V),Z_{k}(W)\rangle.$

Put $c^{\prime}=\int_{\mathbb{C}}|a|^{2}d\mu^{G}(a)$. Since $x\neq y$ we have
$\beta\neq 0$. Then
$\int_{\mathbb{C}^{N_{k}+1}}\frac{a_{j}\overline{a_{l}}}{a_{0}\overline{(\alpha
a_{0}+\beta a_{1})}}d\mu^{G}(a)=0$ for all $0\leq j,l\leq N_{k}$ with $j\neq
l$ and $(j,l)\notin\\{(1,0),(0,1)\\}$. We check that

$\displaystyle II_{k}(x,y)(V,W)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}}+1}\frac{\langle
a,U^{x,y}_{k}Z_{k}(V)\rangle\overline{\langle
a,U^{x,y}Z_{k}(W)\rangle}}{|F_{k}(x)||F_{k}(y)|a_{0}\overline{(\alpha
a_{0}+\beta a_{1})}}d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\left(\int_{\mathbb{C}^{2}}\frac{c^{\prime}d\mu^{G}(a)}{a_{0}\overline{(\alpha
a_{0}+\beta
a_{1})}}\right)\sum_{j=2}^{N_{k}}\frac{\kappa_{j}\overline{\lambda_{j}}}{|F_{k}(x)||F_{k}(y)|}$
$\displaystyle+\int_{\mathbb{C}^{2}}\frac{|a_{0}|^{2}\kappa_{0}\overline{\lambda_{0}}+|a_{1}|^{2}\kappa_{1}\overline{\lambda_{1}}+a_{0}\overline{a_{1}}\kappa_{0}\overline{\lambda_{1}}+a_{1}\overline{a_{0}}\kappa_{1}\overline{\lambda_{0}}}{|F_{k}(x)||F_{k}(y)|a_{0}\overline{(\alpha
a_{0}+\beta a_{1})}}d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\left(\int_{\mathbb{C}^{2}}\frac{c^{\prime}d\mu^{G}(a)}{a_{0}\overline{(\alpha
a_{0}+\beta
a_{1})}}\right)\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{Z_{k}(W)}{|F_{k}(y)|}\right\rangle$
$\displaystyle+\left(\int_{\mathbb{C}^{2}}\frac{c^{\prime}d\mu^{G}(a)}{a_{0}\overline{(\alpha
a_{0}+\beta
a_{1})}}\right)\frac{\overline{\alpha}}{\beta^{2}}\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle\overline{\left\langle\frac{Z_{k}(W)}{|F_{k}(y)|},\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle}$
$\displaystyle+\left(\int_{\mathbb{C}^{2}}\frac{a_{1}\overline{(\beta
a_{0}-\alpha a_{1})}}{a_{0}\overline{(\alpha a_{0}+\beta
a_{1})}}d\mu^{G}(a)\right)\frac{1}{\beta^{2}}\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}\right\rangle\overline{\left\langle\frac{Z_{k}(W)}{|F_{k}(y)|},\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle}.$

With (5.2.5) and since

$\left|\int_{\mathbb{C}^{2}}\frac{a_{1}\overline{(\beta a_{0}-\alpha
a_{1})}}{a_{0}\overline{(\alpha a_{0}+\beta
a_{1})}}d\mu^{G}(a)\right|\leq\int_{\mathbb{C}^{2}}\frac{|a_{1}|^{2}+|a_{0}||a_{1}|}{|a_{0}||\alpha
a_{0}+\beta
a_{1}|}d\mu^{G}(a)\leq\frac{c}{\beta}c^{\prime}+\int_{\mathbb{C}^{2}}\frac{|a_{1}|}{|\alpha
a_{0}+\beta a_{1}|}d\mu^{G}(a)$

we conclude with $\beta=\sqrt{1-h^{F_{k}}(x,y)}$ that there is a constant
$C_{2}>0$ independent of $k$, $x$, $y$ and $V\in T_{x}X$, $W\in T_{y}X$ such
that $\gamma_{k}(x,y)\colon T_{x}X\times T_{y}X\to\mathbb{C}$ defined by
$\gamma_{k}(x,y)=\sqrt{1-h^{F_{k}}(x,y)}II_{k}(x,y)$ satisfies

$\displaystyle|\gamma_{k}(x,y)(V,W)|$ $\displaystyle\leq
C_{2}\left(\left|\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{Z_{k}(W)}{|F_{k}(y)|}\right\rangle\right|+\frac{1}{1-h^{F_{k}}(x,y)}\left|\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}\right\rangle\right|\left|\left\langle\frac{Z_{k}(W)}{|F_{k}(y)|},\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle\right|\right).$

∎

######  Lemma 5.12.

With the notation and assumptions above given $0<\varepsilon<1$ and a volume
form $dV$ on $X\times X$ there exist $C,k_{0}>0$ such that

(5.2.6) $\displaystyle\int_{X\times
X}\left|\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}d\mu^{G}\right|dV\leq
Ck^{2-n-1}(|\kappa(k)|^{2}+k^{\varepsilon})$

for all $k\geq k_{0}$.

###### Proof.

As in the proof of Lemma 5.11 we can find $k_{1},C_{0},C_{1}>0$ such that

(5.2.7) $\displaystyle C_{0}^{-1}k^{n+1}\leq|F_{k}|^{2}\leq C_{0}k^{n+1}$

and

(5.2.8) $\displaystyle|VF_{k}|\leq C_{1}k|F_{k}(x)||V|$

for all $k\geq k_{1}$, $x\in X$ and $V\in T_{x}X$. In addition, we choose
$k_{1}$ large enough such that for all $k\geq k_{1}$ and $x,y\in X$ we have
$h^{F_{k}}(x,y)=1$ if and only if $x=y$ (see Lemma 3.8). Recall that given
$x\neq y$, $V\in T_{x}X$, $W\in T_{y}X$ and $k\geq k_{1}$ we have

$\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}(a,x,y)(V,W)d\mu^{G}(a)=\frac{\gamma_{k}(x,y)(V,W)}{\sqrt{1-h^{F_{k}}(x,y)}}$

where $\gamma_{k}(x,y)(V,W)$ defined in (5.2.2) satisfies the estimate
(5.2.3). Using (5.2.3) we would like to estimate
$\left|\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}(a,x,y)d\mu^{G}(a)\right|$ and its
integral over $X\times X$. First we will show that $k^{-2}|\gamma_{k}(x,y)|$
is uniformly bounded in $x$, $y$ and $k$. Then we will estimate
$|\gamma_{k}(x,y)|$ outside the diagonal in $X\times X$ interms of
$\eta(T_{P})(x,y)$. As consequence we obtain (5.2.6) using Lemma 3.15 and
Lemma 3.16.

One can check that

$\displaystyle\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\frac{\overline{VF_{k}}}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}\right\rangle-\left\langle\frac{\overline{VF_{k}}}{|F_{k}(x)|},\frac{\overline{F_{k}(x)}}{{|F_{k}(x)|}}\right\rangle\left\langle\frac{\overline{F_{k}(x)}}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\frac{\overline{VF_{k}}}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}-\left\langle\frac{\overline{F_{k}(x)}}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right\rangle$

and

$\displaystyle\left|\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}-\left\langle\frac{\overline{F_{k}(x)}}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{|F_{k}(y)|}\right\rangle\frac{\overline{F_{k}(x)}}{|F_{k}(x)|}\right|^{2}=1-h^{F_{k}}(x,y).$

Hence it follows

$\left|\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{\overline{F_{k}(y)}}{{|F_{k}(y)|}}\right\rangle\right|\leq\sqrt{1-h_{k}(x,y)}\frac{|VF_{k}|}{|F_{k}(x)|}\leq\sqrt{1-h_{k}(z,w)}C_{1}k|V|$

and in the same way we obtain

$\left|\left\langle\frac{Z_{k}(W)}{|F_{k}(y)|},\frac{\overline{F_{k}(x)}}{{|F_{k}(x)|}}\right\rangle\right|\leq\sqrt{1-h_{k}(z,w)}C_{1}k|W|.$

With

$\left|\left\langle\frac{Z_{k}(V)}{|F_{k}(x)|},\frac{Z_{k}(W)}{|F_{k}(y)|}\right\rangle\right|\leq\frac{4|VF_{k}||WF_{k}|}{|F_{k}(x)||F_{k}(y)|}\leq
4(C_{1}k)^{2}|V||W|$

we conclude that there exists a constant $C_{3}>0$ such that
$|\gamma_{k}(x,y)(V,W)|\leq C_{3}k^{2}|V||W|$ for all $k\geq k_{1}$, $x,y\in
X$, $x\neq y$, and $V\in T_{x}X$, $W\in T_{x}Y$. The metric on TX induces a
metric on the bilinear forms from $T_{x}X\times T_{y}X$ to $\mathbb{C}$. It
follows that we can choose the constant $C_{3}>0$ such that

(5.2.9) $\displaystyle|\gamma_{k}(x,y)|\leq C_{3}k^{2}$

for all $k\geq k_{1}$, $x,y\in X$, $x\neq y$.

In order to obtain estimates for the integral of
$\left|\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}(a,x,y)d\mu^{G}(a)\right|$ over
$X\times X$ using Lemma 3.16 we need to estimate $|\gamma_{k}(x,y)|$ in terms
of $\eta_{k}(T_{P})(x,y)$.

