Deformation Stability of $p$-SKT and $p$-HS manifolds

Houda Bellitir

Abstract. In this paper, we introduce the notions of $p$-Hermitian-symplectic
and $p$-pluriclosed compact complex manifolds as generalisations for an
arbitrary positive integer $p$ not exceeding the complex dimension of the
manifold of the standard notions of Hermitian-symplectic and SKT manifolds
that correspond to the case $p=1$. We then notice that these two properties
are equivalent on $\partial\bar{\partial}$-manifolds and go on to prove that
in (smooth) complex analytic families of $\partial\bar{\partial}$-manifolds,
they are deformation open. Concerning closedness results, we prove that the
cones $\mathcal{A}_{p}$, resp. $\mathcal{C}_{p}$, of Aeppli cohomology classes
of strictly weakly positive $(p,p)$-forms $\Omega$ that are $p$-pluriclosed,
resp. $p$-Hermitian-symplectic, must be equal on the limit fibre if they are
equal on the other fibres and if some rather weak
$\partial\bar{\partial}$-type assumptions are made on the other fibres.

##  1 Introduction

Let $\mathcal{X}$ be a complex manifold and let $\Delta$ be an open ball
containing the origin in $\mathbb{C}^{m}$ for some $m\in\mathbb{N}^{*}$.
Recall the following standard notion: a complex analytic family of compact
complex manifolds is a proper holomorphic submersion
$\pi:\mathcal{X}\longrightarrow\Delta$.

This means that the following conditions are satisfied :

  1. (i)

For each $t\in\Delta$, $X_{t}:=\pi^{-1}(t)$ is a compact complex connected
submanifold of $\mathcal{X}$.

  2. (ii)

the rank of Jacobian matrix of $\pi$ is equal to $n$ at every point of
$\mathcal{X}$.

  3. (iii)

There is a locally finite open covering $\\{\mathcal{U}_{j}/j=1,2,\cdots\\}$
of $\mathcal{X}$ and complex-valued holomorphic functions
$\xi_{j}^{1}(p),\cdots,\xi_{j}^{n}(p)$, defined on $\mathcal{U}_{j}$ such that
for each $t$ , the set :

$\\{p\in\mathcal{U}_{j}\longrightarrow(\xi_{j}^{1}(p),\cdots,\xi_{j}^{n}(p))/\mathcal{U}_{j}\cap\pi^{-1}(t)\neq\emptyset\\}$

is a system of local holomorphic coordinates of $X_{t}$.

By a result of Ehresmann ([Voi07], Theorem 9.3), all the fibres
$X_{t}:=\pi^{-1}(t)$, for all $t\in\Delta$, are $C^{\infty}$-diffeomorphic to
a fixed $C^{\infty}$ manifold $X$. Therefore, the holomorphic family
$(X_{t})_{t\in\Delta}$ of compact complex manifolds can be viewed as a single
$C^{\infty}$ manifold $X$ endowed with a $C^{\infty}$ family of complex
structures $(J_{t})_{t\in\Delta}$.

The purpose of this paper is to relax the notion of $p$-Kählerianity
introduced by Alessandrini and Andreatta in [AA87] for compact complex
manifolds to two notions that offer analogues at the level of $(p,\,p)$-forms
(satisfying a mild strict positivity assumption) for the classical notions of
Hermitian-symplectic, respectively SKT (or pluriclosed) manifolds that
correspond to the case $p=1$. Specifically, a smooth strictly weakly positive
$(p,\,p)$-form $\Omega$ (cf. Definition 2.1) will be said to be $p$-Hermitian-
symplectic ($p$-HS), respectively $p$-SKT (or $p$-pluriclosed), if $\Omega$ is
the $(p,\,p)$-type component of a real $d$-closed $(2p)$-form, respectively if
$\partial\bar{\partial}\Omega=0$. Compact complex manifolds carrying such
forms will be termed $p$-Hermitian-symplectic ($p$-HS) manifolds, respectively
$p$-SKT (or $p$-pluriclosed) manifolds in Definition 2.2.

Our first main result (cf. Theorems 4.3 and 4.5 for precise statements) places
these two notions in the context of deformations of complex structures.

######  Theorem 1.1.

The notion of $p$-Hermitian-symplectic manifold and the notion of $p$-SKT
$\partial\bar{\partial}$-manifold are both open under holomorphic deformations
of the complex structure.

Recall that the property of $p$-Kählerianity of [AA87] and [AB91] is not
deformation-open.

In the second part of the paper, we will investigate some deformation-
closedness issues. To this end, we introduce the following positivity cones in
the Aeppli cohomology of bidegree $(p,\,p)$ on any compact complex
$n$-dimensional manifold $X$:

$\mathcal{A}_{p}(X)=\\{[\Omega]_{A}/\Omega\mbox{ strictly weakly positive such
that }\partial\bar{\partial}\Omega=0\\}\subset
H_{A}^{p,p}(X,\mathbb{R})\subset H_{A}^{p,p}(X,\mathbb{C}),$
$\mathcal{C}_{p}(X)=\\{[\Omega]_{A}/\Omega\mbox{ strictly weakly positive such
that }\Omega\mbox{ is }p\mbox{-HS}\\}\subset H_{A}^{p,p}(X,\mathbb{R})\subset
H_{A}^{p,p}(X,\mathbb{C}).$

The inclusion $\mathcal{C}_{p}(X)\subset\mathcal{A}_{p}(X)$ always holds
trivially.

We prove (cf. Proposition 5.8) the following behaviour of these cones under
holomorphic deformations of the complex structure.

######  Proposition 1.2.

Let $(X_{t})_{t\in\Delta}$ be a holomorphic family of compact complex
manifolds. If $\mathcal{A}_{p}(X_{t})=\mathcal{C}_{p}(X_{t})$ for all
$t\in\Delta\setminus\\{0\\}$ and if some special cases of the
$\partial\bar{\partial}$-Lemma (that we call the hypotheses
$\tilde{H}_{1},\cdots,\tilde{H}_{p+1}$) are satisfied for all $t\neq 0$ while
the cohomology numbers $h_{A}^{p+k-1,p-k+1}(t):=\dim
H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})$, $h_{BC}^{p+k,p-k+1}(t):=\dim
H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})$ and $h_{\partial}^{p+k,p-k+1}(t):=\dim
H_{\partial}^{p+k,p-k+1}(X_{t},\mathbb{C})$ are independent of $t\in\Delta$
for all $k\in\\{1,\cdots,p+1\\}$, then
$\mathcal{A}_{p}(X_{0})=\mathcal{C}_{p}(X_{0})$

We hope that these results will contribute to a possible resolution of
Barlet’s Properness Conjecture for the Barlet space of relative cycles
associated with a holomorphic family of class $\mathcal{C}$ manifolds. Before
presenting this conjecture let us recall that a compact complex manifold $X$
is said to be in the (Fujiki) class $\mathcal{C}$ if there exists a proper
holomorphic bimeromorphic map (i.e. a modification)
$\mu:\tilde{X}\longrightarrow X$ from a compact Kähler manifold $\tilde{X}$ to
$X$ (see [Var86]).

Fujiki introduced class $\mathcal{C}$ manifolds $X$, in [Fuj78], as
meromorphic images of compact Kähler manifolds. This definition was proved to
be equivalent to the above-stated definition by Varouchas in [Var86].
On the other hand, denote by $\mathcal{C}^{p}(\mathcal{X}\diagup\Delta)$ the
relative Barlet space of effective analytic $p$-cycles contained in the fibres
$X_{t}$, and let:

$\mathcal{C}(\mathcal{X}\diagup\Delta)=\underset{{0\leq p\leq
n}}{\cup}\mathcal{C}^{p}(\mathcal{X}\diagup\Delta)$

######  Conjecture 1.3.

([Pop10], one of the versions of Barlet’s conjecture) Let
$\pi:\mathcal{X}\longrightarrow\Delta$ be a complex analytic family of compact
complex manifolds such that the fibre $X_{t}$ is a class $\mathcal{C}$
manifold for every $t\in\Delta$. Then the irreducible components of the
relative Barlet space $\mathcal{C}(\mathcal{X}\diagup\Delta)$ of cycles on
$\mathcal{X}$ are proper over $\Delta$ in the following sense. Consider the
holomorphic map

$\begin{array}[]{ll}P:&\mathcal{C}(\mathcal{X}/\Delta)\longrightarrow\Delta\\\
&\hskip 36.98866ptZ_{t}\mapsto P(Z_{t})=t\end{array}$

mapping every divisor $Z_{t}\subset X_{t}$ contained in some fibre $X_{t}$ to
the base point $t\in\Delta$. The map $P$ has the property that its
restrictions to the irreducible components of
$\mathcal{C}(\mathcal{X}/\Delta)$ are proper.

In the end, we prove (cf. Corollary 5.11) that on a fixed compact complex
manifold $X$ satisfying some assumptions weaker than the
$\partial\bar{\partial}$-lemma, the natural map:

$H_{DR}^{k}(X,\mathbb{C})\oplus
H_{DR}^{2n-k}(X,\mathbb{C})\longrightarrow\bigoplus_{p+q=k}H_{A}^{p,q}(X,\mathbb{C})\oplus\bigoplus_{p+q=2n-k}H_{A}^{p,q}(X,\mathbb{C})$

is an isomorphism.

Acknowledgements. The author is grateful to her supervisor Dan Popovici for
his continuous guidance, suggestions, support, encouragements during her
thesis and for a very careful reading of the paper as well as to Professor
Ahmed Zeriahi for many interesting discussions.

