text:
\section*{Introduction}

On compact oriented Riemannian manifolds, Hodge theory allows to compute cohomology solving systems of differential equations.
For \emph{nilmanifolds}, namely, compact quotients of connected simply-connected nilpotent Lie groups by co-compact discrete subgroups, K. Nomizu proved in \cite[Theorem 1]{nomizu} that the de Rham complex admits a finite-dimensional subcomplex, defined in Lie theoretic terms, as minimal model.
Furthermore, also the Dolbeault cohomology often reduces to the cohomology of the corresponding finite-dimensional complex of forms on the Lie algebra: this happens, for example, for holomorphically parallelizable complex structures, as proved by Y. Sakane in \cite[Theorem 1]{sakane}, or for rational complex structures, as proved by S. Console and A. Fino in \cite[Theorem 2]{console-fino}; see the surveys \cite{console} and \cite{rollenske-survey} and also \cite{rollenske} for more results in this direction.

Together with the Dolbeault cohomology, the \emph{Bott-Chern cohomology} provides an important tool to study the geometry of complex manifolds: it is the bi-graded algebra defined by
$$ H^{\bullet,\bullet}_{BC}(X) \;:=\; \frac{\ker\partial\cap\ker\overline{\del}}{\imm\partial\overline{\del}} \;,$$
$X$ being a complex manifold. It turns out that there is a Hodge theory also for the Bott-Chern cohomology (see, e.g., \cite[\S2]{schweitzer}) and therefore, if $X$ is compact, then the $\mathbb{C}$-vector space $H^{\bullet,\bullet}_{BC}(X)$ is finite-dimensional; furthermore, if $X$ is a compact K\"ahler manifold or, more in general, if $X$ is in Fujiki class $\mathcal{C}$, then the Dolbeault and Bott-Chern cohomologies coincide and they give a splitting for the de Rham cohomology algebra.

In this paper, we give some tools to compute the Bott-Chern cohomology ring of certain compact complex homogeneous manifolds.\\
More precisely, let $N=\left.\Gamma\right\backslash G$ be a nilmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$ and its complexification by $\mathfrak{g}_\mathbb{C}:=\mathfrak{g}\otimes_\mathbb{R}\mathbb{C}$.
Dealing with $G$-left-invariant objects on $N$, we mean objects whose pull-back to $G$ is invariant for the left-action of $G$ on itself.
For any $k\in\mathbb{N}$ and $p,q\in\mathbb{N}$, we denote the space of smooth global sections of the bundle of real $k$-forms (respectively, complex $k$-forms, $(p,q)$-forms) on $N$ by the symbol $\wedge^kN$ (respectively, $\wedge^k(N;\mathbb{C})$, $\wedge^{p,q}N$).
For any $p,q\in\mathbb{N}$, the $(p,q)$-th Bott-Chern cohomology group $H^{p,q}_{BC}(N)$ of $N$ is computed as the cohomology of the complex
$$
\wedge^{p-1,q-1}N \stackrel{\partial\overline{\del}}{\longrightarrow} \wedge^{p,q}N \stackrel{\de}{\longrightarrow} \wedge^{p+q+1}(N;\mathbb{C}) \;.
$$
Restricting to $G$-left-invariant forms $\wedge^{\bullet,\bullet}_\text{inv}N\simeq \wedge^{\bullet,\bullet}\mathfrak{g}_\mathbb{C}^*$ on $N$, one has the subcomplex
\begin{equation}\label{eq:bc-subcomplex}
\begin{gathered}
\xymatrix{
\wedge^{p-1,q-1}\mathfrak{g}_\mathbb{C}^* \ar[r]^{\partial\overline{\del}} \ar@{^{(}->}[d] & \wedge^{p,q}\mathfrak{g}_\mathbb{C}^* \ar[r]^{\de} \ar@{^{(}->}[d] & \wedge^{p+q+1}\mathfrak{g}_\mathbb{C}^* \ar@{^{(}->}[d] \\
\wedge^{p-1,q-1}N \ar[r]^{\partial\overline{\del}} & \wedge^{p,q}N \ar[r]^{\de} & \wedge^{p+q+1}(N;\mathbb{C})
}
\end{gathered}
\;.
\end{equation}
We prove the following theorem, which extends the results in \cite[Theorem 1]{sakane}, \cite[Main Theorem]{cordero-fernandez-gray-ugarte-2000}, \cite[Theorem 1, Theorem 2, Remark 4]{console-fino}, \cite[Theorem 1.10]{rollenske}, see also \cite{console, rollenske-survey}, to the Bott-Chern case.

\smallskip
\noindent {\bfseries Theorem} (see Theorem \ref{thm:bc-invariant-cor} and Theorem \ref{thm:bc-invariant-open}){\bfseries .}
{\itshape
 Let $N=\left.\Gamma\right\backslash G$ be a nilmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$.
Then, for every $p,q\in\mathbb{N}$, the injective homomorphism in cohomology
$$
i\colon\frac{\ker\left(\de\colon\wedge^{p,q}\mathfrak{g}_\mathbb{C}^*\to\wedge^{p+q+1}\mathfrak{g}_\mathbb{C}^*\right)}{\imm\left(\partial\overline{\del}\colon\wedge^{p-1,q-1}\mathfrak{g}_\mathbb{C}^*\to\wedge^{p,q}\mathfrak{g}_\mathbb{C}^*\right)} \; \hookrightarrow \; H^{p,q}_{BC}(N)
$$
induced by \eqref{eq:bc-subcomplex} is an isomorphism, provided one of the following conditions holds:
\begin{itemize}
 \item $N$ is holomorphically parallelizable;
 \item $J$ is an Abelian complex structure;
 \item $J$ is a nilpotent complex structure;
 \item $J$ is a rational complex structure;
 \item $\mathfrak{g}$ admits a torus-bundle series compatible with $J$ and with the rational structure induced by $\Gamma$.
\end{itemize}
Moreover, the property of $i$ being an isomorphism is open in the space of all $G$-left-invariant complex structures on $N$.
}
\smallskip

\noindent Similar results are obtained for a certain class of solvmanifolds.

Then, we use these tools to explicitly compute the Bott-Chern cohomology for a three-dimensional holomorphically parallelizable nilmanifold, the so called \emph{Iwasawa manifold} (see, e.g., \cite{fernandez-gray}) and for its small deformations. The Iwasawa manifold is one of the simplest example of non-K\"ahler manifold: indeed, being non-formal, see \cite[page 158]{fernandez-gray}, it admits no K\"ahler structure; it has been studied by several authors as a fruitful source of interesting behaviors: see, for example, \cite{fernandez-gray, nakamura, alessandrini-bassanelli, bassanelli, ye, angella-tomassini}.\\
As regards the Dolbeault cohomology, I. Nakamura, in \cite{nakamura}, already computed the Hodge numbers for the Iwasawa manifold and for its small deformations: he used these computations to prove that the Hodge numbers are not invariant under small deformations, \cite[Theorem 2]{nakamura} (compare also \cite[\S4]{ye}), and that small deformations of a holomorphically parallelizable complex structure are not necessarily holomorphically parallelizable, \cite[page 86]{nakamura} (compare also \cite[Theorem 5.1, Corollary 5.2]{rollenske-jems}). The Bott-Chern cohomology and its computation for the Iwasawa manifold appeared in the work by M. Schweitzer, see \cite[\S1.c]{schweitzer}, while the computations of the Bott-Chern cohomology for its small deformations were there announced but never written.\\
In particular, the computations in \S\ref{sec:bott-chern-iwasawa} show that one can use the Bott-Chern cohomology to get a finer classification of small deformations of the Iwasawa manifold than using the Dolbeault cohomology as in \cite[page 96]{nakamura}.

\medskip

\noindent{\itshape Acknowledgments.}
The author would like to thank Adriano Tomassini and Jean-Pierre Demailly for their constant encouragement, their support and for many useful conversations. He would like also to thank Institut Fourier, Université de Grenoble \textsc{i}, for its warm hospitality.
Very interesting conversations with S\"onke Rollenske at \textsc{cirm} in Luminy and with Greg Kuperberg at Institut Fourier in Grenoble gave great motivations for looking at further results on this subject. Many thanks to S\"onke are due also for his comments and remarks which improved the presentation of this paper.
The author is very grateful to the anonymous referee for his/her careful reading and for many suggestions and remarks that highly improved the presentation of the paper.

\section{The Bott-Chern and Aeppli cohomologies of a complex manifold}\label{sec:bott-chern-cohomology}

\subsection{The Bott-Chern cohomology}\label{subsec:hodge-theory-bott-chern}
Let $X$ be a compact complex manifold of complex dimension $n$ and denote its complex structure by $J$. The \emph{Bott-Chern cohomology} groups are defined, for $p,q\in\mathbb{N}$, as
$$ H^{p,q}_{BC}(X) \;:=\; \frac{\ker \left(\de\colon\wedge^{p,q}X\to\wedge^{p+q+1}(X;\mathbb{C})\right)}{\imm\left(\partial\overline{\del}\colon\wedge^{p-1,q-1}X\to\wedge^{p+1,q+1}X\right)} \;.$$
Note that, for every $p,q\in\mathbb{N}$, the conjugation induces an isomorphism
$$ H^{p,q}_{BC}(X)\;\simeq\; H^{q,p}_{BC}(X) \;,$$
unlike in the case of the Dolbeault cohomology groups.\\
For every $k\in\mathbb{N}$ and for every $p,q\in\mathbb{N}$, one has the natural maps
$$ \bigoplus_{r+s=k}H^{r,s}_{BC}(X)\to H^{k}_{dR}(X;\mathbb{C}) \qquad \text{ and } \qquad H^{p,q}_{BC}(X)\to H^{p,q}_{\overline{\del}}(X) \;.$$
In general, these maps are neither injective nor surjective, see, e.g., the examples in \cite[\S1.c]{schweitzer} or \S\ref{sec:bott-chern-iwasawa}; if $X$ satisfies the $\partial\overline{\del}$-Lemma (for example, if $X$ admits a K\"ahler structure or if $X$ is in Fujiki class $\mathcal{C}$), then the above maps are isomorphisms, see \cite[Remark 5.16]{deligne-griffiths-morgan-sullivan}.

We collect here some results on Hodge theory for the Bott-Chern cohomology, referring to \cite[\S2]{schweitzer} (see also \cite[\S5]{bigolin}).
Fix $g$ a $J$-Hermitian metric on $X$ and define the $4$-th order self-adjoint elliptic differential operator
$$ \tilde\Delta_{BC} \;:=\; \partial\overline{\del}\overline{\del}^*\partial^*+\overline{\del}^*\partial^*\partial\overline{\del}+\overline{\del}^*\partial\del^*\overline{\del}+\partial^*\overline{\del}\overline{\del}^*\partial+\overline{\del}^*\overline{\del}+\partial^*\partial \;, $$
see \cite[Proposition 5]{kodaira-spencer-3} and also \cite[\S2.b]{schweitzer}, \cite[\S5.1]{bigolin}. Given $u\in\wedge^{p,q}X$, one has that
$$ \tilde\Delta_{BC}\,u\;=\;0 \qquad \Leftrightarrow \qquad
\left\{
\begin{array}{rcl}
 \overline{\del} u &=& 0 \\[5pt]
 \partial u &=& 0 \\[5pt]
 \overline{\del}^*\partial^* u &=& 0
\end{array}
\right. \;;
$$
moreover, being $\tilde\Delta_{BC}$ a self-adjoint elliptic differential operator, the following result holds, see, e.g., \cite[page 450]{kodaira}.
\begin{thm}[{\cite[Th\'eor\`eme 2.2]{schweitzer}, \cite[Corollaire 2.3]{schweitzer}}]
 Let $X$ be a compact complex manifold and denote its complex structure by $J$; fix $g$ a $J$-Hermitian metric on $X$. Then there are an orthogonal decomposition
$$ \wedge^{\bullet,\bullet}X \;=\; \ker\tilde\Delta_{BC} \,\oplus\, \imm\partial\overline{\del} \,\oplus\, \left(\imm\partial^*\,+\,\imm\overline{\del}^*\right) $$
and an isomorphism
$$ H^{\bullet,\bullet}_{BC}(X) \;\simeq\; \ker\tilde\Delta_{BC} \;.$$
In particular, its Bott-Chern cohomology groups are finite-dimensional $\mathbb{C}$-vector spaces.
\end{thm}

\subsection{The Aeppli cohomology}\label{subsec:hodge-theory-aeppli}
Let $X$ be a compact complex manifold of complex dimension $n$ and denote its complex structure by $J$.
For $p,q\in\mathbb{N}$, one defines the \emph{Aeppli cohomology} group $H^{p,q}_{A}(X)$ as
$$ H^{p,q}_{A}(X) \;:=\; \frac{\ker \left(\partial\overline{\del}\colon\wedge^{p,q}X\to\wedge^{p+1,q+1}X\right)}{\left(\imm\left(\partial\colon\wedge^{p-1,q}X\to\wedge^{p,q}X\right)\right)\,+\,\left(\imm\left(\overline{\del}\colon\wedge^{p,q-1}X\to\wedge^{p,q}X\right)\right)} \;.$$
As for the Bott-Chern cohomology, the conjugation induces an isomorphism
$$ H^{p,q}_{A}(X)\;\simeq\; H^{q,p}_{A}(X) $$
for every $p,q\in\mathbb{N}$.\\
Furthermore, for every $k\in\mathbb{N}$ and for every $p,q\in\mathbb{N}$, one has the natural maps
$$ H^{k}_{dR}(X;\mathbb{C})\to \bigoplus_{r+s=k}H^{r,s}_{A}(X) \qquad \text{ and } \qquad  H^{p,q}_{\overline{\del}}(X)\to H^{p,q}_{A}(X) \;,$$
which are, in general, neither injective nor surjective; once again, the maps above are isomorphisms if $X$ satisfies the $\partial\overline{\del}$-Lemma, \cite[Remark 5.16]{deligne-griffiths-morgan-sullivan}, and hence, in particular, if $X$ admits a K\"ahler structure or if $X$ is in Fujiki class $\mathcal{C}$.