With $|\langle VF_{k},F_{k}(x)\rangle|\leq kC_{1}|F_{k}(x)|^{2}|V|$ (see
5.2.8) we find

$\displaystyle|\langle Z_{k}(V),Z_{k}(W)\rangle|$ $\displaystyle\leq$
$\displaystyle|\langle VF_{k},WF_{k}\rangle|+(C_{1}k)^{2}|\langle
F_{k}(x),F_{k}(y)\rangle|V||W|$ $\displaystyle+C_{1}k|\langle
VF_{k},F_{k}(y)\rangle||W|+C_{1}k|V||\langle F_{k}(x),WF_{k}\rangle|$

and

(5.2.11) $\displaystyle|\langle
Z_{k}(V),\overline{F_{k}(y)}\rangle|\leq|\langle
VF_{k},F_{k}(y)\rangle|+k|\langle F_{k}(x),F_{k}(y)\rangle|.$

We note that

$\displaystyle\langle VF_{k},WF_{k}\rangle=d_{x}\otimes
d_{y}\eta_{k}(T_{p})(x,y).$

Then, by the compactness of $X$, we can find finitely many partial
differential operators
$Q_{j}\colon\mathscr{C}^{\infty}(X)\to\mathscr{C}^{\infty}(X)$, $1\leq j\leq
r$ of order at most two such that

(5.2.12) $\displaystyle|\langle
VF_{k},WF_{k}\rangle|\leq\sum_{j=1}^{r}|Q_{j}\eta_{k}(T_{P})(x,y)||V||W|$

for all $k\geq k_{1}$, $x,y\in X$, $V\in T_{x}X$ and $W\in T_{y}X$. Since

$\displaystyle\langle VF_{k},F_{k}(y)\rangle=d_{x}\eta_{k}(T_{p})(x,y)\text{
and }\langle F_{k}(x),WF_{k}\rangle=d_{y}\eta_{k}(T_{p})(x,y)$

we can find finitely many partial differential operators
$L_{j}\colon\mathscr{C}^{\infty}(X)\to\mathscr{C}^{\infty}(X)$, $1\leq j\leq
s$ of order at most one such that

(5.2.13) $\displaystyle|\langle VF_{k},F_{k}(x)\rangle||W|+|\langle
F_{k}(x),WF_{k}\rangle||V|\leq\sum_{j=1}^{s}|L_{j}\eta_{k}(T_{P})(x,y)||V||W|$

for all $k\geq k_{1}$, $x,y\in X$, $V\in T_{x}X$ and $W\in T_{y}X$. Put

$\gamma^{\prime}_{k}(x,y):=k^{2}|\kappa(k)|^{2}+k^{2}|\eta_{k}(T_{P})(x,y)|+k\sum_{j=1}^{s}|L_{j}\eta_{k}(T_{P})(x,y)|+\sum_{j=1}^{r}|Q_{j}\eta_{k}(T_{P})(x,y)|.$

Recall that $\langle
F_{k}(x),F_{k}(y)\rangle=|\kappa(k)|^{2}+\eta_{k}(T_{P})(x,y)$, $\langle
F_{k}(x),F_{k}(y)\rangle\leq|F_{k}(x)||F_{k}(y)|$, $|\langle
VF_{k},F_{k}(y)\rangle|\leq C_{1}k|F_{k}(x)||F_{k}(y)|$ and $|F_{k}|^{2}\geq
C_{0}^{-1}k^{n+1}$ for all $k\geq k_{1}$ and $x,y\in X$. Then from (5.2.3),
(5.2), (5.2.11), (5.2.12) and (5.2.13) we have that given any $\delta>0$ we
can find a constant $C_{4}>0$ such that

$|\gamma_{k}(x,y)|\leq C_{4}k^{-n-1}\gamma^{\prime}_{k}(x,y).$

for all $x,y\in X$ with $h^{F_{k}}(x,y)\leq 1-\delta$ and all $k\geq k_{1}$.
Given $0<\varepsilon<1$ we can choose $k_{2}\geq k_{1}$ to conclude from Lemma
3.16 that there exists $C_{5}>0$ with

(5.2.14) $\displaystyle\int_{h^{F_{k}}<1-\delta}|\gamma_{k}|dV\leq
C_{4}k^{-n-1}\int_{X\times X}\gamma^{\prime}_{k}dV\leq
C_{5}(|\kappa(k)|^{2}k^{2-n-1}+k^{2+\varepsilon-n-1})$

for all $k\geq k_{2}$. From (5.2.9), (5.2.14) and Lemma 3.15 we conclude that
there exist a constant $C>0$ and $k_{0}>0$ such that

$\int_{X\times
X}\left|\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}d\mu^{G}\right|dV=\int_{X\times
X}\frac{|\gamma_{k}|}{\sqrt{1-h^{F_{k}}}}dV\leq
Ck^{2-n-1}(|\kappa(k)|^{2}+k^{\varepsilon})$

for all $k\geq k_{0}$. ∎

Using Lemma 5.11 and Lemma 5.12 we obtain variance estimate for
$\mathcal{C}_{f}$.

######  Theorem 5.13.

Under the assumptions of Theorem 1.1 and with the notations above put

$\mathbb{V}_{k}(\mathcal{C}_{f}(\psi)):=\int_{A_{k}}\left|\mathcal{C}_{f}(\psi)-\mathbb{E}_{k}(\mathcal{C}_{f})(\psi)\right|^{2}d\mu_{k}(f),\,\,\,\psi\in\Omega^{2n}(X).$

Given $0<\varepsilon<1$ there exist $k_{0}>0$ and $C>0$ such that
$\mathbb{V}_{k}(\mathcal{C}_{f}(\psi))$ is well-defined with

$\mathbb{V}_{k}(\mathcal{C}_{f}(\psi))\leq
Ck^{2-n-1}(|\kappa(k)|^{2}+k^{\varepsilon})\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda_{\mathbb{C}}^{2n}T^{*}X)}$

for all $k\geq k_{0}$ and all $\psi\in\Omega^{2n}(X)$.

###### Proof.

Let $dV$ be a volume form on $X\times X$. Choose $k_{1}>0$ such that
$|F_{k}|>0$ for all $k\geq k_{1}$ and that the statements in Lemma 5.11 hold
true. We have that $\theta_{k}$ as in Definition 5.10 induces an alternating
multiliniear form $\hat{\theta}_{k}(a,x,y)$ on $T_{(x,y)}X\times X$ for all
$x,y\in X$ and $\mu^{G}$-almost every $a\in\mathbb{C}^{N_{k}+1}$. Define
$p_{1},p_{2}\colon X\times X\to X$, $p_{1}(x,y)=x$, $p_{2}(x,y)=y$ and put

(5.2.15) $\displaystyle R_{k}(\psi):=\int_{\mathbb{C}^{N_{k}+1}}\int_{X\times
X}\hat{\theta}_{k}\wedge
p_{1}^{*}\psi\wedge\overline{p_{2}^{*}\psi}d\mu^{G}(a),\,\,\,\,\psi\in\Omega^{2n}(X).$

By Lemma 5.11 and the Fubini-Tonelli theorem we have that $R_{k}(\psi)$ exists
for any $\psi\in\Omega^{2n}(X)$ and $k\geq k_{1}$ with

$\displaystyle|R_{k}(\psi)|=\left|\int_{X\times
X}\int_{\mathbb{C}^{N_{k}+1}}\hat{\theta}_{k}\wedge
p_{1}^{*}\psi\wedge\overline{p_{2}^{*}\psi}\right|\leq
C_{1}\left(\int_{X\times
X}\left|\int_{\mathbb{C}^{N_{k}+1}}\theta_{k}d\mu^{G}\right|dV\right)\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda_{\mathbb{C}}^{2n}T^{*}X)},$

where $C_{1}>0$ is a constant independent of $k$ and $\psi$. Hence, using
Lemma 5.12, we conclude that for any $0<\varepsilon<1$ there exist $k_{0}\geq
k_{1}$ and $C>0$ such that

(5.2.16) $\displaystyle|R_{k}(\psi)|\leq
Ck^{2-n-1}(|\kappa(k)|^{2}+k^{\varepsilon})\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda_{\mathbb{C}}^{2n}T^{*}X)}$

for all $\psi\in\Omega^{2n}(X)$ and $k\geq k_{0}$. Recall that for any $k\geq
k_{0}$ there exists $\tilde{A}_{k}\subset A_{k}$ with
$\mu_{k}(\tilde{A}_{k})=1$ such that for any $f\in\tilde{A}_{k}$ we have

(5.2.17) $\displaystyle\mathcal{C}_{f}(\psi)=\frac{1}{2\pi
i}\int_{X}\frac{df}{f}\wedge\psi,\,\,\,\,\,\psi\in\Omega^{2n}(X).$

By Theorem 5.5 we have
$\mathbb{E}_{k}(\mathcal{C}_{f})(\psi)=\int_{X}\beta_{k}\wedge\psi$ for all
$\psi\in\Omega^{2n}(X)$ with

(5.2.18) $\displaystyle\beta_{k}=\frac{\langle dF_{k},F_{k}\rangle}{2\pi
i|F_{k}|^{2}}.$

Writing $f=\langle a,\overline{F_{k}}\rangle$, $a\in\mathbb{C}^{N_{k}+1}$, we
find

(5.2.19) $\displaystyle\frac{df}{f}-2\pi i\beta_{k}=\frac{\langle
a,\overline{dF_{k}}\rangle-\frac{\langle
dF_{k},F_{k}\rangle}{|F_{k}|^{2}}\langle a,\overline{F_{k}}\rangle}{\langle
a,\overline{F_{k}}\rangle}=\frac{\langle a,Z_{k}\rangle}{\langle
a,\overline{F_{k}}\rangle}$

for all $k\geq k_{0}$ and $\mu^{G}$-almost every $a\in\mathbb{C}^{N_{k}+1}$.
Given $\psi\in\Omega^{2n}(X)$ and $k\geq k_{0}$ we obtain from the definition
of $\theta_{k}$ and (5.2.19) that

$\displaystyle R_{k}(\psi)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\int_{X\times
X}p_{1}^{*}\left(\frac{\langle a,Z_{k}\rangle}{\langle
a,\overline{F_{k}}\rangle}\wedge\psi\right)\wedge\overline{p_{2}^{*}\left(\frac{\langle
a,Z_{k}\rangle}{\langle
a,\overline{F_{k}}\rangle}\wedge\psi\right)}d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\left|\int_{X}\frac{\langle
a,Z_{k}\rangle}{\langle
a,\overline{F_{k}}\rangle}\wedge\psi\right|^{2}d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\int_{A_{k}}\left|\int_{X}\left(\frac{df}{f}-2\pi
i\beta_{k}\right)\wedge\psi\right|^{2}d\mu_{k}(f)$ $\displaystyle=$
$\displaystyle 4\pi^{2}\mathbb{V}_{k}(\mathcal{C}_{f}(\psi)).$

Then the claim follows from (5.2.16). ∎

Theorem 1.7 follows from the next result by taking $\kappa(k)\equiv 0$ and
replacing $\psi$ by $d\psi$ (see Corollary 5.15). Consider $k\in\mathbb{N}$
and put $A_{\infty}=\Pi_{k=1}^{\infty}A_{k}$,
$d\mu_{\infty}=\Pi_{k=1}^{\infty}d\mu_{k}$. We have the following.

