text:
\begin{document}

\title{On the deformed Bott-Chern cohomology}
\author{Wei Xia}
\address{Wei Xia, Mathematical Science Research Center, Chongqing University of Technology, Chongqing, P.R.China, 400054.} \email{xiaweiwei3@126.com}

\thanks{This work was supported by the National Natural Science Foundation of China No. 11901590.}
\date{\today}

\begin{abstract}
Given a compact complex manifold $X$ and a integrable Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$, we introduce a double complex structure on $A^{\bullet,\bullet}(X)$ naturally determined by $\phi$ and study its Bott-Chern cohomology. In particular, we establish a deformation theory for Bott-Chern cohomology and use it to compute the deformed Bott-Chern cohomology for the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold. The $\partial\bar{\partial}_{\phi}$-lemma is studied and we show a compact complex manifold satisfying $\partial\bar{\partial}_{\phi}$-lemma is formal.

\vskip10pt
\noindent
{\bf Key words:} deformation of complex structures, Bott-Chern cohomology, $\partial\bar{\partial}_{\phi}$-lemma.
\vskip10pt
\noindent
{\bf MSC~Classification (2010):} 32G05, 32C35, 32G99
\end{abstract}
\maketitle

\section{Introduction}

The Bott-Chern cohomology are important invariants of complex manifolds~\cite{BC65}. It has been studied by many authors in recent years~\cite{Ang13,AT15a,AT15b,AT17,ADT16,AK17a}. For example, Schweitzer studied the Hodge theory for Bott-Chern cohomology and gave a hypercohomology interpretation to it~\cite{Sch07}. Angella-Tomassini proved Fr\"ohlicher type inequalities for Bott-Chern cohomology and gave a beautiful characterization of the $\partial\bar{\partial}$-lemma~\cite{AT13}. Recently, S. Yang and X. Yang proved a blow-up formula for the Bott-Chern cohomology and they showed that satisfying the $\partial\bar{\partial}$-Lemma is a bimeromorphic invariant for threefolds ~\cite{YY17}, see \cite{RYY19,ASTT17,Ste18a,Ste18b,Meng19} for related works.

Let $X$ be a complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. In this paper, we will study the Bott-Chern cohomology of the double complex $(A^{\bullet,\bullet}(X), \partial, \bar{\partial}_{\phi})$:
\begin{equation}
H^{p,q}_{BC\phi}(X) := \frac{\ker d_{\phi} \cap A^{p,q}(X)}{\Image \partial\bar{\partial}_{\phi} \cap A^{p,q}(X)},
\end{equation}
which we called the \emph{deformed Bott-Chern cohomology}, where $d_{\phi}=\partial+\bar{\partial}_{\phi}$ and $\bar{\partial}_{\phi} = \bar{\partial} - \mathcal{L}_{\phi}^{1,0}$. In Section \ref{hypercohomology interpretation}, we will show that there are similar hypercohomology interpretations to the deformed Bott-Chern cohomology as to the usual Bott-Chern cohomology.

Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a deformation of a compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. Given a Bott-Chern class $[y]\in H_{BC}^{p,q}(X)$, as motivated by our previous work on deformation of Dolbeault cohomology classes~\cite{Xia19dDol}, we try to construct a family of $(p,q)$-forms $\sigma (t)$ (on an analytic subset $T$ of $B$) such that
\begin{itemize}
  \item[1.] $\sigma (t)$ is holomorphic in $t$;
  \item[2.] $\partial\sigma (t) =\bar{\partial}_{\phi(t)}\sigma (t)= 0,~\forall t\in T$;
  \item[3.] $[\sigma (0)] = [y]\in H_{BC}^{p,q}(X)$.
\end{itemize}
We will develop a deformation theory for Bott-Chern cohomology in this respect, see Section~\ref{Deformations of BC c}. Among other things, we show the following
\begin{theorem}[=Theorem \ref{th-analytic subset}]\label{th-analytic subset 0}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a deformation of a compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. Then the set $\{t\in B\mid \dim H_{BC\phi(t)}^{p,q}(X)\geq k\}$ is an analytic subset of $B$ for any nonnegative integer $k$.
\end{theorem}

In \cite[Thm.\,1\,and\,2]{AT15b}, Angella-Tomassini generalized their previous result~\cite{AT13} to arbitrary double complex~\cite{AT15a}. This result, when applied to our situation, will give rise to the following
\begin{theorem}[=Theorem \ref{thm-AT-Frolicher-inequality-deformed}]\label{thm-AT-Frolicher-inequality-deformed 0}
Let $X$ be a compact complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. Then for every $(p,q)\in \mathbb{N}\times \mathbb{N}$, we have
\begin{equation}\label{eq-AT-Frolicher-inequality-deformed 0}
\dim H_{BC\phi}^{p,q}(X)+ \dim H_{A\phi}^{p,q}(X) \geq \dim H_{\bar{\partial}_t}^{p,q}(X_t)+ \dim H_{\partial}^{p,q}(X).
\end{equation}
In particular, for every $k\in \mathbb{N}$, we have
\begin{equation}\label{eq-ddbar-phi-lemma-kehua 0}
\sum_{p+q=k}\dim H_{BC\phi}^{p,q}(X)+ \sum_{p+q=k}\dim H_{A\phi}^{p,q}(X) \geq 2\dim H_{dR}^{k}(X),
\end{equation}
and equality holds if and only if $X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma.
\end{theorem}

Note that when $X_t$ is a trivial deformation, i.e. $\phi=0$, Theorem \ref{eq-AT-Frolicher-inequality-deformed 0} is reduced to the result in ~\cite{AT13}. Combine Theorem \ref{eq-AT-Frolicher-inequality-deformed 0} with Theorem \ref{th-analytic subset 0}, we get
\begin{corollary}\label{coro-deformation-closedness-ddbar-lemma 0}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a small deformation of the compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. Then
the set
\[
T:=\{t\in B\mid X~\text{satisfies the}~\partial\bar{\partial}_{\phi(t)}\text{-lemma} \}
\]
is an analytic open subset (i.e. complement of analytic subset) of $B$. In particular, if $B\subset \mathbb{C}$ is a small open disc with $0\in B$ and $T$ is not empty, then $T=B$ or $T=B\setminus\{0\}$.
\end{corollary}
It is known that satisfying the $\partial\bar{\partial}$-lemma is a deformation open property and not a deformation closed property in the sense of Popovici~\cite{Pop14}, see ~\cite{Wu06,AT13,AK17b} and the references therein. But it is still not clear whether satisfying the $\partial\bar{\partial}$-lemma is an analytically open property, i.e. does the corresponding statement in Corollary \ref{coro-deformation-closedness-ddbar-lemma 0} holds for the $\partial\bar{\partial}$-lemma? On the other hand, we see from Corollary \ref{coro-deformation-closedness-ddbar-lemma 0} that if $X$ satisfies the $\partial\bar{\partial}$-lemma then $X$ also satisfies the $\partial\bar{\partial}_{\phi(t)}$-lemma for small $t$. But conversely, if $X$ satisfies the $\partial\bar{\partial}_{\phi(t)}$-lemma for all small $t\neq 0$ it is possible that $X$ does not satisfy the $\partial\bar{\partial}$-lemma\footnote{Though it is still not known whether such examples exist, we think they should be large in number.}. Hence the following Theorem generalize the corresponding well-known result of Deligne-Griffiths-Morgan-Sullivan~\cite{DGMS75}:
\begin{theorem}\label{thm-formal 0}
Let $X$ be a compact complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. If $X$ satisfies the $\partial\bar{\partial}_\phi$-lemma, then $X$ is formal.
\end{theorem}

The dimensions of the deformed Bott-Chern cohomology is computed for the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold, see Section \ref{The deformed Bott-Chern cohomology of the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold}. Comparing this with the computations of Angella-Kasuya~\cite{AK17b}, we see that there exists compact complex manifold $X$ and its small deformation $X_t$ such that $X_t$ satisfy the $\partial\bar{\partial}$-lemma but $X$ does not satisfy the $\partial\bar{\partial}_{\phi(t)}$-lemma.

There are many questions regarding the $\partial\bar{\partial}_{\phi}$-lemma may be asked:
\begin{question}Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a small deformation of the compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$.
\begin{itemize}
  \item[1.] Is it true that
\begin{equation}\label{eq-dim-comparison}
\dim H_{BC\phi(t)}^{p,q}(X)\geq\dim H_{BC}^{p,q}(X_t)
 \end{equation}
for any $t\in B$ and $(p,q)\in \mathbb{N}\times \mathbb{N}$? If this holds, then $X$ satisfies the $\partial\bar{\partial}_{\phi(t)}$-lemma will imply $X_t$ satisfy the $\partial\bar{\partial}$-lemma. Note that \eqref{eq-dim-comparison} is true for the examples considered in Section \ref{The deformed Bott-Chern cohomology of the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold};
  \item[2.] If $B\subset \mathbb{C}$ is a small open disc with $0\in B$, can we find an example such that $T=B\setminus\{0\}$ (in the notation of Corollary \ref{coro-deformation-closedness-ddbar-lemma 0})? According to Corollary \ref{coro-deformation-closedness-ddbar-lemma 0}, there should be many such examples. In this case, the Fr\"ohlicher spectral sequence on the central fiber $X$ must degenerates at $E_1$, see Remark~\ref{rk-to-AK-Frolicher-ineq};
  \item[3.] If $X_t$ is K\"ahler, is it true that $X$ must satisfy the $\partial\bar{\partial}_{\phi(t)}$-lemma?
\end{itemize}
\end{question}

\section{The deformed double complex $(A^{\bullet,\bullet}(X), \partial, \bar{\partial}_{\phi})$ and its Bott-Chern cohomology}
\label{The deformed double complex and its Bott-Chern cohomology}

Let $X$ be a complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. Recall the following useful facts~\cite{LRY15,Xia19deri}:
\[
e^{-i_{\phi}}de^{i_{\phi}}=d-\mathcal{L}_{\phi}^{1,0}-\mathcal{L}_{\phi}^{0,1} -i_{\frac{1}{2}[\phi,\phi]}~\text{and}~\mathcal{L}_{\phi}^{0,1}=-i_{\bar{\partial}\phi}~.
\]
Since $\phi$ satisfy the Maurer-Cartan equation $\bar{\partial}\phi-\frac{1}{2}[\phi,\phi]=0$, we have
\begin{equation}\label{}
d_{\phi}~: = e^{-i_{\phi}}de^{i_{\phi}} = \partial + \bar{\partial}_{\phi},~~ \text{with}~~ \bar{\partial}_{\phi} = \bar{\partial} - \mathcal{L}_{\phi}^{1,0},
\end{equation}
and
\begin{equation}\label{}
d_{\bar{\phi}}~: = e^{-i_{\bar{\phi}}}de^{i_{\bar{\phi}}} = \partial_{\phi} + \bar{\partial},~~ \text{with}~~ \partial_{\phi} = \partial - \mathcal{L}_{\bar{\phi}}^{0,1}.
\end{equation}
Since $[\partial,\bar{\partial}_{\phi}]=[\partial_{\phi},\bar{\partial}]=0$, the \emph{deformed Bott-Chern cohomology} can be defined as follows:
\begin{equation}\label{}
H^{p,q}_{BC\phi}(X) := \frac{\ker d_{\phi} \cap A^{p,q}(X)}{\Image \partial\bar{\partial}_{\phi} \cap A^{p,q}(X)},~~~~~~
H^{p,q}_{BC\bar{\phi}}(X) := \frac{\ker d_{\bar{\phi}} \cap A^{p,q}(X)}{\Image \partial_{\phi}\bar{\partial} \cap A^{p,q}(X)},~~\forall p,q\geq 0~,
\end{equation}
and $h^{p,q}_{BC\phi}:=\dim H^{p,q}_{BC\phi}(X), h^{p,q}_{BC\bar{\phi}}:=\dim H^{p,q}_{BC\bar{\phi}}(X)$. The conjugation gives a natural isomorphism between $H^{p,q}_{BC\phi}(X)$ and $H^{q,p}_{BC\bar{\phi}}(X)$, we thus have $h^{p,q}_{BC\phi}=h^{q,p}_{BC\bar{\phi}}$.

\section{Hypercohomology interpretations to the deformed Bott-Chern cohomology}
\label{hypercohomology interpretation}

It is clear that the Poincar\'e lemma holds for $d_{\phi}$ and $\bar{\partial}_{\phi}$ (for the latter, see~\cite[Thm.\,3.4]{Xia19dDol}). The sheaf of germs of $\bar{\partial}_{\phi}$-closed $p$-forms will be denoted by $\Omega^{p}_{\phi}$. The following Lemma is essentially proved in~\cite{Sch07}:
\begin{lemma}
\label{lem-local solvability}
Let $U\subset\mathbb{C}^n$ be an open ball.
\begin{itemize}
  \item[1.] Let $\theta\in A^k(U)$ with $k\geq 1$ such that $\theta^{p,q}=0$ except $p_1\leq p\leq p_2(p_1< p_2)$. If $\theta$ is $d_{\phi}$-closed, then $\theta= d_{\phi} \alpha$ for some $\alpha\in A^{k-1}(U)$ with $\alpha^{p,q}=0$ except $p_1\leq p\leq p_2-1$.

  \item[2.] Assume $\theta \in A^{p,q}(U)$ is $d_{\phi}$-closed.
  \item[$i)$] If $p\geq 1$ and $q\geq 1$, then $\theta\in \partial\bar{\partial}_{\phi} A^{p-1,q-1}(U)$.
  \item[$ii)$] If $p\geq 1$ and $q=0$, then $\theta\in \partial \Omega^{p-1}_{\phi}(U)$.
  \item[$\bar{ii})$] If $p=0$ and $q\geq 1$, then $\theta\in \bar{\partial}_{\phi} \bar{\Omega}^{q-1}(U)$.
  \item[$iii)$] If $p=q=0$, then $\theta$ is a constant.

  \item[3.] Assume $\theta \in A^{p,q}(U)$ is $\partial\bar{\partial}_{\phi}$-closed.
  \item[$i)$] If $p\geq 1$ and $q\geq 1$, then $\theta\in \bar{\partial}_{\phi} A^{p,q-1}(U)+\partial A^{p-1,q}(U)$.
  \item[$ii)$] If $p\geq 1$ and $q=0$, then $\theta\in \Omega_{\phi}^{p}(U) + \partial A^{p-1,0}(U)$.
  \item[$\bar{ii})$] If $p=0$ and $q\geq 1$, then $\theta\in \bar{\partial}_{\phi}A^{0,q-1}(U) + \bar{\Omega}^{q}(U)$.
  \item[$iii)$] If $p=q=0$, then $\theta\in \mathcal{O}_{\phi}(U) + \bar{\mathcal{O}}(U)$.

  \item[4.] Let $\theta\in A^k(U)$ with $k\geq 1$ and $p_1,~q_1,~p_2,~q_2$ be two positive integers with $p_1+q_1=p_2+q_2= k$. If $(d_{\phi}\theta)^{p,q}=0$ for $p+q=k+1,~p_1+1\leq p\leq p_2$ and $q_1\geq q\geq q_2+1$, then there exists $\gamma^{p_1,q_1}, \alpha^{p_1,q_1-1}, \alpha^{p_1+1,q_1-2},\cdots, \alpha^{p_2-1,q_2}, \gamma^{p_2,q_2}$, s.t. $\gamma^{p_1,q_1}$ is $\partial$-closed, $\gamma^{p_2,q_2}$ is $\bar{\partial}_{\phi}$-closed and
\begin{align*}
\theta^{p_1,q_1}&=\gamma^{p_1,q_1}+\bar{\partial}_{\phi}\alpha^{p_1,q_1-1},\\
\theta^{p_1+1,q_1-1}&=\partial\alpha^{p_1,q_1-1}+\bar{\partial}_{\phi}\alpha^{p_1+1,q_1-2},\\
 &\cdots,\\
\theta^{p_2-1,q_2+1}&=\partial\alpha^{p_2-2,q_2+1}+\bar{\partial}_{\phi}\alpha^{p_2-1,q_2},\\
\theta^{p_2,q_2}&=\partial\alpha^{p_2-1,q_2}+\gamma^{p_2,q_2},
\end{align*}
in particular, we have
\[
\theta^{p_1,q_1} + \theta^{p_1+1,q_1-1}+ \cdots + \theta^{p_2,q_2} = \gamma^{p_1,q_1} + d_{\phi}\alpha+\gamma^{p_2,q_2},
\]
where $\alpha=\alpha^{p_1,q_1-1}+\alpha^{p_1+1,q_1-2}+\cdots +\alpha^{p_2-1,q_2}$.
\end{itemize}
\end{lemma}

\begin{proof}1.First, by the $d_{\phi}$-Poincar\'e lemma, we can write $\theta= d_{\phi}\beta$ for some $\beta\in A^{k-1}(U)$. If $p_1=0$ and $p_2=k$ there is nothing to prove, so we assume\footnote{We may further assume that $k\geq 2 $ because the case $k= 1$ is trivial.} $p_1>0$ or $p_2<k$. We first consider the case $p_1>0$. We deduce from $\theta= d_{\phi}\beta$ that $\bar{\partial}_{\phi}\beta^{0,k-1}= \theta^{0,k}=0$, and by applying the $\bar{\partial}_{\phi}$-Poincar\'e lemma, one can write $\beta^{0,k-1}= \bar{\partial}_{\phi}\gamma^{0,k-2}$. Set $\tilde{\beta}:=\beta-d_{\phi}\gamma^{0,k-2}$, we have $d_{\phi}\tilde{\beta}=\theta$ but $\tilde{\beta}^{0,k-1}=0$. We can therefore assume that $\beta$ does not have components of type $(0,k-1)$. Now if $p_1>1$, then since $\beta^{0,k-1}=0$ we have $0= \theta^{1,k-1}=\bar{\partial}_{\phi}\beta^{1,k-2}+\partial\beta^{0,k-1}=\bar{\partial}_{\phi}\beta^{1,k-2}$. By the $\bar{\partial}_{\phi}$-Poincar\'e lemma, one can write $\beta^{1,k-2}= \bar{\partial}_{\phi}\gamma^{0,k-3}$. Set $\tilde{\beta}:=\beta-d_{\phi}\gamma^{0,k-3}$, we have $d_{\phi}\tilde{\beta}=\theta$ but $\tilde{\beta}^{1,k-2}=0$. We can therefore assume that $\beta$ does not have components of type $(1,k-2)$. By repeating this reasoning, we can assume that $\beta$ does not have components of type $(p,q)$ for $p<p_1$. The case $p_2<k$ can be proved in the same way by applying the $\partial$-Poincar\'e lemma.