\begin{rem}
 On a K\"ahler manifold $X$, the fundamental $2$-form $\omega$ associated to the metric defines a non-zero class in $H^2_{dR}(X;\mathbb{R})$. For general Hermitian manifolds, special classes of metrics are often defined in terms of closedness of powers of $\omega$ (e.g., a Hermitian metric on a complex manifold of complex dimension $n$ is said \emph{balanced} if $\de\omega^{n-1}=0$, \emph{pluriclosed} if $\partial\overline{\del}\omega=0$, \emph{astheno-K\"ahler} if $\partial\overline{\del}\omega^{n-2}=0$, \emph{Gauduchon} if $\partial\overline{\del}\omega^{n-1}=0$), so they define classes in the Bott-Chern or Aeppli cohomology groups. It would be interesting to see if these classes can play a role similar to the one played by the K\"ahler class in some contexts.
\end{rem}

We refer to \cite[\S2.c]{schweitzer} for the results that follows (for a hypercohomology interpretation of the Bott-Chern and Aeppli cohomologies and its applications, see \cite[\S VI.12.1]{demailly-agbook}, \cite[\S4]{schweitzer}, \cite[\S3.2, \S3.5]{kooistra}).\\
Fixed a $J$-Hermitian metric $g$ on $X$ and defined the $4$-th order self-adjoint elliptic differential operator
$$ \tilde\Delta_{A} \;:=\; \partial\del^*+\overline{\del}\overline{\del}^*+\overline{\del}^*\partial^*\partial\overline{\del}+\partial\overline{\del}\overline{\del}^*\partial^*+\partial\overline{\del}^*\overline{\del}\partial^*+\overline{\del}\partial^*\partial\overline{\del}^* \;, $$
one has an orthogonal decomposition
$$ \wedge^{\bullet,\bullet}X \;=\; \ker\tilde\Delta_{A} \,\oplus\, \left(\imm\partial \,+\, \imm\overline{\del}\right)
\,\oplus\, \imm\left(\partial\overline{\del}\right)^* $$
from which one gets an isomorphism
$$ H^{\bullet,\bullet}_{A}(X) \;\simeq\; \ker\tilde\Delta_{A} \;;$$
this proves that the Aeppli cohomology groups of a compact complex manifold are finite-dimensional $\mathbb{C}$-vector spaces.

In fact, for any $p,q\in\mathbb{N}$, one has that the Hodge-$*$-operator associated to a $J$-Hermitian metric induces an isomorphism
$$ H^{p,q}_{BC}(X) \simeq H^{n-q,n-p}_{A}(X) $$
between the Bott-Chern and the Aeppli cohomologies.

\section{Some results on cohomology computation}\label{sec:cohomology-computation}
In this section, we collect some results about cohomology computation for nilmanifolds and solvmanifolds. Using these tools, one recovers the de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies for the Iwasawa manifold and for its small deformations, see \S\ref{sec:derham-iwasawa}, \S\ref{sec:dolbeault-iwasawa} and \S\ref{sec:bott-chern-iwasawa}.

\smallskip

Let $X=\left. \Gamma \right\backslash G$ be a solvmanifold, that is, a compact quotient of the connected simply-connected solvable Lie group $G$ by a discrete and co-compact subgroup $\Gamma$; the Lie algebra naturally associated to $G$ will be denoted by $\mathfrak{g}$ and its complexification by $\mathfrak{g}_\mathbb{C}:=\mathfrak{g}\otimes_\mathbb{R}\mathbb{C}$. Dealing with $G$-left-invariant objects on $X$, we mean objects on $X$ obtained by objects on $G$ that are invariant under the action of $G$ on itself given by left-translation.

A $G$-left-invariant complex structure $J$ on $X$ is uniquely determined by a linear complex structure $J$ on $\mathfrak{g}$ satisfying the integrability condition
$$ \forall x,y\in\mathfrak{g}\;,\qquad \Nij_J(x,y)\;:=\;\left[x,\,y\right]+J\left[Jx,\,y\right]+J\left[x,\,Jy\right]-\left[Jx,\,Jy\right]\;=\;0 \;,$$
see \cite[Theorem 1.1]{newlander-nirenberg}. Therefore, the set of $G$-left-invariant complex structures on $X$ is given by
$$ \mathcal{C}\left(\mathfrak{g}\right) := \left\{ J\in\End\left(\mathfrak{g}\right) \;:\; J^2=-\id_{\mathfrak{g}} \;\text{ and }\;\Nij_J=0 \right\} \;.$$

Recall that the exterior differential $\de$ on $X$ can be written using only the action of $\Gamma(X;\,TX)$ on $\mathcal{C}^\infty(X)$ and the Lie bracket on the Lie algebra of vector fields on $X$. One has that the complex $\left(\wedge^\bullet\mathfrak{g}^*,\,\de\right)$ is isomorphic, as a differential complex, to the differential subcomplex $\left(\wedge^\bullet_{\text{inv}}X,\,\de\lfloor_{\wedge^\bullet_{\text{inv}}X}\right)$ of $\left(\wedge^\bullet X,\,\de\right)$ given by the $G$-left-invariant forms on $X$.\\
If a $G$-left-invariant complex structure on $X$ is given, then one has also the double complex $\left(\wedge^{\bullet,\bullet}\mathfrak{g}_{\mathbb{C}}^*,\,\partial,\,\overline{\del}\right)$, which is isomorphic, as a double complex, to the double subcomplex $\left(\wedge^{\bullet,\bullet}_{\text{inv}}X,\,\partial\lfloor_{\wedge^{\bullet,\bullet}_{\text{inv}}X},\,\overline{\del}\lfloor_{\wedge^{\bullet,\bullet}_{\text{inv}}X}\right)$ of $\left(\wedge^{\bullet,\bullet}X,\,\partial,\,\overline{\del}\right)$ given by the $G$-left-invariant forms on $X$.\\
Lastly, given a $G$-left-invariant complex structure on $G$ and fixed $p,q\in\mathbb{N}$, one has also the following complexes and the following maps of complexes:
\begin{equation}\label{eq:bc-complessi}
\begin{gathered}
\xymatrix{
\wedge^{p-1,q-1}\mathfrak{g}_\mathbb{C}^* \ar[r]^{\partial\overline{\del}} \ar[d]^{\simeq} & \wedge^{p,q}\mathfrak{g}_\mathbb{C}^* \ar[r]^{\de} \ar[d]^{\simeq} & \wedge^{p+q+1}\mathfrak{g}_\mathbb{C}^* \ar[d]^{\simeq} \\
\wedge^{p-1,q-1}_{\text{inv}}X \ar[r]^{\partial\overline{\del}} \ar@{^{(}->}[d]^{i} & \wedge^{p,q}_{\text{inv}}X \ar[r]^{\de} \ar@{^{(}->}[d]^{i} & \wedge^{p+q+1}_{\text{inv}}(X;\mathbb{C}) \ar@{^{(}->}[d]^{i} \\
\wedge^{p-1,q-1}X \ar[r]^{\partial\overline{\del}} & \wedge^{p,q}X \ar[r]^{\de} & \wedge^{p+q+1}(X;\mathbb{C})
}
\end{gathered}
\;;
\end{equation}
the same can be repeated for the complex used to define the $(p,q)$-th Aeppli cohomology group.

For $\star\in\left\{\overline{\del},\,\partial,\,BC,\,A\right\}$ and $\mathbb{K}\in\left\{\mathbb{R},\mathbb{C}\right\}$, we will write $H^\bullet_{dR}\left(\mathfrak{g};\mathbb{K}\right)$ and $H^{\bullet,\bullet}_{\star}\left(\mathfrak{g}_\mathbb{C}\right)$ to denote the cohomology groups of the corresponding complexes of forms on $\mathfrak{g}$, equivalently, of $G$-left-invariant forms on $X$. The rest of this section is devoted to the problem whether these cohomologies are isomorphic to the corresponding cohomologies on $X$.

\subsection{Classical results on computation for the de Rham and Dolbeault cohomologies}\label{subsec:cohomology-computation-derham-dolbeault}

One has the following theorem by K. Nomizu, saying that the de Rham cohomology of a nilmanifold can be computed as the cohomology of the subcomplex of left-invariant forms.

\begin{thm}[{\cite[Theorem 1]{nomizu}}]\label{thm:nomizu}
 Let $N=\left. \Gamma \right\backslash G$ be a nilmanifold and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. The complex $\left(\wedge^{\bullet}\mathfrak{g}^*,\de\right)$ is a minimal model for $N$. In particular, the map of complexes $\left(\wedge^\bullet\mathfrak{g}^*,\,\de\right)\to\left(\wedge^\bullet N,\,\de\right)$ is a quasi-isomorphism, that is, it induces an isomorphism in cohomology:
$$ i\colon H^{\bullet}_{dR}(\mathfrak{g};\mathbb{R}) \stackrel{\simeq}{\longrightarrow} H^\bullet_{dR}(N;\mathbb{R}) \;.$$
\end{thm}

\noindent The proof rests on an inductive argument, which can be performed since every nilmanifold can be seen as a principal torus-bundle over a lower dimensional nilmanifold.\\
A similar result holds also in the case of completely-solvable solvmanifolds and has been proved by A. Hattori in \cite[Corollary 4.2]{hattori}, as a consequence of the Mostow Structure Theorem (see also \cite[Chapter 3]{tralle-oprea}). We recall that a solvmanifold $X=\left.\Gamma\right\backslash G$ is said \emph{completely-solvable} if, for any $g\in G$, all the eigenvalues of $\Ad g$ are real, equivalently, if, for any $X\in\mathfrak{g}$, all the eigenvalues of $\ad X$ are real.

\begin{thm}[{\cite[Corollary 4.2]{hattori}}]
 Let $X=\left. \Gamma \right\backslash G$ be a completely-solvable solvmanifold and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Then the map of complexes $\left(\wedge^\bullet\mathfrak{g}^*,\,\de\right)\to\left(\wedge^\bullet X,\,\de\right)$ is a quasi-isomorphism, that is, it induces an isomorphism in cohomology:
$$ i\colon H^{\bullet}_{dR}(\mathfrak{g};\mathbb{R}) \stackrel{\simeq}{\longrightarrow} H^\bullet_{dR}(X;\mathbb{R}) \;.$$
\end{thm}

\noindent In general, for non-completely-solvable solvmanifolds the map of complexes $\left(\wedge^\bullet\mathfrak{g}^*,\,\de\right)\to\left(\wedge^\bullet X,\,\de\right)$ is not necessarily a quasi-isomorphism, as the example in \cite[Corollary 4.2, Remark 4.3]{debartolomeis-tomassini} shows (for some results about the de Rham cohomology of solvmanifolds, see \cite{console-fino-2}).

\medskip

Considering nilmanifolds endowed with certain $G$-left-invariant complex structures, there are similar results also for the Dolbeault cohomology, see, e.g.,  \cite{console} and \cite{rollenske-survey} for surveys on the known results. (Some results about the Dolbeault cohomology of solvmanifolds have been recently proved by H. Kasuya, see \cite{kasuya}.)

\begin{thm}[{\cite[Theorem 1]{sakane}, \cite[Main Theorem]{cordero-fernandez-gray-ugarte-2000}, \cite[Theorem 2, Remark 4]{console-fino}, \cite[Theorem 1.10]{rollenske}}]\label{thm:dolbeault-invariant}
 Let $N=\left. \Gamma \right\backslash G$ be a nilmanifold and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Let $J$ be a $G$-left-invariant complex structure on $N$.
Then, for every $p\in\mathbb{N}$, the map of complexes
\begin{equation}\label{eq:dolb-complessi}
\left(\wedge^{p,\bullet}\mathfrak{g}_{\mathbb{C}}^*,\,\overline{\del}\right)
\hookrightarrow\left(\wedge^{p,\bullet}N,\,\overline{\del}\right)
\end{equation}
is a quasi-isomorphism, hence
$$
i \colon H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right) \stackrel{\simeq}{\longrightarrow} H^{\bullet,\bullet}_{\overline{\del}}(N) \;,$$
provided one of the following conditions holds:
\begin{itemize}
 \item $N$ is holomorphically parallelizable, see \cite[Theorem 1]{sakane};
 \item $J$ is an \emph{Abelian} complex structure (i.e., $\left[Jx,\,Jy\right]=\left[x,\,y\right]$ for any $x,y\in\mathfrak{g}$), see \cite[Remark 4]{console-fino};
 \item $J$ is a \emph{nilpotent} complex structure (i.e., there is a $G$-left-invariant co-frame $\left\{\omega^1,\ldots,\omega^n\right\}$ for $\left(T^{1,0}N\right)^*$ with respect to which the structure equations of $N$ are of the form
$$ \de\omega^j\;=\; \sum_{h<k<j}A_{hk}^j\,\omega^h\wedge\omega^k+\sum_{h,k<j}B_{h k}^j\,\omega^h\wedge\bar\omega^k $$
with $\left\{A_{hk}^j,\,B_{h k}^j\right\}_{j,h,k}\subset\mathbb{C}$), see \cite[Main Theorem]{cordero-fernandez-gray-ugarte-2000};
 \item $J$ is a \emph{rational} complex structure (i.e., $J\left(\mathfrak{g}_\mathbb{Q}\right)\subseteq \mathfrak{g}_\mathbb{Q}$ where $\mathfrak{g}_\mathbb{Q}$ is the rational structure for $\mathfrak{g}$ ---that is, a $\mathbb{Q}$-vector space such that $\mathfrak{g}=\mathfrak{g}_\mathbb{Q}\otimes_\mathbb{Q}\mathbb{R}$--- induced by $\Gamma$), see \cite[Theorem 2]{console-fino};
 \item $\mathfrak{g}$ admits a torus-bundle series compatible with $J$ and with the rational structure induced by $\Gamma$, see \cite[Theorem 1.10]{rollenske}.
\end{itemize}
\end{thm}

\medskip

We recall also the following theorem by S. Console and A. Fino.
\begin{thm}[{\cite[Theorem 1]{console-fino}}]\label{thm:dolbeault-invariant-open}
 Let $N=\left.\Gamma\right\backslash G$ be a nilmanifold and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Given any $G$-left-invariant complex structure $J$ on $N$, the map of complexes \eqref{eq:dolb-complessi}
induces an injective homomorphism $i$ in cohomology, \cite[Lemma 9]{console-fino}):
$$ i\colon H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right) \hookrightarrow H^{\bullet,\bullet}_{\overline{\del}}(N) \;.$$
 Let $\mathcal{U}\subseteq\mathcal{C}(\mathfrak{g})$ be the subset containing the $G$-left-invariant complex structures $J$ on $N$ such that the inclusion $i$ is an isomorphism:
$$ \mathcal{U} \;:=\; \left\{ J\in\mathcal{C}\left(\mathfrak{g}\right) \;:\; i\colon H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right)\stackrel{\simeq}{\hookrightarrow}H^{\bullet,\bullet}_{\overline{\del}}(N)\right\} \;\subseteq\; \mathcal{C}\left(\mathfrak{g}\right) \;. $$
Then $\mathcal{U}$ is an open set in $\mathcal{C}\left(\mathfrak{g}\right)$.
\end{thm}

\noindent The strategy of the proof consists in proving that the dimension of the orthogonal of $H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right)$ in $H^{\bullet,\bullet}_{\overline{\del}}(N)$ with respect to a given $J$-Hermitian $G$-left-invariant metric on $N$ is an upper-semi-continuous function in $J\in\mathcal{C}\left(\mathfrak{g}\right)$ and thus if it is zero for a given $J\in\mathcal{C}\left(\mathfrak{g}\right)$, then it remains equal to zero in an open neighborhood of $J$ in $\mathcal{C}\left(\mathfrak{g}\right)$.\\
We will use the same argument in proving Theorem \ref{thm:bc-invariant-open}, which is a slight modification of the previous result in the case of the Bott-Chern cohomology.