######  Theorem 5.14.

Under the same assumptions as in Theorem 1.1 and with the notations above,
assuming that $|\kappa(k)|^{2}/(1+|k|^{n-1})|$ is bounded, there exists
$k_{0}>0$ such that for $\mu_{\infty}$-almost every
$f=(f_{k})_{k\in\mathbb{N}}\in A_{\infty}$ we have that
$\mathcal{C}_{f_{k}}(\psi)$ is well defined for all $\psi\in\Omega^{2n}(X)$
and $k\geq k_{0}$. Furthermore, given $\psi\in\Omega^{2n}(X)$, we have for
$\mu_{\infty}$-almost every $f=(f_{k})_{k\in\mathbb{N}}\in A_{\infty}$ that

$\lim_{k\to\infty}k^{-1}\mathcal{C}_{f_{k}}(\psi)=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}\frac{\xi}{\sigma_{P}(\xi)}\wedge\psi$

where $\operatorname{mv}(|\chi|^{2})$ is given by (5.1.5). In addition, there
exists $C>0$ such that for any $k\geq k_{0}$ one has

(5.2.20) $\displaystyle\,\,\,\,\,\,\mu_{k}\left(\left\\{f\in
A_{k}\colon\left|k^{-1}\mathcal{C}_{f}(\psi)-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}\frac{\xi}{\sigma_{P}(\xi)}\wedge\psi\right|\geq\frac{\|\psi\|_{\mathscr{C}^{0}(X,\Lambda^{2n-1}_{\mathbb{C}}T^{*}X)}}{\sqrt{k}}\right\\}\right)\leq\frac{C}{\sqrt{k}}$

for all $\psi\in\Omega^{2n-1}(X)$.

###### Proof.

Given $\psi\in\Omega^{2n}(X)$ and $k>0$ put

$R_{k}(\psi):=\int_{A_{k}}\left|k^{-1}\mathcal{C}_{f}(\psi)-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}\frac{\xi}{\sigma_{P}(\xi)}\wedge\psi\right|^{2}d\mu_{k}(f).$

We will show that there exist $k_{0}\in\mathbb{N}$ and a constant $C>0$ such
that $R_{k}(\psi)$ exists for all $k\geq k_{0}$ with $R_{k}(\psi)\leq
Ck^{-\frac{3}{2}}\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda^{2n}_{\mathbb{C}}T^{*}X)}$
for all $k\geq k_{0}$ and $\psi\in\Omega^{2n}(X)$. From the Markov inequality,
the last part of the claim follows. Furthermore, considering $k\in\mathbb{N}$,
it follows that $\sum_{k=k_{0}}^{\infty}R_{k}(\psi)\leq
C_{0}\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda^{2n}_{\mathbb{C}}T^{*}X)}$ for
some constant $C_{0}$ (independent of $\psi$) which proves the second part of
the claim. The first part of the claim follows immediately from Theorem 5.5.

Given $\psi\in\Omega^{2n}(X)$ and $k\in\mathbb{N}$ put

$R^{\prime}_{k}(\psi):=\left|\frac{1}{k}\int_{X}\beta_{k}\wedge\psi-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}\frac{\xi}{\sigma_{P}(\xi)}\wedge\psi\right|^{2}$

with $\beta_{k}$ as in (5.1.4). From Theorem 5.5 and Corollary 5.8 it follows
that there exist $k_{1}\in\mathbb{N}$ and $C_{1}>0$ such that
$R^{\prime}_{k}(\psi)\leq
C_{1}k^{-2}\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda^{2n}_{\mathbb{C}}T^{*}X)}$
holds for all $k\geq k_{0}$ and $\psi\in\Omega^{2n}(X)$. With Theorem 5.5 and
$\mathbb{V}_{k}$ as in Theorem 5.13 we obtain

$R_{k}(\psi)\leq
2k^{-2}\mathbb{V}_{k}(\mathcal{C}_{f}(\psi))+2R^{\prime}_{k}(\psi)\int_{A_{k}}d\mu_{k}$

Then it follows from Theorem 5.13 with $\varepsilon=\frac{1}{2}$ that there
are $k_{0}\in\mathbb{N}$ and a constant $C>0$ such that $R_{k}(\psi)\leq
Ck^{-\frac{3}{2}}\|\psi\|^{2}_{\mathscr{C}^{0}(X,\Lambda^{2n}_{\mathbb{C}}T^{*}X)}$
for all $k\geq k_{0}$ and $\psi\in\Omega^{2n}(X)$. ∎

######  Corollary 5.15.

Under the same assumptions as in Theorem 1.1, with the notations above and
assuming $\kappa(k)\equiv 0$ there exists $k_{0}>0$ such that for
$\mu_{\infty}$-almost every $f=(f_{k})_{k\in\mathbb{N}}\in A_{\infty}$ we have
that $(\left[\operatorname{Div}\\!\left(f_{k}\right)\right],\psi)$ is well
defined for all $\psi\in\Omega^{2n-1}(X)$ and $k\geq k_{0}$. Furthermore,
given any $\psi\in\Omega^{2n-1}(X)$, we have for $\mu_{\infty}$-almost every
$f=(f_{k})_{k\in\mathbb{N}}\in A_{\infty}$ that

$\lim_{k\to\infty}\left(k^{-1}\left[\operatorname{Div}\\!\left(f_{k}\right)\right],\psi\right)=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}d\alpha_{P}\wedge\psi$

where $\alpha_{P}=(\sigma_{P}(\xi))^{-1}\xi$. In addition, there exists $C>0$
such that for any $k\geq k_{0}$ one has

$\displaystyle\mu_{k}\left(\left\\{f\in
A_{k}\colon\left|\left(k^{-1}\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}d\alpha_{P}\wedge\psi\right|\geq\frac{\|\psi\|_{\mathscr{C}^{1}(X,\Lambda^{2n-1}_{\mathbb{C}}T^{*}X)}}{\sqrt{k}}\right\\}\right)\leq\frac{C}{\sqrt{k}}$

for all $\psi\in\Omega^{2n-1}(X)$.

###### Proof.

From Theorem 4.21 we obtain
$\left[\operatorname{Div}\\!\left(f\right)\right]=d\mathcal{C}_{f}$ in the
sense of distributions for any $f\in H^{0}_{b}(X)\cap C^{\infty}(X)$ such that
zero is a regular value of the map $f\colon X\to\mathbb{C}$. Then the first
part of the claim follows from Theorem 5.5. Fix $\psi\in\Omega^{2n-1}(X)$.
Using Lemma 5.2 and integration by parts, we obtain from Theorem 5.14 with
$\kappa(k)\equiv 0$ that for $\mu_{\infty}$-almost every
$f=(f_{k})_{k\in\mathbb{N}}\in A_{\infty}$ we have that

$\lim_{k\to\infty}\left(k^{-1}\left[\operatorname{Div}\\!\left(f_{k}\right)\right],\psi\right)=\lim_{k\to\infty}k^{-1}\mathcal{C}_{f_{k}}(d\psi)=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}\alpha_{P}\wedge
d\psi=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{X}d\alpha_{P}\wedge\psi.$

Furthermore, replacing $\psi$ by $d\psi$ in (5.2.20) leads to the last part of
the claim. ∎

##  6\. Some Remarks on Complex Manifolds with Boundary

In this section we will recall basic definitions and notations for complex
manifolds with boundary and formulate Theorem 4.14 in this set-up.

Let $M$ be a real smooth $N$-dimensional manifold with smooth boundary. We
denote by $\text{int}(M)$ the set of points in $M$ which have an open
neighborhood homeomorphic to $\mathbb{R}^{N}$ and put
$bM:=M\setminus\text{int}(M)$ to denote the boundary of $M$. We note that
$\text{int}(M)$ is a smooth $N$-dimensional manifold without boundary and that
$bM$, in case $bM\neq\emptyset$, is a smooth $(N-1)$-dimensional smooth
submanifold of $M$. We call $V$ a complex domain with boundary if there exists
a complex manifold $Y$ (without boundary) and a domain $G\subset Y$ with
smooth boundary such that $V=\overline{G}$. Note that $V$ is automatically a
smooth manifold with boundary with a complex structure on $\text{int}(V)$.

######  Definition 6.1.

Let $M$ be a real smooth connected manifold with boundary. A complex structure
on $M$ is a complex structure on $\text{int}(M)$ such that for any point $p\in
bM$ there exist an open neighborhood $U\subset M$ around $p$ and a
diffeomorphism $F\colon U\to V$ where $V$ is a complex domain with boundary
such that $F$ restricted to $U\cap\text{int}(M)$ is a holomorphic map.
A smooth connected manifold with smooth boundary together with a complex
structure is called complex manifold with boundary of (complex) dimension
$\dim_{\mathbb{C}}M:=\dim_{\mathbb{C}}\text{int}(M)$. Furthermore, we denote
by $\mathcal{O}^{\infty}(M):=\mathcal{O}(\text{int}(M))\cap C^{\infty}(M)$ the
space of holomorphic functions which are smooth up to the boundary.

######  Example 6.2.

Put $M=\\{z\in\mathbb{C}^{2}\mid 1<|z|\leq 2\\}$. Then $M$ (together with
differential and complex structure induced by $\mathbb{C}^{2}$) is a complex
manifold with boundary. We have $\text{int}(M)=\\{z\in\mathbb{C}^{2}\mid
1<|z|<2\\}$ and $bM=\\{z\in\mathbb{C}^{2}\mid|z|=2\\}$. Furthermore, with
$Y=\\{z\in\mathbb{C}^{2}\mid|z|>1\\}$ and $G=\\{z\in Y\mid|z|<2\\}$ we find
that $M=\overline{G}$ is a complex domain with boundary.

For further constructions it is necessary to give an infinitesimal
interpretation of complex manifolds with boundary.

######  Lemma 6.3.

Let $M$ be an $n$-dimensional complex manifold with boundary. There exists a
uniquely defined smooth complex subbundle $T^{1,0}M\subset\mathbb{C}TM$ of
rank $n$ such that $T^{1,0}M|_{\operatorname{int}(M)}$ coincides with the
complex structure on $\operatorname{int}(M)$. Furthermore, $T^{1,0}M$ is
formally integrable and $T^{1,0}M\oplus\overline{T^{1,0}M}=\mathbb{C}TM$.