2.$iii)$ is obvious. We first assume $p\geq 1$. We apply 1. to the form $\theta$ for $p_1=p-1, p_2=p$: there exists $\alpha\in A^{p-1,q}(U)$ s.t. $\theta= d_{\phi}\alpha$ and so $\theta= \partial\alpha$ with $\bar{\partial}_{\phi}\alpha=0$. This is $ii)$. If furthermore $q\geq 1$, by the $\bar{\partial}_{\phi}$-Poincar\'e lemma, we can write $\alpha=\bar{\partial}_{\phi}\beta$ and so $\theta= \partial\bar{\partial}_{\phi}\beta$. This is $i)$. For $\bar{ii})$, we apply 1. to $\theta$ for $p_1=0, p_2=1$: there exists $\alpha\in A^{0,q-1}(U)$ s.t. $\theta= d_{\phi}\alpha$ and so $\theta= \bar{\partial}_{\phi}\alpha$ with $\partial\alpha=0$.

3.Set $\theta^{p+1,q}:= \partial\alpha^{p,q}$ then $\theta^{p+1,q}$ is $d_{\phi}$-closed. By 2.$i)$ and $ii)$, there exists $\alpha\in A^{p,q}(U)$ s.t. $\theta^{p+1,q}= \partial\alpha$ with $\bar{\partial}_{\phi}\alpha=0$. Note that $\partial(\theta-\alpha)=0$ and $\theta=(\theta-\alpha)+ \alpha$. Then 3. follows from the $\bar{\partial}_{\phi}$-Poincar\'e lemma and the $\partial$-Poincar\'e lemma.

4.First from the assumption we see that $(d_{\phi}\theta)^{p_1+1,q_1}=\partial\theta^{p_1,q_1}+ \bar{\partial}_{\phi} \theta^{p_1+1,q_1-1}=0$. In particular, $\theta^{p_1,q_1}$ is $\partial\bar{\partial}_{\phi}$-closed. By 3.$i)$ and $\bar{ii})$ there exists $\gamma^{p_1,q_1}$ s.t. $\gamma^{p_1,q_1}$ is $\partial$-closed\footnote{$\gamma^{p_1,q_1}$ is $\partial$-exact if $p_1\geq 1$.} and $\theta^{p_1,q_1}=\gamma^{p_1,q_1}+\bar{\partial}_{\phi}\alpha^{p_1,q_1-1}$. Note that $\bar{\partial}_{\phi}( \theta^{p_1+1,q_1-1}-\partial\alpha^{p_1,q_1-1})=\partial (-\theta^{p_1,q_1}+\bar{\partial}_{\phi}\alpha^{p_1,q_1-1})=\partial\gamma^{p_1,q_1}=0$, so $\theta^{p_1+1,q_1-1}=\partial\alpha^{p_1,q_1-1}+\gamma^{p_1+1,q_1-1}$ such that $\gamma^{p_1+1,q_1-1}$ is $\bar{\partial}_{\phi}$-closed. Hence we can write $\theta^{p_1+1,q_1-1}=\partial\alpha^{p_1,q_1-1}+\bar{\partial}_{\phi}\alpha^{p_1+1,q_1-2}$.

Again from the assumption we see that $(d_{\phi}\theta)^{p_1+2,q_1-1}=\partial\theta^{p_1+1,q_1-1}+ \bar{\partial}_{\phi} \theta^{p_1+2,q_1-2}=0$, and note that $\bar{\partial}_{\phi}( \theta^{p_1+2,q_1-2}-\partial\alpha^{p_1+1,q_1-2})=\partial (-\theta^{p_1+1,q_1-1}+\bar{\partial}_{\phi}\alpha^{p_1+1,q_1-2})=-\partial^2\alpha^{p_1,q_1-1}=0$, we have $\theta^{p_1+2,q_1-2}=\partial\alpha^{p_1+1,q_1-2}+\bar{\partial}_{\phi}\alpha^{p_1+2,q_1-3}$.

Continuing in this way, we get the desired results. In the last two steps, from $(d_{\phi}\theta)^{p_2-1,q_2+2}=0$ we get $\theta^{p_2-1,q_2+1}=\partial\alpha^{p_2-2,q_2+1}+\bar{\partial}_{\phi}\alpha^{p_2-1,q_2}$ and from $(d_{\phi}\theta)^{p_2,q_2+1}=0$ we get $\theta^{p_2,q_2}=\partial\alpha^{p_2-1,q_2}+\gamma^{p_2,q_2}$.
\end{proof}

Let $X$ be a complex manifold. For fixed $p\geq 1$ and $q\geq 1$, we define a sheaf complex $\mathscr{L}_{\phi}^\bullet$ (which depend on $(p,q)$) as follows:
\begin{equation}
\left\{
\begin{array}{ll}
\mathscr{L}_{\phi}^k= \bigoplus_{r+s=k,r<p,s<q}A^{r,s},~~~\text{for}~~~k\leq p+q-2, & \\
\mathscr{L}_{\phi}^{k-1}= \bigoplus_{r+s=k,r\geq p,s\geq q}A^{r,s},~~~\text{for}~~~k\geq p+q.  & \\
\end{array} \right.
\end{equation}
The differential is given by
\begin{equation*}
\xymatrix@C=0.5cm{
  0 \ar[r] & \mathscr{L}_{\phi}^0 \ar[rr]^{\Pi_{\mathscr{L}_{\phi}^1}d_{\phi}} && \mathscr{L}_{\phi}^1 \ar[rr]^{\Pi_{\mathscr{L}_{\phi}^2}d_{\phi}} && \mathscr{L}_{\phi}^2 \ar[r] & \cdots \\  \ar[r] & \mathscr{L}_{\phi}^{p+q-3} \ar[rr]^{\Pi_{\mathscr{L}_{\phi}^{p+q-2}}d_{\phi}} && \mathscr{L}_{\phi}^{p+q-2} \ar[rr]^{\partial\bar{\partial}_{\phi}} && \mathscr{L}_{\phi}^{p+q-1} \ar[rr]^{d_{\phi}} && \mathscr{L}_{\phi}^{p+q} \ar[rr]^{d_{\phi}} && \cdots ,}
\end{equation*}
where $\Pi_{\mathscr{L}^k}:\bigoplus_{r+s=k}A^{r,s}\longrightarrow \mathscr{L}_{\phi}^k$ is the projection.
In particular, we find that
\begin{equation*}
\xymatrix@C=0.5cm{
\mathscr{L}_{\phi}^{p+q-2}=A^{p-1,q-1} \ar[rr]^{\quad\partial\bar{\partial}_{\phi}} && \mathscr{L}_{\phi}^{p+q-1}=A^{p,q} \ar[rr]^{d_{\phi}\qquad} && \mathscr{L}_{\phi}^{p+q}=A^{p,q+1}\oplus A^{p+1,q},}
\end{equation*}
and so $\mathbb{H}^{p+q-1}(X,\mathscr{L}_{\phi}^\bullet)\cong H^{p+q-1}(\mathscr{L}_{\phi}^\bullet(X))=H_{BC\phi}^{p,q}(X)$. The sheaf complex $\mathscr{L}_{\phi}^\bullet$ has the following subcomplexes
\begin{equation*}
\xymatrix@C=0.5cm{
(\mathscr{S'}_{\phi}^\bullet,\partial):~\mathcal{O}_{\phi}\ar[r]^{\qquad\partial} & \Omega^{1}_{\phi} \ar[r]^{\partial} & \Omega^{2}_{\phi} \ar[r]^{\partial} & \cdots  \ar[r]^{\partial} & \Omega^{p-1}_{\phi} \ar[r] & 0,\\
(\mathscr{S''_{\phi}}^\bullet,\bar{\partial}_{\phi}):~\bar{\mathcal{O}}\ar[r]^{\qquad\bar{\partial}_{\phi}} & \bar{\Omega}^{1} \ar[r]^{\bar{\partial}_{\phi}} & \bar{\Omega}^{2} \ar[r]^{\bar{\partial}_{\phi}} & \cdots \ar[r]^{\bar{\partial}_{\phi}} & \bar{\Omega}^{q-1} \ar[r] & 0,}
\end{equation*}
and
\[
\mathscr{S}_{\phi}^\bullet:=(\mathscr{S'}_{\phi}^\bullet,\partial)+(\mathscr{S''}_{\phi}^\bullet,\bar{\partial}_{\phi})\footnote{The sum is direct except $k=0$ and $\mathcal{O}_{\phi}+\bar{\mathcal{O}}\longrightarrow \Omega^{1}_{\phi}\oplus\bar{\Omega}^{1}: f+g\to (\partial f, \bar{\partial}_{\phi} g)$.}.
\]
Note that by Lemma \ref{lem-local solvability}, the complex $(\mathscr{S'}_{\phi}^\bullet,\partial)$ is exact for $0<k< p-1$ where $\mathscr{S'}_{\phi}^k=\Omega^{k}_{\phi}$.
\begin{proposition}\label{prop-quasiiso-S-L}
The inclusion $\mathscr{S}_{\phi}^\bullet\hookrightarrow \mathscr{L}_{\phi}^\bullet$ induces an isomorphism $\mathscr{H}^k(\mathscr{S}_{\phi}^\bullet)\cong\mathscr{H}^k(\mathscr{L}_{\phi}^\bullet),~\forall k\geq 0$, and we have\footnote{See also~\cite[pp.\,31]{Koo11}.}
\begin{equation*}
\mathscr{H}^k(\mathscr{S}_{\phi}^\bullet)\cong\mathscr{H}^k(\mathscr{L}_{\phi}^\bullet)=\left\{
\begin{array}{ll}
\mathbb{C},~~~\text{for}~~~k=0,~p>1,~q>1, & \\
\mathcal{O}_{\phi},~~~\text{for}~~~k=0,~p=1,~q>1, & \\
\bar{\mathcal{O}},~~~\text{for}~~~k=0,~p>1,~q=1, & \\
\mathcal{O}_{\phi}\oplus\bar{\mathcal{O}},~~~\text{for}~~~k=0,~p=1,~q=1, & \\
\Omega^{p-1}_{\phi}/\partial\Omega^{p-2}_{\phi},~~~\text{for}~~~0<k=p-1~\text{and}~p\neq q,  & \\
\bar{\Omega}^{q-1}/\bar{\partial}_{\phi}\bar{\Omega}^{q-2},~~~\text{for}~~~0<k=q-1~\text{and}~p\neq q,  & \\
\Omega^{p-1}_{\phi}/\partial\Omega^{p-2}_{\phi}\oplus \bar{\Omega}^{p-1}/\bar{\partial}_{\phi}\bar{\Omega}^{p-2},~~~\text{for}~~~0<k=p-1=q-1,  & \\
0,~~~~~~\text{otherwise}.
\end{array} \right.
\end{equation*}
\end{proposition}

\begin{proof}First, we show that $\mathscr{H}^k(\mathscr{L}_{\phi}^\bullet)=0$ for $k\geq \max\{p,q\}$. In fact, for $k\geq p+q$, this follows from Lemma \ref{lem-local solvability} 1.; for $k= p+q-1$, this follows from Lemma \ref{lem-local solvability} 2.; for $k= p+q-2$, this follows from Lemma \ref{lem-local solvability} 3.; for $k< p+q-2$, this follows\footnote{We apply Lemma \ref{lem-local solvability} 4. for $p_1=k-q+1,~q_1=q-1,~p_2=p-1,~q_2=k-p+1$. Note that we have $\theta^{k-q+1,q-1}=\partial\gamma^{k-q,q-1}+\bar{\partial}_{\phi}\alpha^{k-q+1,q-2}$ and $d_{\mathscr{L}^{k-1}}\gamma^{k-q,q-1}=\partial\gamma^{k-q,q-1}$, where $d_{\mathscr{L}^{k-1}}=\Pi_{\mathscr{L}^{k-1}}d_{\phi}$. Similarly, $\theta^{p-1,k-p+1}=\partial\alpha^{p-2,k-p+1}+\bar{\partial}_{\phi}\gamma^{p-1,k-p}$ and $d_{\mathscr{L}^{k-1}}\gamma^{p-1,k-p}=\bar{\partial}_{\phi}\gamma^{p-1,k-p}$.} from Lemma \ref{lem-local solvability} 4.~.

Now we discuss the cases when $k<p$ or $k<q$.

For $k=p-1\geq q$, if $\theta=\theta^{p-q,q-1}+\cdots \theta^{p-1,0}\in\mathscr{L}_{\phi}^{p-1}(U)$ is $d_{\mathscr{L}_{\phi}^{p-1}}$-closed where $U\subset X$ is an open ball. By Lemma \ref{lem-local solvability} 4., we can write
\[
\theta^{p-q,q-1}=\gamma^{p-q,q-1}+\bar{\partial}_{\phi}\alpha^{p-q,q-2},~\cdots~,\theta^{p-1,0}=\partial\alpha^{p-2,0}+\gamma^{p-1,0},
\]
where $\gamma^{p-q,q-1}$ is $\partial$-closed and $\gamma^{p-1,0}$ is $\bar{\partial}_{\phi}$-closed. Since $p-q\geq 1$, we have $\gamma^{p-q,q-1}=\partial \gamma^{p-q-1,q-1}=d_{\mathscr{L}^{p-2}}\gamma^{p-q-1,q-1}$ and so
\[
\theta=d_{\mathscr{L}_{\phi}^{p-2}}(\gamma^{p-q-1,q-1}+\alpha)+\gamma^{p-1,0},~\text{with}~\alpha=\alpha^{p-q,q-2}+\cdots+\alpha^{p-2,0}.
\]
On the other hand, if $\theta$ is $d_{\mathscr{L}_{\phi}^{\bullet}}$-exact, then there exists $u=u^{p-q-1,q-1}+\cdots+ u^{p-2,0}\in\mathscr{L}_{\phi}^{p-2}(U)$ s.t.
\[
d_{\mathscr{L}_{\phi}^{p-2}}u=(d_{\phi}u)^{p-q,q-1}+\cdots+ (d_{\phi}u)^{p-1,0}=\theta=\theta^{p-q,q-1}+\cdots+ \theta^{p-1,0}.
\]
Therefore $\partial u^{p-2,0}=\theta^{p-1,0}=\partial\alpha^{p-2,0}+\gamma^{p-1,0}\Rightarrow~\gamma^{p-1,0}=\partial (u^{p-2,0}-\alpha^{p-2,0})$ and $u^{p-2,0}-\alpha^{p-2,0}$ is $\partial\bar{\partial}_{\phi}$-closed. By Lemma \ref{lem-local solvability} 3.$ii)$, we see that $\gamma^{p-1,0}\in \partial\Omega_{\phi}^{p-2}(U)$. We thus have
\[
\mathscr{H}^{p-1}(\mathscr{L}_{\phi}^\bullet)=\frac{\Image d_{\mathscr{L}_{\phi}^{p-2}} +\Omega^{p-1}_{\phi} }{\Image d_{\mathscr{L}_{\phi}^{p-2}} +\partial\Omega^{p-2}_{\phi} }=\frac{\Omega^{p-1}_{\phi}}{\partial\Omega^{p-2}_{\phi}}=\mathscr{H}^{p-1}(\mathscr{S}_{\phi}^\bullet).
\]