##  2 Preliminaries

In this section, we recall some background that will be needed in the sequel
and introduce our first definitions. Let $X$ be a compact complex manifold of
complex dimension $n$.

The following positivity notion is standard, though a different name has been
used in the literature, and goes back to Lelong. We spell out the details of
the definition in order to dispel the confusion over the English and French
meanings of “positivity”.

######  Definition 2.1.

(see e.g. [Dem], Chapter III and [AB91], Definition 1.1) (i) Let $V$ be a
complex vector space of dimension $n$ and let $V^{*}$ be its dual. Fix any
integer $1\leq p\leq n-1$. A $(p,p)-$form $\alpha\in\Lambda^{p,p}V^{*}$ is
said to be strictly weakly positive if for all linearly independent
$\tau_{1},\dots,\tau_{q}\in V^{*}$, with $q=n-p$, the $(n,n)-$form

$\alpha\wedge i\tau_{1}\wedge\bar{\tau_{1}}\wedge\cdots\wedge
i\tau_{q}\wedge\bar{\tau}_{q}$

is positive (i.e. $>0$) whenever $\alpha\wedge
i\tau_{j}\wedge\overline{\tau_{j}}\neq 0$ for all $j\in\\{1,\cdots,q\\}$.

(ii) Let $X$ be a complex manifold of dimension $n$. A smooth $(p,\,p)$-form
$\alpha\in C^{\infty}_{p,\,p}(X,\,\mathbb{C})$ on $X$ is said to be strictly
weakly positive if, for every point $x\in X$,
$\alpha(x)\in\Lambda^{p,p}T_{x}^{*}X$ is a strictly weakly positive
$(p,p)$-form on the holomorphic tangent space $T_{x}X$ to $X$ at $x$.

(These forms are called transverse in [AB91].)

It is standard that a $(p,\,p)$-form $\alpha\in\Lambda^{p,p}V^{*}$ on a vector
space $V$ is strictly weakly positive if and only if its restriction
$\alpha_{|E}$ to every $p$-dimensional vector subspace $E\subset V$ is a
positive (i.e. $>0$) volume form on $E$. (See e.g. [Dem], Chapter III).

Consequently, a smooth $(p,\,p)$-form $\alpha\in
C^{\infty}_{p,\,p}(X,\,\mathbb{C})$ on a manifold $X$ is strictly weakly
positive if and only if, for every coordinate patch $U\subset X$ and every
$p$-dimensional complex submanifold $Y\subset U$, its restriction
$\alpha_{|Y}$ is a positive (i.e. $>0$) volume form on $Y$. (See e.g. [AB91]).

It is also standard that, for $p=1$ and $p=n-1$, the notion of strict weak
positivity coincides with the usual notion of positive definiteness for
$(1,\,1)$ and $(n-1,\,n-1)$-forms ($=$ the positivity of all the eigenvalues
of the coefficient matrix). Moreover, all strictly weakly positive
$(p,\,p)$-forms $\alpha$ are always real, in the sense that
$\alpha=\overline{\alpha}$. (See e.g. [Dem], Chapter III).

Now, we introduce $p$-Hermitian-symplectic ($p$-HS) and $p$-SKT forms and
manifolds as the following generalisations of the notions of $p$-Kähler forms
and manifolds introduced in [AA87] and further studied in [AB91].

######  Definition 2.2.

Let $X$ be a compact complex manifold of complex dimension $n$ and let
$\Omega$ be a $C^{\infty}$ strictly weakly positive $(p,p)-$form on $X$.

  1. (i)

$\Omega$ is said to be a $p-$Hermitian-symplectic form if there exist forms
$\alpha^{i,2p-i}\in C_{i,2p-i}^{\infty}(X,\mathbb{C})$ for
$i\in\\{0,\cdots,p-1\\}$ such that
$d(\sum_{i=0}^{p-1}\alpha^{i,2p-i}+\Omega+\sum_{i=0}^{p-1}\overline{\alpha^{i,2p-i}})=0$.

  2. (ii)

$\Omega$ is said to be a $p-$SKT form if $\partial\bar{\partial}\Omega=0$.

  3. (iii)

$X$ is said to be a $p-$SKT manifold (resp. a $p-$HS manifold) if there exists
a $p$-SKT (resp. a $p$-HS) strictly weakly positive $(p,p)$-form on $X$.

  4. (iv)

([AB91]) A compact complex manifold $X$ is said to be $p$-Kähler if it
supports a $d$-closed strictly weakly positive $(p,p)-$form $\Omega$. Such an
$\Omega$ is called $p$-Kähler form.

Finally, recall the following standard definitions. For all $p,q=0,\cdots,n$,
the Bott-Chern cohomology group of $X$ of type $(p,q)$ is defined as:

$H^{p,q}_{BC}(X,\mathbb{C})=\dfrac{\ker\\{\partial:C^{\infty}_{p,q}(X)\longrightarrow
C^{\infty}_{p+1,q}(X)\\}\cap\ker\\{\bar{\partial}:C^{\infty}_{p,q}(X)\longrightarrow
C^{\infty}_{p,q+1}(X)\\}}{Im\\{\partial\bar{\partial}:C^{\infty}_{p-1,q-1}(X,\mathbb{C})\longrightarrow
C^{\infty}_{p,q}(X,\mathbb{C})\\}},$

while the Aeppli cohomology group of type $(p,q)$ is defined as:

$H^{p,q}_{A}(X,\mathbb{C})=\dfrac{\ker\\{\partial\bar{\partial}:C^{\infty}_{p,q}(X,\mathbb{C})\longrightarrow
C^{\infty}_{p+1,q+1}(X,\mathbb{C})\\}}{Im\\{\partial:C^{\infty}_{p-1,q}(X)\longrightarrow
C^{\infty}_{p,q}(X)\\}+Im\\{\bar{\partial}:C^{\infty}_{p,q-1}(X)\longrightarrow
C^{\infty}_{p,q}(X)\\}}.$

##  3 Deformations of pluriclosed manifolds

Let us now recall some basic notions in the following

######  Definition 3.1.

Let $\omega>0$ be a Hermitian metric on a complex manifold $X$.

  1. 1-

$\omega$ is said to be pluriclosed (or SKT) if
$\partial\bar{\partial}\omega=0$.

  2. 2-

$\omega$ is called Hermitian-symplectic (H-S for simplicity) (cf. Definition
in [ST10]) if there exists $\alpha^{0,2}\in C_{0,2}^{\infty}(X,\mathbb{C})$
such that

$d(\overline{\alpha^{0,2}}+\omega+\alpha^{0,2})=0.$

This is equivalent to the existence of a $C^{\infty}$ real $2$-form $\Omega$
on $X$ such that $d\Omega=0$ and $\Omega^{1,1}=\omega>0$.

  3. 3-

$X$ is a SKT manifold (resp. H-S manifold) if there exists an SKT (resp. a
H-S) metric on $X$.

  4. 4-

$X$ is a $\partial\bar{\partial}$-manifold (cf. e.g. [Pop14] for this version
of a notion introduced in [DGMS75]) if the $\partial\bar{\partial}$-lemma is
satisfied on $X$, that is for all $p$, $q$ and for every $C^{\infty}$
$(p,q)$-form $u$ on $X$ such that $du=0$, the following properties are
equivalent:

$\mbox{u is }d-exact\Leftrightarrow\mbox{u is }\partial-
exact\Leftrightarrow\mbox{u is }\bar{\partial}-exact\Leftrightarrow\mbox{u is
}\partial\bar{\partial}-exact$

It is proved in [Pop15], that on every $\partial\bar{\partial}$-manifold, the
notion of SKT metric and the notion of Hermitian-symplectic metric are
equivalent:

######  Lemma 3.2.

Let $X$ be a $\partial\bar{\partial}-$manifold. For any Hermitian metric
$\omega$ on $X$, the following equivalence holds:

$\omega\mbox{ is SKT }\Longleftrightarrow\,\omega\mbox{ is Hermitian-
symplectic}.$

Proof. $\Leftarrow)$ This implication holds on any compact complex manifold.
Suppose that there exists $\alpha^{0,2}\in C_{0,2}^{\infty}(X,\mathbb{C})$
such that $d(\alpha^{0,2}+\omega+\overline{\alpha^{0,2}})=0$. This is
equivalent to

$\left\\{\begin{array}[]{ll}\partial\overline{\alpha^{0,2}}=0\\\
\partial\alpha^{0,2}+\bar{\partial}\omega=0.\end{array}\right.$

Applying $\partial$ on the above identity we get
$\partial\bar{\partial}\omega=0$.