###### Proof.

Given $p\in\text{int}(M)$ we choose $T^{1,0}_{p}M$ in accordance with the
complex structure on $\text{int}(M)$, that is,
$T^{1,0}_{p}M=T^{1,0}_{p}\text{int}(M)$. Given $p\in bM$ put

$T_{p}^{1,0}M:=\\{W_{p}\mid
W\in\mathscr{C}^{\infty}(M,\mathbb{C}TM),\,W|_{\text{int}(M)}\in\mathscr{C}^{\infty}(\text{int}(M),T^{1,0}\text{int}(M))\\}.$

We find an open neighborhood $U\subset M$ around $p$ and a diffeomorphism
$F\colon U\to V$ where $V$ is a complex domain with boundary such that $F$
restricted to $U\cap\text{int}(M)$ is a holomorphic map. Furthermore, we can
write $V=\overline{G}$ for some smoothly bounded domain $G$ of some complex
manifold $Y$. Since $F$ is a diffeomorphism and holomorphic on
$U\cap\text{int}(M)$ we observe by continuity reasons
$T^{1,0}_{p}M=(dF_{p})^{-1}T^{1,0}_{F(p)}Y$. Hence $T^{1,0}_{p}M$ is a complex
vector space of dimension $n$ and $T^{1,0}M$ is a smooth subbundle. With
$\overline{T^{1,0}_{p}M}=(dF_{p})^{-1}\overline{T^{1,0}_{F(p)}Y}$ and
$\mathbb{C}TY_{F(p)}=T^{1,0}Y\oplus\overline{T^{1,0}Y}$ we find
$\mathbb{C}TX=T^{1,0}_{p}M\oplus\overline{T^{1,0}_{p}M}$. Since $T^{1,0}Y$ is
formally integrable it follows that $T^{1,0}M$ is formally integrable.
Moreover, by construction we have that $T^{1,0}M$ is contained in any smooth
complex subbundle which coincides with $T^{1,0}\text{int}(M)$ on
$\text{int}(M)$. Since $\text{rank}T^{1,0}M=n$ the uniqueness follows. ∎

Put $T^{0,1}M:=\overline{T^{1,0}M}$. The decomposition
$\mathbb{C}TM=T^{1,0}M\oplus T^{0,1}M$ immediately leads to the complex vector
bundles $\Lambda^{\bullet,\bullet}\mathbb{C}T^{*}M$ over $M$ defined in the
usual way. For any $p,q\in\mathbb{N}_{0}$ we denote by

$\Omega^{p,q}(M):=\mathscr{C}^{\infty}(M,\Lambda^{p,q}\mathbb{C}T^{*}M)$

the space of smooth $(p,q)$-forms on $M$ and let
$\Omega_{c}^{p,q}(M)\subset\Omega^{p,q}(M)$ be the space of those forms with
compact support. The operators $\partial,\overline{\partial}$, well understood
for $(p,q)$ forms on $\text{int}(M)$, uniquely extend to operators

$\partial\colon\Omega^{p,q}(M)\to\Omega^{p+1,q}(M),\,\,\,\overline{\partial}\colon\Omega^{p,q}(M)\to\Omega^{p,q+1}(M).$

One can easily check that those construction coincides with the definitions in
Section 4 when $M$ is a complex domain with boundary. By definition $M$ is
locally isomorphic to complex domains with boundary. Then, the notion of
$f\in\mathcal{O}^{\infty}(M)$ being regular with respect to $bM$ (see
Definition 4.1) as in Section 4 can be extended for complex manifold with
boundary. We immediately observe the following.

######  Lemma 6.4.

Lemma 4.4 and Corollary 4.5 hold when $G$ and $\overline{G}$ are replaced by
$\text{int}(M)$ and $M$ respectively.

In conclusion we obtain the following.

######  Theorem 6.5.

Let $M$ be an $(n+1)$-dimensional complex manifold with boundary,
$f\in\mathcal{O}^{\infty}(M)$ regular with respect to $bM$ and
$\psi\in\Omega^{n,n}_{c}(M)$. We have that

$\text{\,\,\,\,\,\,\,}\int_{M}\partial\overline{\partial}\log(|f|)\wedge\psi:=-\int_{bM}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{bM}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{M}\log(|f|)\wedge\partial\overline{\partial}\psi$

and $\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)$ are
well defined with

(6.0.1)
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=\frac{i}{\pi}\int_{M}\partial\overline{\partial}\log(|f|)\wedge\psi.$

Here, $\iota\colon bM\to M$ denotes the inclusion map.

###### Proof.

It is enough to show that any point $p\in M$ has an open neighborhood
$U\subset M$ such that the statement holds for $\psi\in\Omega^{n,n}_{c}(U)$.
Given $p\in\text{int}(M)$ we can choose $U=\text{int}(M)$ and the claim
follows. Given $p\in bM$ we find an open neighborhood $U\subset M$ around $p$
and a diffeomorphism $F\colon U\to V$ where $V$ is a complex domain with
boundary such that $F$ restricted to $U\cap\text{int}(M)$ is a holomorphic
map. Then the claim follows from Theorem 4.14. ∎

The following lemma shows that $bM$ naturally carries the structure of a CR
manifold.

######  Lemma 6.6.

Let $M$ be an $(n+1)$-dimensional complex manifold with boundary such that
$bM\neq\emptyset$ and $n\geq 1$. Then $T^{1,0}bM:=\mathbb{C}TbM\cap T^{1,0}M$
defines a (codimension one) CR structure on $bM$ such that the restriction
$f|_{bM}$ of any $f\in\mathcal{O}^{\infty}(M)$ is a smooth CR function.

###### Proof.

We first observe that $bM$ is an orientable smooth manifold of real dimension
$\dim_{\mathbb{R}}bM=2n+1$. Basic arguments from linear algebra show that
$\text{rank}_{\mathbb{C}}T^{1,0}bM=n$ and $T^{1,0}bM\cap T^{0,1}bM=\text{'zero
section'}$. Since $TbM$ and $T^{1,0}M$ are formally integrable it follows that
$T^{1,0}bM$ defines a (codimension one) CR structure on $bM$. Given
$f\in\mathcal{O}^{\infty}(M)$ we have that $f|_{bM}$ is smooth and we obtain
for continuity reasons that $\overline{Z}f=0$ for all $Z\in T^{1,0}M$. Hence
$\overline{Z}(f|_{bM})=0$ for all $Z\in T^{1,0}bM$ which proves that $f|_{bM}$
is a CR function. ∎

######  Definition 6.7.

Let $M$ be an $(n+1)$-dimensional complex manifold with boundary. We say that
$p\in bM$ is a strictly pseudoconvex boundary point if there exists a
diffeomorphism $F\colon U\to V$ where $U$ is an open neighborhood around $p$
and $V$ is complex domain with boundary such that $F$ is holomorphic on
$U\cap\text{int}(M)$ and $F(p)$ is a strictly pseudoconvex boundary point of
$V$. We call $M$ a complex manifold with strictly pseudoconvex boundary if any
point $p\in bM$ is a strictly pseudoconvex boundary point.

We observe that if $M$ is a complex manifold with strictly pseudoconvex
boundary it follows that $(bM,T^{1,0}bM)$ is a strictly pseudoconvex CR
manifold. There is the following relation between complex manifolds with
strictly pseudoconvex boundary and complex domains with boundary due to
Heunemann [Heu86] and Ohsawa [O84], see also Catlin [MR1128581], and Hill-
Nacinovich [MR1289628].

######  Theorem 6.8 (cf. [MM07, Theorem 6.3.15]).

Let $M$ be a compact complex manifold with strictly pseudoconvex boundary.
Then $M$ can be realized as a domain with boundary in a larger complex
manifold $Y$. More preciesely, there exists a strictly pseudoconvex domain
with boundary $V\subset Y$ and a diffeomorphism $F\colon M\to V$ which is a
biholomorphism between $\operatorname{int}(M)$ and $\operatorname{int}(V)$,
$F(bM)=bV$.

######  Definition 6.9.

Let $M$ be an $(n+1)$-dimensional complex manifold with boundary. We say that
$M$ has the CR extension property if for any smooth CR function $f$ on $bM$
there exists a function $F\in\mathcal{O}^{\infty}(M)$ with $F|_{bM}=f$.

Let us give some examples for complex manifolds with boundary satisfying the
CR extension property.

######  Example 6.10.

Let $M$ be a complex domain with boundary. As already mentioned, it follows
that $M$ is a complex manifold with boundary. Assume $\dim_{\mathbb{C}}M>1$
and that each point of $p\in bM$ is a strictly pseudoconvex boundary point. If
$bM$ is compact we have that $bM$ is CR embedabble into the complex Euclidean
space (see [HsM17]). If $M$ is compact it follows from results due to Grauert
[Gr58] and Kohn–Rossi [KR65] that $bM$ is connected and that $M$ has the CR
extension property.

######  Example 6.11.

Let $(L,h)\to N$ be a positive holomorphic line bundle with Hermitian metric
$h$ over a compact complex manifold $N$ (without boundary). Put $M=\\{v\in
L^{*}\mid|v|_{h^{*}}\leq 1\\}$ where $L^{*}$ denotes the dual line bundle of
$L$ and $h^{*}$ the Hermitian metric on $L^{*}$ induced by $h$. Then $M$ is a
compact complex manifold with strictly pseudoconvex boundary which has the CR
extension property.

######  Example 6.12.

Let $(N,g)$ be a compact (without boundary) connected real analytic Riemannian
manifold with real analytic metric $g$. Assume $\dim_{\mathbb{R}}N\geq 2$. For
any sufficiently small $r>0$ we have that $M:=\\{v\in TN\mid|v|_{g}\leq r\\}$
is a compact complex manifold with strictly pseudoconvex boundary with respect
to the adapted complex structure induced by $g$ (see Grauert [Gr58] and
Guillemin–Stenzel [GS91]). Furthermore, $bM$ is CR embeddable into the complex
Euclidean space and $M$ has the CR extension property by Example 6.10.

######  Example 6.13.

Let $(X,T^{1,0}X)$ be a compact strictly pseudoconvex CR manifold which is CR
embeddable into the complex Euclidean space. There exists $r>0$ such that
$M:=X\times[0,r)$ carries the structure of a complex manifold with strictly
pseudoconvex boundary such that $(X,T^{1,0}X)$ is CR isomorphic to
$(bM,T^{1,0}bM)$ (see Harvey–Lawson [HL75]). Furthermore, choosing $r>0$ small
enough, it follows that $M$ has the CR extension property.