For $k=p-1<q-1$, if $\theta=\theta^{0,p-1}+\cdots \theta^{p-1,0}\in\mathscr{L}_{\phi}^{p-1}(U)$ is $d_{\mathscr{L}_{\phi}^{p-1}}$-closed, by Lemma \ref{lem-local solvability} 4., we can write
\[
\theta^{0,p-1}=\gamma^{0,p-1}+\bar{\partial}_{\phi}\alpha^{0,p-2},~\cdots,~\theta^{p-1,0}=\partial\alpha^{p-2,0}+\gamma^{p-1,0},
\]
where $\gamma^{0,p-1}$ is $\partial$-closed and $\gamma^{p-1,0}$ is $\bar{\partial}_{\phi}$-closed. Note that since $k=p-1<q-1$, we have $d_{\mathscr{L}_{\phi}^{p-1}}\theta=0\Rightarrow\bar{\partial}_{\phi}\theta^{0,p-1}=(d_{\phi}\theta)^{0,p}=0\Rightarrow\gamma^{0,p-1}\in \bar{\partial}_{\phi}\bar{\Omega}^{p-2}$ by Lemma \ref{lem-local solvability} 2.$\bar{ii})$. Hence $\gamma^{0,p-1}\in \Image d_{\mathscr{L}_{\phi}^{p-2}}$. On the other hand, if $\theta$ is $d_{\mathscr{L}_{\phi}^{\bullet}}$-exact, then one can show as above that $\gamma^{p-1,0}\in \partial\Omega_{\phi}^{p-2}(U)$. We thus have
\[
\mathscr{H}^{p-1}(\mathscr{L}_{\phi}^\bullet)=\frac{\Image d_{\mathscr{L}_{\phi}^{p-2}} +\Omega^{p-1}_{\phi} }{\Image d_{\mathscr{L}_{\phi}^{p-2}} +\partial\Omega^{p-2}_{\phi} }=\frac{\Omega^{p-1}_{\phi}}{\partial\Omega^{p-2}_{\phi}}=\mathscr{H}^{p-1}(\mathscr{S}_{\phi}^\bullet).
\]

 For $k=p-1= q-1$, we have
\[
\mathscr{H}^{p-1}(\mathscr{L}_{\phi}^\bullet)=\frac{\Image d_{\mathscr{L}_{\phi}^{p-2}} +\bar{\Omega}^{p-1}+\Omega^{p-1}_{\phi} }{\Image d_{\mathscr{L}_{\phi}^{p-2}}+\bar{\partial}_{\phi}\bar{\Omega}^{p-2} +\partial\Omega^{p-2}_{\phi}}=\frac{\Omega^{p-1}_{\phi}}{\partial\Omega^{p-2}_{\phi}}\oplus\frac{\bar{\Omega}^{p-1} }{\bar{\partial}_{\phi}\bar{\Omega}^{p-2}}   =\mathscr{H}^{p-1}(\mathscr{S}_{\phi}^\bullet).
\]
\end{proof}

Consider the complex $\mathscr{B}_{\phi}^\bullet$ which is a modification of $\mathscr{S}_{\phi}^\bullet$ given by\footnote{This is the case when $p\geq q$, the case $p< q$ is similar. To make our notations clear and simple, we will only write explicitly one of the cases in what follows.}
\begin{equation*}
\xymatrix@C=0.5cm{
\mathscr{B}_{\phi}^\bullet:~\mathbb{C}\ar[r] &\mathcal{O}_{\phi}\oplus\bar{\mathcal{O}} \ar[r]^{\partial\oplus\bar{\partial}_{\phi}} & \Omega^{1}_{\phi}\oplus\bar{\Omega}^{1} \ar[r]^{\partial\oplus\bar{\partial}_{\phi}} & \Omega^{2}_{\phi}\oplus\bar{\Omega}^{2} \ar[r] & \cdots \\
 \ar[r] & \Omega^{q-1}_{\phi}\oplus\bar{\Omega}^{q-1}\ar[r]^{\qquad \partial\oplus 0} & \Omega^{q}_{\phi}\ar[r]^{\partial} &\cdots  \ar[r]^{\partial} & \Omega^{p-1}_{\phi} \ar[r] & 0,}
\end{equation*}
where the first morphism is defined by
\[
\mathbb{C}\longrightarrow\mathcal{O}_{\phi}\oplus\bar{\mathcal{O}}:a\mapsto (a,-a)~.
\]

\begin{proposition}\label{prop-quasiiso-B-S-phi}
The natural map from $\mathscr{B}_{\phi}^\bullet$ to $\mathscr{S}_{\phi}^\bullet[1]$, where
\[
\mathscr{B}_{\phi}^1=\mathcal{O}_{\phi}\oplus\bar{\mathcal{O}}\longrightarrow\mathscr{S}_{\phi}^1[1]=\mathcal{O}_{\phi}+\bar{\mathcal{O}}:(a,b)\mapsto a-b~,
\]
induces an isomorphism $\mathscr{H}^k(\mathscr{S}_{\phi}^\bullet[1])\cong\mathscr{H}^k(\mathscr{B}_{\phi}^\bullet),~\forall k\geq 0$.
\end{proposition}

\begin{proof}Note that $\mathscr{H}^{1}(\mathscr{B}_{\phi}^\bullet)=\mathbb{C}\oplus \mathbb{C}/\mathbb{C}(1,-1)\longrightarrow \mathbb{C}=\mathscr{H}^{1}(\mathscr{S}_{\phi}^\bullet[1]):(a,b)\mapsto a-b$ is an isomorphism.
\end{proof}

It follows that
\begin{equation}\label{eq-BC-hyper-Bphi}
H_{BC\phi}^{p,q}(X)\cong\mathbb{H}^{p+q}(M,\mathscr{L}^\bullet_{\phi}[1])\cong\mathbb{H}^{p+q}(M,\mathscr{S}^\bullet_{\phi}[1])\cong\mathbb{H}^{p+q}(M,\mathscr{B}_{\phi}^\bullet)~.
\end{equation}
Note that \eqref{eq-BC-hyper-Bphi} and Proposition \ref{prop-quasiiso-B-S-phi} is just a slight generalization of the result obtained by Schweitzer. In fact, Proposition \ref{prop-quasiiso-B-S-phi} reduce to \cite[Prop.\,4.3]{Sch07} when $\phi=0$.

Similarly, for fixed $p\geq 1$ and $q\geq 1$, we define a sheaf complex $\mathscr{L}_{\bar{\phi}}^\bullet$ as follows:
\begin{equation}
\left\{
\begin{array}{ll}
\mathscr{L}_{\bar{\phi}}^k= \bigoplus_{r+s=k,r<p,s<q}A^{r,s},~~~\text{for}~~~k\leq p+q-2, & \\
\mathscr{L}_{\bar{\phi}}^{k-1}= \bigoplus_{r+s=k,r\geq p,s\geq q}A^{r,s},~~~\text{for}~~~k\geq p+q.  & \\
\end{array} \right.
\end{equation}
The differential is given by
\begin{equation*}
\xymatrix@C=0.5cm{
  0 \ar[r] & \mathscr{L}_{\bar{\phi}}^0 \ar[rr]^{\Pi_{\mathscr{L}_{\bar{\phi}}^1}d_{\bar{\phi}}} && \mathscr{L}_{\bar{\phi}}^1 \ar[rr]^{\Pi_{\mathscr{L}_{\bar{\phi}}^2}d_{\bar{\phi}}} && \mathscr{L}_{\bar{\phi}}^2 \ar[r] & \cdots \\  \ar[r] & \mathscr{L}_{\bar{\phi}}^{p+q-3} \ar[rr]^{\Pi_{\mathscr{L}_{\bar{\phi}}^{p+q-2}}d_{\bar{\phi}}} && \mathscr{L}_{\bar{\phi}}^{p+q-2} \ar[rr]^{\partial_{\phi}\bar{\partial}} && \mathscr{L}_{\bar{\phi}}^{p+q-1} \ar[rr]^{d_{\bar{\phi}}} && \mathscr{L}_{\bar{\phi}}^{p+q} \ar[rr]^{d_{\bar{\phi}}} && \cdots .}
\end{equation*}
We have $\mathbb{H}^{p+q-1}(X,\mathscr{L}_{\bar{\phi}}^\bullet)\cong H^{p+q-1}(\mathscr{L}_{\bar{\phi}}^\bullet(X))=H_{BC\bar{\phi}}^{p,q}(X)$. The sheaf complex $\mathscr{L}_{\bar{\phi}}^\bullet$ has the following subcomplex
\begin{equation*}
\xymatrix@C=0.5cm{
\mathscr{S}_{\bar{\phi}}^\bullet:~\mathcal{O}\oplus\bar{\mathcal{O}}_{\phi} \ar[r]^{\quad\partial_{\phi}\oplus\bar{\partial}} & \Omega^{1}\oplus\bar{\Omega}^{1}_{\phi} \ar[r]^{\partial_{\phi}\oplus\bar{\partial}} & \Omega^{2}\oplus\bar{\Omega}^{2}_{\phi} \ar[r] & \cdots \\
 \ar[r] & \Omega^{p-1}\oplus\bar{\Omega}^{p-1}_{\phi}\ar[r]^{\qquad 0\oplus \bar{\partial}} & \bar{\Omega}^{p}_{\phi}\ar[r]^{\bar{\partial}} &\cdots  \ar[r]^{\bar{\partial}} & \bar{\Omega}^{q-1}_{\phi} \ar[r] & 0.}
\end{equation*}
\begin{proposition}\label{prop-quasiiso-S-L-phibar}
The inclusion $\mathscr{S}_{\bar{\phi}}^\bullet\hookrightarrow \mathscr{L}_{\bar{\phi}}^\bullet$ induces an isomorphism $\mathscr{H}^k(\mathscr{S}_{\bar{\phi}}^\bullet)\cong\mathscr{H}^k(\mathscr{L}_{\bar{\phi}}^\bullet),~\forall k\geq 0$, and we have
\begin{equation*}
\mathscr{H}^k(\mathscr{S}_{\bar{\phi}}^\bullet)\cong\mathscr{H}^k(\mathscr{L}_{\bar{\phi}}^\bullet)=\left\{
\begin{array}{ll}
\mathbb{C},~~~\text{for}~~~k=0,~p>1,~q>1, & \\
\mathcal{O},~~~\text{for}~~~k=0,~p=1,~q>1, & \\
\bar{\mathcal{O}}_{\phi},~~~\text{for}~~~k=0,~p>1,~q=1, & \\
\mathcal{O}\oplus\bar{\mathcal{O}}_{\phi},~~~\text{for}~~~k=0,~p=1,~q=1, & \\
\Omega^{p-1}/\partial_{\phi}\Omega^{p-2},~~~\text{for}~~~0<k=p-1\neq q-1,  & \\
\bar{\Omega}^{q-1}_{\phi}/\bar{\partial}\bar{\Omega}^{q-2}_{\phi},~~~\text{for}~~~0<k=q-1\neq p-1,  & \\
\Omega^{p-1}/\partial_{\phi}\Omega^{p-2}\oplus \bar{\Omega}^{p-1}_{\phi}/\bar{\partial}\bar{\Omega}^{p-2}_{\phi},~~~\text{for}~~~0<k=p-1=q-1,  & \\
0,~~~~~~\text{otherwise}.
\end{array} \right.
\end{equation*}
\end{proposition}

Consider the complex $\mathscr{B}_{\bar{\phi}}^\bullet$ which is a modification of $\mathscr{S}_{\bar{\phi}}^\bullet$ given by
\begin{equation*}
\xymatrix@C=0.5cm{
\mathscr{B}_{\bar{\phi}}^\bullet:\mathbb{C} \ar[r] & \mathcal{O}\oplus\bar{\mathcal{O}}_{\phi} \ar[r]^{\partial_{\phi}\oplus\bar{\partial}} & \Omega^{1}\oplus\bar{\Omega}^{1}_{\phi} \ar[r]^{\partial_{\phi}\oplus\bar{\partial}} & \Omega^{2}\oplus\bar{\Omega}^{2}_{\phi} \ar[r] & \cdots \\
 \ar[r] & \Omega^{p-1}\oplus\bar{\Omega}^{p-1}_{\phi}\ar[r]^{\qquad 0\oplus \bar{\partial}} & \bar{\Omega}^{p}_{\phi}\ar[r]^{\bar{\partial}} &\cdots  \ar[r]^{\bar{\partial}} & \bar{\Omega}^{q-1}_{\phi} \ar[r] & 0.}
\end{equation*}
\begin{proposition}\label{prop-quasiiso-B-S-phibar}
The natural map from $\mathscr{B}_{\bar{\phi}}^\bullet$ to $\mathscr{S}_{\bar{\phi}}^\bullet[1]$, where
\[
\mathscr{B}_{\bar{\phi}}^1=\bar{\mathcal{O}}_{\phi}\oplus\mathcal{O}\longrightarrow\mathscr{S}_{\bar{\phi}}^1[1]=\bar{\mathcal{O}}_{\phi}+\mathcal{O}:(a,b)\mapsto a-b~,
\]
induces an isomorphism $\mathscr{H}^k(\mathscr{S}_{\bar{\phi}}^\bullet[1])\cong\mathscr{H}^k(\mathscr{B}_{\bar{\phi}}^\bullet),~\forall k\geq 0$.
\end{proposition}

It follows that
\begin{equation}\label{eq-BCphibar-hyper-Bphibar}
H_{BC\bar{\phi}}^{p,q}(X)\cong\mathbb{H}^{p+q}(M,\mathscr{L}^\bullet_{\bar{\phi}}[1])\cong\mathbb{H}^{p+q}(M,\mathscr{S}^\bullet_{\bar{\phi}}[1])\cong\mathbb{H}^{p+q}(M,\mathscr{B}_{\bar{\phi}}^\bullet)~.
\end{equation}
\begin{remark}There are natural isomorphisms
\[
H_{BC\phi}^{p,0}(X)\cong H_{BC}^{p,0}(X_t):\sigma\mapsto e^{i_{\phi}}\sigma,~~H_{BC\bar{\phi}}^{0,q}(X)\cong H_{BC}^{0,q}(X_t):\sigma\mapsto e^{i_{\bar{\phi}}}\sigma,
\]
and note also that
\[
H_{BC\phi}^{0,q}(X)= H_{BC}^{0,q}(X),~~H_{BC\bar{\phi}}^{p,0}(X)\cong H_{BC}^{p,0}(X).
\]
\end{remark}

\subsection{The Bott-Chern cohomology on $X_t$}
\label{Relations between}

Let $X$ be a complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$, then by \cite[Th.\,4.3]{Xia19dDol} or \cite[Prop.\,2.13]{RZ18} we know that there are isomorphism of sheaves
\[
e^{i_{\phi}}:\Omega^{p}_{\phi}\longrightarrow \Omega^{p}_{X_t},~p=0,1,2,\cdots,n,
\]
which give rise to the following commutative diagram
\[
\xymatrix{
\mathscr{C}^\bullet:\mathbb{C}  \ar[r]^{(+,-)}  & \mathcal{O}_{\phi}\oplus\bar{\mathcal{O}}_{\phi} \ar[r]^{\partial\oplus\bar{\partial}}
\ar[d]^{id} &\Omega^{1}_{\phi}\oplus\bar{\Omega}^{1}_{\phi} \ar[r] \ar[d]^{e^{i_{\phi}}\oplus e^{i_{\bar{\phi}}}} &  \cdots \\
\mathscr{B}_{X_t}^\bullet:\mathbb{C} \ar[r]^{(+,-)}    & \mathcal{O}_{X_t}\oplus\bar{\mathcal{O}}_{X_t} \ar[r]^{\partial_t\oplus\bar{\partial}_{t}} & \Omega^{1}_{X_t}\oplus\bar{\Omega}^{1}_{X_t}  \ar[r] &  \cdots }
\]
\[
\xymatrix{
\ar[r] & \Omega^{q-1}_{\phi}\oplus\bar{\Omega}^{q-1}_{\phi}\ar[r]^{\qquad\partial\oplus 0} \ar[d]^{e^{i_{\phi}}\oplus e^{i_{\bar{\phi}}}} & \Omega^{q}_{\phi}\ar[r] \ar[d]^{e^{i_{\phi}}} & \cdots \ar[r]^{\partial} & \Omega^{p-1}_{\phi}\ar[r] \ar[d]^{e^{i_{\phi}}}& 0 \\
\ar[r] & \Omega^{q-1}_{X_t}\oplus\bar{\Omega}^{q-1}_{X_t}\ar[r]^{\qquad\partial_t\oplus 0} & \Omega^{q}_{X_t}\ar[r] &  \cdots \ar[r]^{\partial_t} & \Omega^{p-1}_{X_t}\ar[r] & 0 . }
\]
We see that
\[
H_{BC}^{p,q}(X_t)= H^{p+q-1}(\mathscr{L}_{X_t}^\bullet(M))\cong \mathbb{H}^{p+q-1}(M,\mathscr{L}_{X_t}^\bullet)\cong \mathbb{H}^{p+q}(M,\mathscr{B}_{X_t}^\bullet)\cong \mathbb{H}^{p+q}(M,\mathscr{C}^\bullet),
\]
where $M$ is the underlying smooth manifold of $X$ and $X_t$.