\medskip

The aforementioned results suggest the following conjecture.

\begin{conj}[{\cite[Conjecture 1]{rollenske-survey}; see also \cite[page 5406]{cordero-fernandez-gray-ugarte-2000}, \cite[page 112]{console-fino}}]\label{conj:dolbeault}
 Let $N=\left. \Gamma \right\backslash G$ be a nilmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$.
Then the maps of complexes \eqref{eq:dolb-complessi} are quasi-isomorphisms, that is, they induce an isomorphism in cohomology:
$$ i\colon H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right) \stackrel{\simeq}{\longrightarrow} H^{\bullet,\bullet}_{\overline{\del}}(N) \;.$$
\end{conj}

\noindent Note that, since $i$ is always injective by \cite[Lemma 9]{console-fino}, this is equivalent to ask that
$$ \dim_\mathbb{C} \left(H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right)\right)^\perp=0 \;,$$
where the orthogonality is meant with respect to the scalar product induced by a given $J$-Hermitian $G$-left-invariant metric $g$ on $N$.

\subsection{Some results on computation for the Bott-Chern cohomology}\label{subsec:cohomology-computation-bott-chern}
We prove here some results about Bott-Chern cohomology computation for nilmanifolds and solvmanifolds.

The first result is a slight modification of \cite[Lemma 9]{console-fino} proved by S. Console and A. Fino for the Dolbeault cohomology: we repeat here their argument in the case of the Bott-Chern cohomology.

\begin{lemma}\label{lemma:inj}
 Let $X=\left. \Gamma \right\backslash G$ be a solvmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$.
 The map of complexes \eqref{eq:bc-complessi} induces an injective homomorphism between cohomology groups
$$ i\colon H^{\bullet,\bullet}_{BC}\left(\mathfrak{g}_\mathbb{C}\right) \hookrightarrow H^{\bullet,\bullet}_{BC}(X) \;.$$
\end{lemma}

\begin{proof}
 Fix $p,q\in\mathbb{N}$. Let $g$ be a $J$-Hermitian $G$-left-invariant metric on $X$ and consider the induced scalar product $\left\langle\left.\cdot\right|\cdot\right\rangle$ on $\wedge^{\bullet,\bullet} X$. Hence both $\partial$, $\overline{\del}$ and their adjoints $\partial^*$, $\overline{\del}^*$ preserve the $G$-left-invariant forms on $X$ and therefore also $\tilde\Delta_{BC}$ does. In such a way, we get a Hodge decomposition also at the level of $G$-left-invariant forms:
\begin{eqnarray*}
\wedge^{p,q}\mathfrak{g}_\mathbb{C}^* &=& \ker\tilde\Delta_{BC}\lfloor_{\wedge^{p,q}\mathfrak{g}_\mathbb{C}^*} \,\oplus\, \imm\partial\overline{\del}\lfloor_{\wedge^{p-1,q-1}\mathfrak{g}_\mathbb{C}^*} \\[5pt]
 && \oplus\, \left(\imm\partial^*\lfloor_{\wedge^{p+1,q}\mathfrak{g}_\mathbb{C}^*} \,+\, \imm\overline{\del}^*\lfloor_{\wedge^{p,q+1}\mathfrak{g}_\mathbb{C}^*}\right) \;.
\end{eqnarray*}
 Now, take $[\omega]\in H^{p,q}_{BC}\left(\mathfrak{g}_\mathbb{C}\right)$ such that $i\left[\omega\right]=0$ in $H^{p,q}_{BC}(X)$, that is, $\omega$ is a $G$-left-invariant $(p,q)$-form on $X$ and there exists a (possibly non-$G$-left-invariant) $(p-1,q-1)$-form $\eta$ on $X$ such that $\omega=\partial\overline{\del}\,\eta$. Up to zero terms in $H^{p,q}_{BC}\left(\mathfrak{g}_\mathbb{C}\right)$, we may assume that $\eta\in \left(i\left(\wedge^{p,q}\mathfrak{g}_\mathbb{C}^*\right)\right)^\perp\subseteq \wedge^{p,q}X$. Therefore, since $\overline{\del}^*\partial^*\partial\overline{\del}\eta$ is a $G$-left-invariant form (being $\partial\overline{\del}\eta$ a $G$-left-invariant form), we have that
$$ 0=\left\langle \left. \overline{\del}^*\partial^*\partial\overline{\del}\eta \right| \eta\right\rangle = \left\|\partial\overline{\del}\eta\right\|^2=\left\|\omega\right\|^2 $$
and therefore $\omega=0$.
\end{proof}

The second general result says that, if the Dolbeault and de Rham cohomologies of a solvmanifold are computed using just left-invariant forms, then also the Bott-Chern cohomology is computed using just left-invariant forms. The idea of the proof is inspired by \cite[\S1.c]{schweitzer}, where a similar argument is used to explicitly compute the Bott-Chern cohomology in the special case of the Iwasawa manifold.

\begin{thm}\label{thm:bc-invariant}
 Let $X=\left.\Gamma\right\backslash G$ be a solvmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Suppose that
$$ i\colon H^\bullet_{dR}(\mathfrak{g};\mathbb{C})\stackrel{\simeq}{\hookrightarrow} H^\bullet_{dR}(X;\mathbb{C}) \qquad \text{ and } \qquad i\colon H^{\bullet,\bullet}_{\overline{\del}}\left(\mathfrak{g}_\mathbb{C}\right)\stackrel{\simeq}{\hookrightarrow} H^{\bullet,\bullet}_{\overline{\del}}(X) \;.$$
Then also
$$ i\colon H^{\bullet,\bullet}_{BC}\left(\mathfrak{g}_\mathbb{C}\right) \stackrel{\simeq}{\hookrightarrow}H^{\bullet,\bullet}_{BC}(X) \;. $$
\end{thm}

\begin{proof}
 Fix $p,q\in\mathbb{N}$. We prove the theorem as a consequence of the following steps.
\smallskip
\paragraph{\bfseries Step 1} {\itshape We may reduce to study if $\frac{\imm\de\cap\wedge^{p,q}X}{\imm\partial\overline{\del}}$ can be computed using just $G$-left-invariant forms.}\\
Indeed, we have the exact sequence
$$ 0 \to \frac{\imm\de\cap\wedge^{p,q}X}{\imm\partial\overline{\del}} \to H^{p,q}_{BC}(X) \to H^{p+q}_{dR}(X;\mathbb{C}) $$
and, by hypothesis, $H^\bullet_{dR}(X;\mathbb{C})$ can be computed using just $G$-left-invariant forms.
\smallskip
 \paragraph{\bfseries Step 2} {\itshape Under the hypothesis that the Dolbeault cohomology is computed using just $G$-left-invariant forms, if $\psi$ is a $G$-left-invariant $\overline{\del}$-closed form then every solution $\phi$ of $\overline{\del}\phi=\psi$ is $G$-left-invariant up to $\overline{\del}$-exact terms.}\\
Indeed, since $[\psi]=0$ in $H^{\bullet,\bullet}_{\overline{\del}}(X)$, there is a $G$-left-invariant form $\alpha$ such that $\psi=\overline{\del}\alpha$. Hence, $\phi-\alpha$ defines a class in $H^{\bullet,\bullet}_{\overline{\del}}(X)$ and hence $\phi-\alpha$ is $G$-left-invariant up to a $\overline{\del}$-exact form, and so $\phi$ is.
\smallskip
 \paragraph{\bfseries Step 3} {\itshape Under the hypothesis that the Dolbeault cohomology is computed using just $G$-left-invariant forms, the space $\frac{\imm\de\cap\wedge^{p,q}X}{\imm\partial\overline{\del}}$ can be computed using just $G$-left-invariant forms.}\\
Consider
\begin{equation}\label{eq:proof-step-3}
\omega^{p,q}\;=\;\de\eta\mod\imm\partial\overline{\del}\;\in\;\frac{\imm\de\cap\wedge^{p,q}X}{\imm\partial\overline{\del}} \;.
\end{equation}
Decomposing $\eta=:\sum_{p,q}\eta^{p,q}$ in pure-type components, the equality \eqref{eq:proof-step-3} is equivalent to the system
$$
\left\{
\begin{array}{cccccccc}
 && \partial\eta^{p+q-1,0} &=&0 & \mod \imm\partial\overline{\del} && \\[5pt]
\overline{\del}\eta^{p+q-\ell,\ell-1} &+& \partial\eta^{p+q-\ell-1,\ell} &=& 0 &\mod\imm\partial\overline{\del} & \text{ for } & \ell\in\{1,\ldots,q-1\} \\[5pt]
\overline{\del}\eta^{p,q-1} &+& \partial\eta^{p-1,q} &=& \omega^{p,q} & \mod\imm\partial\overline{\del} && \\[5pt]
\overline{\del}\eta^{\ell,p+q-\ell-1} &+& \partial\eta^{\ell-1,p+q-\ell} &=& 0 &\mod\imm\partial\overline{\del} & \text{ for } & \ell\in\{1,\ldots,p-1\} \\[5pt]
\overline{\del}\eta^{0,p+q-1} &&&=&0 &\mod\imm\partial\overline{\del} &&
\end{array}
\right. \;.
$$
Applying several times {\itshape Step 2}, we may suppose that, for $\ell\in\{0,\ldots, p-1\}$, the forms $\eta^{\ell,p+q-\ell-1}$ are $G$-left-invariant: indeed, they are $G$-left-invariant up to $\overline{\del}$-exact terms, but $\overline{\del}$-exact terms give no contribution in the system, since it is modulo $\imm\partial\overline{\del}$. Analogously, using the conjugate version of {\itshape Step 2}, we may suppose that, for $\ell\in\{0,\ldots,q-1\}$, the forms $\eta^{p+q-\ell-1,\ell}$ are $G$-left-invariant. Then we may suppose that $\omega^{p,q}=\overline{\del}\eta^{p,q-1}+\partial\eta^{p-1,q}$ is $G$-left-invariant.
\end{proof}

As a corollary of Theorem \ref{thm:nomizu}, Theorem \ref{thm:dolbeault-invariant} and Theorem \ref{thm:bc-invariant}, we get the following result.
\begin{thm}\label{thm:bc-invariant-cor}
 Let $N=\left.\Gamma\right\backslash G$ be a nilmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Suppose that one of the following conditions holds:
\begin{itemize}
 \item $N$ is holomorphically parallelizable;
 \item $J$ is an Abelian complex structure;
 \item $J$ is a nilpotent complex structure;
 \item $J$ is a rational complex structure;
 \item $\mathfrak{g}$ admits a torus-bundle series compatible with $J$ and with the rational structure induced by $\Gamma$.
\end{itemize}
Then the de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies can be computed as the cohomologies of the corresponding subcomplexes given by the space of $G$-left-invariant forms on $N$; in other words, the inclusions of the several subcomplexes of $G$-left-invariant forms on $N$ into the corresponding complexes of forms on $N$ are quasi-isomorphisms:
$$ i\colon H^\bullet_{dR}\left(\mathfrak{g};\mathbb{R}\right) \;\stackrel{\simeq}{\hookrightarrow}\; H^\bullet_{dR}(N;\mathbb{R}) \qquad \text{ and }\qquad i\colon H^{\bullet,\bullet}_{\star}\left(\mathfrak{g}_\mathbb{C}\right) \;\stackrel{\simeq}{\hookrightarrow}\; H^{\bullet,\bullet}_{\star}\left(N\right) \;,$$
for $\star\in\{\partial,\,\overline{\del},\,BC,\,A\}$.
\end{thm}

\medskip

A slight modification of \cite[Theorem 1]{console-fino} by S. Console and A. Fino gives the following result, which says that the property of computing the Bott-Chern cohomology using just left-invariant forms is open in the space of left-invariant complex structures on solvmanifolds.