##  7\. Equidistribution on Complex Manifolds with Strictly Pseudoconvex
Boundary

In this section we are going to prove Theorem 1.10, Theorem 1.11 and Theorem
1.12 (see Theorem 7.9, Theorem 7.3 and Theorem 7.11, respectively). Let $M$ be
an $(n+1)$-dimensional complex manifold with strictly pseudoconvex boundary,
$n\geq 1$, such that $M$ has the CR extension property (see Section 6). Put
$(X,T^{1,0}X):=(bM,T^{1,0}bM)$ and assume that $X$ is non-empty, compact and
CR embeddable. Then $X$ (seen as the boundary of $M$) is a compact oriented
strictly pseudoconvex CR manifold which satisfies the assumptions in Theorem
1.1. We denote by $\iota\colon bM\to M$ the inclusion map. Choose a volume
form $dV$ on $X$. Denote by $\xi$ a contact form on $X$ such that the
respective Levi form is positive definite and let $\mathcal{T}$ be the
respective Reeb vector field uniquely determined by
$\iota_{\mathcal{T}}\xi\equiv 1$ and $\iota_{\mathcal{T}}d\xi\equiv 0$. Let
$P\in L^{1}_{\mathrm{cl}}(X)$ be a first order formally self-adjoint classical
pseudodifferential operator with $\sigma_{P}(\xi)>0$ where $\sigma_{P}$
denotes the principal symbol of $P$. Denote by $T_{P}=\Pi P\Pi$ the
corresponding Toeplitz operator. Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$
denote the the positive eigenvalues of $T_{P}$ and denote by
$\\{f_{j}\\}_{j\in\mathbb{N}}\subset H_{b}^{0}(X)\cap\mathscr{C}^{\infty}(X)$
an orthonormal system of eigenfunctions. Given a function
$\eta\in\mathscr{C}^{\infty}_{c}\left(\mathbb{R}_{+},[0,\infty)\right)$ we
have that the integral kernel of $\eta_{k}(T_{P})$ can be written as
$\eta_{k}(T_{P})(x,y)=\sum_{j=1}^{\infty}\eta_{k}(\lambda_{j})f_{j}(x)\overline{f_{j}(y)}$
where $\eta_{k}(t)=\eta(k^{-1}t)$, $t,k>0$. From the CR extension property
assumption on $M$ any smooth CR function $f$ on $X$ extends holomorphically to
$\text{int}(M)$. In particular, since $M$ is connected, there exists a
uniquely determined function
$\mathcal{F}\in\mathcal{O}^{\infty}(M):=\mathcal{O}(\text{int}(M))\cap\mathscr{C}^{\infty}(M)$
such that $f=\mathcal{F}$ on $X$. Hence we have a linear mapping

(7.0.1) $\displaystyle L\colon
H_{b}^{0}(X)\cap\mathscr{C}^{\infty}(X)\to\mathcal{O}^{\infty}(M),\,\,\,\,L(f)=\mathcal{F}.$

We define

(7.0.2) $\displaystyle B^{\eta}_{k}\colon
M\to\mathbb{R},\,\,\,\,\,B^{\eta}_{k}(z)=\sum_{j=1}^{\infty}\eta_{k}(\lambda_{j})L(f_{j})(z)\overline{L(f_{j})(z)}.$

Then $B^{\eta}_{k}$ is smooth and for any $c>0$ we have that
$k^{-1}\partial\overline{\partial}\log(c+B^{\eta}_{k})$ defines a non negative
$(1,1)$-form on $\text{int}(M)$ which is smooth up to the boundary. Let
$\chi\in\mathscr{C}^{\infty}_{c}(\mathbb{R}_{+})$, $\chi\not\equiv 0$, be a
smooth function with $\text{supp}(\chi)\subset(\delta_{1},\delta_{2})$ for
some $0<\delta_{1}<\delta_{2}$. Put
$N_{k}=\\#\\{j\in\mathbb{N}\colon\lambda_{j}\leq\delta_{2}k\\}$. For $k>0$ put
$A_{k}=\text{span}\left(\\{1\\}\cup\\{L(f_{j})\mid\lambda_{j}\leq\delta_{2}k\\}\right)$.
On $A_{k}$ we consider the probability measure $\mu_{k}$ induced by the
standard complex Gaussian measure $\mu^{G}$ on $\mathbb{C}^{N_{k}+1}$ and the
map

$\mathbb{C}^{N_{k}+1}\ni a\mapsto
a_{0}+\sum_{j=1}^{N_{k}}a_{j}\chi_{k}(\lambda_{j})L(f_{j})\in A_{k}$

where $\chi_{k}(t)=\chi(k^{-1}t)$. We use the same notation $(A_{k},\mu_{k})$
for the probability spaces as in Section 5 since the constructions coincides.
In particular, compared to Section 5, we choose $\kappa(k)\equiv 1$ in this
section.

###  7.1. Proof of Theorem 1.11

To simplify the notations in the proofs we consider the following object.

######  Definition 7.1.

Let $f\in\mathcal{O}^{\infty}(M)$ such that $\partial f_{p}\neq 0$ for all
$p\in bM$ with $f(p)=0$. We define

$\mathcal{Z}_{f}\colon\Omega_{c}^{n,n}(M)\to\mathbb{C},\,\,\,\mathcal{Z}_{f}(\psi)=\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right).$

where $\left[\operatorname{Div}\\!\left(f\right)\right]$ denotes the zero
divisor of $f$.

We note that under the assumptions on $f$ in Definition 7.1 we have that
$\mathcal{Z}_{f}$ is well defined by Lemma 4.13 (cf. Section 6.4).
Furthermore, using Theorem 6.5, $Z_{f}(\psi)$ can be computed via the formula

$\displaystyle\mathcal{Z}_{f}(\psi)$ $\displaystyle:=$
$\displaystyle\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)$
$\displaystyle=$
$\displaystyle\frac{i}{\pi}\left(-\int_{bM}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{bM}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{M}\log(|f|)\wedge\partial\overline{\partial}\psi\right)$

for all $\psi\in\Omega_{c}^{n,n}(M)$. Since we wish to understand
$\mathcal{Z}_{f}$ as a random variable on $A_{k}$, we have to make sure that
it is defined on a set $\tilde{A}_{k}\subset A_{k}$ with
$\mu_{k}(\tilde{A_{k}})=1$. We have the following.

######  Lemma 7.2.

Put $\tilde{A}_{k}:=\\{f\in A_{k}\mid\text{0 is a regular value of
}f|_{bM}\colon bM\to\mathbb{C}\\}$. We have that $\tilde{A}_{k}$ is
$\mu_{k}$-measurable with $\mu_{k}(A_{k}\setminus\tilde{A}_{k})=0$.

###### Proof.

We have $1+\sum_{j=1}^{N_{k}}|\chi_{k}(\lambda_{j})|^{2}|f_{j}(x)|^{2}\geq
1>0$ for all $k>0$ and all $x\in bD$. Then the claim follows from Lemma 5.3. ∎

We note that given $f\in\tilde{A}_{k}$ with $\tilde{A}_{k}$ as in Lemma 7.2 it
follows that $\partial f_{p}\neq 0$ for all $p\in bM$ with $f(p)=0$. Hence
$\mathcal{Z}_{f}$ is well defined and we can apply (7.1) for its computation.
Theorem 1.10 follows from the next result by taking $c=1$ and assuming that
$M$ is compact (see Example 6.10).

######  Theorem 7.3.

Let $M$ be an $(n+1)$-dimensional complex manifold with strictly pseudoconvex
boundary, $n\geq 1$, such that $M$ has the CR extension property (see Section
6). Put $X:=bM$ and assume that $X$ is non-empty, compact and CR embeddable.
Let $\xi$ be a contact form for $X$ such that its induced Levi form is
positive definite. Choose a volume form $dV$ on $X$ and consider a Toeplitz
operator $T_{P}=\Pi P\Pi$ on $X$ as in the set-up of Theorem 1.1. With the
notations above we have for each $k>0$ that $\mathcal{Z}_{f}$ is a well-
defined distribution for all $f$ outside a subset in $A_{k}$ of
$\mu_{k}$-measure zero and the expectation value
$\mathbb{E}_{k}\left(\mathcal{Z}_{f}\right)$ defined by

$\left(\mathbb{E}_{k}\left(\mathcal{Z}_{f}\right),\psi\right):=\int_{A_{k}}\mathcal{Z}_{f}(\psi)d\mu_{k}(f),\,\,\,\,\,\psi\in\Omega^{n,n}_{c}(M)$

exists with

$\mathbb{E}_{k}\left(\mathcal{Z}_{f}\right)=\frac{i}{2\pi}\partial\overline{\partial}\log(1+B^{|\chi|^{2}}_{k}).$

Here, $B^{|\chi|^{2}}_{k}$ is given by (7.0.2) where $\chi$ appears in the
definition of $A_{k}$.

Similar to Section 5 we introduce the following notation. For $d\in\mathbb{N}$
we denote by $\langle\cdot,\cdot\rangle$ the standard inner product on
$\mathbb{C}^{d+1}$ that is $\langle
a,b\rangle=\sum_{j=0}^{d}a_{j}\overline{b_{j}}$ for
$a=(a_{0},\ldots,a_{d}),b=(b_{0},\ldots,b_{d})\in\mathbb{C}^{d+1}$.
Furthermore, given $a\in\mathbb{C}^{d+1}$ we put $|a|=\sqrt{\langle
a,a\rangle}$. For $k>0$ consider the map

(7.1.2) $\displaystyle F_{k}\colon
M\to\mathbb{C}^{N_{k}+1},\,\,\,\,F_{k}=(1,\chi_{k}(\lambda_{1})L(f_{1}),\ldots,\chi_{k}(\lambda_{N_{k}})L(f_{N_{k}})).$

Define
$S_{k}(z,w)=\sum_{j=1}^{\infty}|\chi_{k}(\lambda_{j})|^{2}L(f_{j})(z)\overline{L(f_{j})(w)}$,
$z,w\in M$ and put $B_{k}:=B_{k}^{|\chi|^{2}}$. It turns out that
$|F_{k}|^{2}=1+B_{k}$, $S_{k}(z,z)=B_{k}(z)$ and $\langle
F_{k}(w),F_{k}(w)\rangle=1+S_{k}(z,w)$ for all $z,w\in M$ and $k>0$. With
$\eta_{k}=|\chi_{k}|^{2}$ we find
$S_{k}(z,w):=\eta_{k}(T_{P})(z,w)=\sum_{j=1}^{N_{k}}\eta_{k}(\lambda_{j})f_{j}(z)\overline{f_{j}(w)}$
for all $z,w\in bM$ and $k>0$. We further consider $dF_{k}$ component wise,
that is,

$dF_{k}=(0,\chi_{k}(\lambda_{1})df_{1},\ldots,\chi_{k}(\lambda_{N_{k}})df_{N_{k}}).$

Then $\langle a,\overline{dF_{k}}\rangle$ is a one form on $M$ for any
$a\in\mathbb{C}^{N_{k}+1}$. We have for example $\langle
F_{k}(z),(dF_{k})_{w}\rangle=d_{w}S_{k}(z,w)$ and for $f=\langle
a,\overline{F_{k}}\rangle$ we obtain $df=\langle a,\overline{dF_{k}}\rangle$.

We will prove Theorem 7.3 using the decomposition of $\mathcal{Z}_{f}$ given
by (7.1) and consider each term in the sum separately. The first term in the
right-hand side of (7.1) can be handled using the following lemma which is a
direct consequence of the results in Section 5.