\subsection{The case of Aeppli cohomology}
The \emph{deformed Aeppli cohomology} can be defined as follows:
\begin{equation}\label{}
H^{p,q}_{A\phi}(X) := \frac{\ker \partial\bar{\partial}_{\phi} \cap  A^{p,q}(X)}{\Image d_{\phi} \cap A^{p,q}(X)},~~~~~~
H^{p,q}_{A\bar{\phi}}(X) := \frac{\ker \partial_{\phi}\bar{\partial} \cap A^{p,q}(X)}{\Image d_{\bar{\phi}} \cap A^{p,q}(X)},~~\forall p,q\geq 0~,
\end{equation}
and $h^{p,q}_{A\phi}:=\dim H^{p,q}_{A\phi}(X), h^{p,q}_{A\bar{\phi}}:=\dim H^{p,q}_{A\bar{\phi}}(X)$. The conjugation gives a natural isomorphism between $H^{p,q}_{A\phi}(X)$ and $H^{q,p}_{A\bar{\phi}}(X)$, we thus have $h^{p,q}_{A\phi}=h^{q,p}_{A\bar{\phi}}$.

For fixed $p\geq 0$ and $q\geq 0$, similar to the constructions for the Bott-Chern cohomology we define a sheaf complex which still denoted by $\mathscr{L}_{\phi}^\bullet$ as follows:
\begin{equation}
\left\{
\begin{array}{ll}
\mathscr{L}_{\phi}^k= \bigoplus_{r+s=k,r<p+1,s<q+1}A^{r,s},~~~\text{for}~~~k\leq p+q, & \\
\mathscr{L}_{\phi}^{k-1}= \bigoplus_{r+s=k,r\geq p+1,s\geq q+1}A^{r,s},~~~\text{for}~~~k\geq p+q+2.  & \\
\end{array} \right.
\end{equation}
The differential is given by
\begin{equation*}
\xymatrix@C=0.5cm{
  0 \ar[r] & \mathscr{L}_{\phi}^0 \ar[rr]^{\Pi_{\mathscr{L}_{\phi}^1}d_{\phi}} && \mathscr{L}_{\phi}^1 \ar[rr]^{\Pi_{\mathscr{L}_{\phi}^2}d_{\phi}} && \mathscr{L}_{\phi}^2 \ar[r] & \cdots \\  \ar[r] & \mathscr{L}_{\phi}^{p+q-1} \ar[rr]^{\Pi_{\mathscr{L}_{\phi}^{p+q}}d_{\phi}} && \mathscr{L}_{\phi}^{p+q} \ar[rr]^{\partial\bar{\partial}_{\phi}} && \mathscr{L}_{\phi}^{p+q+1} \ar[rr]^{d_{\phi}} && \mathscr{L}_{\phi}^{p+q+2} \ar[rr]^{d_{\phi}} && \cdots ,}
\end{equation*}
In particular, we find that
\begin{equation*}
\xymatrix@C=0.5cm{
\mathscr{L}_{\phi}^{p+q-1}=A^{p,q-1}\oplus A^{p-1,q} \ar[rr]^{\qquad\Pi_{\mathscr{L}_{\phi}^{p+q}}d_{\phi}} && \mathscr{L}_{\phi}^{p+q}=A^{p,q} \ar[rr]^{\partial\bar{\partial}_{\phi}\qquad} && \mathscr{L}_{\phi}^{p+q+1}=A^{p+1,q+1},}
\end{equation*}
and so $\mathbb{H}^{p+q}(X,\mathscr{L}_{\phi}^\bullet)\cong H^{p+q}(\mathscr{L}_{\phi}^\bullet(X))=H_{A\phi}^{p,q}(X)$. The other hypercohomology interpretations of the deformed Bott-Chern cohomology holds similarly for the deformed Aeppli cohomology.
The Hodge star operator induces the following duality between the deformed Bott-Chern cohomology and the deformed Aeppli cohomology~\cite[pp.\,10]{Sch07}:
\begin{equation}
H_{BC\phi}^{p,q}(X)\cong H_{A\phi}^{n-q,n-p}(X),~~\text{and}~~H_{BC\bar{\phi}}^{p,q}(X)\cong H_{A\bar{\phi}}^{n-q,n-p}(X)~.
\end{equation}

\section{Deformations of Bott-Chern classes}\label{Deformations of BC c}
Let $\pi: (\mathcal{X}, X) \to (B, 0)$ be a small deformation of a compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. In this section, power series will always be written in homogenous form, e.g. we write $\phi(t)=\sum_k\phi_k$ where each $\phi_k$ is a homogeneous polynomial of degree $k$ with coefficients in $A^{0,1}(X,T^{1,0})$. The Bott-Chern Laplacian operator is defined as
\begin{equation}
\square_{BC}:=(\partial\bar{\partial})(\partial\bar{\partial})^*+(\partial\bar{\partial})^*(\partial\bar{\partial})+ (\bar{\partial}^*\partial)(\bar{\partial}^*\partial)^*+(\bar{\partial}^*\partial)^*(\bar{\partial}^*\partial)+ \bar{\partial}^*\bar{\partial} + \partial^*\partial,
\end{equation}
and the deformed Bott-Chern Laplacian operator is defined as
\begin{equation}
\square_{BC\phi}:=(\partial\bar{\partial}_{\phi})(\partial\bar{\partial}_{\phi})^*+(\partial\bar{\partial}_{\phi})^*(\partial\bar{\partial}_{\phi})+ (\bar{\partial}_{\phi}^*\partial)(\bar{\partial}_{\phi}^*\partial)^*+(\bar{\partial}_{\phi}^*\partial)^*(\bar{\partial}_{\phi}^*\partial)+ \bar{\partial}_{\phi}^*\bar{\partial}_{\phi} + \partial^*\partial,
\end{equation}
where $\phi=\phi(t)$.
Both $\square_{BC}$ and $\square_{BC\phi}$ are $4$-th order self-adjoint elliptic differential operator~\cite{Sch07,MK71}. We have
\begin{equation}
\mathcal{H}_{BC}:=\ker\square_{BC}= \ker\partial\cap\ker\bar{\partial}\cap\ker (\partial\bar{\partial})^*
\end{equation}
and the following orthogonal direct sum decomposition holds:
\begin{equation}\label{eq-BC-Hodgedecomposition}
A^{\bullet,\bullet}(X)=\ker\square_{BC}\oplus \Image\partial\bar{\partial}\oplus (\Image\partial^*+\Image\bar{\partial}^*),
\end{equation}
which is equivalent to the existence of the Green operator $G_{BC}$ such that
\[
1=\mathcal{H}_{BC}+\square_{BC}G_{BC}.
\]
The same is true for the deformed Bott-Chern Laplacian operator $\square_{BC\phi}$.
It follows from \eqref{eq-BC-Hodgedecomposition} that
\begin{equation}\label{eq-ker-ddbar*}
\ker (\partial\bar{\partial})^*=\mathcal{H}_{BC}\oplus(\Image \partial^*+\Image\bar{\partial}^*)~\text{and}~\ker d=\mathcal{H}_{BC}\oplus \Image\partial\bar{\partial}~.
\end{equation}
The Aeppli Laplacian operator is defined as
\begin{equation}
\square_{A}:=(\partial\bar{\partial})^*(\partial\bar{\partial})+(\partial\bar{\partial})(\partial\bar{\partial})^*+ (\bar{\partial}\partial^*)^*(\bar{\partial}^*\partial)+(\bar{\partial}\partial^*)(\bar{\partial}\partial^*)^*+ \bar{\partial}\bar{\partial}^* + \partial\partial^*,
\end{equation}
and we have correspondingly
\begin{equation}\label{eq-A-Hodgedecomposition}
A^{\bullet,\bullet}(X)=\ker\square_{A}\oplus \Image(\partial\bar{\partial})^*\oplus (\Image\partial+\Image\bar{\partial}),
\end{equation}
or $1=\mathcal{H}_{A}+\square_{A}G_{A}$ where $G_{A}$ is the Green operator for $\square_{A}$. Since for any $x\in A^{\bullet,\bullet}(X)$,
we have $\square_{BC}G_{BC}\partial\bar{\partial}x=\partial\bar{\partial}x$ and $\square_{BC}\partial\bar{\partial}G_{A}x=\partial\bar{\partial}x$ which implies
\begin{equation}\label{eq-GBC-ddbar}
G_{BC}\partial\bar{\partial}=\partial\bar{\partial}G_{A}~.
\end{equation}
Similarly, we have
\begin{equation}\label{eq-GA-ddbar*}
(\partial\bar{\partial})^*G_{BC}=G_{A}(\partial\bar{\partial})^*~.
\end{equation}

Let $\varphi\in A^{p,q}(X)$ and $G_{BC}: A^{p,q}(X)\to A^{p,q}(X)$ be the Green operator, then for $k\geq 2$ we have
\begin{equation}
\|G_{BC}\varphi\|_{k+\alpha}\leq C \|\varphi\|_{k-4+\alpha} ,
\end{equation}
where $C>0$ is independent of $\varphi$ and $\|\cdot\|_{k+\alpha}$ is the H\"older norm.

We have the following observation:
\begin{proposition}\label{prop-unique small solution}
$1$. $\forall \sigma \in A^{p,q}(X)$, if $d_{\phi(t)} \sigma = d\sigma - \mathcal{L}_{\phi(t)}^{1,0} \sigma =0$ and $(\partial\bar{\partial})^*\sigma=0$, then we must have
\[
\sigma = \mathcal{H}_{BC}\sigma - G_{BC}A \partial i_{\phi(t)} \sigma ,
\]
where $\mathcal{H}_{BC}: A^{p,q}(X)\to \mathcal{H}^{p,q}_{BC}(X)$ is the projection operator to harmonic space and $A:=\bar{\partial}^*\partial\partial^*+\bar{\partial}^*$.\\
$2$. For any fixed $\sigma_0\in \mathcal{H}^{p,q}_{BC}(X)$, the equation
\begin{equation}\label{Kuranishi eq}
\sigma=\sigma_0 - G_{BC}A \partial i_{\phi(t)} \sigma ,
\end{equation}
has an unique solution given by $\sigma=\sum_{k} \sigma_k\in A^{p,q}(X)$ and $\sigma_{k}=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i $ for $|t|$ small where each $\sigma_k$ is a homogeneous polynomial of degree $k$ with coefficients in $A^{p,q}(X)$.
\end{proposition}

\begin{proof}The first assertion follows from the Hodge decomposition:
\[
\sigma = \mathcal{H}_{BC}\sigma + G_{BC}\square_{BC} \sigma
       = \mathcal{H}_{BC}\sigma + G_{BC}A \mathcal{L}_{\phi(t)}^{1,0} \sigma
       = \mathcal{H}_{BC}\sigma - G_{BC}A \partial i_{\phi(t)} \sigma~,
\]
where we have used the fact that $d_{\phi(t)} \sigma =0 \Leftrightarrow  \partial\sigma= \bar{\partial}\sigma + \partial i_{\phi(t)} \sigma =0$.

For the second assertion, substitute $\sigma=\sigma(t)=\sum_{k}$ in \eqref{Kuranishi eq}, we have
\begin{equation}\label{formal solutions}
\left\{
\begin{array}{ll}
\sigma_1 &= - G_{BC}A \partial i_{\phi_1} \sigma_0,  \\
\sigma_2 &= - G_{BC}A (\partial i_{\phi_2} \sigma_0 + \partial i_{\phi_1} \sigma_1 ),\\
         &\cdots,  \\
\sigma_k &= - G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i ,~~\forall k > 0.  \\
\end{array} \right.
\end{equation}
For the convergence of $\sigma(t)$, we note that
\begin{equation}
\|\sigma_{j}\|_{k+\alpha}=\| G_{BC}A\sum_{a+b=j} \partial i_{\phi_a} \sigma_b \|_{k+\alpha} \leq C \sum_{a+b=j} \|\phi_{a}\|_{k+\alpha} \|\sigma_{b}\|_{k+\alpha},
\end{equation}
for some constant $C$ depends only on $k$ and $\alpha$.
Now it is left to show the uniqueness. Let $\sigma$ and $\sigma'$ be two solutions to $\sigma=\sigma_0 - G_{BC}A \partial i_{\phi} \sigma$ and set $\tau=\sigma-\sigma'$. Then $\tau=-G_{BC}A \partial i_{\phi} \tau$, we have
\begin{equation}
\|\tau\|_{k+\alpha}\leq c \|\phi (t)\|_{k+\alpha}  \|\tau\|_{k+\alpha},
\end{equation}
for some constant $c>0$. When $|t|$ is sufficiently small, $\|\phi (t)\|_{k+\alpha}$ is also small. Hence we must have $\tau=0$.
For smoothness of the solution, note that we have
\[
\square_{BC}\sigma=-\square_{BC}G_{BC}A \partial i_{\phi(t)} \sigma=-(1-\mathcal{H}_{BC\phi(t)} )A \partial i_{\phi(t)} \sigma=-A \partial i_{\phi(t)} \sigma,
\]
which implies
\begin{equation}\label{regularity eq}
\square_{BC}\sigma + (\bar{\partial}^*\partial\partial^*+\bar{\partial}^*)\partial i_{\phi(t)} \sigma= 0,
\end{equation}
which is a standard elliptic equation for small $t$.
\end{proof}
Note that the solution $\sigma$ of \eqref{Kuranishi eq} automatically satisfies $(\partial\bar{\partial})^*\sigma=0$ in view of \eqref{eq-GA-ddbar*}.

\begin{lemma}\label{lem-ddbar*-closed-rep of deformed-BC-coho}
The natural map
\begin{equation}\label{eq-natural map}
\frac{\ker(\partial\bar{\partial})^*\cap\ker d_{\phi(t)} \cap A^{p,q}(X)}{\ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)} \cap A^{p,q}(X)}\longrightarrow H_{BC\phi(t)}^{p,q}(X)
\end{equation}
is an isomorphism.
\end{lemma}

\begin{proof}By (the deformed version of) \eqref{eq-BC-Hodgedecomposition} and \eqref{eq-ker-ddbar*}, we have the following orthogonal direct sum decomposition
\[
A^{p,q}(X)=\left(\ker d_{\phi(t)}\cap\ker(\partial\bar{\partial})^*\right)\oplus
(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*+\Image\partial\bar{\partial})
\]
which implies
\[
\ker d_{\phi(t)}\cap A^{p,q}(X)=\left(\ker d_{\phi(t)}\cap\ker(\partial\bar{\partial})^*\right)\oplus
\left(\ker d_{\phi(t)}\cap(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*+\Image\partial\bar{\partial})\right),
\]
and
\[
\Image\partial\bar{\partial}_{\phi(t)}\cap A^{p,q}(X)=\left(\Image\partial\bar{\partial}_{\phi(t)}\cap\ker(\partial\bar{\partial})^*\right)\oplus
\left(\Image\partial\bar{\partial}_{\phi(t)}\cap(\ker(\partial\bar{\partial}_{\phi(t)})^*+\Image\partial\bar{\partial})\right).
\]
Moreover, for any $x\in \Image\partial\bar{\partial}$, there exists unique $y\in\ker d_{\phi(t)}$ and unique $z\in (\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*)$ such that $x=y+z$. This defines a surjective homomorphism
\[
\Image\bar{\partial}\bar{\partial}\longrightarrow\ker d_{\phi(t)}\cap(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*+\Image\partial\bar{\partial}):
x\longmapsto y,
\]
with kernel equal to $\Image\partial\bar{\partial}\cap(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*)$. It follows that
\[
\ker d_{\phi(t)}\cap(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*+\Image\partial\bar{\partial})
\cong\frac{\Image\partial\bar{\partial}}{\Image\partial\bar{\partial}\cap(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*)}.
\]
Similarly, we have
\[
\Image\partial\bar{\partial}_{\phi(t)}\cap(\ker(\partial\bar{\partial}_{\phi(t)})^*+\Image\partial\bar{\partial})
\cong\frac{\Image\partial\bar{\partial}}{\Image\partial\bar{\partial}\cap\ker(\partial\bar{\partial}_{\phi(t)})^*}.
\]
Hence,
\[
H_{BC\phi(t)}^{p,q}(X)\cong
\frac{\ker(\partial\bar{\partial})^*\cap\ker d_{\phi(t)}}{\ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)} } \oplus
\frac{\Image\partial\bar{\partial}\cap\ker(\partial\bar{\partial}_{\phi(t)})^*}{\Image\partial\bar{\partial}\cap(\Image\partial^*+\Image\bar{\partial}_{\phi(t)}^*)}
.\]
We claim $\Image\partial\bar{\partial}\cap\ker(\partial\bar{\partial}_{\phi(t)})^*=0$. Indeed, let $\sigma\in\ker\partial\bar{\partial}\cap\ker(\partial\bar{\partial}_{\phi(t)})^*$, then it follows from the same proof of Proposition \ref{prop-unique small solution} that $\sigma$ is the solution of the equation
\[
\sigma=\sigma_0+G_{BC}\partial\bar{\partial}(\partial\mathcal{L}_{\phi(t)}^{1,0})^*\sigma,\quad \sigma_0:=\mathcal{H}_{BC}\sigma
\]
which is uniquely determined by $\sigma_0$. If $\sigma\in \Image\partial\bar{\partial}\cap\ker(\partial\bar{\partial}_{\phi(t)})^*$, then $\sigma_0=\mathcal{H}_{BC}\sigma=0\Rightarrow\sigma=0$.
\end{proof}