\begin{thm}\label{thm:bc-invariant-open}
 Let $X=\left. \Gamma \right\backslash G$ be a solvmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Let $\star\in\{\partial,\,\overline{\del},\,BC,\,A\}$. Suppose that
$$ i\colon H^{\bullet,\bullet}_{\star_J}\left(\mathfrak{g}_\mathbb{C}\right) \;\stackrel{\simeq}{\hookrightarrow}\; H^{\bullet,\bullet}_{\star_J}\left(X\right) \;.$$
Then there exists an open neighbourhood $\mathcal{U}$ of $J$ in $\mathcal{C}\left(\mathfrak{g}\right)$ such that any $\tilde J\in \mathcal{U}$ still satisfies
$$ i\colon H^{\bullet,\bullet}_{\star_{\tilde{J}}}\left(\mathfrak{g}_\mathbb{C}\right) \;\stackrel{\simeq}{\hookrightarrow}\; H^{\bullet,\bullet}_{\star_{\tilde{J}}}\left(X\right) \;.$$
In other words, the set
$$ \mathcal{U} \;:=\; \left\{ J\in\mathcal{C}\left(\mathfrak{g}\right) \;:\; i\colon H^{\bullet,\bullet}_{\star_J}\left(\mathfrak{g}_\mathbb{C}\right) \;\stackrel{\simeq}{\hookrightarrow}\; H^{\bullet,\bullet}_{\star_J}\left(X\right) \right\} $$
is open in $\mathcal{C}\left(\mathfrak{g}\right)$.
\end{thm}

\begin{proof}
 As a matter of notation, for $\varepsilon>0$ small enough, we consider
 $$ \left\{\left(X,\,J_t\right) \;:\; t\in \Delta(0,\varepsilon)\right\} \twoheadrightarrow \Delta(0,\varepsilon) $$
 a complex-analytic family of $G$-left-invariant complex structures on $X$, where $\Delta(0,\varepsilon):=\left\{t\in\mathbb{C}^m\;:\; \left|t\right|<\varepsilon\right\}$ for some $m\in\mathbb{N}\setminus\{0\}$; moreover, let $\left\{g_t\right\}_{t\in \Delta(0,\varepsilon)}$ be a family of $J_t$-Hermitian $G$-left-invariant metrics on $X$ depending smoothly on $t$. We will denote by $\overline{\del}_t:=\overline{\del}_{J_t}$ and $\overline{\del}_t^*:=-*_{g_t}\partial_{J_t}*_{g_t}$ the delbar operator and its $g_t$-adjoint respectively for the Hermitian structure $\left(J_t,\,g_t\right)$ and we set $\Delta_t:=\Delta_{\star_{J_t}}$ one of the differential operators involved in the definition of the Dolbeault, conjugate Dolbeault, Bott-Chern or Aeppli cohomologies with respect to $\left(J_t,\,g_t\right)$; we remark that $\Delta_t$ is a self-adjoint elliptic differential operator for all the considered cohomologies.

 By hypothesis, we have that $\left(H^{\bullet,\bullet}_{\star_{J_0}}\left(\mathfrak{g}_\mathbb{C}\right)\right)^\perp=\{0\}$, where the orthogonality is meant with respect to the scalar product induced by $g_0$, and we have to prove the same replacing $0$ with $t\in\Delta(0,\varepsilon)$. Therefore, it will suffice to prove that
$$ \Delta(0,\varepsilon)\ni t \mapsto \dim_\mathbb{C}\left(H^{\bullet,\bullet}_{\star_{J_t}}\left(\mathfrak{g}_\mathbb{C}\right)\right)^\perp\in\mathbb{N} $$
is an upper-semi-continuous function at $0$.\\
 For any $t\in\Delta(0,\varepsilon)$, being $\Delta_t$ a self-adjoint elliptic differential operator, there exists a complete orthonormal basis $\{e_i(t)\}_{i\in I}$ of eigenforms for $\Delta_t$ spanning $\left(\wedge^{\bullet,\bullet}_{J_t}\mathfrak{g}_\mathbb{C}^*\right)^\perp$, the orthogonal complement of the space of $G$-left-invariant forms, see \cite[Theorem 1]{kodaira-spencer-3}. For any $i\in I$ and $t\in\Delta(0,\varepsilon)$, let $a_i(t)$ be the eigenvalue corresponding to $e_i(t)$; $\Delta_t$ depending differentiably on $t\in\Delta(0,\varepsilon)$, for any $i\in I$, the function $\Delta(0,\varepsilon)\ni t \mapsto a_i(t)\in\mathbb{C}$ is continuous, see \cite[Theorem 2]{kodaira-spencer-3}. Therefore, for any $t_0\in\Delta(0,\varepsilon)$, choosing a constant $c>0$ such that $c\not\in \overline{\left\{a_i(t_0) \;:\; i\in I\right\}}$, the function
 $$ \Psi_c\colon\Delta(0,\varepsilon)\to\mathbb{N} \;,\qquad t \mapsto \dim \Span\left\{e_i(t) \;:\; a_i(t)<c\right\} $$
is locally constant at $t_0$; moreover, for any $t\in\Delta(0,\varepsilon)$ and for any $c>0$, we have
$$ \Psi_c(t) \;\geq\; \dim_\mathbb{C} \left(H^{\bullet,\bullet}_{\star_{J_t}}\left(\mathfrak{g}_\mathbb{C}\right)\right)^\perp \;.$$
Since the spectrum of $\Delta_{t_0}$ has no accumulation point for any $t_0\in\Delta(0,\varepsilon)$, see \cite[Theorem 1]{kodaira-spencer-3}, the theorem follows choosing $c>0$ small enough so that $\Psi_c(0)=\dim_\mathbb{C}\left(H^{\bullet,\bullet}_{\star_{J_0}}\left(\mathfrak{g}_\mathbb{C}\right)\right)^\perp$.
\end{proof}

\medskip

In particular, Theorem \ref{thm:bc-invariant-cor} and Theorem \ref{thm:bc-invariant-open} say that the following conjecture, which generalizes Conjecture \ref{conj:dolbeault}, holds for certain left-invariant complex structures on nilmanifolds.
\begin{conj}\label{conj:BC}
 Let $N=\left. \Gamma \right\backslash G$ be a nilmanifold endowed with a $G$-left-invariant complex structure $J$ and denote the Lie algebra naturally associated to $G$ by $\mathfrak{g}$. Then the de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies can be computed as the cohomologies of the corresponding subcomplexes given by the space of $G$-left-invariant forms on $N$, that is,
$$ \dim_\mathbb{R} \left(H^\bullet_{dR}\left(\mathfrak{g};\mathbb{R}\right)\right)^{\perp} \;=\; 0 \qquad \text{ and }\qquad \dim_\mathbb{C} \left(H^{\bullet,\bullet}_{\star}\left(\mathfrak{g}_\mathbb{C}\right)\right)^{\perp}\;=\;0 \;,$$
where $\star\in\{\partial,\,\overline{\del},\,BC,\,A\}$ and the orthogonality is meant with respect to the scalar product induced by a given $J$-Hermitian $G$-left-invariant metric $g$ on $N$.
\end{conj}

\section{The Iwasawa manifold and its small deformations}\label{part:iwasawa}

\subsection{The Iwasawa manifold}\label{sec:iwasawa}
Let $\mathbb{H}(3;\mathbb{C})$ be the $3$-dimensional \emph{Heisenberg group} over $\mathbb{C}$ defined by
$$
\mathbb{H}(3;\mathbb{C}) := \left\{
\left(
\begin{array}{ccc}
 1 & z^1 & z^3 \\
 0 &  1  & z^2 \\
 0 &  0  &  1
\end{array}
\right) \in \mathrm{GL}(3;\mathbb{C}) \;:\; z^1,\,z^2,\,z^3 \in\mathbb{C} \right\}
\;,
$$
where the product is the one induced by matrix multiplication.
It is straightforward to prove that $\mathbb{H}(3;\mathbb{C})$ is a connected simply-connected complex $2$-step nilpotent Lie group, that is, the Lie algebra $\left(\mathfrak{h}_3,\,\left[\cdot,\cdot\right]\right)$ naturally associated to $\mathbb{H}(3;\mathbb{C})$ satisfies $\left[\mathfrak{h}_3,\mathfrak{h}_3\right]\neq 0$ and $\left[\mathfrak{h}_3,\left[\mathfrak{h}_3,\mathfrak{h}_3\right]\right]=0$.\\
One finds that
$$
\left\{
\begin{array}{rcl}
 \varphi^1 &:=& \de z^1 \\[5pt]
 \varphi^2 &:=& \de z^2 \\[5pt]
 \varphi^3 &:=& \de z^3-z^1\,\de z^2
\end{array}
\right.
$$
is a $\mathbb{H}(3;\mathbb{C})$-left-invariant co-frame for the space of $(1,0)$-forms on $\mathbb{H}(3;\mathbb{C})$ and that the structure equations with respect to this co-frame are
$$
\left\{
\begin{array}{rcl}
 \de\varphi^1 &=& 0 \\[5pt]
 \de\varphi^2 &=& 0 \\[5pt]
 \de\varphi^3 &=& -\varphi^1\wedge\varphi^2
\end{array}
\right. \;.
$$
For the sake of completeness, we write also
$$
\left\{
\begin{array}{rcl}
 \partial\,\varphi^1 &=& 0 \\[5pt]
 \partial\,\varphi^2 &=& 0 \\[5pt]
 \partial\,\varphi^3 &=& -\varphi^1\wedge\varphi^2
\end{array}
\right. \qquad \text{ and } \qquad
\left\{
\begin{array}{rcl}
 \overline{\del}\,\varphi^1 &=& 0 \\[5pt]
 \overline{\del}\,\varphi^2 &=& 0 \\[5pt]
 \overline{\del}\,\varphi^3 &=& 0
\end{array}
\right. \;.
$$

\smallskip

Consider the action on the left of $\mathbb{H}\left(3;\mathbb{Z}\left[\im\right]\right):=\mathbb{H}(3;\mathbb{C})\cap\mathrm{GL}\left(3;\mathbb{Z}\left[\im\right]\right)$ on $\mathbb{H}\left(3;\mathbb{C}\right)$ and take the compact quotient
$$ \mathbb{I}_3 := \left. \mathbb{H}\left(3;\mathbb{Z}\left[\im\right]\right) \right\backslash \mathbb{H}(3;\mathbb{C})\;. $$
One gets that $\mathbb{I}_3$ is a $3$-dimensional complex nilmanifold, whose ($\mathbb{H}(3;\mathbb{C})$-left-invariant) complex structure $J_{\mathbf{0}}$ is the one inherited by the standard complex structure on $\mathbb{C}^3$; $\mathbb{I}_3$ is called the \emph{Iwasawa manifold}.\\
Since the forms $\varphi^1$, $\varphi^2$ and $\varphi^3$ are $\mathbb{H}(3;\mathbb{C})$-left-invariant, they define a co-frame also for $\left(T^{1,0}\mathbb{I}_3\right)^*$. Note that $\mathbb{I}_3$ is a holomorphically parallelizable manifold, that is, its holomorphic tangent bundle is holomorphically trivial.

Since, for example, $\varphi^3$ is a non-closed holomorphic form, it follows that $\mathbb{I}_3$ admits no K\"ahler metric. In fact, one can show that $\mathbb{I}_3$ is not formal, see \cite[page 158]{fernandez-gray}, therefore the underlying smooth manifold of $\mathbb{I}_3$ has no complex structure admitting K\"ahler metrics, even though all the topological obstructions concerning the Betti numbers are satisfied.

We sketch in Figure \ref{fig:iwasawa} the structure of the finite-dimensional double complex $\left(\wedge^{\bullet,\bullet}\left(\mathfrak{h}_3\otimes_\mathbb{R}\mathbb{C}\right)^*,\,\partial,\,\overline{\del}\right)$: horizontal arrows are meant as $\partial$, vertical ones as $\overline{\del}$ and zero arrows are not depicted.

\smallskip
\begin{center}
\begin{figure}[ht]
 \includegraphics[width=6.5cm]{iwasawa.1}
 \caption{The double complex $\left(\wedge^{\bullet,\bullet}\left(\mathfrak{h}_3\otimes_\mathbb{R} \mathbb{C}\right)^*,\,\partial,\,\overline{\del}\right)$.}
 \label{fig:iwasawa}
\end{figure}
\end{center}
\smallskip

\subsection{Small deformations of the Iwasawa manifold}\label{sec:deformations-iwasawa}
I. Nakamura classified in \cite[\S2]{nakamura} the three-dimensional holomorphically parallelizable solvmanifolds into four classes by numerical invariants, giving $\mathbb{I}_3$ as an example in the second class. Moreover, he explicitly constructed the Kuranishi family of deformations of $\mathbb{I}_3$, showing that it is smooth, \cite[pages 94--95]{nakamura}, compare also \cite[Corollary 4.9]{rollenske-jems}; in particular, he computed the Hodge numbers of the small deformations of $\mathbb{I}_3$ proving that they have not to remain invariant along a complex-analytic family of complex structures, \cite[Theorem 2]{nakamura}, compare also \cite[\S4]{ye}; moreover, he proved in this way that the property of being holomorphically parallelizable is not stable under small deformations, \cite[page 86]{nakamura}, compare also \cite[Theorem 5.1, Corollary 5.2]{rollenske-jems}. I. Nakamura divided the small deformations of $\mathbb{I}_3$ into three classes according to their Hodge diamond, see \cite[page 96]{nakamura} (see
also \cite[\S17]{ueno}).