Conversely, suppose that $\partial\bar{\partial}\omega=0$ and $X$ is a
$\partial\bar{\partial}-$manifold, then $\partial\omega\in\ker\bar{\partial}$.
Meanwhile $\partial\omega$ is a $(2,1)$-form which is $d$-closed and
$\partial$-exact, hence by the $\partial\bar{\partial}$-lemma $\partial\omega$
must be $\bar{\partial}-$exact.
This means that there exists $\alpha^{2,0}\in C_{2,0}^{\infty}(X,\mathbb{C})$
such that
$\partial\omega=-\bar{\partial}\alpha^{2,0}=-\overline{\partial\alpha^{0,2}}$
with $\alpha^{0,2}:=\overline{\alpha^{2,0}}$. We get:

$\displaystyle d(\alpha^{0,2}+\omega+\overline{\alpha^{0,2}})$
$\displaystyle=\partial\alpha^{0,2}+\bar{\partial}\alpha^{0,2}+\partial\omega+\bar{\partial}\omega+\partial\overline{\alpha^{0,2}}+\bar{\partial}\overline{\alpha^{0,2}}$
$\displaystyle=(\partial\omega+\bar{\partial}\overline{\alpha^{0,2}})+(\overline{\partial\omega+\bar{\partial}\overline{\alpha^{0,2}}})+\bar{\partial}\alpha^{0,2}+\partial\overline{\alpha^{0,2}}$
$\displaystyle=\bar{\partial}\alpha^{0,2}+\partial\overline{\alpha^{0,2}}.$

It remains to show that $\partial\alpha^{2,0}=0$. Indeed, we have
$\bar{\partial}\partial\alpha^{2,0}=-\partial\bar{\partial}\alpha^{2,0}=\partial^{2}\omega=0$.
So $\partial\alpha^{2,0}$ is a $d$-closed and $\partial$-exact $(3,0)$-form,
therefore by the $\partial\bar{\partial}$-lemma it is $\bar{\partial}$-exact.
For bidegree reasons, this means that $\partial\alpha^{2,0}=0$. This proves
that $\omega$ is H-S. $\Box$

In the rest of this section, we will show that the property of a manifold
being Hermitian-symplectic is open under holomorphic deformations.

######  Theorem 3.3.

Let $(X_{t})_{t\in\Delta}$ be a holomorphic family of compact complex
manifolds over an open disc $\Delta\subset\mathbb{C}$ containing the origin.

If there exists an H-S metric $\omega_{0}$ on $X_{0}$, then after possibly
shrinking $\Delta$ about $0$, there exists a $C^{\infty}$ family
$(\omega_{t})_{t\in\Delta}$ of H-S metrics on the fibres
$(X_{t})_{t\in\Delta}$ whose term corresponding to $t=0$ is the original
$\omega_{0}$.

Proof. Suppose that there exists a H-S metric $\omega_{0}$ on $X_{0}$. Then,
there exists $\alpha_{0}^{0,2}\in C_{0,2}^{\infty}(X_{0},\mathbb{C})$ such
that

$d(\alpha_{0}^{0,2}+\omega_{0}+\overline{\alpha_{0}^{0,2}})=0$

Putting $\Omega:=\alpha_{0}^{0,2}+\omega_{0}+\overline{\alpha_{0}^{0,2}}$, we
get $\Omega$ a $d-$closed real $2-$form on $X_{0}$. Since, by Ehresmann,
$\Omega$ does not depend on the complex structure of the fibre. Let
$(\Omega_{t}^{1,1})_{t\in\Delta}$ be the $C^{\infty}$ family of components of
$\Omega$ of $J_{t}$-type $(1,1)$, namely
$\Omega_{t}^{1,1}=(\Omega)_{t}^{1,1}$, so $(\Omega_{t}^{1,1})_{t\in\Delta}$
vary smoothly with $t$. Thus, by the continuity of the family
$(\Omega_{t}^{1,1})_{t\in\Delta}$, the strict positivity of $\Omega_{0}^{1,1}$
implies the strict positivity of $\Omega_{t}^{1,1}$ for all $t\in\Delta$
sufficiently close to 0. So, we obtain a H-S metric $\omega_{t}$ on $X_{t}$
for every $t\in\Delta$ close to $0$. $\Box$

Now, recall the following theorem of Wu [Wu06] according to which the
$\partial\bar{\partial}$-property of compact complex manifolds is open under
holomorphic deformations of the complex structure:

$X_{0}\mbox{ is }\partial\bar{\partial}\mbox{-manifold
}\Longrightarrow\,X_{t}\mbox{ is }\partial\bar{\partial}\mbox{-manifold for
}t\in\Delta,\,t\thicksim 0.$

As a consequence of Wu’s theorem and of our Theorem 3.3, we get the following

######  Corollary 3.4.

Let $(X_{t})_{t\in\Delta}$ be a holomorphic family of compact complex
manifolds over an open disc $\Delta\subset\mathbb{C}$ containing the origin.

If $X_{0}$ is an SKT $\partial\bar{\partial}$-manifold, then $X_{t}$ is an SKT
$\partial\bar{\partial}$-manifold for all $t\in\Delta$ sufficiently close to
$0$.

Moreover, if $X_{0}$ is an SKT $\partial\bar{\partial}$-manifold, any SKT
metric $\omega_{0}$ on $X_{0}$ deforms in a $C^{\infty}$ way to a family
$(\omega_{t})_{t}$ of SKT metrics on the nearby fibres $(X_{t})_{t}$.

Proof. On $\partial\bar{\partial}$-manifolds, the notions of H-S and SKT
metrics are equivalent, as noticed in Lemma 3.2. $\Box$

##  4 Deformations of $p$-HS and $p$-SKT manifolds

In this section, we will prove that the property of a compact complex
$\partial\bar{\partial}$-manifold carrying $p$-SKT form is deformation open.

######  Lemma 4.1.

Let $X$ be a $\partial\bar{\partial}-$manifold with $\dim_{\mathbb{C}}X=n$.
Fix $p\in\\{1,\cdots,n-1\\}$.
If there exists a $p-$SKT form $\omega$ on $X$, then $\omega$ is a
$p$-Hermitian-symplectic form.

Proof. Suppose there exists a $p$-pluriclosed form $\omega$ on $X$. Namely,
there exists a $C^{\infty}$ strictly weakly positive real $(p,p)-$form
$\omega$ such that $\partial\bar{\partial}\omega=0$. Notice that
$\partial\omega$ is a $d$-closed $(p+1,p)-$form. By the
$\partial\bar{\partial}-$lemma, $\partial\omega$ is $\bar{\partial}-$exact,
that is there exists $\alpha^{p+1,p-1}\in C_{p+1,p-1}^{\infty}(X,\mathbb{C})$
such that $\partial\omega=-\bar{\partial}\alpha^{p+1,p-1}$. Note that
$\partial\alpha^{p+1,p-1}$ is $d-$closed, so by $\partial\bar{\partial}-$lemma
there exists $\alpha^{p+2,p-2}\in C_{p+2,p-2}^{\infty}(X,\mathbb{C})$ such
that $\partial\alpha^{p+1,p-1}=-\bar{\partial}\alpha^{p+2,p-2}$.

We continue until we find a form $\alpha^{2p,0}\in
C_{2p,0}^{\infty}(X,\mathbb{C})$ such that
$\partial\alpha^{2p-1,1}=-\bar{\partial}\alpha^{2p,0}$. Thus,
$\partial\alpha^{2p,0}$ is $d$-closed hence by $\partial\bar{\partial}-$lemma
it must also be $\bar{\partial}-$exact. However, the only
$\bar{\partial}-$exact $(2p,0)-$form is zero, thus $\partial\alpha^{2p,0}=0$.
As a consequence, we have:

$\begin{array}[]{cccc}&d(\sum_{i=0}^{p-1}\alpha^{i,2p-i}+\omega+\sum_{i=0}^{p-1}\overline{\alpha^{i,2p-i}})\\\
&=\partial\alpha^{0,2p}+\bar{\partial}\alpha^{0,2p}+\partial\overline{\alpha^{0,2p}}+\bar{\partial}\overline{\alpha^{0,2p}}+\partial\alpha^{1,2p-1}+\bar{\partial}\alpha^{1,2p-1}+\partial\overline{\alpha^{1,2p-1}}\\\
&+\bar{\partial}\overline{\alpha^{1,2p-1}}+\cdots+\partial\alpha^{p-3,p+3}+\bar{\partial}\alpha^{p-3,p+3}+\partial\overline{\alpha^{p-3,p+3}}+\bar{\partial}\overline{\alpha^{p-3,p+3}}\\\
&+\partial\alpha^{p-2,p+2}+\bar{\partial}\alpha^{p-2,p+2}+\partial\overline{\alpha^{p-2,p+2}}+\bar{\partial}\overline{\alpha^{p-2,p+2}}+\bar{\partial}\alpha^{p-1,p+1}\\\
&+\partial\alpha^{p+1,p-1}=0.\end{array}$

Thus, $\omega$ is a $p$-HS form. $\Box$

A key observation for us will be the following deformation-openness property
of the strict weak positivity for $(p,\,p)$-forms.

######  Lemma 4.2.

Let $(X_{t})_{t\in\Delta}$ be a holomorphic family of compact complex
$n$-dimensional manifolds over an open disc $\Delta\subset\mathbb{C}$
containing the origin. Fix any $p\in\\{1,\dots,n-1\\}$. Let
$(\Omega_{t})_{t\in\Delta}$ be a $C^{\infty}$ family of real $(2p)$-forms on
the fibres $X_{t}$ such that every $\Omega_{t}$ is of type $(p,\,p)$ for the
complex structure $J_{t}$ of $X_{t}$.

If $\Omega_{0}$ is strictly weakly positive for the complex structure $J_{0}$
of $X_{0}$, then $\Omega_{t}$ is strictly weakly positive for the complex
structure $J_{t}$ of $X_{t}$ for all $t\in\Delta$ sufficiently close to
$0\in\Delta$.