######  Lemma 7.4.

With the notations and assumptions above given $k>0$ and
$\psi\in\Omega_{c}^{n,n}(M)$ we have that

$I_{k}(\psi):=\int_{A_{k}}\int_{bM}\iota^{*}\left(\frac{\partial
f}{f}\wedge\psi\right)d\mu_{k}(f)$

exists with

$I_{k}(\psi)=\int_{bM}\iota^{*}\left((\partial\log(1+B_{k}))\wedge\psi\right).$

###### Proof.

Fix $k>0$ and $\psi\in\Omega_{c}^{n,n}(M)$. We note that $|F_{k}|^{2}\geq
1>0$. Given $f\in\mathcal{O}^{\infty}(M)$ we have $\iota^{*}(\partial
f)=\iota^{*}(df)=d(f\circ\iota)$ with $f\circ\iota\in H_{b}^{0}(X)$, $X=bM$.
Then it follows from Lemma 7.2 and Lemma 6.4 that

$\int_{bM}\iota^{*}\left(\frac{\partial
f}{f}\wedge\psi\right)=\int_{bM}\iota^{*}\left(\frac{\partial
f}{f}\right)\wedge\iota^{*}\psi=\int_{bM}\frac{d(f\circ\iota)}{f\circ\iota}\wedge\iota^{*}\psi$

holds for $\mu_{k}$-almost every $f\in A_{k}$. Since
$\iota^{*}\psi\in\Omega^{2n}(X)$ and $|F_{k}|^{2}>0$ we conclude from Theorem
5.5 that $I_{k}(\psi)$ exists with

$I_{k}(\psi)=\int_{bM}2\pi i\beta_{k}\wedge\iota^{*}\psi$

where

$2\pi i\beta_{k}=\frac{\langle
d(F_{k}\circ\iota),F_{k}\circ\iota\rangle}{|F_{k}\circ\iota|^{2}}.$

Since $d(F_{k}\circ\iota)=\iota^{*}dF_{k}=\iota^{*}\partial F_{k}$ and
$\langle\partial F_{k},F_{k}\rangle=\partial|F_{k}|^{2}=\partial B_{k}$ we
obtain

$2\pi i\beta_{k}=\iota^{*}\left(\frac{\langle\partial
F_{k},F_{k}\rangle}{|F_{k}|^{2}}\right)=\iota^{*}\left(\frac{\partial
B_{k}}{1+B_{k}}\right)=\iota^{*}\left(\partial\log(1+B_{k})\right).$

The claim follows. ∎

In order to handle the log-terms in (7.1) we need the following.

######  Lemma 7.5.

With the notations and assumptions above we have that for each $k>0$ and any
compact set $K\subset M$ there exists a constant $C_{k,K}>0$ such that

(7.1.3) $\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\left|\log(|\langle
a,\overline{F_{k}(z)}\rangle|)\right|d\mu^{G}(a)\leq C_{k,K}$

holds for all $z\in K$. Furthermore, we have that

$\int_{\mathbb{C}^{N_{k}+1}}\log|\langle
a,\overline{F_{k}(z)}\rangle|d\mu^{G}(a)-\frac{1}{2}\log(1+B_{k}(z))=\int_{\mathbb{C}}\log|a_{0}|d\mu^{G}(a_{0})$

for all $k>0$, $z\in M$.

###### Proof.

We have that $c:=\int_{\mathbb{C}^{N_{k}+1}}|\log|a_{0}||d\mu^{G}(a)<\infty$
is independent of $k>0$ and $z\in\overline{D}$. Since $F_{k}(z)/|F_{k}(z)|$
has unit length we obtain from the unitary invariance of $d\mu^{G}$ that

$\displaystyle c$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\left|\log\left(\left\langle
a,\frac{\overline{F_{k}(z)}}{|F_{k}(z)|}\right\rangle\right)\right|d\mu^{G}(a)$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\left|\log(\langle
a,\overline{F_{k}(z)}\rangle)\right|d\mu^{G}(a)-|\log(|F_{k}(z)|)|.$

Since $|F_{k}(z)|^{2}=1+B_{k}(z)\geq 1$ we obtain (7.1.3). Similarly, we find

$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\log(\langle
a,\overline{F_{k}(z)}\rangle)\ d\mu^{G}(a)-\log(|F_{k}(z)|)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\log\left(\left\langle
a,\frac{\overline{F_{k}(z)}}{|F_{k}(z)|}\right\rangle\right)d\mu^{G}(a)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{C}}\log|a_{0}|d\mu^{G}(a_{0}).$

With $2\log(|F_{k}(z)|)=\log(1+B_{k}(z))$ the claim follows. ∎

Now we can consider the last two terms in the right-hand side of (7.1). We
have the following.

######  Lemma 7.6.

With the notations and assumptions given $k>0$ and
$\psi\in\Omega_{c}^{n,n}(M)$ we have that

$II_{k}(\psi):=\int_{A_{k}}\left(-\int_{bM}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{M}\log(|f|)\wedge\partial\overline{\partial}\psi\right)d\mu_{k}(f)$

exists with

$II_{k}(\psi)=-\frac{1}{2}\int_{M}(\partial\log(1+B_{k}))\wedge\overline{\partial}\psi.$

###### Proof.

Fix $k>0$ and $\psi\in\Omega_{c}^{n,n}(M)$. From (7.1.3) it follows by the
Fubini-Tonelli theorem that $II_{k}(\psi)$ exists. Put
$c:=\int_{\mathbb{C}}\log|a_{0}|d\mu^{G}(a_{0})=\int_{\mathbb{C}^{N_{k}+1}}\log|a_{0}|d\mu^{G}(a)$.
From Lemma 7.5, the Fubini theorem and the definition of $\mu_{k}$ we obtain

$\displaystyle\int_{M}\left(\frac{1}{2}\log(1+B_{k})+c\right)\partial\overline{\partial}\psi$
$\displaystyle=$ $\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\int_{M}\log|\langle
a,F_{k}\rangle|\partial\overline{\partial}\psi d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\int_{A_{k}}\int_{M}\log(|f|)\wedge\partial\overline{\partial}\psi
d\mu_{k}(f).$

With the same arguments we find

$\displaystyle\int_{bM}\iota^{*}\left(\left(\frac{1}{2}\log(1+B_{k})+c\right)\overline{\partial}\psi\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\int_{bM}\iota^{*}\left(\log|\langle
a,F_{k}\rangle|\overline{\partial}\psi\right)d\mu^{G}(a)$ $\displaystyle=$
$\displaystyle\int_{A_{k}}\int_{bM}\iota^{*}\left(\log(|f|)\overline{\partial}\psi\right)d\mu_{k}(f).$

Using Stokes theorem we obtain
$\int_{bM}\iota^{*}(\overline{\partial}\psi)=\int_{M}\partial\overline{\partial}\psi$
and

$\displaystyle\int_{bM}\iota^{*}\left(\log(1+B_{k})\overline{\partial}\psi\right)$
$\displaystyle=$
$\displaystyle\int_{M}\left(d\log(1+B_{k})\right)\wedge\overline{\partial}\psi+\int_{M}\log(1+B_{k})\wedge
d\overline{\partial}\psi$ $\displaystyle=$
$\displaystyle\int_{M}\left(\partial\log(1+B_{k})\right)\wedge\overline{\partial}\psi+\int_{M}\log(1+B_{k})\wedge\partial\overline{\partial}\psi.$

We conclude

$\displaystyle 2II_{k}(\psi)$ $\displaystyle=$
$\displaystyle-\int_{bM}\iota^{*}\left(\log(1+B_{k})\overline{\partial}\psi\right)+\int_{M}\log(1+B_{k})\wedge\partial\overline{\partial}\psi+2c\left(-\int_{bM}\iota^{*}(\overline{\partial}\psi)+\int_{M}\partial\overline{\partial}\psi\right)$
$\displaystyle=$
$\displaystyle-\int_{M}\left(\partial\log(1+B_{k})\right)\wedge\overline{\partial}\psi.$

∎

Using (7.1) we obtain Theorem 7.3 as a consequence of Lemma 7.4 and Lemma 7.6.

###### Proof of Theorem 7.3.

Fix $k>0$ and $\psi\in\Omega_{c}^{n,n}(M)$. From Lemma 7.4 and Lemma 7.6 it
follows that

$III_{k}(\psi):=\frac{i}{\pi}\int_{A_{k}}\left(-\int_{bM}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{bM}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{M}\log(|f|)\wedge\partial\overline{\partial}\psi\right)d\mu_{k}(f)$

exists with $III_{k}(\psi)<\infty$. Using Lemma 7.2 we find a set
$\tilde{A}_{k}\subset A_{k}$ with $\mu_{k}(\tilde{A_{k}})=1$ such that Theorem
6.5 can be applied for all $f\in\tilde{A}_{k}$. We find that

$\displaystyle III_{k}(\psi)$ $\displaystyle=$
$\displaystyle\frac{i}{\pi}\int_{\tilde{A_{k}}}\left(-\int_{bM}\iota^{*}(\frac{1}{2f}\partial
f\wedge\psi)-\int_{bM}\iota^{*}(\log(|f|)\wedge\overline{\partial}\psi)+\int_{M}\log(|f|)\wedge\partial\overline{\partial}\psi\right)d\mu_{k}(f)$
$\displaystyle=$
$\displaystyle\int_{\tilde{A_{k}}}\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)d\mu_{k}(f)$
$\displaystyle=$
$\displaystyle\int_{A_{k}}\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)d\mu_{k}(f)=\mathbb{E}_{k}(\mathcal{Z}_{f})(\psi).$

Hence $\mathbb{E}_{k}(\mathcal{Z}_{f})(\psi)<\infty$ exists. From Stokes
theorem we find

$\int_{bM}\iota^{*}\left(\partial\log(1+B_{k})\wedge\psi\right)=-\int_{M}\left(\partial\overline{\partial}\log(1+B_{k})\right)\wedge\psi-\int_{M}\left(\partial\log(1+B_{k})\right)\wedge\overline{\partial}\psi.$

Since $III_{k}(\psi)=\pi^{-1}i(-I_{k}(\psi)/2+II_{k}(\psi))$ we conclude from
Lemma 7.4 and Lemma 7.6 that

$\displaystyle III_{k}(\psi)$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\left(-\int_{bM}\iota^{*}((\partial\log(1+B_{k}))\wedge\psi)-\int_{M}(\partial\log(1+B_{k}))\wedge\overline{\partial}\psi\right)$
$\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int_{M}\left(\partial\overline{\partial}\log(1+B_{k})\right)\wedge\psi.$

∎

###  7.2. A Variance Estimate for $\mathcal{Z}_{f}$

The proof of Theorem 1.12 will mainly rely on a variance Estimate for the
distribution valued random variable $\mathcal{Z}_{f}$ which we are going to
explain now. We start with an elementary lemma.