\smallskip

\begin{thm}[{\cite[pages 94--96]{nakamura}}]
There exists a locally complete complex-analytic family of complex structures $\left\{X_\mathbf{t}=\left(\mathbb{I}_3,\,J_\mathbf{t}\right)\right\}_{\mathbf{t}\in \Delta(\mathbf{0},\varepsilon)}$, deformations of $\mathbb{I}_3$, depending on six parameters
$$ \mathbf{t}\;=\;\left(t_{11},\,t_{12},\,t_{21},\,t_{22},\,t_{31},\,t_{32}\right)\;\in\;\Delta(\mathbf{0},\varepsilon)\;\subset\;\mathbb{C}^6 \;,$$
where $\varepsilon>0$ is small enough, $\Delta(\mathbf{0},\varepsilon):=\left\{\mathbf{s}\in\mathbb{C}^6 \;:\; \left|\mathbf{s}\right|<\varepsilon\right\}$ and $X_\mathbf{0}=\mathbb{I}_3$.\\
A set of holomorphic coordinates for $X_\mathbf{t}$ is given by
$$
\left\{
\begin{array}{rcccl}
 \zeta^1 &:=& \zeta^1(\mathbf{t}) &:=& z^1+\sum_{k=1}^{2}t_{1k}\,\bar z^k \\[5pt]
 \zeta^2 &:=& \zeta^2(\mathbf{t}) &:=& z^2+\sum_{k=1}^{2}t_{2k}\,\bar z^k \\[5pt]
 \zeta^3 &:=& \zeta^3(\mathbf{t}) &:=& z^3+\sum_{k=1}^{2}\left(t_{3k}+t_{2k}\,z^1\right)\bar z^k+A\left(\bar z^1,\,\bar z^2\right)-D\left(\mathbf{t}\right)\,\bar z^3
\end{array}
\right.
$$
where
$$ D\left(\mathbf{t}\right):=\det\left(
\begin{array}{cc}
 t_{11} & t_{12} \\
 t_{21} & t_{22}
\end{array}
\right)
\qquad
$$
and
$$ A\left(\bar z^1,\,\bar z^2\right):=\frac{1}{2}\left(t_{11}\,t_{21}\,\left(\bar z^1\right)^2+2\,t_{11}\,t_{22}\,\bar z^1\,\bar z^2+t_{12}\,t_{22}\,\left(\bar z^2\right)^2\right) \;. $$
For every $\mathbf{t}\in \Delta(\mathbf{0},\varepsilon)$, the universal covering of $X_\mathbf{t}$ is $\mathbb{C}^3$; more precisely,
$$ X_\mathbf{t} \;=\; \left. \Gamma_\mathbf{t} \right \backslash\mathbb{C}^3 \;,$$
where $\Gamma_\mathbf{t}$ is the subgroup generated by the transformations
$$ \left(\zeta^1,\,\zeta^2,\,\zeta^3\right) \stackrel{\left(\omega^1,\,\omega^2,\,\omega^3\right)}{\mapsto} \left(\tilde\zeta^1,\,\tilde\zeta^2,\,\tilde\zeta^3\right) \;,$$
varying $\left(\omega^1,\,\omega^2,\,\omega^3\right)\in\left(\mathbb{Z}\left[\im\right]\right)^3$, where
$$
\left\{
\begin{array}{rcl}
 \tilde\zeta^1 &:=& \zeta^1+\left(\omega^1+t_{11}\,\bar\omega^1+t_{12}\,\bar\omega^2\right) \\[10pt]
 \tilde\zeta^2 &:=& \zeta^2+\left(\omega^2+t_{21}\,\bar\omega^1+t_{22}\,\bar\omega^2\right) \\[10pt]
 \tilde\zeta^3 &:=& \zeta^3+\left(\omega^3+t_{31}\,\bar\omega^1+t_{32}\,\bar\omega^2\right)+\omega^1\,\zeta^2\\[5pt]
&&+\left(t_{21}\,\bar\omega^1+ t_{22}\,\bar\omega^2\right)\left(\zeta^1+\omega^1\right)+A\left(\bar \omega^1,\,\bar\omega^2\right)-D\left(\mathbf{t}\right)\,\bar\omega^3
\end{array}
\right. \;.
$$
\end{thm}

Bott-Chern cohomology distinguishes a finer classification of small deformations of $\mathbb{I}_3$ than Nakamura's classification into three classes, see \S\ref{sec:bott-chern-iwasawa}; the classes and subclasses of this classification are characterized by the following values of the parameters:\\
\begin{description}
 \item[class {\itshape (i)}] $t_{11}=t_{12}=t_{21}=t_{22}=0$; \\
 \item[class {\itshape (ii)}] $D\left(\mathbf{t}\right)=0$ and $\left(t_{11},\,t_{12},\,t_{21},\,t_{22}\right)\neq \left(0,\,0,\,0,\,0\right)$:
    \begin{description}
     \item[subclass {\itshape (ii.a)}] $D\left(\mathbf{t}\right)=0$ and $\rk S=1$;
     \item[subclass {\itshape (ii.b)}] $D\left(\mathbf{t}\right)=0$ and $\rk S=2$;\\
    \end{description}
 \item[class {\itshape (iii)}] $D\left(\mathbf{t}\right)\neq 0$:
    \begin{description}
     \item[subclass {\itshape (iii.a)}] $D\left(\mathbf{t}\right)\neq 0$ and $\rk S=1$;
     \item[subclass {\itshape (iii.b)}] $D\left(\mathbf{t}\right)\neq 0$ and $\rk S=2$;\\
    \end{description}
\end{description}
the matrix $S$ is defined by
$$ S \;:=\;
\left(
\begin{array}{cccc}
 \overline{\sigma_{1\bar1}} & \overline{\sigma_{2\bar2}} & \overline{\sigma_{1\bar2}} & \overline{\sigma_{2\bar1}} \\
 \sigma_{1\bar1} & \sigma_{2\bar2} & \sigma_{2\bar1} & \sigma_{1\bar2}
\end{array}
\right)
$$
where $\sigma_{1\bar1},\,\sigma_{1\bar2},\,\sigma_{2\bar1},\,\sigma_{2\bar2}\in\mathbb{C}$ and $\sigma_{12}\in\mathbb{C}$ are complex numbers depending only on $\mathbf{t}$ such that
$$ \de\varphi^3_\mathbf{t} \;=:\; \sigma_{12}\,\varphi^1_{\mathbf{t}}\wedge\varphi^2_{\mathbf{t}}+\sigma_{1\bar1}\,\varphi^1_{\mathbf{t}}\wedge\bar\varphi^1_{\mathbf{t}}+\sigma_{1\bar2}\,\varphi^1_{\mathbf{t}}\wedge\bar\varphi^2_{\mathbf{t}}+\sigma_{2\bar1}\,\varphi^2_{\mathbf{t}}\wedge\bar\varphi^1_{\mathbf{t}}+\sigma_{2\bar2}\,\varphi^2_{\mathbf{t}}\wedge\bar\varphi^2_{\mathbf{t}} \;,$$
being
$$ \varphi^1_\mathbf{t} \;:=\; \de\zeta^1_{\mathbf{t}}\;,\quad \varphi^2_\mathbf{t} \;:=\; \de\zeta^2_{\mathbf{t}}\;,\quad \varphi^3_{\mathbf{t}} \;:=\; \de\zeta^3_{\mathbf{t}}-z_1\,\de\zeta^2_{\mathbf{t}}-\left(t_{21}\,\bar z^1+t_{22}\,\bar z^2\right)\de\zeta^1_{\mathbf{t}} \;,$$
see \S\ref{sec:structure-equations-iwasawa}.
The first order asymptotic behaviour of $\sigma_{12},\,\sigma_{1\bar1},\,\sigma_{1\bar2},\,\sigma_{2\bar1},\,\sigma_{2\bar2}$ for $\mathbf{t}$ near $0$ is the following:
\begin{equation}\label{eq:struttura-asintotica}
\left\{
\begin{array}{rcl}
\sigma_{12} &=& -1 +\opiccolo{\mathbf{t}} \\[5pt]
\sigma_{1\bar1} &=& t_{21} +\opiccolo{\mathbf{t}}  \\[5pt]
\sigma_{1\bar2} &=& t_{22} +\opiccolo{\mathbf{t}}  \\[5pt]
\sigma_{2\bar1} &=& -t_{11} +\opiccolo{\mathbf{t}} \\[5pt]
\sigma_{2\bar2} &=& -t_{12} +\opiccolo{\mathbf{t}} \\[5pt]
\end{array}
\right.
\qquad \text{ for } \qquad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}
\;,
\end{equation}
and, more precisely, for deformations in class {\itshape (ii)} we actually have that
\begin{equation}\label{eq:struttura-asintotica-ii}
\left\{
\begin{array}{rcl}
\sigma_{12} &=& -1 +\opiccolo{\mathbf{t}} \\[5pt]
\sigma_{1\bar1} &=& t_{21} \left(1+\mathrm{o}\left(1\right)\right)  \\[5pt]
\sigma_{1\bar2} &=& t_{22} \left(1+\mathrm{o}\left(1\right)\right)  \\[5pt]
\sigma_{2\bar1} &=& -t_{11} \left(1+\mathrm{o}\left(1\right)\right) \\[5pt]
\sigma_{2\bar2} &=& -t_{12} \left(1+\mathrm{o}\left(1\right)\right) \\[5pt]
\end{array}
\right. \qquad \text{ for } \qquad \mathbf{t}\in\text{ class {\itshape (ii)}}\;.
\end{equation}
\medskip

The complex manifold $X_\mathbf{t}$ is endowed with the $J_\mathbf{t}$-Hermitian $\mathbb{H}(3;\mathbb{C})$-left-invariant metric $g_\mathbf{t}$, which is defined as follows:
$$ g_\mathbf{t} \;:=\; \sum_{j=1}^3\varphi_\mathbf{t}^j\odot\bar\varphi_\mathbf{t}^j \;. $$

\subsection{Computation of the structure equations for small deformations of the Iwasawa manifold}\label{sec:structure-equations-iwasawa}
In this section, we give the structure equations for the small deformations of the Iwasawa manifold; we will use these computations in \S\ref{sec:dolbeault-iwasawa} and \S\ref{sec:bott-chern-iwasawa}.

Consider
$$
\left\{
\begin{array}{rcl}
 \varphi^1_{\mathbf{t}} &:=& \de\zeta^1_{\mathbf{t}} \\[5pt]
 \varphi^2_{\mathbf{t}} &:=& \de\zeta^2_{\mathbf{t}} \\[5pt]
 \varphi^3_{\mathbf{t}} &:=& \de\zeta^3_{\mathbf{t}}-z_1\,\de\zeta^2_{\mathbf{t}}-\left(t_{21}\,\bar z^1+t_{22}\,\bar z^2\right)\de\zeta^1_{\mathbf{t}}
\end{array}
\right. \;.
$$
as a co-frame of $(1,0)$-forms on $X_{\mathbf{t}}$. We want to write the structure equations for $X_{\mathbf{t}}$ with respect to this co-frame.\\
For the complex structures in the class {\itshape (i)}, one checks that the structure equations are the same as the ones for $\mathbb{I}_3$, that is
$$
\left\{
\begin{array}{rcl}
 \de\varphi^1_{\mathbf{t}} &=& 0 \\[10pt]
 \de\varphi^2_{\mathbf{t}} &=& 0 \\[10pt]
 \de\varphi^3_{\mathbf{t}} &=& -\varphi^1_{\mathbf{t}}\wedge\varphi^2_{\mathbf{t}}
\end{array}
\right. \qquad \text{ for } \quad \mathbf{t}\in\text{ class {\itshape (i)}}
\;.
$$
For small deformations in classes {\itshape (ii)} and {\itshape (iii)}, we have that
$$
\left\{
\begin{array}{rcl}
 \de\varphi^1_{\mathbf{t}} &=& 0 \\[10pt]
 \de\varphi^2_{\mathbf{t}} &=& 0 \\[10pt]
 \de\varphi^3_{\mathbf{t}} &=& \sigma_{12}\,\varphi^1_{\mathbf{t}}\wedge\varphi^2_{\mathbf{t}}\\[5pt]
 &&+\sigma_{1\bar1}\,\varphi^1_{\mathbf{t}}\wedge\bar\varphi^1_{\mathbf{t}}+\sigma_{1\bar2}\,\varphi^1_{\mathbf{t}}\wedge\bar\varphi^2_{\mathbf{t}}\\[5pt]
 &&+\sigma_{2\bar1}\,\varphi^2_{\mathbf{t}}\wedge\bar\varphi^1_{\mathbf{t}}+\sigma_{2\bar2}\,\varphi^2_{\mathbf{t}}\wedge\bar\varphi^2_{\mathbf{t}}
\end{array}
\right. \qquad \text{ for } \qquad \mathbf{t}\in\text{ classes {\itshape (ii)} and {\itshape (iii)}} \;,
$$
where $\sigma_{12},\,\sigma_{1\bar1},\,\sigma_{1\bar2},\,\sigma_{2\bar1},\,\sigma_{2\bar2}\in\mathbb{C}$ are complex numbers depending only on $\mathbf{t}$; their explicit values in the case of class {\itshape (ii)} have been written down by the author and A. Tomassini in \cite[page 416]{angella-tomassini} and in the case of class {\itshape (iii)} are a bit more involved; here, we need just the first order asymptotic behavior for $\mathbf{t}$ near $0$ given in \eqref{eq:struttura-asintotica} and \eqref{eq:struttura-asintotica-ii}.

\section{The cohomologies of the Iwasawa manifold and of its small deformations}\label{part:cohomology-iwasawa}

\subsection{The de Rham cohomology of the Iwasawa manifold and of its small deformations}\label{sec:derham-iwasawa}
By a result by C. Ehresmann, every complex-analytic family of compact complex manifolds is trivial as a differentiable family of compact differentiable manifolds, see, e.g., \cite[Theorem 4.1]{kodaira-morrow}. Therefore the de Rham cohomology of small deformations of the Iwasawa manifold is the same as the de Rham cohomology of $\mathbb{I}_3$, which one can compute with the aid of Nomizu's theorem, \cite[Theorem 1]{nomizu}.\\
In the table below, we list the harmonic representatives with respect to the metric $g_\mathbf{0}$ instead of their classes and we shorten the notation (as we will do also in the following) writing, for example, $\varphi^{A\bar B}:=\varphi^A\wedge\bar\varphi^B$.