Proof. Let ${\cal X}=\cup_{t\in\Delta}X_{t}$ be the total space of the family
and let ${\cal X}=U^{(1)}\cup\dots\cup U^{(N)}$ be an open covering by
coordinate patches of ${\cal X}$ such that $X_{0}\subset U^{(1)}\cup\dots\cup
U^{(N)}$. For every $\nu=1,\dots,N$, let
$(z_{1}^{(\nu)},\cdots,z_{n}^{(\nu)},\,t)$ be a local holomorphic coordinate
system on $U^{(\nu)}\subset{\cal X}$, where $t$ is a holomorphic coordinate on
a neighbourhood of $0$ in $\Delta$. Put $U^{(\nu)}_{t}:=U^{(\nu)}\cap
X_{t}\subset X_{t}$ and $z_{j}^{(\nu)}(t):=z_{j|X_{t}}^{(\nu)}$.

Now, the strict weak positivity for $(p,\,p)$-forms on a complex manifold is a
pointwise, hence local, property. For a fixed $t^{\prime}\in\Delta$ and a
fixed index $\nu\in\\{1,\dots,N\\}$, let
$\tau_{1}(t^{\prime}),\dots,\tau_{q}(t^{\prime})$ (where $q=n-p$) be lineraly
independent $C^{\infty}$ $J_{t^{\prime}}$-$(1,\,0)$-forms on
$U^{(\nu)}_{t^{\prime}}\subset X_{t^{\prime}}$. We extend the forms
$\tau_{j}(t^{\prime})$ to $C^{\infty}$ $(1,\,0)$-forms
$\tau_{1},\dots,\tau_{q}$ on $U^{(\nu)}\subset{\cal X}$. (This is always
possible for forms defined on a coordinate patch. For example, we can extend
their coefficients by requiring them to remain constant in the $t$-direction.)
For every $t\in\Delta$, the restrictions $\tau_{1}(t),\dots,\tau_{q}(t)$ of
the forms $\tau_{1},\dots,\tau_{q}$ to $U^{(\nu)}_{t}\subset X_{t}$ are of
type $(1,\,0)$ for $J_{t}$. They are also linearly independent if $t$ lies in
a sufficiently small neighbourhood (depending on the forms
$\tau_{1}(t^{\prime}),\dots,\tau_{q}(t^{\prime})$) of $0$. Since $\Omega_{0}$
is strictly weakly positive for $J_{0}$, we have

$\Omega_{0}\wedge i\tau_{1}(0)\wedge\overline{\tau_{1}}(0)\wedge\cdots\wedge
i\tau_{n-p}(0)\wedge\overline{\tau_{n-p}}(0)>0\hskip 11.62495pt\mbox{on}\hskip
3.87498ptU^{(\nu)}_{0}\subset X_{0}.$

By continuity, we still have $\Omega_{t}\wedge
i\tau_{1}(t)\wedge\overline{\tau_{1}}(t)\wedge\cdots\wedge
i\tau_{n-p}(t)\wedge\overline{\tau_{n-p}}(t)>0$ for all $t$ lying in a
sufficiently small neighbourhood (depending on the forms
$\tau_{1}(t^{\prime}),\dots,\tau_{q}(t^{\prime})$) of $0$.

Meanwhile, the strict weak positivity at a point has to be tested on all the
$p$-dimensional complex vector subspaces of an $n$-dimensional space. These
subspaces form a compact complex manifold, the Grassmannian $G_{p,\,n}$.
Thanks to the compactness of this Grassmannian, there exists a uniform open
neighbourhood $\Delta_{0}$ of $0$ in $\Delta$ such that $\Omega_{t}$ is
strictly weakly positive for the complex structure $J_{t}$ of $X_{t}$ for all
$t\in\Delta_{0}$. $\Box$

The first main result of this section is the following

######  Theorem 4.3.

Let $X$ be a compact complex manifold of dimension $n$. Fix
$p\in\\{1,\cdots,n-1\\}$. If there exists a $p$-HS form $\omega_{0}$ on
$X_{0}$, then there exists a $p$-HS form $\omega_{t}$ on $X_{t}$ for
$t\in\Delta$ sufficiently close to $0$.

Proof. Assume that there exists a $p-$HS form $\omega_{0}$ on $X_{0}$. This
means that there exist $\alpha^{i,2p-i}\in
C_{i,2p-i}^{\infty}(X_{0},\mathbb{C})$ for $i=0,\cdots,2p$ such that
$d(\sum_{i=0}^{p-1}\alpha^{i,2p-i}+\omega_{0}+\sum_{i=0}^{p-1}\overline{\alpha^{i,2p-i}})=0$.

Put
$\Omega:=\sum_{i=0}^{p-1}\alpha^{i,2p-i}+\omega+\sum_{i=0}^{p-1}\overline{\alpha^{i,2p-i}}$.
This is a $d$-closed real $(2p)$-form on the $C^{\infty}$ manifold $X$
underlying the fibres $X_{t}$. Let $(\Omega_{t}^{p,p})_{t\in\Delta}$ be the
$C^{\infty}$ family of components of $\Omega$ of $J_{t}$-type $(p,p)$. The
forms $\Omega_{t}^{p,p}$ vary smoothly with $t\in\Delta$.

Thanks to Lemma 4.2, the weak strict positivity of
$\Omega^{p,p}_{0}=\omega_{0}$ implies the weak strict positivity of
$\Omega_{t}^{p,p}$ for all $t\in\Delta$ sufficiently close to 0. Thus,
$\omega_{t}:=\Omega_{t}^{p,p}$ is a $p$-HS form on $X_{t}$ for all
$t\in\Delta$ sufficiently close to 0. $\Box$

As a consequence we get

######  Conclusion 4.4.

If $X_{0}$ is a $p$-SKT $\partial\bar{\partial}$-manifold, then $X_{0}$ is a
$p$-HS manifold. Therefore, $X_{t}$ is a $p$-HS manifold, hence also a $p$-SKT
manifold, for $t\in\Delta$ sufficiently close to 0.

On the other hand, by [Wu06], the $\partial\bar{\partial}$-lemma property is
open under holomorphic deformations.

So, we have proved the following:

######  Theorem 4.5.

Let $\pi:\mathcal{X}\longrightarrow\Delta$ be a holomorphic family of compact
complex n-dimensional manifolds. Fix $p\in\\{1,\cdots,n-1\\}$. If $X_{0}$ is a
$p-$SKT $\partial\bar{\partial}$-manifold, then $X_{t}$ is a $p-$SKT
$\partial\bar{\partial}$-manifold for $t\in\Delta$ sufficiently close to $0$.

##  5 Deformation limits of positive cones

Let us start by defining the notions of $E_{k}$-closedness and
$\overline{E_{k}}$-closedness.

######  Definition 5.1.

A smooth $(r,s)$-form $\alpha^{r,s}$ is called $E_{k}$-closed (resp.
$\overline{E_{k}}$-closed) if $\bar{\partial}\alpha^{r,s}=0$,
$\partial\alpha^{r,s}=\bar{\partial}\alpha^{r+1,s-1}$, $\cdots$,
$\partial\alpha^{r+k-1,s-k+1}=\bar{\partial}\alpha^{r+k,s-k}$ (resp.
$\partial\alpha^{r,s}=0$,
$\bar{\partial}\alpha^{r,s}=\partial\alpha^{r+1,s-1}$, $\cdots$,
$\bar{\partial}\alpha^{r+k-1,s-k+1}=\partial\alpha^{r+k,s-k}$) with
$\alpha^{r+l,s-l}\in C^{\infty}_{r+l,s-l}(X,\mathbb{C})$ for
$l\in\\{1,\cdots,k\\}$.

Recall that a $C^{\infty}$ strictly weakly positive $(p,p)-$form $\Omega$ is
$p-$Hermitian-symplectic ($p-$HS) if and only if there exist
$\alpha^{i,2p-i}\in C_{i,2p-i}^{\infty}(X,\mathbb{C})$ for $i=0,\cdots,2p$
such that

$d(\sum_{i=0}^{p-1}\alpha^{i,2p-i}+\Omega+\sum_{i=0}^{p-1}\overline{\alpha^{i,2p-i}})=0.$

Therefore,

$\Omega$ is $p$-HS $\Longleftrightarrow$
$\partial\overline{\alpha^{0,2p}}=0,\,\partial\alpha^{2p-1,1}+\bar{\partial}\alpha^{2p,0}=0,\,\cdots,\,\partial\alpha^{p+1,p-1}+\bar{\partial}\alpha^{p+2,p-2}=0\,\mbox{
and }\,\partial\Omega+\bar{\partial}\alpha^{p+1,p-1}=0$

$\Longleftrightarrow$ $\alpha^{2p,0}$ is $\overline{E}_{p+1}$-closed and
$\overline{d}_{p+1}(\\{\alpha^{2p,0}\\}_{\overline{E}_{p+1}})=\\{\bar{\partial}\Omega\\}_{\overline{E}_{p+1}}$

$\Longleftrightarrow$ $\alpha^{0,2p}$ is $E_{p+1}$-closed and
$d_{p+1}(\\{\alpha^{0,2p}\\}_{E_{p+1}})=\\{\partial\Omega\\}_{E_{p+1}}$.

So, given a strictly weakly positive $(p,p)$-form $\Omega$, we have:

$\Omega$ is $p$-HS $\Longleftrightarrow$ $\partial\Omega$ is $E_{p+1}$-closed
and $\\{\partial\Omega\\}_{E_{p+1}}\in\mbox{ Im }d_{p+1}$
$\Longleftrightarrow$ $\partial\Omega$ is $E_{p+2}$-exact.