######  Lemma 7.7.

For all $\alpha,\beta\in\mathbb{C}$ with $|\alpha|^{2}+|\beta|^{2}=1$ we have

$\int_{\mathbb{C}^{2}}|\log(|a_{0}|)\log(|\alpha a_{0}+\beta
a_{1}|)|d\mu^{G}(a)\leq\int_{\mathbb{C}}|\log(|a|)|^{2}d\mu^{G}(a).$

###### Proof.

Put $c:=\int_{\mathbb{C}}|\log(|a|)|^{2}d\mu^{G}(a)$ by the unitary invariance
of $\mu^{G}$ we obtain

$\int_{\mathbb{C}^{2}}|\log(|\alpha a_{0}+\beta a_{1}|)|^{2}d\mu^{G}(a)=c.$

By the Cauchy-Schwarz inequality we conclude

$\displaystyle\int_{\mathbb{C}^{2}}|\log(|a_{0}|)\log(|\alpha a_{0}+\beta
a_{1}|)|d\mu^{G}(a)$ $\displaystyle\leq$
$\displaystyle\sqrt{\int_{\mathbb{C}^{2}}|\log(|a_{0}|)|^{2}d\mu^{G}(a)}\sqrt{\int_{\mathbb{C}^{2}}|\log(|\alpha
a_{0}+\beta a_{1}|)|^{2}d\mu^{G}(a)}$ $\displaystyle=$ $\displaystyle c.$

∎

We have the following result for the variance of $\mathcal{Z}_{f}$.

######  Theorem 7.8.

Under the assumptions of Theorem 7.3, with the notations above put

$\mathbb{V}_{k}(\mathcal{Z}_{f}(\psi)):=\int_{A_{k}}\left|\mathcal{Z}_{f}(\psi)-\mathbb{E}_{k}(\mathcal{Z}_{f})(\psi)\right|^{2}d\mu_{k}(f),\,\,\,\psi\in\Omega_{c}^{n,n}(M).$

Given $0<\varepsilon<1$ there exist $k_{0}>0$ such that for any compact set
$K\subset M$ there is a constant $C_{K}>0$ with

$\mathbb{V}_{k}(\mathcal{Z}_{f}(\psi))\leq
C_{K}k^{\varepsilon}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$

for all $k\geq k_{0}$ and all $\psi\in\Omega_{c}^{n,n}(M)$,
$\operatorname{supp}\psi\subset K$.

###### Proof.

Let $0<\varepsilon<1$ be arbitrary. For $k>0$ and $\psi\in\Omega_{c}^{n,n}(M)$
put

$R_{k}(\psi):=\int_{A_{k}}\left|\frac{i}{\pi}\int_{M}\partial\overline{\partial}\log|f|\wedge\psi-\frac{i}{2\pi}\int_{M}\left(\partial\overline{\partial}\log(1+B_{k})\right)\wedge\psi\right|^{2}d\mu_{k}(f)$

and

$\displaystyle R^{(1)}_{k}(\psi)$ $\displaystyle=$
$\displaystyle\int_{A_{k}}\left|\int_{bM}\iota^{*}\left(\frac{\partial
f}{f}-\partial\log(1+B_{k})\right)\wedge\iota^{*}\psi\right|^{2}d\mu_{k}(f),$
$\displaystyle R^{(2)}_{k}(\psi)$ $\displaystyle=$
$\displaystyle\int_{A_{k}}\left|\int_{bM}\iota^{*}\left(\log\left(\frac{|f|}{\sqrt{1+B_{k}}}\right)\overline{\partial}\psi\right)\right|^{2}d\mu_{k}(f),$
$\displaystyle R^{(3)}_{k}(\psi)$ $\displaystyle=$
$\displaystyle\int_{A_{k}}\left|\int_{M}\log\left(\frac{|f|}{\sqrt{1+B_{k}}}\right)\partial\overline{\partial}\psi\right|^{2}d\mu_{k}(f).$

We have
$\frac{i}{\pi}\int_{M}\partial\overline{\partial}\log|f|\wedge\psi=\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)=:\mathcal{Z}_{f}(\psi)$
for all $f$ in a subset $\tilde{A}_{k}$ of $A_{k}$ with
$\mu_{k}(\tilde{A}_{k})=1$ by Lemma 7.2. Then, using Theorem 7.3, we find

(7.2.1) $\displaystyle R_{k}(\psi)=\mathbb{V}_{k}(Z_{f}(\psi)).$

Furthermore, from the definition of
$\int_{M}\partial\overline{\partial}\log|f|\wedge\psi$ and Stokes theorem we
find

(7.2.2) $\displaystyle
R_{k}(\psi)\leq\frac{1}{2\pi^{2}}R^{(1)}_{k}(\psi)+\frac{4}{\pi^{2}}\left(R^{(2)}_{k}(\psi)+R^{(3)}_{k}(\psi)\right).$

Hence, it is enough to show that there is $k_{0}>0$ and, given a compact set
$K\subset M$, a constant $C_{K}>0$ such that $R^{(j)}_{k}(\psi)\leq
C_{K}k^{\varepsilon}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$
for all $k\geq k_{0}$, $\psi\in\Omega^{n,n}_{c}(M)$ with
$\operatorname{supp}\psi\subset K$ and $j=1,2,3$. Then the claim in Theorem
7.8 follows immediately.

We will estimate the term $R^{(1)}_{k}(\psi)$ using Theorem 5.13. With respect
to Theorem 5.5 we have that

$\beta_{k}(x)=\frac{d_{x}\eta_{k}(T_{P})(x,y)|_{x=y}}{2\pi
i(1+\eta_{k}(T_{P})(x,x))},\,\,\,x\in bM$

defines a smooth one form on $bM$ for all $k>0$ where $\eta=|\chi|^{2}$.
Recall that

$\eta_{k}(T_{P})(z,w)=\sum_{j=1}^{\infty}|\chi_{k}(\lambda_{j})|^{2}f_{j}(z)\overline{f_{j}(w)}=\langle
F_{k}(z),F_{k}(w)\rangle-1,\,\,\,z,w\in bM.$

The components of $F_{k}$ are functions in $\mathcal{O}^{\infty}(M)$. Hence we
obtain $\partial F_{k}=dF_{k}$ and in addition we find with
$|F_{k}|^{2}=1+B_{k}$ that

$2\pi i\beta_{k}=\frac{\iota^{*}\partial
B_{k}}{1+B_{k}\circ\iota}=\iota^{*}\partial\log\left(1+B_{k}\right).$

Using Theorem 5.5 we obtain in conclusion

$R^{(1)}_{k}(\psi)=\int_{A_{k}}\left|\int_{bM}\left(\iota^{*}\left(\frac{df}{f}\right)-2\pi
i\beta_{k}\right)\wedge\iota^{*}\psi\right|^{2}d\mu_{k}(f)=4\pi^{2}\mathbb{V}_{k}(\mathcal{C}_{f}(\iota^{*}\psi))$

for all $k>0$ and all $\psi\in\Omega^{n,n}_{c}(M)$, where
$\mathbb{V}_{k}(\mathcal{C}_{f}(\iota^{*}\psi))$ is as in Theorem 5.13 with
$X=bM$ and $\kappa(k)\equiv 1$. Then, by Theorem 5.13, there exist
$k_{0},C_{1}>0$ such that

(7.2.3) $\displaystyle R_{k}^{(1)}(\psi)\leq
C_{1}k^{\varepsilon}\|\psi\|^{2}_{\mathscr{C}^{0}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$

holds for all $k\geq k_{0}$ and $\psi\in\Omega^{n,n}_{c}(M)$. It remains to
estimate $R_{k}^{(2)}(\psi)$ and $R_{k}^{(3)}(\psi)$.

Fix a compact set $K\subset M$. Recall that $|F_{k}|\geq 1>0$ for all $k>0$.
For $z,w\in M$ we can choose a unitary transformation $U^{z,w}_{k}$ of
$\mathbb{C}^{N_{k}+1}$ such
$U^{z,w}_{k}\frac{\overline{F_{k}(z)}}{|F_{k}(z)|}=(1,0,\ldots,0)$ and
$U^{z,w}_{k}\frac{\overline{F_{k}(w)}}{|F_{k}(w)|}=(\alpha,\beta,0\ldots,0)$
with $\alpha,\beta\in\mathbb{C}$, $|\alpha|^{2}+|\beta|^{2}=1$. Denote by
$p_{1},p_{2}\colon M\times M\to M$ the projections on the first and second
component. We write

$R^{(3)}_{k}=\int_{A_{k}}\int_{M\times M}\log\left(\frac{|f\circ
p_{1}|}{\sqrt{1+B_{k}\circ p_{1}}}\right)\log\left(\frac{|f\circ
p_{2}|}{\sqrt{1+B_{k}\circ
p_{2}}}\right)p_{1}^{*}(\partial\overline{\partial}\psi)\wedge\overline{p_{2}^{*}(\partial\overline{\partial}\psi)}d\mu_{k}(f).$

Using $U^{z,w}_{k}$ we find by the invariance of $\mu^{G}$ under unitary
transformation and Lemma 7.7 that

$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}\left|\log\left|\left\langle
a,\frac{\overline{F_{k}(z)}}{|F_{k}(z)|}\right\rangle\right|\log\left|\left\langle
a,\frac{\overline{F_{k}(w)}}{|F_{k}(w)|}\right\rangle\right|\right|d\mu^{G}(a)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{C}^{N_{k}+1}}|\log(|a_{0}|)\log(|\alpha
a_{0}+\beta a_{1}|)|d\mu^{G}(a)$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{C}}|\log(|a|)|^{2}d\mu^{G}(a).$

With $|F_{k}|^{2}=1+B_{k}$, $f=\langle a,\overline{F_{k}}\rangle$ for
$a\in\mathbb{C}^{N_{k}+1}$, and $p_{1}(z,w)=z$, $p_{2}(z,w)=w$ it follows from
the Fubini-Tonelli theorem that there is a constant $C_{3,K}>0$ such that
$R_{k}^{(3)}(\psi)\leq
C_{3}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$ holds
for all $k>0$ and $\psi\in\Omega^{n,n}_{c}(M)$ with
$\operatorname{supp}\psi\subset K$. With the same arguments we obtain
$R_{k}^{(2)}(\psi)\leq
C_{2,K}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$ for
some constant $C_{2,K}>0$ independent of $k$ and $\psi$,
$\operatorname{supp}\psi\subset K$. Combining those estimates for
$R_{k}^{(2)}(\psi)$ and $R_{k}^{(3)}(\psi)$ with the estimate for
$R_{k}^{(1)}(\psi)$ in (7.2.3) the claim of the theorem follows from (7.2.1)
and (7.2.2). ∎

###  7.3. Proof of Theorem 1.10

Theorem 1.10 follows from the next result by putting $c=1$.