\smallskip
\begin{center}
\begin{tabular}{c||c|c}
$H^k_{dR}\left(\mathbb{I}_3;\mathbb{C}\right)$ & $g_{\mathbf{0}}$-harmonic representatives & dimension\\[5pt]
\hline
\hline
$k=1$ & $\varphi^1$, $\varphi^2$, $\bar{\varphi}^1$, $\bar{\varphi}^2$ & $4$\\[5pt]
$k=2$ & $\varphi^{13}$, $\varphi^{23}$, $\varphi^{1\bar{1}}$, $\varphi^{1\bar{2}}$, $\varphi^{2\bar{1}}$, $\varphi^{2\bar{2}}$,$\varphi^{\bar{1}\bar{3}}$, $\varphi^{\bar{2}\bar{3}}$ & $8$\\[5pt]
$k=3$ & $\varphi^{123}$, $\varphi^{13\bar{1}}$, $\varphi^{13\bar{2}}$, $\varphi^{23\bar{1}}$, $\varphi^{23\bar{2}}$, $\varphi^{1\bar{1}\bar{3}}$, $\varphi^{1\bar{2}\bar{3}}$, $\varphi^{2\bar{1}\bar{3}}$, $\varphi^{2\bar{2}\bar{3}}$, $\varphi^{\bar{1}\bar{2}\bar{3}}$ & $10$\\[5pt]
$k=4$ & $\varphi^{123\bar{1}}$, $\varphi^{123\bar{2}}$, $\varphi^{13\bar{1}\bar{3}}$, $\varphi^{13\bar{2}\bar{3}}$, $\varphi^{23\bar{1}\bar{3}}$, $\varphi^{23\bar{2}\bar{3}}$, $\varphi^{1\bar{1}\bar{2}\bar{3}}$, $\varphi^{2\bar{1}\bar{2}\bar{3}}$ & $8$\\[5pt]
$k=5$ & $\varphi^{123\bar{1}\bar{3}}$, $\varphi^{123\bar{2}\bar{3}}$, $\varphi^{13\bar{1}\bar{2}\bar{3}}$, $\varphi^{23\bar{1}\bar{2}\bar{3}}$ & $4$
\end{tabular}
\end{center}
\smallskip

Note that all the $g_\mathbf{0}$-harmonic representatives of $H^\bullet_{dR}(\mathbb{I}_3;\mathbb{R})$ are of pure type with respect to $J_{\mathbf{0}}$, that is, they are in $\left(\wedge^{p,q}\mathbb{I}_3\oplus \wedge^{q,p}\mathbb{I}_3\right) \cap \wedge^{p+q}\mathbb{I}_3$ for some $p,q\in\{0,\,1,\,2,\,3\}$. This holds no more true for $J_\mathbf{t}$ with $\mathbf{t}\neq\mathbf{0}$ small enough, see \cite[Theorem 5.1]{angella-tomassini}.

\subsection{The Dolbeault cohomology of the Iwasawa manifold and of its small deformations}\label{sec:dolbeault-iwasawa}
The Hodge numbers of the Iwasawa manifold and of its small deformations have been computed by I. Nakamura in \cite[page 96]{nakamura}. The $g_\mathbf{t}$-harmonic representatives for $H^{\bullet,\bullet}_{\overline{\del}}\left(X_{\mathbf{t}}\right)$, for $\mathbf{t}$ small enough, can be computed using the considerations in \S\ref{subsec:cohomology-computation-derham-dolbeault} and the structure equations given in \S\ref{sec:structure-equations-iwasawa}. We collect here the results of the computations.\\
In order to reduce the number of cases under consideration, note that, on a compact complex Hermitian manifold $X$ of complex dimension $n$, for any $p,q\in\mathbb{N}$, the Hodge-$*$-operator induces an isomorphism between
$$ H^{p,q}_{\overline{\del}}(X) \stackrel{\simeq}{\longrightarrow} H^{n-q,n-p}_{\partial}(X)\simeq \overline{H^{n-p,n-q}_{\overline{\del}}(X)} \;.$$

\medskip

As regards $1$-forms, we have that
$$
\dim_\mathbb{C} H^{1,0}_{\overline{\del}}(X_\mathbf{t})\;=\;
\left\{
\begin{array}{lcl}
3 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (i)}}\\[5pt]
2 & \quad \text{ for } \quad & \mathbf{t}\in\text{ classes {\itshape (ii)} and {\itshape (iii)}}
\end{array}
\right. \;,
$$
which in particular means that $X_{\mathbf{t}}$ is not holomorphically parallelizable for $\mathbf{t}$ in classes {\itshape (ii)} and {\itshape (iii)}, see \cite[pages 86, 96]{nakamura},
and that
$$
\dim_\mathbb{C} H^{0,1}_{\overline{\del}}(X_\mathbf{t})\;=\;2 \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;.
$$

\medskip

\noindent As regards $2$-forms, we have that
$$
\dim_\mathbb{C} H^{2,0}_{\overline{\del}}(X_\mathbf{t})\;=\;
\left\{
\begin{array}{lcl}
 3 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (i)}} \\[5pt]
 2 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (ii)}} \\[5pt]
 1 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (iii)}}
\end{array}
\right.
$$
and
$$
\dim_\mathbb{C} H^{1,1}_{\overline{\del}}(X_\mathbf{t})\;=\;
\left\{
\begin{array}{lcl}
 6 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (i)}} \\[5pt]
 5 & \quad \text{ for } \quad & \mathbf{t}\in\text{ classes {\itshape (ii)} and {\itshape (iii)}}
\end{array}
\right.
$$
and
$$
\dim_\mathbb{C} H^{0,2}_{\overline{\del}}(X_\mathbf{t})\;=\;2 \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;.
$$

\medskip

\noindent Finally, as regards $3$-forms, we have that
$$
\dim_\mathbb{C} H^{3,0}_{\overline{\del}}(X_{\mathbf{t}}) \;=\; 1 \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}
$$
and
$$
\dim_\mathbb{C} H^{2,1}_{\overline{\del}}(X_{\mathbf{t}})\;=\;
\left\{
\begin{array}{lcl}
6 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (i)}} \\[5pt]
5 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (ii)}} \\[5pt]
4 & \quad \text{ for } \quad & \mathbf{t}\in\text{ class {\itshape (iii)}}
\end{array}
\right. \;.
$$

\subsection{The Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations}\label{sec:bott-chern-iwasawa}
In this section, using the results of Theorem \ref{thm:bc-invariant-cor} and Theorem \ref{thm:bc-invariant-open}, we explicitly compute the dimensions and we compute the $g_\mathbf{t}$-harmonic representatives for $H_{BC}^{\bullet,\bullet}(X_{\mathbf{t}})$, for $\mathbf{t}$ small enough.
\\
In order to reduce the number of cases under consideration, note that, on a compact complex Hermitian manifold $X$ of complex dimension $n$, for every $p,q\in\mathbb{N}$, the conjugation induces an isomorphism between
$$ H^{p,q}_{BC}(X) \;\simeq\; H^{q,p}_{BC}(X) $$
and the Hodge-$*$-operator induces an isomorphism between
$$ H^{p,q}_{BC}(X) \stackrel{\simeq}{\longrightarrow} H^{n-q,n-p}_{A}(X) \;;$$
furthermore, note that
$$ H^{p,0}_{BC}(X) \;\simeq\; \ker\left(\de\colon \wedge^{p,0}X\to\wedge^{p+1}(X;\mathbb{C})\right) $$
and
$$ H^{n,0}_{BC}(X) \;\simeq\; H^{n,0}_{\overline{\del}}(X) \;.$$

\medskip

We summarize the results of the computations below in the following theorem.
\begin{thm}\label{thm:bc-iwasawa}
 Let $\mathbb{I}_3$ be the Iwasawa manifold and consider its small deformations $\left\{X_\mathbf{t}=\left(\mathbb{I}_3,\,J_\mathbf{t}\right)\right\}_{\mathbf{t}\in\Delta(\mathbf{0},\varepsilon)}$, where $\varepsilon>0$ is small enough and $X_\mathbf{0}=\mathbb{I}_3$. Then the dimensions $h^{p,q}_{BC}:=h^{p,q}_{BC}\left(X_\mathbf{t}\right):=\dim_\mathbb{C} H^{p,q}_{BC}\left(X_\mathbf{t}\right)=\dim_\mathbb{C} H^{3-p,3-q}_{A}\left(X_\mathbf{t}\right)$ does not depend on $\mathbf{t}\in\Delta(\mathbf{0},\varepsilon)$ if $p+q$ is odd or $(p,q)\in\left\{(1,1),\,(3,1),\,(1,3)\right\}$ and they are equal to
$$
\begin{array}{ccccccccccc}
 h^{1,0}_{BC} &=& h^{0,1}_{BC} &=& 2 \;, & & & & & & \\[5pt]
 h^{2,0}_{BC} &=& h^{0,2}_{BC} &\in& \left\{1,\,2,\,3\right\}\;, & \qquad & & & h^{1,1}_{BC} & = & 4\;, \\[5pt]
 h^{3,0}_{BC} &=& h^{0,3}_{BC} &=& 1 \;, & \qquad & h^{2,1}_{BC} & = & h^{1,2}_{BC} & = & 6\;, \\[5pt]
 h^{3,1}_{BC} &=& h^{1,3}_{BC} &=& 2 \;, & \qquad & & & h^{2,2}_{BC} & \in & \left\{6,\,7,\,8\right\} \;, \\[5pt]
 h^{3,2}_{BC} &=& h^{2,3}_{BC} &=& 3 \;. & & & & & &
\end{array}
$$
\end{thm}

\begin{rem}
 As a consequence of the computations below, we notice that the Bott-Chern cohomology allows to give a finer classification of the small deformations of $\mathbb{I}_3$ than the Dolbeault cohomology: indeed, note that $\dim_\mathbb{C} H^{2,2}_{BC}(X_\mathbf{t})$ assumes different values according to different parameters in the same class {\itshape (ii)} or {\itshape (iii)}; in a sense, this says that the Bott-Chern cohomology ``carries more informations'' about the complex structure that the Dolbeault one. Note also that most of the dimensions of Bott-Chern cohomology groups are invariant under small deformations: this happens for example for the odd-degree Bott-Chern cohomology groups.
\end{rem}

\medskip

\subsubsection*{$1$-forms}\label{subsec:bott-chern-iwasawa-1}
It is straightforward to check that
$$ H^{1,0}_{BC}(X_{\mathbf{t}})\;=\:\mathbb{C}\left\langle \phit{1},\,\phit{2}\right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;. $$

\subsubsection*{$2$-forms}\label{subsec:bott-chern-iwasawa-2}
It is straightforward to compute
$$ H^{2,0}_{BC}(X_{\mathbf{t}}) \;=\; \mathbb{C}\left\langle \phit{12},\,\phit{13},\,\phit{23} \right\rangle \quad \text{ for }\quad \mathbf{t}\in\text{ class {\itshape (i)}} \;.$$
The computations for $H^{2,0}_{BC}(X_{\mathbf{t}})$ reduce to find $\psi=A\,\phit{12}+B\,\phit{13}+C\,\phit{23}$ where $A,\,B,\,C\in\mathbb{C}$ satisfy the linear system
$$
\left(
\begin{array}{ccc}
 0 & 0 & 0 \\
 0 & -\sigma_{2\bar1} & \sigma_{1\bar1} \\
 0 & -\sigma_{2\bar2} & \sigma_{1\bar2}
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
 A \\
 B \\
 C
\end{array}
\right)
=
\left(
\begin{array}{c}
 0 \\
 0 \\
 0
\end{array}
\right) \;,
$$
whose matrix has rank $0$ for $\mathbf{t}\in\text{ class {\itshape (i)}}$, rank $1$ for $\mathbf{t}\in\text{ class {\itshape (ii)}}$ and rank $2$ for $\mathbf{t}\in\text{ class {\itshape (iii)}}$; so, in particular, we get that
$$ \dim_\mathbb{C} H^{2,0}_{BC}(X_{\mathbf{t}}) \;=\; 2 \quad \text{ for } \quad \mathbf{t}\in\text{ class {\itshape (ii)}} $$
and
$$ \dim_\mathbb{C} H^{2,0}_{BC}(X_{\mathbf{t}}) \;=\; 1 \quad \text{ for } \quad \mathbf{t}\in\text{ class {\itshape (iii)}} $$
(more precisely, for $\mathbf{t}\in\text{ class {\itshape (iii)}}$ we have $H^{2,0}_{BC}(X_{\mathbf{t}})=\mathbb{C}\left\langle \phit{12}\right\rangle$).\\
It remains to compute $H^{1,1}_{BC}(X_\mathbf{t})$ for $\mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}$. First of all, it is easy to check that
$$ H^{1,1}_{BC}(X_{\mathbf{t}}) \;\supseteq\; \mathbb{C}\left\langle \phit{1\bar1},\,\phit{1\bar2},\,\phit{2\bar1},\,\phit{2\bar2} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;,$$
and equality holds if $\mathbf{t}\in\text{ class {\itshape (i)}}$, hence, in particular, if $\mathbf{t}=\mathbf{0}$: this immediately implies that
$$ H^{1,1}_{BC}(X_{\mathbf{t}}) \;=\; \mathbb{C}\left\langle \phit{1\bar1},\,\phit{1\bar2},\,\phit{2\bar1},\,\phit{2\bar2} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;:$$
indeed, being $H^{1,1}_{BC}(X_\mathbf{t})$ isomorphic to the kernel of the self-adjoint elliptic differential operator $\tilde\Delta_{BC_{J_\mathbf{t}}}$, the function $\mathbf{t}\mapsto \dim_\mathbb{C} H^{1,1}_{BC}(X_\mathbf{t})$ is upper-semi-continuous at $0$. (One can explain this argument saying that the new parts appearing in the computations for $\mathbf{t}\neq \mathbf{0}$ are ``too small'' to balance out the lack for the $\partial$-closure or the $\overline{\del}$-closure.) From another point of view, we can note that $(1,1)$-forms of the type $\psi=A\,\phit{1\bar3}+B\,\phit{2\bar3}+C\,\phit{3\bar1}+D\,\phit{3\bar2}$ are $\tilde\Delta_{BC_{J_\mathbf{t}}}$-harmonic if and only if $A,\,B,\,C,\,D\in\mathbb{C}$ satisfy the linear system
$$
\left(
\begin{array}{cccc}
 -\overline{\sigma_{12}} & 0 & -\sigma_{1\bar2} & -\sigma_{1\bar1} \\
 0 & -\overline{\sigma_{12}} & -\sigma_{2\bar2} & -\sigma_{2\bar1} \\
\hline
 \overline{\sigma_{1\bar2}} & -\overline{\sigma_{1\bar1}} & \sigma_{12} & 0 \\
 \overline{\sigma_{2\bar2}} & -\overline{\sigma_{2\bar1}} & 0 & \sigma_{12}
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
 A \\
 B \\
 C \\
 D
\end{array}
\right)
\;=\;
\left(
\begin{array}{c}
 0 \\
 0 \\
 0 \\
 0
\end{array}
\right) \;,
$$
whose matrix has rank $4$ for every $\mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}$.

