# HKT manifolds: Hodge Theory, formality and balanced metrics

Giovanni Gentili  Dipartimento di Matematica “G. Peano”
Università degli studi di Torino
Via Carlo Alberto 10
10123 Torino, Italy<EMAIL_ADDRESS>and  Nicoletta Tardini
Dipartimento di Scienze Matematiche, Fisiche e Informatiche
Unità di Matematica e Informatica
Università degli Studi di Parma
Parco Area delle Scienze 53/A
43124 Parma, Italy<EMAIL_ADDRESS>
###### Abstract.

Let $(M,I,J,K,\Omega)$ be a compact HKT manifold and denote with $\partial$
the conjugate Dolbeault operator with respect to $I$,
$\partial_{J}:=J^{-1}\overline{\partial}J$,
$\partial^{\Lambda}:=[\partial,\Lambda]$ where $\Lambda$ is the adjoint of
$L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the
complexes $(A^{\bullet,0},\partial,\partial_{J})$ and
$(A^{\bullet,0},\partial,\partial^{\Lambda})$ showing a similar behavior to
Kähler manifolds. In particular, several relations among the Laplacians, the
spaces of harmonic forms and the associated cohomology groups, together with
Hard Lefschetz properties, are proved. Moreover, we show that for a compact
HKT $\mathrm{SL}(n,\mathbb{H})$-manifold the differential graded algebra
$(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for
the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$-structure $(I,J,K,\Omega)$
on a compact complex manifold $(M,I)$. Finally, balanced HKT structures on
solvmanifolds are studied.

###### Key words and phrases:

HKT; Hodge theory; formality; balanced metric; solvmanifold.

###### 2020 Mathematics Subject Classification:

53C26, 58A14, 22E25

##  1\. Introduction

Let $(M,I,J,K)$ be a _hypercomplex manifold_ , i.e. a smooth manifold $M$
equipped with three complex structures $I,J,K$ that anticommute with each
other, and such that $IJ=K$. On a hypercomplex manifold there is a
distinguished connection $\nabla$, called the Obata connection [25], that is
torsion-free and preserves the hypercomplex structure, in the sense that

$\nabla I=0\,,\qquad\nabla J=0\,,\qquad\nabla K=0\,.$

A Riemannian metric that is Hermitian with respect to all three complex
structures $I,J,K$ is said to be _hyperhermitian_ , and, accordingly,
$(M,I,J,K,g)$ is called an hyperhermitian manifold.

On a hyperhermitian manifold the $2$-form

$\Omega:=\frac{g(J\cdot,\cdot)+ig(K\cdot,\cdot)}{2}$

is non-degenerate of type $(2,0)$ with respect to $I$ and completely
determines the hyperhermitian metric via the relation
$\Omega(Z,J\bar{Z})=g(Z,\bar{Z})$ for every $Z\in T^{1,0}M$. Furthermore there
is a bijective correspondence between hyperhermitian metrics and q-positive
q-real $(2,0)$-forms, where a $(2,0)$-form $\Omega$ is called q-real if
$J\Omega=\bar{\Omega}$ and it is called q-positive if additionally satisfies
$\Omega(Z,J\bar{Z})>0$ for every non-zero $Z\in T^{1,0}M$.

A hyperhermitian manifold is _HKT_ , namely _hyperkähler with torsion_ , if
the associated $(2,0)$-form satisfies

$\partial\Omega=0\,,$

where $\partial$ is the Dolbeault operator with respect to $I$. The HKT
condition is at many levels the hypercomplex analogue of the Kähler condition
(see e.g. [2, 5, 20, 21, 24, 33, 35]). The main purpose of this work is
twofold: on one hand we explore the analogies with the Kähler setting from a
cohomological point of view, as done in [20, 24, 33], on the other hand we
extend some results proved on hypercomplex nilmanifolds in [6] to hypercomplex
solvmanifolds.