######  Theorem 7.9.

Under the assumptions of Theorem 7.3 and with the notation above, assuming
that $M$ is compact, choose
$\eta\in\mathscr{C}^{\infty}(\mathbb{R}_{+},[0,\infty))$, $\eta\not\equiv 0$,
and $c>0$. With $B_{k}^{\eta}$ as in (7.0.2), for all
$\psi\in\Omega^{n,n}(M)$, we have

(7.3.1)
$\lim_{k\to\infty}\int_{M}k^{-1}i\partial\overline{\partial}\log(c+B^{\eta}_{k})\wedge\psi=\operatorname{mv}(\eta)\int_{bM}\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi,$

where

(7.3.2)
$\displaystyle\operatorname{mv}(\eta)=\frac{\int_{0}^{+\infty}t^{n+1}\eta(t)dt}{\int_{0}^{+\infty}t^{n}\eta(t)dt}.$

Here $\iota\colon bM\to M$ denotes the inclusion map. More precisely, there
exists a constant $C>0$ such that

$\left|\int_{M}k^{-1}i\partial\overline{\partial}\log(c+B^{\eta}_{k})\wedge\psi-\operatorname{mv}(\eta)\int_{bM}\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi\right|\leq
Ck^{-1}(\log(k)+1)\|\psi\|_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$

for all $k\geq 1$ and $\psi\in\Omega^{n,n}(M)$.

For the proof of Theorem 7.9 we need the following lemma.

######  Lemma 7.10.

In the situation of Theorem 7.9 there exists a constant $C>0$ such that for
all $k>0$ and $z\in M$ we have $|\log(c+B_{k}^{\eta}(z))|\leq C(\log(k)+1)$.

###### Proof.

Given $z\in bM$ and $k>0$ we have $B_{k}^{\eta}(z)=\eta_{k}(T_{P})(z,z)$.
Furthermore, using Theorem 2.10 we obtain a constant $R>0$ such that
$0\leq\eta_{k}(T_{P})(z,z)\leq Rk^{n+1}$ holds for all $z\in bM$ and $k>0$.
Since $B^{\eta}_{k}\in\mathscr{C}^{\infty}(M)$ is plurisubharmonic on
$\text{int}(M)$ it follows from the maximum principle for plurisubharmonic
functions that $0\leq B^{\chi}_{k}(z)\leq Rk^{n+1}$ for all $z\in M$ and and
$k>0$. The statement follows from $\log(c+B^{\chi}_{k})\geq\log(c)$ with
$c>0$. ∎

###### Proof of Theorem 7.9.

Recall the definition of $L$ in (7.0.1). Given a smooth CR function $f$ on
$X=bM$ we have $L(f)\in\mathcal{O}^{\infty}(M)$ and hence
$\partial(L(f))=d(L(f))$ on $M$. It follows that

$(\iota^{*}\overline{\partial}B^{\eta}_{k})(z)=\sum_{j=1}^{\infty}\eta_{k}(\lambda_{j})f_{j}(z)d\overline{f_{j}}(z)=d_{w}\eta_{k}(T_{P})(z,w)|_{z=w}$

for all $z\in bM$. Using Lemma 2.11 and Theorem 2.10 we conclude

(7.3.3)
$k^{-1}\iota^{*}\overline{\partial}\log(c+B^{\eta}_{k})=k^{-1}\iota^{*}\left(\frac{\overline{\partial}B^{\eta}_{k}}{c+B^{\eta}_{k}}\right)=-i\operatorname{mv}(\eta)\frac{\xi}{\sigma_{P}(\xi)}+O(k^{-1})$

on $bM$ in $\mathscr{C}^{\infty}$-norm. It follows that there exists a
constant $C_{1}>0$ such that

(7.3.4)
$\left|\int_{bM}k^{-1}\iota^{*}\left(\frac{\overline{\partial}B^{\eta}_{k}}{c+B^{\eta}_{k}}\wedge\psi\right)+\int_{bM}i\operatorname{mv}(\eta)\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi\right|\leq
C_{1}k^{-1}\|\psi\|_{\mathscr{C}^{0}(\overline{D},\Lambda^{n,n}T^{*}\mathbb{C}^{n+1})}$

for all $\psi\in\Omega^{n,n}(M)$ and $k>0$. Let $\psi\in\Omega^{n,n}(M)$ be
arbitrary. By Stokes theorem we find that

$\displaystyle\int_{M}\overline{\partial}\log(c+B^{\eta}_{k})\wedge d\psi=$
$\displaystyle\int_{M}\overline{\partial}\log(c+B^{\eta}_{k})\wedge\partial\psi=\int_{M}d\log(c+B^{\eta}_{k})\wedge\partial\psi$
$\displaystyle=$
$\displaystyle\int_{bM}\log(c+B^{\eta}_{k}\circ\iota)\wedge\iota^{*}(\partial\psi)+\int_{M}\log(c+B^{\eta}_{k})\wedge\partial\overline{\partial}\psi.$

Then it follows from Lemma 7.10 that there exists a constant $C_{2}>0$ such
that

(7.3.5) $\left|\int_{M}\overline{\partial}\log(c+B^{\eta}_{k})\wedge
d\psi\right|\leq
C_{2}(\log(k)+1)\|\psi\|_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$

holds for all $\psi\in\Omega^{n,n}(M)$ and $k>0$. Using Stokes theorem again
we find

(7.3.6)
$\int_{M}\partial\overline{\partial}\log(c+B^{\eta}_{k})\wedge\psi=\int_{bM}\iota^{*}(\overline{\partial}\log(c+B^{\eta}_{k})\wedge\iota^{*}\psi+\int_{M}\overline{\partial}\log(c+B^{\eta}_{k})\wedge
d\psi.$

Finally, the claim follows from (7.3.3), (7.3.5) and (7.3.6). ∎

###  7.4. Proof of Theorem 1.12

Consider $k\in\mathbb{N}$ and put $A_{\infty}=\Pi_{k=1}^{\infty}A_{k}$,
$d\mu_{\infty}=\Pi_{k=1}^{\infty}d\mu_{k}$. Theorem 1.12 follows from the next
result.

######  Theorem 7.11.

Under the assumptions of Theorem 7.3 and with the notation above, assuming
that $M$ is compact, we have for $\mu_{\infty}$-almost every
$u=(u_{k})_{k\in\mathbb{N}}\in A_{\infty}$ that
$(\left[\operatorname{Div}\\!\left(u_{k}\right)\right],\psi)$ is well-defined
for all $\psi\in\Omega^{n,n}(M)$ and $k\in\mathbb{N}$. Furthermore, given
$\psi\in\Omega^{n,n}(M)$, we have

$\lim_{k\to\infty}\left(k^{-1}\left[\operatorname{Div}\\!\left(u_{k}\right)\right],\psi\right)=\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{bD}\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi$

for $\mu_{\infty}$-almost every $u=(u_{k})_{k\in\mathbb{N}}\in A_{\infty}$. In
addition, there exist $C,k_{0}>0$ such that for any $k\geq k_{0}$ one has

$\displaystyle\mu_{k}\left(\left\\{f\in
A_{k}\colon\left|\left(k^{-1}\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{bM}\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi\right|\geq\frac{\|\psi\|_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}}{\sqrt{k}}\right\\}\right)\leq\frac{C}{\sqrt{k}}$

for all $\psi\in\Omega^{n,n}(M)$.

###### Proof.

From Theorem 6.5 and Lemma 7.2 it follows that
$(\left[\operatorname{Div}\\!\left(u_{k}\right)\right],\psi)$,
$k\in\mathbb{N}$, is well-defined for all $\psi\in\Omega^{n,n}(M)$ and
$\mu_{\infty}$-almost every $u=(u_{k})_{k\in\mathbb{N}}\in A_{\infty}$ which
proves the first part of the claim. Given $\psi\in\Omega^{n,n}(M)$ and
$k\in\mathbb{N}$ put

$R_{k}(\psi):=\int_{A_{k}}\left|\left(k^{-1}\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{bM}\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi\right|^{2}d\mu_{k}(f).$

We will show that there exist $k_{0}\in\mathbb{N}$ and a constant $C>0$ such
that $R_{k}(\psi)$ exists for all $k\geq k_{0}$ with $R_{k}(\psi)\leq
Ck^{-\frac{3}{2}}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$
for all $k\geq k_{0}$ and $\psi\in\Omega^{n,n}(M)$. From the Markov
inequality, the last part of the claim follows. Furthermore, considering
$k\in\mathbb{N}$, it follows that $\sum_{k=k_{0}}^{\infty}R_{k}(\psi)\leq
C_{0}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$ for some
constant $C_{0}$ (independent of $\psi$) which leads to the second part of the
claim.

Given $\psi\in\Omega^{n,n}(M)$ and $k\in\mathbb{N}$ put

$R^{\prime}_{k}(\psi):=\left|\frac{i}{2\pi
k}\int_{M}\left(\partial\overline{\partial}\log(1+B_{k})\right)\wedge\psi-\frac{\operatorname{mv}(|\chi|^{2})}{2\pi}\int_{bM}\frac{\xi}{\sigma_{P}(\xi)}\wedge\iota^{*}\psi\right|^{2}.$

From Theorem 7.9 it follows that there exists a constant $C_{1}>0$ such that

$R^{\prime}_{k}(\psi)\leq
C_{1}k^{-\frac{3}{2}}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$

holds for all $k\in\mathbb{N}$ and $\psi\in\Omega^{n,n}(M)$. Since
$\mathcal{Z}_{f}(\psi)=\left(\left[\operatorname{Div}\\!\left(f\right)\right],\psi\right)$
we find with Theorem 7.3 and $\mathbb{V}_{k}$ as in Theorem 7.8 that

$R_{k}(\psi)\leq
2k^{-2}\mathbb{V}_{k}(\mathcal{Z}_{f}(\psi))+2R^{\prime}_{k}(\psi)\int_{A_{k}}d\mu_{k}.$

Then it follows from Theorem 7.8 with $\varepsilon=\frac{1}{2}$ that there are
$k_{0}\in\mathbb{N}$ and a constant $C>0$ such that $R_{k}(\psi)\leq
Ck^{-\frac{3}{2}}\|\psi\|^{2}_{\mathscr{C}^{2}(M,\Lambda^{n,n}\mathbb{C}T^{*}M)}$
for all $k\geq k_{0}$ and $\psi\in\Omega^{n,n}(M)$. ∎

## References