\subsubsection*{$3$-forms}\label{subsec:bott-chern-iwasawa-3}
Since the special form of the structure equations, it is straightforward to compute
$$ H^{3,0}_{BC}(X_{\mathbf{t}})\;=\;\mathbb{C} \left\langle \phit{123} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;.$$
Moreover,
\begin{eqnarray*}
H^{2,1}_{BC}(X_{\mathbf{t}})&=&\mathbb{C} \left\langle \phit{12\bar1},\, \phit{12\bar2},\, \phit{13\bar1}-\frac{\sigma_{2\bar2}}{\overline{\sigma_{12}}}\,\phit{12\bar3},\, \phit{13\bar2}+\frac{\sigma_{2\bar1}}{\overline{\sigma_{12}}}\,\phit{12\bar3},\, \phit{23\bar1}+\frac{\sigma_{1\bar2}}{\overline{\sigma_{12}}}\,\phit{12\bar3},\right.\\[5pt]
&&\left. \phit{23\bar2}-\frac{\sigma_{1\bar1}}{\overline{\sigma_{12}}}\,\phit{12\bar3} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}\;;
\end{eqnarray*}
in particular,
$$ H^{2,1}_{BC}(X_{\mathbf{t}})\;=\;\mathbb{C} \left\langle \phit{12\bar1},\, \phit{12\bar2},\, \phit{13\bar1},\, \phit{13\bar2},\, \phit{23\bar1},\, \phit{23\bar2} \right\rangle  \quad \text{ for } \quad \mathbf{t}\in\text{ class {\itshape (i)}} \;.$$
From another point of view, one can easily check that
$$
H^{2,1}_{BC}(X_{\mathbf{t}}) \;\supseteq\; \mathbb{C} \left\langle \phit{12\bar1},\, \phit{12\bar2} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}
$$
and that the $(2,1)$-forms of the type $\psi=A\,\phit{12\bar3}+B\,\phit{13\bar1}+C\,\phit{13\bar2}+D\,\phit{23\bar1}+E\,\phit{23\bar2}$ are $\tilde\Delta_{BC_{J_\mathbf{t}}}$-harmonic if and only if $A,\,B,\,C,\,D,\,E\in\mathbb{C}$ satisfy the equation
$$
\left(
\begin{array}{ccccc}
 \overline{\sigma_{12}} & \sigma_{2\bar2} & -\sigma_{2\bar1} & \sigma_{1\bar2} & \sigma_{1\bar1}
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
 A \\
 B \\
 C \\
 D \\
 E
\end{array}
\right)
\;=\;
 0 \;,
$$
whose matrix has rank $1$ for every $\mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}}$. Note in particular that the dimensions of $H^{3,0}_{BC}(X_{\mathbf{t}})$ and of $H^{2,1}_{BC}(X_{\mathbf{t}})$ do not depend on $\mathbf{t}$.

\subsubsection*{$4$-forms}\label{subsec:bott-chern-iwasawa-4}
It is straightforward to compute
$$ H^{3,1}_{BC}(X_\mathbf{t}) \;=\; \mathbb{C} \left\langle \phit{123\bar1},\,\phit{123\bar2} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} $$
and
\begin{eqnarray*}
H^{2,2}_{BC}(X_\mathbf{t}) &=& \mathbb{C}\left\langle \phit{12\bar1\bar3},\, \phit{12\bar2\bar3},\, \phit{13\bar1\bar2},\, \phit{13\bar1\bar3},\, \phit{13\bar2\bar3},\, \phit{23\bar1\bar2},\right.\\[5pt]
&& \left.\phit{23\bar1\bar3},\, \phit{23\bar2\bar3} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ class {\itshape (i)}} \;.
\end{eqnarray*}
Moreover, one can check that
$$ H^{2,2}_{BC}(X_\mathbf{t}) \;\supseteq\; \mathbb{C}\left\langle \phit{12\bar1\bar3},\, \phit{12\bar2\bar3},\, \phit{13\bar1\bar2},\, \phit{23\bar1\bar2} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;.$$
For $H^{2,2}_{BC}(X_\mathbf{t})$ with $\mathbf{t}\in\text{ class {\itshape (ii)} and {\itshape (iii)}}$, we get a new behavior: there are subclasses in both class {\itshape (ii)} and class {\itshape (iii)}, which can be distinguished by the dimension of $H^{2,2}_{BC}(X_\mathbf{t})$. Indeed, consider $(2,2)$-forms of the type $\psi=A\,\phit{13\bar1\bar3}+B\,\phit{13\bar2\bar3}+C\,\phit{23\bar1\bar3}+D\,\phit{23\bar2\bar3}$; a straightforward computation shows that such a $\psi$ is $\tilde\Delta_{BC_{J_\mathbf{t}}}$-harmonic if and only if $A,\,B,\,C,\,D\in\mathbb{C}$ satisfy the linear system
$$
\left(
\begin{array}{cccc}
 \overline{\sigma_{2\bar2}} & -\overline{\sigma_{1\bar2}} & -\overline{\sigma_{2\bar1}} & \overline{\sigma_{1\bar1}} \\
 \sigma_{2\bar2} & -\sigma_{2\bar1} & -\sigma_{1\bar2} & \sigma_{1\bar1}
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
 A \\
 B \\
 C \\
 D
\end{array}
\right)
=
\left(
\begin{array}{c}
 0 \\
 0
\end{array}
\right) \;:
$$
as one can easily note, the rank of the matrix involved is $0$ for $\mathbf{t}\in\text{ class {\itshape (i)}}$, while it is $1$ or $2$ depending on the values of the parameters and independently from $\mathbf{t}$ belonging to class {\itshape (ii)} or class {\itshape (iii)}. Therefore
$$ \dim_\mathbb{C} H^{2,2}_{BC}(X_\mathbf{t}) \;=\; 7 \quad \text{ for } \quad \mathbf{t}\in\text{ subclasses {\itshape (ii.a)} and {\itshape (iii.a)}} $$
and
$$ \dim_\mathbb{C} H^{2,2}_{BC}(X_\mathbf{t}) \;=\; 6 \quad \text{ for } \quad \mathbf{t}\in\text{ subclasses {\itshape (ii.b)} and {\itshape (iii.b)}} \;.$$

\subsubsection*{$5$-forms}\label{subsec:bott-chern-iwasawa-5}
Lastly, let us compute $H^{3,2}_{BC}(X_\mathbf{t})$.\\
It is straightforward to check that
$$ H^{3,2}_{BC}(X_{\mathbf{t}}) \;=\; \mathbb{C}\left\langle \phit{123\bar1\bar2},\, \phit{123\bar1\bar3},\, \phit{123\bar2\bar3} \right\rangle \quad \text{ for } \quad \mathbf{t}\in\text{ classes {\itshape (i)}, {\itshape (ii)} and {\itshape (iii)}} \;:$$
in particular, it does not depend on $\mathbf{t}\in\Delta(\mathbf{0},\varepsilon)$.

redpajama_set_name: RedPajamaArXiv


Remote execution of the command given from the Routing Engine (RE) to the PIC failed. There was no reply from the PIC. This message is generated by the mspinfo process, which is responsible for RPC (Remote Procedure Call) communication between the Routing Engine and the Multiservices PICs. This only applies to M-series and T-series routers, and MX-series routers with an MS-DPC in use.
The LIBMSPRPC_CLIENT_NO_REPLY message is logged each time the mspinfo process does not receive an expected reply from a Multiservices PIC.
The message reports the command that was sent to the Multiservices PIC.
Look for any related events that occurred at or just before the LIBMSPRPC_CLIENT_NO_REPLY message.
1. If this issue is time critical or if you will need a root cause analysis then open a case with your technical support representative to troubleshoot further. Otherwise the following steps can be used to attempt to resolve the issue.
Move the PIC to another FPC slot, and if possible on a different FPC.
Replace the PIC with a spare.
Otherwise reset the FPC the PIC is in.
3. If the failure messages continue, open a case with Juniper technical support.


Image Title: Home Decorators Bar Stools Ameri Swivel Stool With Regard To Depot Inspirations 3. Filename: home-decorators-bar-stools-ameri-swivel-stool-with-regard-to-depot-inspirations-3.jpg. Image Dimension: 400 x 329 pixels. Images Format: jpg/jpeg. Publisher/Author: Hubert McGlynn. Uploaded Date: Friday - May 04th. 2018 20:46:07 PM. Category: Architecture. Image Source: housetohome.co.uk.
Tap The Thumbnail Bellow to See Related Gallery of "Home Decorators Bar Stools Ameri Swivel Stool With Regard To Depot Inspirations 3"


Would this be a good Spec Savers ad ?
Click for the song with French sub-titles BUT watch out, there are spelling mistakes in the sub-titles. For the correct text, see the words below.
Paroles de la chanson "Le petit pain au chocolat"
Et pour qu'ils soient heureux.


Today is the first day of spring, 2013, and it's a sunny day in Chicago. And 14 degrees. Fahrenheit. This makes me just crabby enough to trot out the only portrait I've taken that, to my knowledge at least, was not the person I was assigned to photograph. On another sunny day, this one in July 10, 2003 I headed to the Goodman theater to photograph comedy legend Del Close for an article that would soon appear in Chicago magazine. I set up lights and camera, and waited. That he'd been dead for four years, I thought, might make for an interesting image.
Before Close moved on in 1999, he requested that his skull be preserved, then donated to the Goodman theater's prop department. Above all other uses in performance, he wanted it to be cast as the jester Yorick in Shakespeare's Hamlet. In my case he made his entrance in a clear plexiglass box, resting on a red pillow in a small office adjacent to the theater.
Printed from a Polaroid Type 54 negative, the above photograph was an outtake that I've always liked. With its rough edges, processing aberrations, and unbelievably fine grain, the whole of the picture has always felt more than a little surreal to me.
Three years later Close's friend Charna Halpern, charged with delivering his skull to the Goodman, admitted to the New Yorker that, because logistics kept her from producing the actual item, a substitute was purchased from an anatomical supply company. She did mention that a lot of care was taken in finding one that would resemble Close, donating what I hope is now considered to be, at very least, his stunt double.
A brief search shows me that on Myspace, Del Close's Skull has 1197 friends.
Love this post…! To have that many virtual friends in his condition….


They system had been able to go for weeks without an incident. More recently, as more users have been migrated onto the system, we would a series of days with spikes on this wait event, though they were normally quite short-lived. The incident shown in this chart was unusual in that it lasted most of a day. The chart also shows that a reasonable amount of time was lost on db file sequential read, this is consistent with blocks being loaded into the buffer cache, requiring access to the cache buffer chains, and hence requiring access to the latch that protects these chain.
Using a similar technique I was graph the number of database sessions over time, and I found a similar series of spikes.
By eye I could see that the spikes in the latch contention seemed to correspond to the spikes in the number of concurrent user sessions, and they had roughly the same shape.
I then graphed time waited for this latch against the number of concurrent user sessions and I got a strong, and surprisingly linear, correlation.
It is very clear that whenever we had more than about 90 concurrent user sessions, the system also lost time waiting on the cache buffers chains latch. Basically, this application won't scale any further!
So, why was the latch being held for such a long time? The buffer cache is 1Gb (with a block size of 8Kb), so it is large, but not excessively so. At this point somebody (and I can't claim the credit for this) thought to check for migrated rows on the tables involved in the longest running SQL statements during the periods of high latch contention, and discovered that a number of the most heavily use tables had a significant quantity of migrated rows.
A row migrates when the data is updated and there is no free space left in the data block to store the new data values. Oracle puts the row into a new block, but it does not update the indexes to point to the new block. Instead, it puts a forwarding pointer into the original block. Therefore, to read the data from a migrated row requires visits to two blocks. When retrieving data into the buffer cache your process need to update two buffer blocks, and must acquire the buffer chains latch twice. Row migration causes poor performance because Oracle must do twice as much work.
A process inserts a row, and then subsequently updates columns on that row, often from null to not null values, but sometimes with just more data. In my case, the application was storing XML structures in a clob (and most of the clobs were small enough to be stored in-line with the rest of the row).
There is insufficient free space left in the data blocks to allow for the subsequent updates to be stored in the original data block.
In this particular case, I was dealing with a third-party packaged application. So there was no possibility to change the way the application inserts and updats the data. However, we could and did rebuild the object to eliminate migration of existing rows and set a realistic PCTFREE to reserve free space for future rows.
This chart shows the same data as the first one, except that I have changed the scale on the y-axis. The tables with the worst row migration were reorganised on the Tuesday night and the system ran on Wednesday under a normal load without any problem.
Contention on the buffer cache chains latch fell to trivial levels, just 408 seconds in 24 hours. Interestingly, the amount of db file sequential read also fell by 70%. This is due to better cache efficiency. Since eliminating much of the row migration, fewer blocks need to be loaded into cache, and so blocks stay in the buffer cache for longer.
and what are the potential implications of not doing so!
I often use data collected by the Oracle Automatic Workload Repository (AWR) to help to diagnose performance problems. However, I often work on performance problems with application teams, rather than the DBAs. It is surprising how often that I don't have access to Oracle Enterprise Manager.
Somebody might say that the system was slow at a particular time. I want to get an overview of the database at that time, and I might also want to compare it to another time when the system wasn't slow. Later I may generate AWR reports using particular pairs of snapshots, but I need something to direct me to when an issue occurred, and hence which snapshots to compare.
When dealing with performance problems the question is nearly always the same. "How long did I spend doing what?". Amongst lots of metrics, AWR collects snapshots of how much time system-wide has been spent waiting on each wait event.
To give me an overview of that data, I have created an excel spreadsheet that will query the AWR repository and retrieve that data for up to a week into am Excel pivot table, and will then graph the top wait events. Click on the thumbnail on the left to see an example.
First; don't drown in numbers. The eye is very good at picking out patterns and irregularities. You can then drill into the numbers behind the chart.
The chart provides a quick overview of the behaviour of the database. You can quickly see if there is anything to examine in more detail, or whether the database is exhibiting its normal characteristics.
The AWR data is extracted into the spreadsheet, so I can keep a copy of the spreadsheet for reference. I generally like to collect AWR snapshots every 15 minutes. One week of data, at that frequency, turns into about 6Mb of spreadsheet.
Excel charts can easily be put into documents, presentations or e-mails. Excel is very flexible, it is easy to format the chart to show only what is important.
It's free (that is to say that you won't need to go and buy anything else). All you need is Excel and an Oracle client.
Over time I have produced a variety of spreadsheets that collect all sorts of metrics from the Oracle database and others that collect specific metrics from PeopleSoft applications.
To be fair, AWR's forerunner, Statspack collects a lot of similar data to AWR, the essential difference being that AWR is a licensed option. This technique could also be used to extract data from the statspack repository, but most of my customers are licensed for AWR.
You can download a sample Excel workbook and query from my website. Go to one of the sheets with data and 'Refresh Data'. The first time you will probably be prompted to select and ODBC source. You will be prompted for the login credentials.
In a subsequent blog posting I will discuss a real example of where this technique helped me to resolve a problem.
It is fairly straightforward to create spreadsheets that query databases, but here are some links that should help to get you going if you are not familiar with the process.
Querying External Data in Excel. This is a straightforward recipe for using MSQuery.
Update 16.6.2010: I have released a new version with an adjustment to the query used in the spreadsheet. AWR snapshots happen at slightly different times on different RAC nodes, although they have the same SNAP_ID. This was causing multiple rows for the same snapshot in the pivot tables, and so the charts did not show correctly aggregated wait events across RAC nodes.