\begin{proposition}\label{prop-kerddbar*-imageddbarphi}
$1$. For any fixed $t\in B$, the following homomorphism
\[
g_t:\ker\partial\bar{\partial}\cap A^{p,q}(X)\longrightarrow \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)}\cap A^{p+1,q+1}(X):x_0\longmapsto \partial\bar{\partial}_{\phi(t)}x(t),
\]
is surjective with $\ker g_t=\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\cap A^{p,q}(X)$, where $x(t)$ is the unique solution of $x(t)=x_0 + (\partial\bar{\partial})^*G_{BC} \partial i_{\phi(t)} \partial x(t)$.\\
$2$. Let $\hat{g}_t:\mathcal{H}^{p,q}_{BC}(X)\longrightarrow \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)}\cap A^{p+1,q+1}(X)$ be the restriction of $g_t$ on $\mathcal{H}^{p,q}_{BC}(X)$, then $\hat{g}_t$ is surjective with
$\ker \hat{g}_t=\mathcal{H}^{p,q}_{BC}(X)\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)$. Moreover, we have
\begin{align*}
\dim\mathcal{H}^{p,q}_{BC}(X)=&\dim\ker\partial\bar{\partial}_{\phi(t)}\cap(\mathcal{H}^{p,q}_{BC}(X)+\Image(\partial\bar{\partial})^*)\cap A^{p,q}(X)\\+
&\dim \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)}\cap A^{p+1,q+1}(X).
\end{align*}
\end{proposition}
\begin{proof}$1$. Let $x\in A^{p,q}(X)$, then by Hodge decomposition we have
\[
\partial\bar{\partial}_{\phi(t)}x=\partial\bar{\partial}x-\partial i_{\phi(t)}\partial x
=\partial\bar{\partial}x-\mathcal{H}_{BC}\partial i_{\phi(t)}\partial x-\square_{BC}G_{BC}\partial i_{\phi(t)}\partial x,
\]
thus
\[
\partial\bar{\partial}_{\phi(t)}x\in\ker(\partial\bar{\partial})^*\Leftrightarrow \partial\bar{\partial}x-\partial\bar{\partial}(\partial\bar{\partial})^*G_{BC}\partial i_{\phi(t)}\partial x=0.
\]
Set $x_0=x-(\partial\bar{\partial})^*G_{BC}\partial i_{\phi(t)}\partial x$, then $x$ is a solution to the equation $x=x_0 + (\partial\bar{\partial})^*G_{BC}\partial i_{\phi(t)}\partial x$ which is uniquely determined by $x_0$ in view of the proof of Proposition \ref{prop-unique small solution}.

It is left to show $\ker g_t=\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\cap A^{p,q}(X)$. In fact, obviously we have
$\ker g_t\subseteq\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\cap A^{p,q}(X)$. Conversely, let us consider the following surjective homomorphism
\begin{align*}
\ker&\partial\bar{\partial}_{\phi(t)}\longrightarrow \ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\\
&x\longmapsto x-(\partial\bar{\partial})^*\partial\bar{\partial}G_Ax
=x-(\partial\bar{\partial})^*G_{BC}\partial\bar{\partial}x=x-(\partial\bar{\partial})^*G_{BC}\partial i_{\phi(t)}\partial x,
\end{align*}
whose kernel is $\ker\partial\bar{\partial}_{\phi(t)}\cap\Image(\partial\bar{\partial})^*=0$ by Proposition \ref{prop-unique small solution}. Its inverse is given by
\[
\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)
\longrightarrow \ker\partial\bar{\partial}_{\phi(t)}:
x_0\longmapsto x(t),
\]
where $x(t)$ is the unique solution of $x(t)=x_0 + (\partial\bar{\partial})^*G_{BC}\partial i_{\phi(t)}\partial x(t)$.
So let $x_0\in\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)$ and $x(t)$ be the unique solution of $x(t)=x_0 + (\partial\bar{\partial})^*G_{BC}\partial i_{\phi(t)}\partial x(t)$, we must have $x(t)\in \ker\partial\bar{\partial}_{\phi(t)}\Rightarrow x_0\in\ker g_t$. \\

$2$. It can be proved in the same way that $\ker \hat{g}_t=\mathcal{H}^{p,q}_{BC}(X)\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)$. To show $\hat{g}_t$ is surjective it is enough to show
\[
\frac{\mathcal{H}^{p,q}_{BC}(X)}{\mathcal{H}^{p,q}_{BC}(X)\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)}\cong
\frac{\ker\partial\bar{\partial}\cap A^{p,q}(X)}{\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\cap A^{p,q}(X)}.
\]
Indeed, we have
\[
\ker\partial\bar{\partial}=\mathcal{H}^{p,q}_{BC}(X)\oplus\left\{\ker\partial\bar{\partial}\cap\left(\Image(\partial\bar{\partial})+ \Image\partial^*+ \Image\bar{\partial}^*\right)\right\},
\]
and
\begin{align*}
&\ker\partial\bar{\partial}\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\cong\ker\partial\bar{\partial}_{\phi(t)}\\
=&\left\{\ker\partial\bar{\partial}_{\phi(t)}\cap (\mathcal{H}^{p,q}_{BC}(X)+\Image(\partial\bar{\partial})^*)\right\}\oplus\\
&\left\{\ker\partial\bar{\partial}_{\phi(t)}\cap\left[\Image(\partial\bar{\partial}_{\phi(t)})^*+ \ker\partial\bar{\partial}\cap(\Image\partial\bar{\partial} +\Image\partial^*+\Image\bar{\partial}^*)\right]\right\}\\
\cong& \left\{\mathcal{H}^{p,q}_{BC}(X)\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)\right\}
\oplus \left\{\ker\partial\bar{\partial}\cap\left(\Image(\partial\bar{\partial})+ \Image\partial^*+ \Image\bar{\partial}^*\right)\right\}.
\end{align*}
\end{proof}
\begin{remark}It can be proved in a similar way that $g_t$ when restricted on $\mathcal{H}^{p,q}_{A}(X)$ is also surjective with kernel equal to
$\mathcal{H}^{p,q}_{A}(X)\cap\left(\ker\partial\bar{\partial}_{\phi(t)}+\Image(\partial\bar{\partial})^*\right)$ and
\begin{align*}
\dim\mathcal{H}^{p,q}_{A}(X)=&\dim\ker\partial\bar{\partial}_{\phi(t)}\cap(\mathcal{H}^{p,q}_{A}(X)+\Image(\partial\bar{\partial})^*)\cap A^{p,q}(X)\\+
&\dim \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)}\cap A^{p+1,q+1}(X).
\end{align*}
\end{remark}

\begin{definition}\label{def-V_t}
For any $t\in B$ and a vector subspace $V=\mathbb{C}\{ \sigma_0^1, \cdots, \sigma_0^N \}\subseteq \mathcal{H}_{BC}^{p,q}(X)$, we set
\begin{align*}\label{V_E,t}
V_{t}:=
&\{ \sum_{l=1}^N a_l\sigma_0^l\in V \mid  (a_1, \cdots, a_N)\in \mathbb{C}^{N}~\text{s.t.}~\sigma(t)\in\ker d_{\phi(t)},\\
&\text{where}~\sigma (t)=\sum_{k} \sigma_k~\text{with}~ \sigma_0=\sum_l a_l\sigma_0^l~\text{and}~ \sigma_{k}=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i,~\forall k\neq 0 \}.
\end{align*}
Note that $V_{t}$ consists of those vectors of the form $\sum_l a_l\sigma_0^l$ such that the coefficients $a_l$ satisfy the following linear equation:
\[
\sum_{l=1}^N a_ld_{\phi(t)}\sigma^l(t)=0,
\]
where $\sigma^l(t)= \sum_{k} \sigma_k^l$ with $\sigma_{k}^l=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j}\sigma_i,~\forall k\neq 0$.
\end{definition}

\begin{definition}\label{def-f_t-g_t}
We set
\begin{align*}
f_t: &V_{t} \longrightarrow \frac{\ker(\partial\bar{\partial})^*\cap\ker d_{\phi(t)} \cap A^{p,q}(X)}{\ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)} \cap A^{p,q}(X)}\cong H_{BC\phi(t)}^{p,q}(X),\\
&\sigma_0\longmapsto \sigma(t)=\sum_{k} \sigma_k,~\text{where}~\sigma_{k}=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i,~\forall k\neq 0.
\end{align*}
\end{definition}

\begin{proposition}\label{prop V-E-t}
If $V= \mathcal{H}_{BC}^{p,q}(X)$, then $f_t$ is surjective.
\end{proposition}

\begin{proof}By Proposition \ref{prop-unique small solution}, The map
\begin{align*}
\tilde{f}_t: &V_{t} \longrightarrow \ker(\partial\bar{\partial})^*\cap\ker d_{\phi(t)} \cap A^{p,q}(X),\\
&\sigma_0\longmapsto \sigma(t)=\sum_{k} \sigma_k,~\text{where}~\sigma_{k}=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i,~\forall k\neq 0,
\end{align*}
is an isomorphism.
\end{proof}

\begin{theorem}\label{Deformation of B-C cohomology classes}
Let $X$ be a compact complex manifold and $\pi: (\mathcal{X}, X) \to (B, 0)$ a small deformation of $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. For any $p,q\geq 0$, let $V=\mathbb{C}\{ \sigma_0^1, \cdots, \sigma_0^N \}$ be a linear subspace of $\mathcal{H}_{BC}^{p,q}(X)$ and $\sigma^l(t)=\tilde{f}_t\sigma_{0}^l,~l=1, \cdots, N$. Define a subset $B(V)$ of $B$ by
\[
B(V):=\{t\in \mathcal{B}\mid d_{\phi(t)}\sigma^l(t)=0, l=1, \cdots, N\},
\]
Then $B(V)$ are analytic subsets of $B$ and we have
\begin{equation}\label{B(V)-discription}
B(V)=\{t\in B\mid \dim V = \dim \Image f_t +\dim \ker f_t\}.
\end{equation}
In particular, we have
\begin{equation}\label{B'-discription}
B'=B(\mathcal{H}_{BC}^{p,q}(X))=\{t\in B\mid \dim H_{BC}^{p,q}(X)=\dim H_{BC\phi(t)}^{p,q}(X)+\dim \ker f_t\}.
\end{equation}
\end{theorem}

\begin{proof}First, let $\{U_{\alpha}\} $ be a finite open cover of $X$ and $ u^\alpha_1, u^\alpha_{2}, \cdots, $ a local unitary frames of $p+q+1$-forms on the $U_{\alpha}$, then $\forall l=1, \cdots, N$, we have
\[
d_{\phi(t)}\sigma^l(t)=0 \Leftrightarrow a_{j}^{l\alpha}(t):=< d_{\phi(t)}\sigma^l(t)\mid_{U_{\alpha}} ,u^\alpha_j>=0,~\forall j,~\alpha,
\]
where $<\cdot,\cdot>$ is the $L^2$-inner product on the space $A^{p+q+1}(U_{\alpha})$. We see that each $a_{j}^{l\alpha}(t)$ is holomorphic in $t$ and so
\[
B(V)=\{t\in B\mid a_{j}^{l\alpha}(t)=0, \forall j, l, \alpha\}
\]
is an analytic subset of $B$.

Furthermore, note that
\[
 t\in B(V) \Leftrightarrow V_{t}=V .
\]
So \eqref{B(V)-discription} follows from the fact that $\dim V_{t}=\dim \Image f_t +\dim \ker f_t$. If $V= \mathcal{H}^{0,q}(X,E)$, then $f_t: V_{t} \to H^{0,q}_{\bar{\partial}_{\phi(t)}}(X,E)$ is surjective by Proposition \ref{prop V-E-t} and \eqref{B'-discription} follows.
\end{proof}
\begin{remark}\label{rk-analyticity-Vt}
From the above proof, we can see that $V_{t}\subseteq V$ varies holomorphically with $t$. In fact, in the notations of Definition \ref{def-V_t}, $V_{t}$ consists of those vectors of the form $\sum_l a_l\sigma_0^l$ such that the coefficients $a_l$ satisfy
\[
\sum_{l=1}^N a_l\cdot a_{j}^{l\alpha}(t)=0,\quad j=1,\cdots,m;~\forall \alpha~,
\]
where $a_{j}^{l\alpha}(t)$ are holomorphic functions in $t$. In particular, $\{t\in B\mid \dim V_t\geq k\}$ is an analytic subset of $B$ for any nonnegative integer $k$.

For the same reason, $\{t\in B\mid\dim\ker\partial\bar{\partial}_{\phi(t)}\cap(\mathcal{H}^{p,q}_{BC}(X)+\Image(\partial\bar{\partial})^*)\cap A^{p,q}(X)\geq k\}$ is also an analytic subset of $B$ for any nonnegative integer $k$. It follows from this and Proposition \ref{prop-kerddbar*-imageddbarphi} that $\{t\in B\mid\dim \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)}\cap A^{p,q}(X)\leq k\}$ is an analytic subset of $B$ for any nonnegative integer $k$.
\end{remark}

\begin{definition}\label{def-deformation-Bott-Chern}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a deformation of a compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. Given $y\in \ker d\cap A^{p,q}(X)$ and $T\subseteq B$, which is a complex subspace of $B$ containing $0$, a (Bott-Chern) \emph{deformation} of $y$ (w.r.t. $\pi: (\mathcal{X}, X)\to (B,0)$ ) on $T$ is a family of $(p,q)$-forms $\sigma (t)$ such that
\begin{itemize}
  \item[1.] $\sigma (t)$ is holomorphic in $t\in T$ and $\sigma (0) = y$;
  \item[2.] $d_{\phi(t)}\sigma (t) = 0,~\forall t\in T$.
\end{itemize}
A deformation of $[y]\in \mathcal{H}_{BC}^{p,q}(X)$ (w.r.t. $\pi$) on $T$ is a triple $(y,\sigma (t),T)$ which consisting of a representative $y\in[y]$ and a deformation $\sigma (t)$ of $y$ (w.r.t. $\pi$) on $T$. Two deformations $(y,\sigma (t),T)$ and $(y',\sigma' (t),T)$ of $[y]$ on $T$ are \emph{equivalent} if
\[
[\sigma (t) - \sigma' (t)] = 0 \in H_{BC}^{p,q}(X),~\forall t\in T.
\]
A deformation $\sigma (t)$ of $y$ on $T$ is called \emph{canonical} if
\[
\sigma(t)=\sigma_0 - G_{BC}(\bar{\partial}^*\partial\partial^*+\bar{\partial}^*) \partial i_{\phi(t)} \sigma(t),\quad \forall t\in T .
\]
By Proposition \ref{prop-unique small solution}, canonical deformation is unique on its existence domain.