This is equivalent to the property

$(E_{k}):\quad\bar{\partial}\alpha^{p+k,p-k}=-\partial\alpha^{p+k-1,p-k+1}\hskip
28.45274pt\forall k=1,\cdots,p$ (1)

We now introduce our main objects of study in this section.

######  Definition 5.2.

Let $X$ be a compact $n$-dimensional complex manifold. Let
$p\in\\{1,\dots,n\\}$. The cones $\mathcal{A}_{p}(X)$ and $\mathcal{C}_{p}(X)$
are defined as:

$\mathcal{A}_{p}(X)=\\{[\Omega]_{A}/\Omega\mbox{ strictly weakly positive such
that }\partial\bar{\partial}\Omega=0\\}\subset
H_{A}^{p,p}(X,\mathbb{R})\subset H_{A}^{p,p}(X,\mathbb{C}),$ (2)
$\mathcal{C}_{p}(X)=\\{[\Omega]_{A}/\Omega\mbox{ strictly weakly positive such
that }\Omega\mbox{ is }p\mbox{-HS}\\}\subset H_{A}^{p,p}(X,\mathbb{R})\subset
H_{A}^{p,p}(X,\mathbb{C}).$ (3)

Note that $\mathcal{C}_{p}(X)\subset\mathcal{A}_{p}(X)$.

######  Lemma 5.3.

The subsets $\mathcal{C}_{p}(X)\subset\mathcal{A}_{p}(X)$ are open convex
cones in $H_{A}^{p,p}(X,\mathbb{R})$.

Proof. To see that $\mathcal{A}_{p}(X)$ is a convex cone, let
$[\Omega]_{A},[\tilde{\Omega}]_{A}\in\mathcal{A}_{p}(X)$ and $\lambda>0$. Then
$\Omega+\lambda\tilde{\Omega}$ remains strictly weakly positive and
$\partial\bar{\partial}(\Omega+\lambda\tilde{\Omega})=0$, i.e.
$[\Omega+\lambda\tilde{\Omega}]_{A}\in\mathcal{A}_{p}(X)$.

To prove openness for $\mathcal{A}_{p}(X)$, let
$[\Omega]_{A}\in\mathcal{A}_{p}(X)$ and $[\gamma]_{A}\in
H^{p,p}_{A}(X,\mathbb{C})$ be arbitrary. We will prove that
$[\Omega]_{A}+\varepsilon[\gamma]_{A}\in\mathcal{A}_{p}(X)$ for all
sufficiently small $\varepsilon>0$.

Pick arbitrary representatives $\Omega$ and $\gamma$ of their respective
Aeppli cohomology classes such that $\Omega$ is strictly weakly positive.
Notice that $\partial\bar{\partial}(\Omega+\varepsilon\gamma)=0$ for every
$\varepsilon$ since $\Omega$ and $\gamma$ represent Aeppli classes.

It remains to prove that $\Omega+\varepsilon\gamma$ is strictly weakly
positive for all sufficiently small $\varepsilon>0$. Since this positivity is
a pointwise property, we can reason locally. Let $x_{0}\in X$ be an arbitrary
point.

By hypothesis, $(\Omega\wedge
i\alpha_{1}\wedge\overline{\alpha_{1}}\wedge\cdots\wedge
i\alpha_{n-p}\wedge\overline{\alpha_{n-p}})(x_{0})>0$ for every choice of
locally defined, linearly independent, smooth $(1,\,0)$-forms
$\alpha_{1},\cdots\alpha_{n-p}$. Equivalently, the restriction of
$\Omega(x_{0})$ to the $p$-dimensional vector subspace
$E_{0}=E_{x_{0}}=\ker\alpha_{1}(x_{0})\cap\dots\cap\ker\alpha_{n-p}(x_{0})\subset
T^{1,\,0}_{x_{0}}X$ is a positive (i.e. $>0$) volume form on
$E_{0}=E_{x_{0}}$. Now, the quantity $(\Omega+\varepsilon\gamma)(x)_{|E_{x}}$
depends in a $C^{\infty}$ way on the triple $(\varepsilon,\,x,\,E)$.
Therefore, by continuity, there exist a constant $\varepsilon_{0}>0$ and open
neighbourhoods $U_{x_{0}}$ of $x_{0}$ in $X$ and ${\cal U}_{E_{0}}$ of $E_{0}$
in the Grassmannian $G_{p,\,n}$ of $p$-dimensional vector subspaces of
$\mathbb{C}^{n}$, such that

$(\Omega+\varepsilon\gamma)_{|E_{x}}>0\hskip 11.62495pt\mbox{for all}\hskip
3.87498pt(\varepsilon,\,x,\,E)\in(0,\,\varepsilon_{0})\times
U_{x_{0}}\times{\cal U}_{E_{0}}.$

Since both $X$ and the Grasssmannian $G_{p,\,n}$ are compact manifolds, we
conclude that there exists a uniform constant $\varepsilon_{0}>0$ such that
the $(p,\,p)$-form $\Omega+\varepsilon\gamma$ is strictly weakly positive for
all $0<\varepsilon<\varepsilon_{0}$.

Therefore, the cone $\mathcal{A}_{p}(X)$ is open. The same arguments apply to
$\mathcal{C}_{p}(X)$. $\Box$

We now show that the cones $\mathcal{A}_{p}(X)$ and $\mathcal{C}_{p}(X)$ are
equal if the hypotheses $(H_{k})$ spelt out in (4) below for $k=1,\cdots,p+1$
are satisfied. These hypotheses are a collection of special cases of the
$\partial\bar{\partial}$-lemma in a few select bidegrees. Thus, it is not
necessary to assume the validity of the $\partial\bar{\partial}$-lemma in full
generality.

######  Proposition 5.4.

Let $X$ be a compact complex manifold with $\dim_{\mathbb{C}}X=n$. For a fixed
$p\in\\{1,\dots,n-1\\}$ and a fixed $k\in\\{1,\dots,p+1\\}$, let us consider
the following hypothesis $(H_{k})$:

$(H_{k}):\quad\forall\,\Gamma\in C_{p+k,p-k+1}^{\infty}(X,\mathbb{C})\mbox{
such that }d\Gamma=0,\,\Gamma\in Im\,\partial\Rightarrow\Gamma\in
Im\,\bar{\partial}.$ (4)

  1. (i)

If the hypotheses $(H_{1}),\cdots,\mbox{ and }(H_{p+1})$ are satisfied, then
$\mathcal{A}_{p}(X)=\mathcal{C}_{p}(X)$.

  2. (ii)

If $\mathcal{A}_{p}(X)=\mathcal{C}_{p}(X)$, the hypothesis $(H_{1})$ holds.

Proof.

  1. (i)

Assume that the hypotheses $(H_{1}),\cdots,\mbox{ and }(H_{p+1})$ are
satisfied. Let $[\Omega]_{A}\in\mathcal{A}_{p}(X)$. Since
$\partial\bar{\partial}\Omega=0$, $\partial\Omega$ is a $d$-closed and
$\partial$-exact $(p+1,p)-$form and by the hypothesis $(H_{1})$ there exists
$\alpha^{p+1,p-1}\in C_{p+1,p-1}^{\infty}(X,\mathbb{C})$ that solves equation
$(E_{1}):$ $\bar{\partial}\alpha^{p+1,p-1}=-\partial\Omega$. Let
$\alpha^{p+1,p-1}\in C_{p+1,p-1}^{\infty}(X,\mathbb{C})$ be an arbitrary
solution of the equation $(E_{1})$. On the other hand we have
$\partial\alpha^{p+1,p-1}\in C_{p+2,p-1}^{\infty}(X,\mathbb{C})$. This implies
that $\partial\alpha^{p+1,p-1}$ is a $d$-closed and $\partial$-exact
$(p+2,p-1)$-form. Hence, under the $(H_{2})$ assumption we have
$\partial\alpha^{p+1,p-1}\in Im\,\bar{\partial}$, so there exists
$\alpha^{p+2,p-2}\in C^{\infty}_{p+1,p-1}(X,\mathbb{C})$ that solves $(E_{2})$
i.e. $\partial\alpha^{p+1,p-1}=-\bar{\partial}\alpha^{p+2,p-2}$.
We follow the same process until we obtain, under the hypothesis $(H_{p})$,
the equation $(E_{p})$ had a solution $\alpha^{2p,0}$ such that
$\bar{\partial}\alpha^{2p,0}=-\partial\alpha^{2p-1,1}$. This implies that
$\bar{\partial}\partial\alpha^{2p,0}=0$, so $\partial\alpha^{2p,0}$ is a
$(2p+1,0)$-form that is $d$-closed and $\partial$-exact. This implies that
$\partial\alpha^{2p,0}\in Im\,\bar{\partial}$, hence
$\partial\alpha^{2p,0}=0$.
Therefore
$d(\sum_{i=0}^{p-1}\alpha^{i,2p-i}+\Omega+\sum_{i=0}^{p-1}\overline{\alpha^{i,2p-i}})=0$,
i.e $[\Omega]_{A}\in\mathcal{C}_{p}(X)$.
It follows that $\mathcal{A}_{p}(X)=\mathcal{C}_{p}(X)$.