text:
Football News (Sport)
Nigerian Boxers
Sports has always been a space of utmost engagement for a lot of people all over the world. In North Africa, especially in the country of Nigeria, sports has a place deep in every person's heart.
The country treats sports as a way forward towards progress, development of the youth and the creation of idols and stories that encapsulate all the values that Nigerians stand for. In 4000 BC, boxing was prevalent as a sport in Nigeria, but it was known by a different name. Boxing was known as Pugilism which refers to sweet science. People in these highly skilled combat sports used to wear protective gloves and throw punches at their opponents, but make sure that they do so in a way that is not fatal and within a stipulated period of time, inside a ring.
This primitive form has not changed much and is now called boxing all over the world. Being a very popular sport in Nigeria, the sports enthusiasts love to cheer their favourite players when they are in the ring and bet on them too. Check out the Sporto Betting Guide now. For more sports based articles, keep following onpointy.com.
The following is a list of some of the most famous and spectacular Nigerian boxers of all time:
1. Anthony Joshua: Born in 1989 on the 15th of October, Anthony Joshua is a Nigerian Boxer by birth and has been a world boxing champion many times but has given a severe ache in the hearts of almost all Nigerian citizens because he has never represented his country of birth at the national level or the international level. Joshua currently represent the United Kingdom or Great Britain and he is a unified heavyweight world champion, being the proud holder of the IBF title since 2016 and the WBA (Super) and the IBO titles since April 2017;
2. Samuel Peter: Samuel Okon Peter is another Nigerian Boxing Champion who has represented Nigeria at the international level with the successful way in which he has handled his active days in the boxing ring. He has a string of knockout wins on his radar with a host of heavyweight titles to his name in a time when heavyweight titles were very fragmented and mostly distributed. In 2008, Samuel Okon Peter also won a part of the world heavyweight championship by winning against Oleg Maskaev to get the WBC title;
3. Tiger: This man is not only a Nigerian Boxing Champion but one of the most famous Nigerians of all time. Tiger was born on the 14th of November 1929. He was a professional boxer who won the world middleweight title when he punched his way to beat Gene Fulmer in 1962. He also dethroned José Torres of Puerto Rico in the lightweight section in 1966.
Tiger was also an ethnic Igbo man and a soldier who served in the Biafran army during the Nigerian Civil War.
So, what are you waiting for? Go ahead and check out more facts about these amazing Nigerian boxers now because they are the worthy idols you want to look up to!
Nigerian Businessman Arrested For Flaunting Cash On IG Allegedly Dies In Dubai Police Custody Apart from Burna Boy and Wizkid, See Another Nigerian Musician Who Won Grammy Yesterday Nigerian man clashes with his girlfriend after finding out she saved another man's number on her phone with "my love" and his with 'big head' (video) Ghana Deports Nigerian Singer, Bella Shmurda For Attacking Physician Who Announced His COVID-19 Outcome
Previous Barcelona Appoint Sergi Barjuan Interim Manager
Next Tottenham Sack Nuno Espirito Santo

redpajama_set_name: RedPajamaCommonCrawl


Ahead of today's Microsoft Surface Studio announcement, there had been rumors floating around that the company would launch an all-in-one desktop. We didn't know it's have a 4500 x 3000 pixel touchscreen display, support for a new accessory called the Surface Dial, or a hinge that lets you adjust the screen for writing or drawing. But we were pretty sure some sort of desktop was coming.
One reason? A new Surface Keyboard showed up at the FCC website earlier this month.
Now that keyboard is available for pre-order for $130 $100. But it's just one of the new Surface-branded accessories Microsoft is launching. Update: The price is now showing as $100.
There's also a $130 Microsoft Surface Ergonomic Keyboard, for folks that prefer an angled, split-key layout and a large palm rest area. And there's a new Surface Mouse that sells for $50.
These Bluetooth accessories have a soft grey finish that makes them look like they were built to work with Surface products (because they were).
It's interesting to see that the leaked keyboard looks pretty much identical to the one that's now available for purchase.
Each item is designed to pair with a Windows, Mac, or Android device over Bluetooth 4.1 Low Energy and each is powered by 2 AAA batteries which should provide up to a year of battery life.
Do you actually ever look at the keyboard while typing? I rarely do and only when I somehow manage to mistype a password 3-4 times in a row.
Actually, I don't. But a lot of people do. The average user do.
I have to imagine that the products don't photograph well and look better in person.
Per the link the ergonomic keyboard only works with 3 surface devices. I wonder why they would limit it in such a manner? I'd like to have one to use with my notebook.
I imagine it would work with more, but few notebooks (and certainly none I've ever had) support Bluetooth 4.0 low energy.
I hadn't thought about that, or really researched BT much, but does it matter? I've seen BT LE devices work with non-LE devices, but those were things that would just connect for a short time. Maybe it would matter more if connected for long periods, increasing the battery drain? Still if that were the case wouldn't Microsoft would say it would work with any Windows 10 device with BT LE?
Yes, it does matter, but only sometimes. It looks like the standard for Low Energy is not backwards compatible with earlier Bluetooth standards, but allows for implementation support for the old standard sort of side by side in a device, which would explain why some devices would work. But this may be an indication that MS decided not to include legacy BT support in order to increase battery life to a year, as claimed.
On the battery life issue starting about 10 years ago I use to use a Microsoft BT keyboard on my HTPC, and the battery life was quite good. But although that computer was on quite a bit, I wasn't really doing a lot of keystrokes, so probably not the best test of battery life.


text:
Фонд свердловин (  ) — загальна кількість свердловин, пробурених на експлуатаційному об'єкті (родовищі, підприємстві), закріплених за нафтогазовидобувним підприємством на кінець звітного періоду (кварталу, року).

Фонд свердловин основний — видобувні і нагнітальні свердловини нафтового експлуатаційного об'єкту, які бурять на першій стадії його розробки за рівномірною сіткою, раціональною для режиму роботи в умовах однорідного пласта (з врахуванням розміру лінз і зон виклинювання). Ці свердловини повинні забезпечити високий рівень видобутку нафти в перші роки і охопити розробкою основну, найбільш монолітну частину пласта.

ФОНД СВЕРДЛОВИН РЕЗЕРВНИЙ — свердловини другої черги, які бурять після свердловин основного фонду як нагнітальні та видобувні і в кількості від декількох до 100 % відносно до основного фонду (в залежності від геолого-фізичних та інших особливостей експлуатаційного об'єкта) на ділянках (окремі лінзи, зони виклинювання і застійні зони), не залучених до вироблення основним фондом свердловин.

Література

Свердловини

redpajama_set_name: RedPajamaWikipedia


Today, Bernhard Law Firm won dismissal of two Brazilian shareholders in a million $ dollar guaranty lawsuit for leasing of South Beach commercial space. (Florida Bar disclaimer: results may not be typical. You may not have as beneficial a result). This dismissal came on the heels of Bernhard Law Firm's success in appealing the initial attempted service on these Brazilian shareholders at corporate depositions in Miami. In that appeal, the Third District Court of Appeal quashed attempted service, given the Brazilians' immunity at corporate depositions. Click HERE for that decision in Queiroz v. Bentley Bay Retail, LLC, 237 So. 3d 1108 (Fla. 3d DCA 2018).
After the Court issued its appellate mandate, no proper service was obtained on the Brazilian shareholders. Bernhard Law Firm ultimately requested that the Court dismiss and drop the shareholders from the lawsuit, given the unreasonable delay in service of process. Today, the Court agreed with Bernhard Law Firm and dismissed the shareholders from the lawsuit altogether. For more on the Florida rules requiring dismissal of defendants for unreasonable delay in service, keep reading below. If you have any questions on defense of South Florida companies, shareholders, contract actions, or guaranty demands, please contact Bernhard Law Firm at www.bernhardlawfirm.com, 786-871-3349, abernhard@bernhardlawfirm.com. Bernhard Law Firm handles trials and appeals in business and financial disputes.
As to the Florida law, under Florida Rule of Civil Procedure 1.070(j), the Court should drop defendants Luiz Queiroz and Karine Queiroz, given failure to serve process in nearly two years without good cause. Fla. R. Civ. P. 1.070(j) (Summons; Time Limit); Gondal v. Martinez, 606 So. 2d 490, 491 (Fla. 3d DCA 1992) (filing affidavit of diligent search over a year after complaint filed did not preclude dropping defendants for failure to serve within 120 days); Hernandez v. Page, 580 So. 2d 793, 795 (Fla. 3d DCA 1991) (aff'ing dismissal under Rule 1.070(j) on holding settlement negotiations did not amount to good cause excusing failure to serve defendants within 120 days); Powell v. Madison Cty. Sheriff's Dept., 100 So. 3d 753, 754–55 (Fla. 4th DCA 2012) (aff'ing dismissal for failure to serve within 120 days under Rule 1.070(j)); see also Del Amico v. Fishermen's Hosp., Inc., 715 So. 2d 1053, 1053 (Fla. 3d DCA 1998) ("dismissing a case is required when there is a failure to obtain service of process within 120 days and no good cause for this failure is demonstrated"); Uvanni v. Gallenga, 619 So. 2d 31, 31 (Fla. 3d DCA 1993) (aff'g dismissal for failure to serve within 120 days); Taco Bell Corp. v. Costanza, 686 So. 2d 773, 773 (Fla. 4th DCA 1997) (rev'g with instructions to enter order dismissing defendants from suit for non-compliance with Rule 1.070(j)); Gen. Motors Acceptance Corp. v. Lanman, 630 So. 2d 682, 682 (Fla. 4th DCA 1994) (counsel attempts to settle was not sufficient good cause to justify failure to serve within 120 days, and thus dismissal of defendants was required). Inadvertence or mistake of counsel does not constitute good cause to avoid dismissal Rule 1.070(j). Id.
If you have any questions on defense of South Florida companies, shareholders, contract actions, or guaranty demands, please contact Bernhard Law Firm at www.bernhardlawfirm.com, 786-871-3349, abernhard@bernhardlawfirm.com. Bernhard Law Firm handles trials and appeals in business and financial disputes.
Next Post Get Prevailing Party Fees for Winning, After Losing the Case by Dismissing?


text:
Two Manchester United players leave the club on loan
Man Utd players Cameron Borthwick-Jackson and Matty Willock have gone out on loan in search of regular first-team football.
Ciaran Kelly
Matty Willock and Cameron Borthwick-Jackson have gone out on loan again - moving to St Mirren and Scunthorpe United respectively.
Willock, who made two matchday squads for the first team in 2016-17, spent last season at FC Utrecht and St Johnstone as he sampled life on loan for the first time in his career.
Borthwick-Jackson, a favourite of former boss Louis van Gaal, played just six games on loan at Leeds before returning to United's reserves in January.
Both players are now 21 so they need these loan spells to go right for them as Borthwick-Jackson alluded to after completing his move to Glanford Park.
"I need to play regular games and I'm at the age where I should be playing regular first-team football," he told the club's website.
Manchester United transfer news LIVE Liverpool team news and mufc fixtures updates
"I know it'll be more a lot more physical in League One, so I'll have to see how I adapt to that and hopefully I can impress."
Willock helped St Johnstone to an eighth-placed finish in the Scottish Premiership last season and will hope for more of the same with newly-promoted St Mirren.
"I'm really delighted. I spoke to the gaffer and he really sold me on the way he wants the team to progress this season," he told the club's website.
"I know what I am coming into. I know the tempo of the game is really quick so hopefully my experience last season stands me in good stead.
"I'm a midfielder, I work hard, I try and drive forward with the ball and will hopefully chip in with some goals and assists."
Get all the latest Manchester United news first with our new app. Download it here now.
Cameron Borthwick-Jackson
Matty Willock
Manchester United FCAnthony Elanga youth coach reveals he almost joined Man City over Manchester UnitedBoth Manchester clubs came calling while he was playing two years above his age category at Hattersley FC
Rio Ferdinand issues response to Cristiano Ronaldo storming off for Manchester United
Cristiano RonaldoThe five-time Ballon d'Or winner was frustrated when Ralf Rangnick subbed him off against Brentford.
Amir Khan fires 'jealousy' jibe after Kell Brook 'disrespect' claim ahead of Manchester fight
Amir KhanAmir Khan and Kell Brook have both detailed their dislike for each other ahead of their February 19th bout in Manchester.
AppleFraggle Rock: Back to the Rock release date and trailer as show returns through Apple TV +The reboot is said to honour the original show


Correa, 23, left Wednesday's game with discomfort in his side, and will be held of the lineup Thursday against the Rangers. However, it appears Correa avoided a long-term injury.
After getting off to a hot start, Correa struggled in May, batting under .200 for the month. He's currently slashing .264/.348/.464 this season with 10 home runs and 39 RBIs.
The Astros (38-25) entered Thursday's action one game behind the Mariners in the American League West.