For a given small deformation $\pi: (\mathcal{X}, X)\to (B,0)$ with smooth $B$, we say $y\in \ker d\cap A^{p,q}(X)$ is \emph{(canonically) unobstructed w.r.t. $\pi$} if a (canonical) deformation of $y$ (w.r.t. $\pi$) exists on $B$ and a class $\alpha\in H_{BC}^{p,q}(X)$ is \emph{(canonically) unobstructed w.r.t. $\pi$} if there is a $y\in\alpha$ such that $y$ is (canonically) unobstructed w.r.t. $\pi$. If every Bott-Chern classes in $H_{BC}^{p,q}(X)$ have canonically unobstructed deformation w.r.t. $\pi$, then we say the \emph{deformations of classes in $H_{BC}^{p,q}(X)$ is canonically unobstructed w.r.t. $\pi$}. If these holds for any small deformation of $X$, we will drop the term ``w.r.t. $\pi$''. For example, we say $y\in \ker d\cap A^{p,q}(X)$ is \emph{(canonically) unobstructed} if for any small deformation $\pi: (\mathcal{X}, X)\to (B,0)$ with smooth $B$ there is a (canonical) deformation of $y$ on $B$.
\end{definition}

Although a Bott-Chern deformation $\sigma (t)$ of $y\in \ker d\cap A^{p,q}(X)$ can also be viewed as a Dolbeault deformation with the additional requirement $\partial\sigma (t)=0$, the ways we identify deformations in these two cases is very different.
We want to point out another difference between the deformation theory of Dolbeault cohomology~\cite{Xia19dDol} and that of Bott-Chern cohomology. Let $\sigma^{BC} (t)$ and $\sigma^{D} (t)$ be the canonical Bott-Chern/Dolbeault deformation of $y\in \ker d\cap A^{p,q}(X)$ respectively, it is known that $\mathcal{L}_{\phi(t)}^{1,0}\sigma^{D}(t)\in\ker\bar{\partial}$ for any $t\in B$, see \cite[Prop.\,5.2]{Xia19dDol}. This seems does not hold for the Bott-Chern deformation in general. More precisely, it is not guaranteed that $\mathcal{L}_{\phi(t)}^{1,0}\sigma^{BC} (t)\in\ker d_{\phi(t)}$.

In the remainder of this section, We confine ourselves to sketching the essential points of the deformation theory of Bott-Chern cohomology. Since this part of the theory is very similar to the case of Dolbeault cohomology, the proofs will be omitted.

A notable consequence of the deformation theory for Bott-Chern classes is the following
\begin{theorem}\label{th-analytic subset}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a deformation of a compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. Then the set $\{t\in B\mid \dim H_{BC\phi(t)}^{p,q}(X)\geq k\}$ is an analytic subset of $B$ for any nonnegative integer $k$.
\end{theorem}
\begin{proof}It follows from Proposition \ref{prop V-E-t} that
\begin{align*}
&\{t\in B\mid \dim H_{BC\phi(t)}^{p,q}(X)\geq k\}\\
=&\{t\in B\mid \dim V_t/\ker f_t\geq k\}\\
=&\{t\in B\mid \dim V_t-\dim\left(\ker\partial\bar{\partial}^*\cap\Image\partial\bar{\partial}_{\phi(t)}\right)\geq k\}.
\end{align*}
The conclusion then follows from Remark \ref{rk-analyticity-Vt}.
\end{proof}

The canonical deformations has the following properties:
\begin{theorem}\label{universal and uniqueness of the canonical deformation}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a deformation of a compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$.
\begin{itemize}
	\item[$(i)$] Assume $S$ is an analytic subset of $B$ with $0\in S$ and $y\in \ker d\cap A^{p,q}(X)$. If the canonical deformation of $y$ exists on $S$ then we must have $S\subseteq B(\mathbb{C}\mathcal{H}_{BC}y)$;
	\item[$(ii)$] For any deformed Bott-Chern cohomology class $[u]\in H_{BC\phi(t)}^{p,q}(X)$, there exists $\sigma_0\in \mathcal{H}_{BC}^{p,q}(X)$ such that $[u]=[\sigma(t)]$ where $\sigma(t)$ is the canonical deformation of $\sigma_0$.
\end{itemize}
\end{theorem}
\begin{proof}$(i)$ follows from Theorem \ref{Deformation of B-C cohomology classes} and $(ii)$ follows from Lemma \ref{lem-ddbar*-closed-rep of deformed-BC-coho}.
\end{proof}

We end this section with the following result which is of particular interests.
\begin{theorem}\label{thm-2nd-main}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a small deformation of the compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. For each $(p,q)\in \mathbb{N}\times \mathbb{N}$, set
\[
v^{p,q}_t:=\dim H_{BC}^{p,q}(X)-\dim \ker d_{\phi(t)}\cap\ker(\partial\bar{\partial})^*\cap A^{p,q}(X)\geq 0,
\]
and
\[
u^{p,q}_t:=\dim H_{BC}^{p,q}(X)-\dim \ker \partial\bar{\partial}_{\phi(t)}\cap\left(\mathcal{H}_{BC}^{p,q}(X)+\Image(\partial\bar{\partial})^*\right)\cap A^{p,q}(X)\geq 0,
\]
then we have
\begin{equation}\label{eq-1461}
\dim H_{BC}^{p,q}(X)=\dim H_{BC\phi(t)}^{p,q}(X)+v^{p,q}_t+u^{p-1,q-1}_t.
\end{equation}
\end{theorem}
\begin{proof}This follows immediately from Lemma \ref{lem-ddbar*-closed-rep of deformed-BC-coho} and Proposition \ref{prop-kerddbar*-imageddbarphi}.
\end{proof}

\section{The deformed Fr\"ohlicher spectral sequences and the $\partial\bar{\partial}_{\phi}$-lemma}
\label{The deformed Frohlicher spectral sequences and ddbar-lemma}

Let $X$ be a complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. Set the deformed de Rahm cohomology as
\[
H^{\bullet}_{d_\phi}(X):=\ker d_\phi/\Image d_\phi,
\]
then it is clear that $e^{i_{\phi}}:H^{\bullet}_{d_\phi}(X)\to H^{\bullet}_{dR}(X)$ is an isomorphism and the identity map induces the following commutative diagram:
\begin{equation}\label{dia-deofrmed 131}
\xymatrix{
 & H^{\bullet,\bullet}_{BC\phi}(X) \ar[ld] \ar[d] \ar[rd] & \\
 H^{\bullet,\bullet}_{\partial}(X) \ar[rd] & H^{\bullet,\bullet}_{d_\phi}(X) \ar[d] & H^{\bullet,\bullet}_{\bar{\partial}_\phi}(X) \ar[ld] \\
 & H^{\bullet,\bullet}_{A\phi}(X) & \\
}
\end{equation}
\begin{definition}\label{def-deformed Frohlicher spectral sequences and ddbar-lemma}
The spectral sequence associated to the double complex $(A^{\bullet,\bullet}(X), \partial, \bar{\partial}_{\phi})$ will be called the \emph{deformed Fr\"ohlicher spectral sequence} and we say $X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma if the homomorphism $H^{\bullet,\bullet}_{BC\phi}(X)\to H^{\bullet,\bullet}_{d_\phi}(X)$ in \eqref{dia-deofrmed 131} is injective, i.e.
\[
\ker \partial \cap \ker \bar{\partial}_{\phi} \cap \Image d_\phi = \Image\partial\bar{\partial}_{\phi}.
\]
\end{definition}
Set $d_\phi^c:=J^{-1}d_\phi J=\sqrt{-1}(\bar{\partial}_{\phi}-\partial)$, where $J$ is the almost complex structure on $X$. It is easy to see that $\ker\partial\cap\ker\bar{\partial}_{\phi}=\ker d_\phi\cap \ker d_\phi^c$ and $\Image\partial\bar{\partial}_{\phi}=\Image d_\phi d_\phi^c$. Hence, $X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma if and only if
\begin{equation}\label{eq-dphi-dphic-lemma-1}
\ker d_\phi \cap \ker d_\phi^c \cap \Image d_\phi = \Image d_\phi d_\phi^c~,
\end{equation}
or
\begin{equation}\label{eq-dphi-dphic-lemma-2}
\ker d_\phi \cap \ker d_\phi^c \cap \Image d_\phi^c = \Image d_\phi d_\phi^c~.
\end{equation}
There are two natural filtrations on $A^{\bullet,\bullet}(X)$:
\[
F^pA^{k}(X)=\bigoplus_{p\leq r\leq k}A^{r,k-r}(X),~\bar{F}^pA^{k}(X)=\bigoplus_{p\leq s\leq k}A^{k-s,s}(X),
\]
which induces two filtrations on the deformed de Rahm cohomology $H^{k}_{d_\phi}(X)$ for each $k\geq 0$:
\[
F^pH^{k}_{d_\phi}(X)=\{\alpha\in H^{k}_{d_\phi}(X)\mid \exists u\in F^pA^{k}(X)~s.t.~\alpha=[u] \},
\]
and
\[
\bar{F}^pH^{k}_{d_\phi}(X)=\{\alpha\in H^{k}_{d_\phi}(X)\mid \exists u\in \bar{F}^pA^{k}(X)~s.t.~\alpha=[u] \}.
\]
As usual, there are many ways to characterize the $\partial\bar{\partial}_{\phi}$-lemma:
\begin{proposition}\label{prop-char-ddbar-phi}The following statements are equivalent:
\begin{itemize}
  \item[1.]$X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma;
  \item[2.]The maps in \eqref{dia-deofrmed 131} induced by the identity map are all isomorphisms;
  \item[3.]The deformed Fr\"ohlicher spectral sequence degenerates at $E_1$ and there is a Hodge decomposition
\[
H^{k}_{d_\phi}(X;\mathbb{C})=\bigoplus_{p+q=k}F^pH^{k}_{d_\phi}(X)\cap \bar{F}^qH^{k}_{d_\phi}(X)~,~\forall k .
\]
\end{itemize}
\end{proposition}
\begin{proof}This follows directly from ~\cite[pp.\,268]{DGMS75}.
\end{proof}

\begin{theorem}\label{thm-AT-Frolicher-inequality-deformed}
Let $X$ be a compact complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. Then for every $(p,q)\in \mathbb{N}\times \mathbb{N}$, we have
\begin{equation}\label{eq-AT-Frolicher-inequality-deformed}
\dim H_{BC\phi}^{p,q}(X)+ \dim H_{A\phi}^{p,q}(X) \geq \dim H_{\bar{\partial}_t}^{p,q}(X_t)+ \dim H_{\partial}^{p,q}(X).
\end{equation}
In particular, for every $k\in \mathbb{N}$, we have
\begin{equation}\label{eq-ddbar-phi-lemma-kehua}
\sum_{p+q=k}\dim H_{BC\phi}^{p,q}(X)+ \sum_{p+q=k}\dim H_{A\phi}^{p,q}(X) \geq 2\dim H_{dR}^{k}(X),
\end{equation}
and equality holds if and only if $X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma.
\end{theorem}
\begin{proof}This follows from similar arguments as in~\cite{AT13}. In fact, this theorem is a direct consequence of ~\cite[Thm.\,1\,and\,2]{AT15b} by noting that $\dim H_{\bar{\partial}_{\phi}}^{p,q}(X)=\dim H_{\bar{\partial}_t}^{p,q}(X_t)$~\cite[Thm.\,4.4]{Xia19dDol}.
\end{proof}
\begin{remark}\label{rk-to-AK-Frolicher-ineq}
\begin{itemize}
  \item[1.] From the work of Angella-Tardini~\cite[Thm.\,3.1]{AT17} we know that $X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma if and only if
\[
\sum_{p+q=k}\dim H_{BC\phi}^{p,q}(X)= \sum_{p+q=k}\dim H_{A\phi}^{p,q}(X)~;
\]
  \item[2.] From Proposition \ref{prop-char-ddbar-phi} we see that if $X$ satisfies the $\partial\bar{\partial}_{\phi}$-lemma, then for every $(p,q)\in \mathbb{N}\times \mathbb{N}$, we have
\[
\dim H_{BC\phi}^{p,q}(X)= \dim H_{A\phi}^{p,q}(X) = \dim H_{\bar{\partial}_t}^{p,q}(X_t)= \dim H_{\partial}^{p,q}(X).
\]
In particular, by Theorem \ref{thm-AT-Frolicher-inequality-deformed} we have $h_{BC\phi}^{k}=h_{A\phi}^{k}=h_{\bar{\partial}_t}^{k}(X_t)=h_{\bar{\partial}}^{k}=b_k$\footnote{We follow the notations as given in \cite{AT13}, e.g. $h_{BC\phi}^{k}:=\sum_{p+q=k}\dim H_{BC\phi}^{p,q}(X)$ and $b_k$ is the $k$-th Betti number.}, namely, the Fr\"ohlicher spectral sequence of $(A^{\bullet,\bullet}(X), \partial, \bar{\partial})$ degenerates at $E_1$.
\end{itemize}
\end{remark}

\begin{corollary}\label{coro-deformation-closedness-ddbar-lemma}
Let $\pi: (\mathcal{X}, X)\to (B,0)$ be a small deformation of the compact complex manifold $X$ such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi(t)$. Then
the set
\[
T:=\{t\in B\mid X~\text{satisfies the}~\partial\bar{\partial}_{\phi(t)}\text{-lemma} \}
\]
is an analytic open subset (i.e. complement of analytic subset) of $B$. In particular, if $B\subset \mathbb{C}$ is a small open disc with $0\in B$ and $T$ is not empty, then $T=B$ or $T=B\setminus\{0\}$.
\end{corollary}
\begin{proof}First, by Theorem~\ref{thm-AT-Frolicher-inequality-deformed}, $X$ satisfies the $\partial\bar{\partial}_{\phi(t)}$-lemma if and only if
\begin{equation}\label{eq-ddbar-phi-lemma-kehua=}
h_{BC\phi(t)}^{k}+ h_{A\phi(t)}^{k} = 2b_k.
\end{equation}
We note that by Theorem~\ref{th-analytic subset} the set $\{t\in B\mid \eqref{eq-ddbar-phi-lemma-kehua=}~\text{holds}\}$ is an analytic open subset of $B$ since
\[
\{t\in B\mid h_{BC\phi(t)}^{k}+ h_{A\phi(t)}^{k} = 2b_k,~\forall~k\}= B\setminus \{t\in B\mid h_{BC\phi(t)}^{k}+ h_{A\phi(t)}^{k}\geq 2b_k+1,~\forall~k\}.
\]
\end{proof}
In particular, if $X$ satisfies the $\partial\bar{\partial}$-lemma, then by the above corollary $X$ also satisfies the $\partial\bar{\partial}_{\phi(t)}$-lemma for any small $t\in B$. Combining this with Remark~\ref{rk-to-AK-Frolicher-ineq}, we get that the Hodge numbers $\dim H_{\bar{\partial}_t}^{p,q}(X_t)$ and $\dim H_{BC\phi(t)}^{p,q}(X)$ are independent of $t$.

Recall that a smooth manifold $X$ is called \emph{formal} if its de Rahm complex $(A^{\bullet}(X),d)$ is formal as a differential graded algebra (DGA for short). The later means that there is a sequence of quasi-isomorphisms from $(A^{\bullet}(X),d)$ to its cohomology algebra $(H_{dR}^{\bullet}(X),0)$\footnote{Here, $(H^{\bullet}(X),0)$ is considered as a differential graded algebra with trivial differential.}, see~\cite{DGMS75,FHT01}.
\begin{theorem}\label{thm-formal}
Let $X$ be a compact complex manifold and $X_t$ a small deformation (of $X$) whose complex structure is represented by a Beltrami differential $\phi\in A^{0,1}(X, T_{X}^{1,0})$. If $X$ satisfies the $\partial\bar{\partial}_\phi$-lemma, then $X$ is formal.
\end{theorem}
\begin{proof}Consider the following homomorphisms of DGA
\begin{equation*}
\xymatrix@C=0.5cm{
(A^{\bullet}(X),d_\phi) && \ar[ll]_{i\quad}  (A^{\bullet}(X)\cap\ker d_\phi^c ,d_\phi)   \ar[rr]^{\quad p} && (H^{\bullet}_{d_\phi^c}(X),d_\phi=0),}
\end{equation*}
where $i$ is the inclusion and $p$ is the projection. We claim that the induced map $i^*$ is an isomorphism on cohomology. Indeed, $\forall x\in \ker d_\phi\cap \ker d_\phi^c$ if $x\in \Image d_\phi$ then by \eqref{eq-dphi-dphic-lemma-1}, $x\in\Image d_\phi d_\phi^c\Rightarrow i^*$ is injective; on the other hand, by \eqref{eq-dphi-dphic-lemma-2} $\forall x\in \ker d_\phi$ there exist $y\in A^{\bullet}(X)$ such that $x-d_\phi y\in \ker d_\phi\cap \ker d_\phi^c$, this shows that $i^*$ is surjective. Similarly, one shows that $p^*$ is an isomorphism on cohomology and $d_\phi=0$ on $H^{\bullet}_{d_\phi^c}(X)$. The conclusion then follows since $(A^{\bullet}(X),d_\phi)$ is isomorphic to $(A^{\bullet}(X),d)$ and $H^{\bullet}_{d_\phi^c}(X)\cong H^{\bullet}_{d_\phi}(X)\cong H^{\bullet}_{dR}(X)$.
\end{proof}