  2. (ii)

Consider the following linear map :

$\begin{array}[]{ll}&H_{A}^{p,p}(X,\mathbb{C})\overset{T}{\longrightarrow}H_{\bar{\partial}}^{p+1,p}(X,\mathbb{C})\\\
&\hskip
31.2982pt[\Omega]_{A}\longmapsto[\partial\Omega]_{\bar{\partial}}\end{array}$

Let us show that this map is well defined. Let $[\Omega]_{A}\in
H_{A}^{p,p}(X,\mathbb{C})$, hence $\partial\bar{\partial}\Omega=0$, i.e
$\bar{\partial}(\partial\Omega)=0$, thus $\partial\Omega$ defines a class
$[\partial\Omega]_{\bar{\partial}}\in H_{\bar{\partial}}^{p,p}(X,\mathbb{C})$.
If $[\Omega_{1}]_{A}=[\Omega_{2}]_{A}$, then there exists a $(p-1,p)$-form $u$
and a $(p,p-1)$-form $v$ such that $\Omega_{1}-\Omega_{2}=\partial
u+\bar{\partial}v$. Thus
$\partial(\Omega_{1}-\Omega_{2})=\partial\bar{\partial}v=\bar{\partial}(-\partial
v)\in Im\,\bar{\partial}$. This proves that
$[\partial\Omega_{1}]_{\bar{\partial}}=[\partial\Omega_{2}]_{\bar{\partial}}$.
Consequently the map $T$ is well defined.

Now, we want to prove that the map $T$ vanishing identically is equivalent to
the hypothesis $(H_{1})$. Indeed, let $\Gamma\in
C_{p+1,p}^{\infty}(X,\mathbb{C})$ such that $\Gamma=\partial\Omega\in\ker d$,
so
$\bar{\partial}\Gamma=\bar{\partial}\partial\Omega=-\partial\bar{\partial}\Omega$.
But $\Gamma$ is a $d$-closed form of pure type $(p+1,p)$, i.e
$\partial\Gamma=0$ and $\bar{\partial}\Gamma=0$, then
$\partial\bar{\partial}\Omega=0$.

On the other hand $T\equiv 0$ means that
$T([\Omega]_{A})=[\partial\Omega]_{\bar{\partial}}=0\in
H_{\bar{\partial}}^{p+1,p}(X,\mathbb{C})$, for every $[\Omega]_{A}\in
H^{p,p}_{A}(X,\mathbb{C})$. Hence $\partial\Omega\in Im\,\bar{\partial}$.

Conversely, assume that for all $\Gamma\in C_{p+1,p}^{\infty}(X,\mathbb{C})$
such that $d\Gamma=0$, we have the implication $\Gamma\in Im\,\partial$
$\Rightarrow$ $\Gamma\in Im\,\bar{\partial}$. Let $[\Omega]_{A}\in
H^{p,p}_{A}(X,\mathbb{C})$. Then $\Gamma=\partial\Omega$ implies that
$\Gamma=\partial\Omega\in Im\,\bar{\partial}$, so
$[\partial\Omega]_{\bar{\partial}}=0=T([\Omega]_{A})$. This proves $T\equiv
0$.

Now, since $\mathcal{A}_{p}(X)=\mathcal{C}_{p}(X)$, we have

$\mathcal{A}_{p}(X)\,\cap\,\ker T=\\{[\Omega]_{A}/\Omega\mbox{ strictly weakly
positive such that }\partial\bar{\partial}\Omega=0\mbox{ and
}\partial\Omega\in Im\,\bar{\partial}\\}\supset\mathcal{C}_{p}(X).$

So $\mathcal{C}_{p}(X)\subset\mathcal{A}_{p}(X)\,\cap\,\ker
T\subset\mathcal{A}_{p}(X)$. Since $\mathcal{A}_{p}(X)=\mathcal{C}_{p}(X)$,
this implies that $\mathcal{A}_{p}(X)\cap\ker T=\mathcal{A}_{p}(X)$.
Hence $\ker T=H_{A}^{p,p}(X,\mathbb{C})$, which is equivalent to the $(H_{1})$
assumption.

$\Box$

We shall now consider, for $k=1,\cdots,p+1$, the linear map:

$\begin{array}[]{ll}&H_{A}^{p+k-1,p-k+1}(X,\mathbb{C})\overset{\widehat{T}_{k}}{\longrightarrow}H_{BC}^{p+k,p-k+1}(X,\mathbb{C})\\\
&\hskip 76.82234pt[\Omega]_{A}\longrightarrow[\partial\Omega]_{BC}\end{array}$

This map is well defined. Indeed.
Suppose that $\partial\bar{\partial}\Omega=0$, i.e.
$\bar{\partial}(\partial\Omega)=0$, so $\partial\Omega\in\ker\bar{\partial}$
this is equivalent to $\partial\Omega\in\ker\partial\cap\ker\bar{\partial}$.
Moreover, if $\Omega=\partial u+\bar{\partial}v$ with $u$ and $v$ are
$(p+k-2,p-k+1)$-form and $(p+k-1,p-k)$-form respectively, hence
$\partial\Omega=\partial\bar{\partial}v\in Im\,\partial\bar{\partial}$. Thus
the map $\widehat{T}_{k}$ is well defined.

Consider the linear map :

$\begin{array}[]{ll}I^{p+k,p-k+1}:&H_{BC}^{p+k,p-k+1}(X,\mathbb{C})\longrightarrow
H_{\partial}^{p+k,p-k+1}(X,\mathbb{C})\\\ &\hskip
56.9055pt[\Gamma]_{BC}\longrightarrow[\Gamma]_{\partial}\end{array}$

We have $\partial\Gamma=0$, and $\bar{\partial}\Gamma=0$. To show that the map
$I^{p+k,p-k+1}$ is well defined, we still have to show that the definition is
independent of the choice of representative of the class $[\Gamma]_{BC}$. In
other words, if $\Gamma=\partial\bar{\partial}u$ with $u$ a
$(p+k-1,p-k)$-form, then $\Gamma\in Im\,\partial$. This is obvious, so the map
$I^{p+k,p-k+1}$ is well defined.

We will infer the following

######  Remark 5.5.

The map $\widehat{T}_{k}$ vanishes identically if and only if for all
$\Omega\in
C_{p+k-1,p-k+1}^{\infty}(X,\mathbb{C})\cap\ker\partial\bar{\partial}$, we have
$\partial\Omega\in Im\,\partial\bar{\partial}$. This is equivalent to:

$(\tilde{H}_{k}):\quad\forall\Gamma\in
C_{p+k,p-k+1}^{\infty}(X,\mathbb{C})\mbox{ such that }d\Gamma=0,\mbox{ we have
}\Gamma\in Im\,\partial\Rightarrow\Gamma\in Im\,\partial\bar{\partial}.$ (5)

This is further equivalent to $I^{p+k,p-k+1}$ is injective, hence to $\ker
I^{p+k,p-k+1}=\\{0\\}$.

Consider the diagram:

$\textstyle{H_{A}^{p+k-1,p-k+1}(X,\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\widehat{T}_{k}}$$\textstyle{H_{BC}^{p+k,p-k+1}(X,\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I^{p+k,p-k+1}}$$\textstyle{H_{\partial}^{p+k,p-k+1}(X,\mathbb{C})}$

where $g=I^{p+k,p-k+1}\circ\widehat{T}_{k}$. We have $\ker
I^{p+k,p-k+1}=\\{[\Gamma]_{BC}/[\Gamma]_{\partial}=0\\}$ and
$Im\,\widehat{T}_{k}=\\{[\partial\Omega]_{BC}/\Omega\in\ker\partial\bar{\partial}\subset
C_{p+k-1,p-k+1}^{\infty}(X,\mathbb{C})\\}\subset\ker I^{p+k,p-k+1}$.

Conversely, if $[\Gamma]_{BC}\in\ker I^{p+k,p-k+1}$, then there exists a
$(p+k-1,p-k+1)$-form $\Omega$ such that $\Gamma=\partial\Omega$. Since
$[\Gamma]_{BC}\in H_{BC}^{p+k,p-k+1}(X,\mathbb{C})$, then
$0=\bar{\partial}\Gamma=\bar{\partial}\partial\Omega$. This implies that
$[\Gamma]_{BC}=[\partial\Omega]_{BC}=\widehat{T}_{k}([\Omega]_{A})$. Therefore
we always have:

$\ker I^{p+k,p-k+1}=Im\,\widehat{T}_{k}$

Furthermore, the hypothesis $(\tilde{H}_{k})$ (cf. 5) is satisfied if and only
if $\ker I^{p+k,p-k+1}=Im\,\widehat{T}_{k}=\\{0\\}$.

We can now prove the following

######  Theorem 5.6.

Let $(X_{t})_{t\in\Delta}$ be a holomorphic family of compact complex
manifolds with $\dim_{\mathbb{C}}X_{t}=n$ and $(\omega_{t})_{t\in\Delta}$ a
smooth family of Hermitian metrics on $(X_{t})_{t\in\Delta}$ for $t\in\Delta$.
If for all $t$ sufficiently close to $0$ we have :

$\left\\{\begin{array}[]{rl}&h_{A}^{p+k-1,p-k+1}(0)=h_{A}^{p+k-1,p-k+1}(t):=dim_{\mathbb{C}}H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\\\
&\\\
&h_{BC}^{p+k,p-k+1}(0)=h_{BC}^{p+k,p-k+1}(t):=dim_{\mathbb{C}}H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})\\\
&\\\
&h_{\partial}^{p+k,p-k+1}(0)=h_{\partial}^{p+k,p-k+1}(t):=dim_{\mathbb{C}}H_{\partial}^{p+k,p-k+1}(X_{t},\mathbb{C}),\end{array}\right.$
$\mbox{ then \hskip 14.22636pt }\left\\{\begin{array}[]{rl}&\Delta\ni
t\overset{\mathcal{H}_{A}}{\longmapsto}H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\\\
&\\\ &\Delta\ni
t\overset{\mathcal{H}_{BC}}{\longmapsto}H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})\\\
&\\\ &\Delta\ni
t\overset{\mathcal{H}_{\partial}}{\longmapsto}H_{\partial}^{p+k,p-k+1}(X_{t},\mathbb{C})\end{array}\right.$

are $C^{\infty}$ vector bundles. Moreover, the linear maps:

$\textstyle{H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{k}(t)}$$\scriptstyle{\widehat{T}_{k}(t)}$$\textstyle{H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I^{p+k,p-k+1}(t)}$$\textstyle{H_{\partial_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C})}$

vary in a $C^{\infty}$ way with $t\in\Delta$, where
$g_{k}=I^{p+k,p-k+1}\circ\widehat{T}_{k}$.

Proof. The Laplace-type operators

$\left\\{\begin{array}[]{rl}&\Delta_{A,t}=\partial_{t}\partial_{t}^{*}+\bar{\partial}_{t}\bar{\partial}_{t}^{*}+\bar{\partial}_{t}^{*}\partial_{t}^{*}\partial_{t}\bar{\partial}_{t}+\partial_{t}\bar{\partial}_{t}\bar{\partial}_{t}^{*}\partial_{t}^{*}+\partial_{t}\bar{\partial}_{t}^{*}\bar{\partial}_{t}\partial_{t}^{*}+\bar{\partial}_{t}\partial_{t}^{*}\partial_{t}\bar{\partial}_{t}^{*}\hskip
56.9055pt(cf.\quad\cite[cite]{[\@@bibref{}{Sch07}{}{}]})\\\
&\Delta_{BC,t}=\partial_{t}\bar{\partial}_{t}\bar{\partial}_{t}^{*}\partial_{t}^{*}+\bar{\partial}_{t}^{*}\partial_{t}^{*}\partial_{t}\bar{\partial}_{t}+\bar{\partial}_{t}^{*}\partial_{t}\partial_{t}^{*}\bar{\partial}_{t}+\partial_{t}^{*}\bar{\partial}_{t}\bar{\partial}_{t}^{*}\partial_{t}+\bar{\partial}_{t}^{*}\bar{\partial}_{t}+\partial_{t}^{*}\partial_{t}\hskip
56.9055pt(cf.\quad\cite[cite]{[\@@bibref{}{KS60}{}{}]})\\\
&\Delta_{t}^{\prime}=\partial_{t}\partial_{t}^{*}+\partial_{t}^{*}\partial_{t}\end{array}\right.$

are elliptic (cf. [KS60], [Sch07]), so we have the Hodge isomorphisms:

$\left\\{\begin{array}[]{rl}&H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta_{A,t}}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\\\
&\\\
&H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta_{BC,t}}^{p+k,p-k+1}(X_{t},\mathbb{C})\\\
&\\\
&H_{\partial_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta^{\prime}_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C}).\end{array}\right.$

where
$\mathcal{H}_{\Delta_{BC,t}}^{p+k,p-k+1}(X_{t},\mathbb{C})=\ker\Delta_{BC,t}$
and
$\mathcal{H}_{\Delta_{A,t}}^{p+k-1,p-k+1}(X_{t},\mathbb{C})=\ker\Delta_{A,t}$
stand for the Bott-Chern and Aeppli harmonic spaces (spaces of Bott-Chern and
Aeppli harmonic forms ) respectively and
$\mathcal{H}_{\Delta^{\prime}_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C})=\ker\Delta^{\prime}_{t}$.
On the otherhand, since $h_{A}^{p+k-1,p-k+1}(0)=h_{A}^{p+k-1,p-k+1}(t)$,
$h_{BC}^{p+k,p-k+1}(0)=h_{BC}^{p+k,p-k+1}(t)$ and
$h_{\partial}^{p+k,p-k+1}(0)=h_{\partial}^{p+k,p-k+1}(t)$ for all $t$
sufficiently close to $0$, by Kodaira-spencer (cf. [K86], theorem 7.4) we have
:

$\left\\{\begin{array}[]{rl}&t\overset{\mathcal{H}_{A}}{\longmapsto}H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta_{A,t}}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\\\
&\\\
&t\overset{\mathcal{H}_{BC}}{\longmapsto}H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta_{BC,t}}^{p+k,p-k+1}(X_{t},\mathbb{C})\\\
&\\\
&t\overset{\mathcal{H}_{\partial}}{\longmapsto}H_{\partial}^{p+k,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta^{\prime}_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C})\end{array}\right.$

are $C^{\infty}$ vector bundles. Recall that the linear maps
$\widehat{T}_{k}(t)$ and $I^{p+k,p-k+1}(t)$ are well defined as shown above.
Let $h_{t}$ and $F_{t}$ be the orthogonal projection of
$C_{p+k-1,p-k+1}^{\infty}(X_{t})$, resp. $C_{p+k,p-k+1}^{\infty}(X_{t})$ onto
$\mathcal{H}_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})$, resp.
$\mathcal{H}_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})$ respectively.

$\textstyle{C_{p+k-1,p-k+1}^{\infty}(X_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{t}}$$\scriptstyle{f_{t}=\partial_{t}}$$\textstyle{C_{p+k,p-k+1}^{\infty}(X_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{t}}$$\textstyle{H_{A}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\simeq\mathcal{H}_{\Delta_{A}}^{p+k-1,p-k+1}(X_{t},\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{T}_{k}(t)}$$\textstyle{\mathcal{H}_{\Delta_{BC}}^{p+k,p-k+1}(X_{t},\mathbb{C})\simeq
H_{BC}^{p+k,p-k+1}(X_{t},\mathbb{C})}$

We have $F_{t}\circ f_{t}=\widehat{T}_{k}(t)\circ h_{t}$ , $f_{t}$ is
$C^{\infty}$ and by Kodaira-Spencer as in [K86] $F_{t}$ and $h_{t}$ vary
smoothly with $t\in\Delta$. Therefore $\widehat{T}_{k}(t)$ varies smoothly
with $t$. Let $P_{t}$ the orthogonal projection of
$C_{p+k,p-k+1}^{\infty}(X_{t})$ onto
$\mathcal{H}_{\Delta^{\prime}_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C})$.

$\textstyle{C_{p+k,p-k+1}^{\infty}(X_{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{t}}$$\scriptstyle{F_{t}}$$\textstyle{\mathcal{H}_{\Delta_{BC}}^{p+k,p-k+1}(X_{t},\mathbb{C})\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I^{p+k,p-k+1}(t)}$$\textstyle{\quad\mathcal{H}_{\Delta^{\prime}_{t}}^{p+k,p-k+1}(X_{t},\mathbb{C})}$

We have $P_{t}=I^{p+k,p-k+1}(t)\circ F_{t}$ and by Kodaira-Spencer as in [K86]
$P_{t}$ and $F_{t}$ are $C^{\infty}$ with $t\in\Delta$, then
$I^{p+k,p-k+1}(t)$ is also $C^{\infty}$ with $t\in\Delta$. Consequently $g(t)$
varies smoothly with $t\in\Delta$. $\Box$

One obtains sections $\widehat{T}_{k}=(\widehat{T}_{k}(t))_{t\in\Delta}\in
C^{\infty}(\Delta,End(\mathcal{H}_{A},\mathcal{H}_{BC}))$ and
$I^{p+k,p-k+1}=(I^{p+k,p-k+1}(t))_{t\in\Delta}\in
C^{\infty}(\Delta,End(\mathcal{H}_{BC},\mathcal{H}_{\partial}))$.

As a consequence, we obtain the following conclusion on the deformation limit
of the $X_{t}$’s that satisfy the $(\tilde{H}_{k})$ assumption.

######  Corollary 5.7.

Let $(X_{t})_{t\in\Delta}$ be a holomorphic family of compact complex
manifolds and $(\omega_{t})_{t\in\Delta}$ a smooth family of metrics on
$(X_{t})_{t\in\Delta}$, $\dim_{\mathbb{C}}X_{t}=n$, $t\in\Delta$.

$\mbox{Assume that :\hskip 14.22636pt
}\left\\{\begin{array}[]{rl}&h_{A}^{p+k-1,p-k+1}(0)=h_{A}^{p+k-1,p-k+1}(t)\\\
&\\\ &h_{BC}^{p+k,p-k+1}(0)=h_{BC}^{p+k,p-k+1}(t)\hskip 56.9055pt\forall t\sim
0\\\ &\\\
&h_{\partial}^{p+k,p-k+1}(0)=h_{\partial}^{p+k,p-k+1}(t)\end{array}\right.$

If $X_{t}$ satisfies the hypothesis $(\tilde{H}_{k})$, $\forall
t\in\Delta\setminus\\{0\\}$, then $X_{0}$ also satisfies the hypothesis
$(\tilde{H}_{k})$ .

Proof. Recall that $X_{t}$ satisfying $(\tilde{H}_{k})$ is equivalent to the
map $\widehat{T}_{k}(t)$ vanishing identically.

Moreover, if $\widehat{T}_{k}(t)\equiv 0$, for all
$t\in\Delta\setminus\\{0\\}$, then by continuity of $\widehat{T}_{k}(t)$,
$\widehat{T}_{k}(0)\equiv 0$ which is equivalent to $X_{0}$ satisfying the
$(\tilde{H}_{k})$ assumption. $\Box$

Hence we obtain the following:

######  Proposition 5.8.

Fix $p\in\\{1,\cdots,n-1\\}$ and let $k\in\\{1,\cdots,p+1\\}$ such that
$p+k\leq n$.