\section{The deformed Bott-Chern cohomology of the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold}
\label{The deformed Bott-Chern cohomology of the Iwasawa manifold and the holomorphically parallelizable Nakamura manifold}

\begin{example}
Case III-(2). Let $G$ be the matrix Lie group defined by
\[
G := \left\{
\left(
\begin{array}{ccc}
 1 & z^1 & z^3 \\
 0 &  1  & z^2 \\
 0 &  0  &  1
\end{array}
\right) \in \mathrm{GL}(3;\mathbb{C}) \mid z^1,\,z^2,\,z^3 \in\mathbb{C} \right\}\cong \mathbb{C}^3
\;.
\]
Consider the discrete subgroup $\Gamma$ defined by
\[
\Gamma := \left\{
\left(
\begin{array}{ccc}
 1 & \omega^1 & \omega^3 \\
 0 &  1  & \omega^2 \\
 0 &  0  &  1
\end{array}
\right) \in G \mid \omega^1,\,\omega^2,\,\omega^3 \in\mathbb{Z}[\sqrt{-1}] \right\}
\;,
\]
The quotient $X=G/\Gamma$ is called the \emph{Iwasawa manifold}. A basis of $H^0(X,\Omega^1)$ is given by
\[
\varphi^1 = d z^1,~ \varphi^2 = d z^2,~ \varphi^3 = d z^3-z^1\,d z^2,
\]
and a dual basis $\theta^1, \theta^2, \theta^3\in H^0(X,T_X^{1,0})$ is given by
\[
\theta^1=\frac{\partial}{\partial z^1},~\theta^2=\frac{\partial}{\partial z^2} + z^1\frac{\partial}{\partial z^3},~\theta^3=\frac{\partial}{\partial z^3}.
\]
$X$ is equipped with the Hermitian metric $\sum_{i=1}^3\varphi^i\otimes\bar{\varphi}^i$. The Beltrami differential of the Kuranishi family of $X$ is
\[
\phi(t) = t_{i\lambda}\theta^i\bar{\varphi}^{\lambda} - D(t)\theta^3\bar{\varphi}^{3},~\text{with}~D(t)=t_{11}t_{22}-t_{21}t_{12},
\]
and the Kuranishi space of $X$ is
\[
\mathcal{B}=\{t=(t_{11}, t_{12}, t_{21}, t_{22}, t_{31}, t_{32})\in \mathbb{C}^6\mid |t_{i\lambda}|<\epsilon, i=1, 2, 3, \lambda=1,2 \},
\]
where $\epsilon>0$ is sufficiently small. Set
\[
\phi_1=\sum_{i=1}^3\sum_{\lambda=1}^2t_{i\lambda}\theta^i\bar{\varphi}^{\lambda},~ \phi_2 = D(t)\theta^3\bar{\varphi}^{3},
\]
and write the canonical deformation of $\sigma_0\in H_{BC}^{p,q}(X)$ by $\sigma(t)=\sum_k \sigma_k$ with each
\[
\sigma_k= - G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i ,
\]
being the homogeneous term of degree $k>0$ in $t\in \mathcal{B}$. We will use the isomorphism $H_{BC\phi(t)}^{p,q}(X)\cong\dim V_t/\ker f_t$ proved in Proposition \ref{prop V-E-t} to compute $\dim H_{BC\phi(t)}^{p,q}(X)$. Since $\mathcal{B}$ is a polydisc, it is sufficient to check the coefficients of $d_{\phi(t)}\sigma(t)=0$, that is,
\begin{equation}\label{obstruction eq in example}
\partial\sigma_k= \bar{\partial}\sigma_k+ \sum_{j=1}^{k} \partial i_{\phi_j} \sigma_{k-j}=0 , ~~k>0~ .
\end{equation}
\end{example}

Let us now consider Bott-Chern deformations of forms in the harmonic space:
\[
\mathcal{H}_{BC}^{2,2}(X)=\mathbb{C}\{\varphi^{12\overline{13}}, \varphi^{12\overline{23}}, \varphi^{13\overline{12}}, \varphi^{13\overline{13}}, \varphi^{13\overline{23}}, \varphi^{23\overline{12}}, \varphi^{23\overline{13}}, \varphi^{23\overline{23}}\}.
\]
Set $\sigma_0 = \sum a_{ijkl}\varphi^{ij\overline{kl}}\in \mathcal{H}_{BC}^{2,2}(X)$, then
\[
\partial i_{\phi_1}\sigma_0= (-t_{12}a_{13\overline{13}} + t_{11}a_{13\overline{23}} - t_{22}a_{23\overline{13}} + t_{21}a_{23\overline{23}}) \varphi^{12\overline{123}}
\]
is $\bar{\partial}$-exact if and only if
\begin{equation}\label{a_{23}}
t_{12}a_{13\overline{13}} - t_{11}a_{13\overline{23}} + t_{22}a_{23\overline{13}} - t_{21}a_{23\overline{23}}=0 ,
\end{equation}
and in this case
\[
\sigma_{1}=-G_{BC}A \partial i_{\phi_1}\sigma_0 =0~ .
\]
But
\[
\partial i_{\phi_2}\sigma_0 = 0 \Longrightarrow \sigma_{2}=-G_{BC}A \partial  ( i_{\phi_2}\sigma_0 + i_{\phi_1}\sigma_1 ) =0~ ,
\]
and $\phi_k=0,~ k>2$ we thus have $\sigma_{k}=0,~ k>2$.

Therefore, for $V=\mathcal{H}_{BC}^{2,2}(X)$ we have (see Definition \ref{def-V_t})
\begin{align*}
V_{t}=
&\{ \sum a_{ijkl}\varphi^{ij\overline{kl}}\in \mathcal{H}_{BC}^{2,2}(X) \mid  (a_{1213}, a_{1223}, a_{1312}, a_{1313}, a_{1323}, a_{2312}, a_{2313}, a_{2323})\in \mathbb{C}^8\\&~\text{s.t.}~\sigma(t)\in\ker d_{\phi(t)},
\text{where}~\sigma (t)=\sum_{k} \sigma_k~\text{with}~ \sigma_0=\sum a_{ijkl}\varphi^{ij\overline{kl}}\\
&~\text{and}~ \sigma_{k}=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i,~\forall k\neq 0 \}\\
=&\{\sum a_{ijkl}\varphi^{ij\overline{kl}} \mid (a_{1213}, a_{1223}, a_{1312}, a_{1313}, a_{1323}, a_{2312}, a_{2313}, a_{2323})\in \mathbb{C}^8~\text{satisfy}~ \eqref{a_{23}}\}.
\end{align*}
On the other hand, $\Image\partial\bar{\partial}_{\phi(t)}=\mathbb{C}\{\partial\bar{\partial}_{\phi(t)}\varphi^{3\bar{3}}\}=\mathbb{C}\{\varphi^{12\overline{12}}\}$ (Since $X$ is parallelizable, we only need to consider left invariant forms. See the discussions in the last paragraph of this section) and
\[
(\partial\bar{\partial})^*\varphi^{12\overline{12}}=(*\partial*)(*\bar{\partial}*)\varphi^{12\overline{12}}
=-(*\partial\bar{\partial}*)\varphi^{12\overline{12}}=-*\partial\bar{\partial}\varphi^{3\bar{3}}=\varphi^{3\bar{3}}\neq 0,
\]
implies
\[
\ker f_t\cong \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)} \cap A^{2,2}(X)=0.
\]
By Proposition \ref{prop V-E-t} we have
\begin{equation}
\dim H_{BC\phi(t)}^{2,2}(X)=\dim V_t-\dim\ker f_t=
\left\{
\begin{array}{rcl}
8,~ &(t_{11}, t_{12}, t_{21}, t_{22})=0 \\[5pt]
7,~ &(t_{11}, t_{12}, t_{21}, t_{22})\neq0 ~.
\end{array}
\right.
\end{equation}

The other deformed Bott-Chern cohomology can be computed in the same way. Write $(i), (ii), (iii)$ for the three cases when $(t_{11}, t_{12}, t_{21}, t_{22})=0$, $(t_{11}, t_{12}, t_{21}, t_{22})\neq0$ and $D(t)= 0$, $D(t)\neq 0$, respectively. Then we have the following (where $h^{p,q}:=\dim H_{BC\phi(t)}^{p,q}(X)$ and $t\in (i), (ii), (iii)$, respectively)
\renewcommand\arraystretch{1.5}
\begin{table}[!htbp]
\centering
\begin{center}
\begin{tabular}{|cc|ccc|cccc|ccc|cc|}
\hline
 $h^{1,0}$ & $h^{0,1}$ & $h^{2,0}$ & $h^{1,1}$ & $h^{0,2}$ & $h^{3,0}$ & $h^{2,1}$ & $h^{1,2}$ & $h^{0,3}$ & $h^{3,1}$ & $h^{2,2}$ & $h^{1,3}$ & $h^{3,2}$ & $h^{2,3}$\\
\hline
 $2$ & $2$ & $3$ & $4$ & $3$ & $1$ & $6$ & $6$ & $1$ & $2$ & $8$ & $2$ & $3$ & $3$ \\
\hline
 $2$ & $2$ & $2$ & $4$ & $3$ & $1$ & $6$ & $6$ & $1$ & $2$ & $7$ & $2$ & $3$ & $3$\\
\hline
 $2$ & $2$ & $1$ & $4$ & $3$ & $1$ & $6$ & $6$ & $1$ & $2$ & $7$ & $2$ & $3$ & $3$\\
\hline
\end{tabular}
\end{center}
\end{table}

Comparing this with the computations made by Angella~\cite{Ang13} we see that $\dim H_{BC\phi(t)}^{p,q}(X)=\dim H_{BC}^{p,q}(X_t)$ is not true in general for $p=q$.

\begin{example}
Case III-(3b). Let $X=\mathbb{C}^3/\Gamma$ be the solvable manifold constructed by Nakamura in Example III-(3b) of~\cite{Nak75}. We have
\begin{align*}
H^0(X,\Omega_X^{1})~ =& ~\mathbb{C}\{\varphi^1 = d z^1,~ \varphi^2 = e^{z_1} d z^2,~ \varphi^3 = e^{-z_1} d z^3\} ~,\\
H^0(X,T_X^{1,0})~  =& ~\mathbb{C}\{\theta^1=\frac{\partial}{\partial z^1},~\theta^2= e^{-z_1} \frac{\partial}{\partial z^2},~\theta^3= e^{z_1} \frac{\partial}{\partial z^3}\} ~,\\
\mathcal{H}^{0,1}(X)~ =& ~\mathbb{C}\{ \psi^{\bar{1}} = d z^{\bar{1}},~ \psi^{\bar{2}} = e^{z_1} d z^{\bar{2}},~ \psi^{\bar{3}} = e^{-z_1} d z^{\bar{3}}\} ~,\\
\mathcal{H}^{0,1}(X, T_X^{1,0})~ =& ~\mathbb{C}\{\theta^i\psi^{\bar{\lambda}} ,~ i=1, 2, 3, \lambda=1, 2, 3\} ~,
\end{align*}
where $X$ is equipped with the Hermitian metric $\sum_{i=1}^3\varphi^i\otimes\bar{\varphi}^i$. The Beltrami differential of the Kuranishi family of $X$ is
\[
\phi(t) = \phi_1 = t_{i\lambda}\theta^i\psi^{\bar{\lambda}}
\]
and the Kuranishi space of $X$ is
\[
\mathcal{B}=\{t=(t_{11}, t_{12}, t_{13}, t_{21}, t_{22}, t_{23}, t_{31}, t_{32}, t_{33})\in \mathbb{C}^9\mid |t_{i\lambda}|<\epsilon, i=1, 2, 3, \lambda=1, 2, 3 \},
\]
where $\epsilon>0$ is sufficiently small. We will restrict to the one parameter family defined by $t_{12}=t_{13}=t_{21}=t_{22}=t_{23}=t_{31}=t_{32}=t_{33}=0$ and in this case the Beltrami differential is $\phi=\phi(t)=t\frac{\partial}{\partial z^1}dz^{\bar{1}}$ where $t=t_{11}$.

Let us consider the Bott-Chern deformations of forms in
\begin{align*}
\mathcal{H}_{BC}^{2,1}(X)=\mathbb{C}\{&e^{z^1}dz^{12\overline{1}}, e^{2z^1}dz^{12\overline{2}}, dz^{12\overline{3}}, e^{-z^1}dz^{13\overline{1}}, dz^{13\overline{2}}, e^{-2z^1}dz^{13\overline{3}}, dz^{23\overline{1}},\\ &e^{z^{\bar{1}}}dz^{13\overline{1}}, e^{z^{\bar{1}}}dz^{12\overline{1}}\}.
\end{align*}
Set
\begin{align*}
\sigma_0 =& a_{121}e^{z^1}dz^{12\overline{1}}+a_{122}e^{2z^1}dz^{12\overline{2}}+a_{123}dz^{12\overline{3}}+a_{131}e^{-z^1}dz^{13\overline{1}}+a_{132}dz^{13\overline{2}}\\&+a_{133}e^{-2z^1}dz^{13\overline{3}} +a_{231}dz^{23\overline{1}}+b_{131}e^{z^{\bar{1}}}dz^{13\overline{1}}+b_{121}e^{z^{\bar{1}}}dz^{12\overline{1}},
\end{align*}
then
\[
\partial i_{\phi_1}\sigma_0= -2a_{122}te^{2z^1}dz^{12\overline{12}} + 2a_{133}te^{-2z^1}dz^{13\overline{13}}
\]
is $\bar{\partial}$-exact if and only if $t=0$.
Therefore, for $V=\mathcal{H}_{BC}^{2,1}(X)$ and $t\neq 0$ we have
\begin{align*}
V_{t}=
&\{ \sigma_0\in \mathcal{H}_{BC}^{2,1}(X) \mid  (a_{121}, a_{122}, a_{123}, a_{131}, a_{132}, a_{133}, a_{231}, b_{131}, b_{121} )\in \mathbb{C}^9~\text{s.t.}~\\&\sigma(t)\in\ker d_{\phi(t)},
\text{where}~\sigma (t)=\sum_{k} \sigma_k~\text{with}~\sigma_{k}=-G_{BC}A\sum_{i+j=k} \partial i_{\phi_j} \sigma_i,~\forall k\neq 0 \}\\
=&\mathbb{C}\{e^{z^1}dz^{12\overline{1}}, dz^{12\overline{3}}, e^{-z^1}dz^{13\overline{1}}, dz^{13\overline{2}}, dz^{23\overline{1}}, e^{z^{\bar{1}}}dz^{13\overline{1}}, e^{z^{\bar{1}}}dz^{12\overline{1}}\}.
\end{align*}
On the other hand,
\begin{align*}
\ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)} \cap A^{2,1}(X)=&\mathbb{C}\{\partial\bar{\partial}_{\phi(t)}e^{z^1}dz^{2}, \partial\bar{\partial}_{\phi(t)}e^{-z^1}dz^{3}\}\\=&\mathbb{C}\{te^{z^1}dz^{12\bar{1}}, te^{-z^1}dz^{13\bar{1}}\}
\end{align*}
and
\[
\dim\ker f_t= \dim \ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)} \cap A^{2,2}(X)=2 .
\]
By Proposition \ref{prop V-E-t} we have
\begin{equation}
\dim H_{BC\phi(t)}^{2,1}(X)=\dim V_t-\dim\ker f_t=
\left\{
\begin{array}{rcl}
9,~ &t=0 \\[5pt]
5,~ &t\neq0 ~.
\end{array}
\right.
\end{equation}
We summarise the computations of the deformed Bott-Chern cohomology in this case as follows (where $h^{p,q}:=\dim H_{BC\phi(t)}^{p,q}(X)$ and $t=0,~\neq 0$, respectively):
\renewcommand\arraystretch{1.5}
\begin{table}[!htbp]
\centering
\begin{center}
\begin{tabular}{|cc|ccc|cccc|ccc|cc|}
\hline
 $h^{1,0}$ & $h^{0,1}$ & $h^{2,0}$ & $h^{1,1}$ & $h^{0,2}$ & $h^{3,0}$ & $h^{2,1}$ & $h^{1,2}$ & $h^{0,3}$ & $h^{3,1}$ & $h^{2,2}$ & $h^{1,3}$ & $h^{3,2}$ & $h^{2,3}$\\
\hline
 $1$ & $1$ & $3$ & $7$ & $3$ & $1$ & $9$ & $9$ & $1$ & $3$ & $11$ & $3$ & $5$ & $5$ \\
\hline
 $1$ & $1$ & $1$ & $5$ & $3$ & $1$ & $5$ & $7$ & $1$ & $1$ & $7$  & $3$ & $3$ & $3$\\
\hline
\end{tabular}
\end{center}
\end{table}

From this table and~\cite{AK17b}, we notice that $X_t$ satisfy the $\partial\bar{\partial}$-lemma but $X$ does not satisfy the $\partial\bar{\partial}_{\phi(t)}$-lemma for any $t\neq 0$ .
\end{example}

We need to point out that in the above computations (especially those concerning $\ker(\partial\bar{\partial})^*\cap\Image\partial\bar{\partial}_{\phi(t)}$), only invariant forms is considered. This is valid because the Bott-Chern cohomology of complex parallelizable manifold may be computed by left invariant forms~\cite{Ang13} and given a family of deformations $\{X_t\}_{t\in B}$ of such manifolds the set of $t$ for which the deformed Bott-Chern cohomology may be computed by left invariant forms is an open subset of $B$ (this will be proved in \cite{Xia20g}).