  1. (1)

$\forall k\in\\{1,\cdots,p+1\\}$, $(\tilde{H}_{k})$ $\Rightarrow$ $(H_{k})$

  2. (2)

$(\tilde{H}_{1})+\cdots+(\tilde{H}_{p+1})$ $\Rightarrow$
$(H_{1})+\cdots+(H_{p+1})$ $\Rightarrow$
$\mathcal{A}_{p}(X)=\mathcal{C}_{p}(X)$

  3. (3)

Suppose that $\mathcal{A}_{p}(X_{t})=\mathcal{C}_{p}(X_{t})$ (cf. (2),(3))
$\forall t\in\Delta\setminus\\{0\\}$. Then:

$\left.\begin{array}[]{rl}&\forall k\in\\{1,\cdots,p+1\\}\\\
&(\tilde{H}_{1})+\cdots+(\tilde{H}_{p+1})\hskip 56.9055pt\forall
t\in\Delta\setminus\\{0\\}\\\ &\\\
&h_{A}^{p+k-1,p-k+1}(0)=h_{A}^{p+k-1,p-k+1}(t)\\\ &\\\
&h_{BC}^{p+k,p-k+1}(0)=h_{BC}^{p+k,p-k+1}(t)\hskip 28.45274pt\forall t\sim
0\\\ &\\\
&h_{\partial}^{p+k,p-k+1}(0)=h_{\partial}^{p+k,p-k+1}(t)\end{array}\right\\}\Rightarrow\mathcal{A}_{p}(X_{0})=\mathcal{C}_{p}(X_{0})$

######  Remark 5.9.

In the case where $n=3$ and $p=2$, $k\in\\{1,2,3\\}$ and $2+k\leq 3$ we must
have $k=1$.
So, $\mathcal{A}_{2}(X)=\\{[\Omega]_{A}/\Omega>0\mbox{ such that
}\partial\bar{\partial}\Omega=0\\}=\mathcal{G}_{X}$ Gauduchon cone
$\mathcal{C}_{2}(X)=\\{[\Omega]_{A}/\Omega>0\mbox{ and there exists
}\alpha^{1,3}\in C_{1,3}^{\infty}(X,\mathbb{C})\mbox{ such that
}d(\alpha^{1,3}+\Omega+\overline{\alpha^{1,3}})=0\\}=\mathcal{SG}_{X}$ sG cone
and the following equivalence holds:

$\mathcal{A}_{2}(X)=\mathcal{C}_{2}(X)\Leftrightarrow X\mbox{ is sGG i.e
}\mathcal{SG}_{X}=\mathcal{G}_{X}$

######  Proposition 5.10.

Let $p,q\in\\{1,\cdots,n\\}$ be fixed. Suppose that the implication :

$u\in\mbox{Im }\,\partial\quad\Rightarrow\quad u\in\mbox{Im
}\,\partial\bar{\partial}$

holds for all $d$-closed forms of types $(p,q)$, $(q,p)$, $(p+1,q)$, and
$(q+1,p)$ for all $p,q$ such that $p+q=k$. Then, there exists a canonical
injective linear map :

$\begin{array}[]{ll}&H_{A}^{p,q}(X,\mathbb{C})\hookrightarrow
H_{DR}^{p+q}(X,\mathbb{C})\\\ &\hskip
31.2982pt[\alpha]_{A}\mapsto\\{\alpha\\}_{DR}\end{array}$

where $\alpha$ is any $d$-closed representative of the class $[\alpha]_{A}$
(such an $\alpha$ exists due to the hypothesis)

Proof. The map

$\begin{array}[]{ll}&H_{A}^{p,q}(X,\mathbb{C})\longrightarrow
H_{DR}^{p+q}(X,\mathbb{C})\\\ &\hskip
31.2982pt[\alpha]_{A}\mapsto\\{\alpha\\}_{DR}\end{array}$

is well defined since if $\partial\bar{\partial}\alpha=0$, then by the
hypothesis $Im\,\partial\subset Im\,\partial\bar{\partial}$, there exists
$(p,q-1)$-form $v$ such that $\partial\alpha=-\partial\bar{\partial}v$. On the
other hand, $\bar{\alpha}$ is a $(q,p)$-form and $\partial\bar{\alpha}$ is a
$\partial$-exact $(q+1,p)$-form. Hence, by assumption, $\partial\bar{\alpha}$
is $\partial\bar{\partial}$-exact. Then, by conjugation,
$\bar{\partial}\alpha$ stills $\partial\bar{\partial}$-exact, i.e. there
exists some $(p-1,q)$-form $u$ such that
$\bar{\partial}\alpha=\partial\bar{\partial}u$. This implies that
$d(\alpha+\partial u+\bar{\partial}v)=0$, it means that every Aeppli
cohomology class contains a $d$-closed representative.

Let $\alpha$ to be a $(p,q)$-form such that $d\alpha=0$ and $\alpha=\partial
u+\bar{\partial}v$. Then $\partial\alpha=0=\partial\bar{\partial}v$ and
$\bar{\partial}\alpha=0=\bar{\partial}\partial u$. Note that $\partial u$ is a
$d$-closed $\partial$-exact $(p,q)$-form, so $\partial u\in
Im\,\partial\bar{\partial}$. Thus $\partial u\in Im\,d$. Meanwhile,
$\bar{\partial}v$ is a $d$-closed $\bar{\partial}$-exact $(p,q)$-form, so
$\bar{\partial}v\in Im\,\partial\bar{\partial}$, then $\bar{\partial}v\in
Im\,d$. Therefore $\alpha\in Im\,d$, thus the map above is well defined.

Injectivity: for all $\alpha\in C_{p,q}^{\infty}(X,\mathbb{C})$ such that
$\partial\bar{\partial}\alpha=0$ and $\alpha\in Im\,d$, we have $\alpha\in
Im\,\partial+Im\,\bar{\partial}$. Hence the map is injective once it is well
defined. $\Box$

We can now infer the following

######  Corollary 5.11.

  1. (1)

Fix $k\in\\{0,1,\cdots,2n\\}$, suppose that the implication

$u\in\mbox{Im }\partial\quad\Rightarrow\quad u\in\mbox{Im
}\partial\bar{\partial}$

holds for all $d$-closed forms of types $(p,q)$, $(q,p)$, $(p+1,q)$, and
$(q+1,p)$ for all $p,q$ such that $p+q=k$.
Then there exists a canonical injection:

$\bigoplus_{p+q=k}H_{A}^{p,q}(X,\mathbb{C})\hookrightarrow
H_{DR}^{k}(X,\mathbb{C})$

  2. (2)

Fix $k\in\\{0,1,\cdots,2n\\}$. Suppose that $(*_{k})$ holds, where

$\begin{array}[]{ll}(*_{k}):&\mbox{ the implication }u\in
Im\,\partial\Rightarrow u\in Im\,\partial\bar{\partial}\mbox{ holds for all
}d-closed\mbox{ forms of types }\\\ &(p,q),(q,p),(p+1,q),\mbox{ and
}(q+1,p)\\\ &\mbox{ for all }p,q\mbox{ such that }p+q=k\mbox{ or
}p+q=2n-k\end{array}$ (6)

Then there exists a canonical injection :

$\bigoplus_{p+q=k}H_{A}^{p,q}(X,\mathbb{C})\oplus\bigoplus_{p+q=2n-k}H_{A}^{p,q}(X,\mathbb{C})\hookrightarrow
H_{DR}^{k}(X,\mathbb{C})\oplus H_{DR}^{2n-k}(X,\mathbb{C})$

Due to Angella-Tomassini [AT13], we have :

$2b_{k}\leq\sum_{p+q=k}h^{p,q}_{A}+\sum_{p+q=2n-k}h^{p,q}_{A}$

this map is an isomorphism and
$2b_{k}=\sum_{p+q=k}h_{A}^{p,q}+\sum_{p+q=2n-k}h_{A}^{p,q}$.

Another immediate consequence is the following degeneration at $E_{1}$ of the
Frölicher spectral sequence.

######  Corollary 5.12.

If $(*_{k})$ (cf. (6)) assumption is satisfied for all
$k\in\\{0,1,\cdots,2n\\}$, then $E_{1}(X)=E_{\infty}(X)$ (i.e. the Frölicher
spectral sequence degenerate at $E_{1}$).

In the end, the above results (cf. Corollary 5.11) leads to the following

######  Proposition 5.13.

It is clear that $(*_{2p})$ (cf. (6)) implies the $(\tilde{H}_{k})$ (cf. (5))
assumption for all $k\in\\{1,\cdots,p+1\\}$. If $X_{0}$ satisfies $(*_{2p})$
and $(*_{2p+1})$, then:

$\left\\{\begin{array}[]{rl}&\forall k\in\\{1,\cdots,p+1\\}\\\
&h_{A}^{p+k-1,p-k+1}(0)=h_{A}^{p+k-1,p-k+1}(t)\\\ &\\\
&h_{BC}^{p+k,p-k+1}(0)=h_{BC}^{p+k,p-k+1}(t)\hskip 28.45274pt\forall t\sim
0\\\ &\\\
&h_{\partial}^{p+k,p-k+1}(0)=h_{\partial}^{p+k,p-k+1}(t)\end{array}\right.$

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Ibn Tofail University, Faculty of Sciences, Departement of Mathematics PO 242
Kenitra, Morocco

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