\vskip 1\baselineskip \textbf{Acknowledgements.} I would like to thank Prof. Kefeng Liu for his constant encouragement and many useful discussions. Many thanks to Shengmao Zhu, Sheng Rao and Daniele Angella for useful communications. I would also like to thank Prof. Bing-Long Chen for his constant support.

\bibliographystyle{alpha}


TITLE: Showing the existence of a right-inverse in a von Neumann algebra
QUESTION [1 upvotes]: In this paper, specifically Theorem 4.1 on page 10, one of the last steps in the proof involves saying that a particular right-inverse exists. I'll try to restate what I think are the relevant pieces of info here, instead of just rewriting the whole proof.
Let $\mathfrak{A}$ be a von Neuman algebra with a faithful normal state $\rho$. In the last paragraph, the author looks at a self-adjoint, positive element $y \in \mathfrak{A}$, and then defines $q = \chi_{(1/2, 1]}(y)$, where this latter expression is meant in the sense of the Borel functional calculus on a self-adjoint $y \in \mathfrak{A}$. This $q$ satisfies
$$q \leq 2y \leq q + 1 \; (\dagger).$$
He then claims that there must exist $z \in \mathfrak{A}$ such that $\| z \|\leq 2$ and
$$1 - q = (1- y) z \; (\dagger \dagger) .$$
I'm confused by how he gets the right-hand inequality in $(\dagger)$, and how he conjures up the $z$ in $(\dagger \dagger)$. If anyone can clear this up, I'd greatly appreciate it!

REPLY [2 votes]: The first inequality is most likely a typo, it should be
$$
q\leq 2yq\leq 1+q.
$$
This is simply functional calculus on the inequalities
$$
1_{\big(\tfrac12,1\big]}(t)\leq 2t\,1_{\big(\tfrac12,1\big]}(t)\leq 2\,1_{\big(\tfrac12,1\big]}(t)\leq 1+1_{\big(\tfrac12,1\big]}(t).
$$
For $z$, take
$$
h(t)=\frac1{1-t}\,1_{\big[0,\tfrac12\big]}(t)
$$
and define $z=h(y)$. Since $|h(t)|\leq2$, you get that $\|z\|\leq2$.


TITLE: Why $(\log n)^k = O(n)$
QUESTION [0 upvotes]: If I have $(\log_2(n))^{100}   $ , then what is $C$ and $k$,so that $(\log_2(n))^{100}≤ C\times n?$
I know that $(\log_2(n))^{100}   $ ≤ $n^{100}$, but I cannot find the $C$ and $k$ of $(\log_2(n))^{100}≤ C\times n$,since the leftside is always bigger than the right.

REPLY [0 votes]: Let $x=\log_2n$ Then (at least if $n\ge0$) $n=\exp(x\ln 2)=1+x\ln 2+(x\ln 2)^2/2!
+\cdots+(x\ln 2)^{100}/100!+\cdots$, so take $C=100!/(\ln2)^{100}$.

REPLY [0 votes]: It is mostly based on limit; polynomially defined function is eventually surpassing $lg$ function starting from some value.


TITLE: Troublesome Limit
QUESTION [0 upvotes]: Let $x \geq 1.$ I am trying to find
$$\lim_{n \to \infty} \sum_{i=1}^n \frac{1}{n + i^x}$$
My attempt: For $x = 1$ this is a Riemann sum
$$\lim_{n \to \infty} \sum_{i=1}^n \frac{1}{n + i} = \lim_{n \to \infty} \sum_{i=1}^n \frac{\frac{1}{n}}{1 + \frac{i}{n}} = \int_0^1 \frac{dx}{1+x} = \ln2.$$
For $x > 1$, I'm not sure how to proceed or if the limit exists for all $x$.
Thanks for any help.

REPLY [0 votes]: For any $n\gt0$ and any $x\gt0$, the function $f(u)=1/(n+u^x)$ is strictly decreasing for $u\gt0$. Therefore
$$0\lt\sum_{i=1}^n{1\over n+i^x}\lt\int_0^n{dt\over n+t^x}=\int_0^1{n\,du\over n+n^xu^x}=\int_0^1{du\over1+n^{x-1}u^x}$$
For any $\epsilon\gt0$ we have
$$\begin{align}
\int_0^1{du\over1+n^{x-1}u^x}
&=\int_0^\epsilon{du\over1+n^{x-1}u^x}+\int_\epsilon^1{du\over1+n^{x-1}u^x}\\
&\lt\int_0^\epsilon du+\int_\epsilon^1{du\over n^{x-1}u^x}\\
&=\epsilon+{1\over(x-1)n^{x-1}}(\epsilon^{1-x}-1)
\end{align}$$
and thus, if $x\gt1$,
$$0\le\lim_{n\to\infty}\sum_{i=1}^n{1\over n+i^x}\le\lim_{n\to\infty}\left(\epsilon+{1\over(x-1)n^{x-1}}(\epsilon^{1-x}-1)\right)=\epsilon$$
for all $\epsilon\gt0$, hence the limit must be $0$.


TITLE: Projection in $\ell_1$ norm onto linear subspace of $\mathbb{R}^n$.
QUESTION [3 upvotes]: In the context of $\mathbb{R}^n$, fix a linear subspace $\mathcal{U}$ or arbitrary dimension $r \leq n$ and some vector $\mathbf{x} \in \mathbb{R}^n$.
What is the most efficient way to compute the projection of $\mathbf{x}$ onto $\mathcal{U}$ in $\ell_1$, that is, how had one ought to solve the optimization problem:
$\mathbf{u}^* = \min_{\mathbf{u} \in \mathcal{U}} ||\mathbf{u} - \mathbf{x}||_1$
I have a hard time believing this is not a well understood problem but I am having difficulty with Google.
One could clearly formulate this as a linear program, but I am primarily concerned with high dimensional problems for $n$ large. Is there a more efficient way?

REPLY [2 votes]: Well, $\displaystyle\min_{\mathbf{u}\in\mathcal{U}}||\mathbf{u}-\mathbf{x}||_1=\min\{r\in\mathbb{R}_{\geqslant 0} : \mathcal{U}\cap B_1(\mathbf{x},r)\neq\emptyset\}$, where
$$B_1(\mathbf{x},r)=\{\mathbf{y}\in\mathbb{R}^n : ||\mathbf{y}-\mathbf{x}||_1\leqslant r\}$$
is the closed $l_1$-ball with center $\mathbf{x}$ and radius $r$. It is the convex hull of
$$E_1(\mathbf{x},r)=\{\mathbf{x}\pm r\mathbf{e}_k : 1 \leqslant k \leqslant n\},$$
where $(\mathbf{e}_1,\ldots,\mathbf{e}_n)$ is the standard basis of $\mathbb{R}^n$. It follows that
$$\displaystyle\min_{\mathbf{u}\in\mathcal{U}}||\mathbf{u}-\mathbf{x}||_1=r^\ast\implies\mathcal{U}\cap E_1(\mathbf{x},r^\ast)\neq\emptyset,$$
the minimum is attained at any (not necessarily unique) element $\mathbf{u}^\ast$ of the last set, and the convex hull of all such $\mathbf{u}^\ast$ is the solution to the original problem.
So the whole problem reduces to finding the minimum of at most $n$ real numbers
$$r_k = \inf\{|r| : \mathbf{x}+r\mathbf{e}_k\in\mathcal{U}\},\quad 1\leqslant k\leqslant n$$
(with $\inf\ \emptyset$ considered to be $+\infty$).


TITLE: Oka-Grauert principle, up to the boundary
QUESTION [7 upvotes]: Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of transition functions $(U_{\alpha\beta}:h_{\alpha\beta}\rightarrow GL(m,\mathbb{C}))$ which are holomorphic in $U_{\alpha\beta}\backslash \partial Z$ and smooth up to the boundary.
Suppose $E\rightarrow \bar Z$ is a holomorphic vector bundle, which is trivial as continuous vector bundle. The Oka-Grauert principle then implies that $E\vert_{Z}$ is holomorphically trivial, but does not say anything about whether there exists a global holomorphic frame that extends smoothly up to $\partial Z$. As Donaldson remarks in this paper from 1992, 'the result is almost certainly true', but like him I have 'unfortunately not been able to find such a result in the literature' (he gives an ad hoc proof in $n=2$).
In this paper by Leiterer (1990) there is actually almost the right thing: Theorem 10.1 gives the result for holomorphic vector bundles, which are continuous up to $\partial Z$. That means in the situation above we obtain a global frame that extends continuously to $\partial Z$, but not smoothly.
Questions.

Has somebody since made the effort to write up some sort of Oka-Grauert principle for holomorphic vector bundles that are smooth up to the boundary?
Is Leiterer's paper (or his original article in German, referenced in there) the best reference for the continuous case?
Are there nice elementary approaches for special cases of $Z$'s? (E.g. Donaldson gives an argument for $n=2$ and for $Z$ homeomorphic to a ball.)

REPLY [3 votes]: This is not a full answer, but sketches an approach that works for $n=1$ and likely generalises to higher dimensions with a finite amount of work. Note however, that for $n=1$ there is also an easy shortcut that I will explain first:
Shortcut for $n=1$: In one complex dimension, any holomorphic vector bundle on $\bar Z$ can be extended to a neighbourhood of $\bar Z$ in $\mathbb C$. One can then apply the classical Oka-Grauert principle on the neighbourhood and restrict the resulting trivialisation to $\bar Z$. It will automatically be smooth. (In higher dimensions this does not work, because holomorphic vector bundles cannot simply be extended, as there is a nontrivial integrability condition that the extension would need to satisfy).
Overview of the general approach: The main idea is to view the Oka-Grauert principle as transitivity of a certain group action and try to prove this using an inverse function theorem. This requires a priori estimates for a $\bar \partial$-problem on manifolds with boundary. If $n=1$ these should become standard elliptic estimates. For $n\ge 2$ one has to work harder, but all the right tools should be contained in Hamilton's series of papers on "Deformation of complex structures on manifolds with boundary."

The group action: Let us write $X=\bar Z$ for the closure of $Z$ and assume that this is a compact manifold with smooth boundary. A holomorphic vector bundle structure on the trivial bundle $X\times \mathbb C^m$ is a $(0,1)$-form $\alpha\in \Omega^{0,1}(X,\mathbb C^{m\times m})$, understood to be smooth up to the boundary, that satisfies the integrability condition
$$
F_\alpha = \bar \partial \alpha + \alpha\wedge\alpha = 0.
$$
The holomorphic vector bundle $E_\alpha=(X\times \mathbb C^m,\alpha)$ is trivial, if there exists a function $f\in C^\infty(X,GL(m,\mathbb C))$ such that $\bar \partial f + \alpha f = 0$. Note that $\mathbb G = C^\infty(X,GL(m,\mathbb C))$ is a group that acts on the vector space $\mathcal X = \Omega^{0,1}(X,\mathbb C^{m\times m})$ from the right by the rule
\begin{equation}
\alpha.f=f^{-1}(\bar \partial f + \alpha f).
\end{equation}
One can check that this action leaves the subspace $\mathcal X_0=\{\alpha \in \mathcal X: F_\alpha =0\}$ invariant. Further, the assertion that all vector bundles $E_\alpha$ (with $\alpha\in \mathcal X_0$) are trivial is equivalent to the group $\mathbb G$ acting transitively on $\mathcal X_0$.

The linearisation: To show transitivity one can consider the derivative of $\mathbb G \ni f\mapsto \alpha.f$ for fixed $\alpha \in \mathcal X_0$, which is a linear map
$$
 D_\alpha \colon T_{\mathrm{Id}} \mathbb G=C^\infty(X,\mathbb C^{m\times m})\rightarrow T_{\alpha}\mathcal X_0\subset \Omega^{0,1}(X,\mathbb C^{m\times m}).
$$
If $D_\alpha$ is onto and we have a nice function theorem at our disposal, then we can conclude that the map $f\mapsto \alpha.f$ is locally surjective near $\mathrm{Id}$, which is to say that small perturbations of $\alpha$ within $\mathcal X_0$ stay within the same $\mathbb G$-orbit. If this is true for all $\alpha \in \mathcal X_0$ and additionally $\mathcal X_0$ is connected, then there can only be one orbit and thus the action has to be transitive. We can describe $D_\alpha$ more explicity: Define the holomorphic vector bundle $E_\alpha^{\mathrm{ad}}$ as the bundle $X\times \mathbb C^{n\times n}$ with holomorphic structure induced by the $\bar \partial$-operator
$$
\bar \partial_{\alpha}^{\mathrm{ad}}\colon \Omega^{p,q}(X,E_\alpha^\mathrm{ad}) \rightarrow \Omega^{p,q+1}(X,E_\alpha^\mathrm{ad}),\quad \beta \mapsto   \bar \partial \beta  + [\alpha,\beta].
$$
One can then identify (at least formally)
$$
T_{\mathrm{Id}}\mathbb G\cong \Omega^0(X,E_\alpha^{\mathrm{ad}}),~ T_{\alpha}\mathcal X_0 = \mathrm{ker}(\bar \partial_{\alpha}^{\mathrm{ad}} )\subset  \Omega^{0,1}(X,E^{\mathrm{ad}}_\alpha),\quad D_\alpha= \bar \partial_{\alpha}^{\mathrm{ad}},
$$
such that the surjectivity of $D_\alpha$ is equivalent to the vanishing result
$$
H^{0,1}(X,E_\alpha^{\mathrm{ad}})=0,
$$
where the cohomology is defined in terms of the $\bar \partial_\alpha^{\mathrm{ad}}$-complex for differential forms that are smooth up to the boundary.
This vanishing result is also mentioned as Proposition 4 in Donaldson's paper from the question above.

The inverse function theorem: If we don't want to leave the smooth setting, we can try to apply Theorem 2.4.1 in Hamilton's paper on the Nash-Moser inverse function theorem. This requires that $\mathbb G$, $\mathcal X_0$ and the action itself are smooth tame and that $D_\alpha$ has a tame right inverse $R_\alpha$. Note that $R_\alpha$ ought to map any $\beta\in\mathrm{ker}~\bar \partial_{\alpha}^{\mathrm{ad}}\subset \Omega^{0,1}(X,E_\alpha^{\mathrm{ad}})$ to a solution $u\in \Omega^{0}(X,E_\alpha^{\mathrm{ad}})$ of $\bar \partial_{\alpha}^{\mathrm{ad}}u=\beta$ and tameness means that
$$
    \Vert R_\alpha \beta \Vert_{s} \lesssim \Vert \beta \Vert_{s+r} \tag{$\dagger$}
$$
for all $s\in \mathbb N$ and some fixed $r\in \mathbb N$, where $\Vert \cdot \Vert_s$ denotes, say, a Sobolev norm of order $s$. The fact that we can indeed solve the equation $\bar \partial_{\alpha}^{\mathrm{ad}}u=\beta$ just comes from  $H^{0,1}(X,E_\alpha^{\mathrm{ad}})=0$, but the estimates require extra work.

In general, it is not clear to me whether $\mathcal X_0$ is indeed a manifold at all and one might have to use a a more complicated inverse function theorem, such as Theorem 3 in Hamilton's deformation paper (with $Q$ being the map $\alpha \mapsto F_\alpha$).

The use of the Nash-Moser theorem can probably be avoided by working instead with $\mathbb G^{s+1}$ and $\mathcal X^s$ defined as above, but now containing elements of finite regularity $s+1$ and $s$, respectively. The resulting gauge $f$ can then be shown to be smooth using elliptic regularity.

The case $n=1$ Here the integrability condition disappears and the linearised problem reduces to the following: given $\alpha\in \mathcal X_0=\mathcal X\cong C^\infty(X,\mathbb C^{m\times m})$, find a solution operator $R_\alpha \colon C^\infty(X,\mathbb C^{m})\rightarrow C^\infty(X,\mathbb C^{m})$ such that $u=R_\alpha \beta$ solves $(\partial_{\bar z} + \alpha)u=\beta$ and such that tame estimates as in $(\dagger)$ are satisfied. Let $\Delta_\alpha=(\partial_{z}+\bar \alpha)(\partial_{\bar z} +\alpha)$, then $\Delta_\alpha u = (\partial_z +\bar \alpha)\beta$, so $u$ can be found by solving a (perturbed) Laplace equation and the required estimates should all be standard.