In [34] Verbitsky proved that a compact HKT manifold has trivial canonical
bundle if and only if the holonomy group of the Obata connection is contained
in $\mathrm{SL}(n,\mathbb{H})$. Manifolds with the latter property are called
$\mathrm{SL}(n,\mathbb{H})$-manifolds. If one removes the HKT hypothesis, then
it is still true that an $\mathrm{SL}(n,\mathbb{H})$-manifold has
holomorphically trivial canonical bundle. The converse has been recently
disproved by Andrada-Tolcachier [4], however it is, for instance, true for
hypercomplex nilmanifolds [6]. We prove here that on hypercomplex
solvmanifolds the $\mathrm{SL}(n,\mathbb{H})$ condition is equivalent to the
existence of an invariant holomorphic trivialization of the canonical bundle
(Theorem 5.4).

Another very relevant property in this context is the balanced condition.
Indeed, it turns out [35] that for a compact HKT manifold $(M,I,J,K,g)$ one
has that $(M,I,g)$ is balanced if and only if $\partial\bar{\Omega}^{n}=0$. In
particular, if a compact HKT manifold is balanced it is necessarily
$\mathrm{SL}(n,\mathbb{H})$. A very interesting conjecture posed by Alesker
and Verbitsky predicts that this condition is also sufficient, in the sense
that an HKT $\mathrm{SL}(n,\mathbb{H})$-manifold admits a balanced HKT metric
(which does not necessarily coincide with the initial one). Again, evidence
for this conjecture is provided by nilmanifolds with invariant hypercomplex
structure [6] and we shall extend this fact to solvmanifolds with invariant
hypercomplex structure (Theorem 5.6).

As explained in [35], one way to approach this problem is by studying the
quaternionic Calabi-Yau conjecture proposed by Alesker and Verbitsky in [3].
The solvability of the relative equation would allow to prescribe the complex
volume of the HKT manifold. More precisely, for any q-positive $(2n,0)$-form
$\Theta$, one could find a new HKT form $\Omega^{\prime}$, compatible with the
given hypercomplex structure, satisfying

$(\Omega^{\prime})^{n}=\Theta\,.$

Choosing $\Theta$ to be holomorphic would imply that $\Omega^{\prime}$ is
balanced HKT. For further information on the (still unsolved in general)
quaternionic Calabi-Yau conjecture see e.g. [1, 8, 13, 17, 18, 19, 30, 38] and
references therein.
Other sufficient conditions on solvmanifolds with an invariant HKT structure
to have a balanced metric are given in Theorems 5.1, 5.5.

On a hypercomplex manifold $(M,I,J,K)$ endowed with a HKT structure $\Omega$
there are three important differential operators. Denote with
$A^{p,q}(M)=A^{p,q}_{I}(M)$ the space of $(p,q)$-forms with respect to $I$.
Then we are interested in the usual (conjugate) Dolbeault operator
$\partial\colon A^{p,q}(M)\to A^{p+1,q}(M)$ and the twisted differential
operator $\partial_{J}\colon A^{p,q}(M)\to A^{p+1,q}(M)$ defined as
$\partial_{J}:=J^{-1}\bar{\partial}J$. Because of the integrability of
$I,J,K$, these operators anticommute and both square to zero. Therefore we
obtain a cochain complex $(A^{\bullet,q},\partial,\partial_{J})$ for every
fixed $q$. This differs from the complex case, as here we obtain a single
complex, while in the complex setting $\partial$ and $\bar{\partial}$ give
rise to a double complex. Here, we restrict to study the case $q=0$.
Moreover, the existence of $\Omega$ leads to the definition of the operator
$\partial^{\Lambda}:=[\partial,\Lambda]$ where $\Lambda$ is the adjoint of
$L:=\Omega\wedge-$. Therefore we obtain a cochain complex
$(A^{\bullet,q},\partial,\partial^{\Lambda})$ for every fixed $q$, and again
we will restrict to $q=0$. Hence one can study cohomology groups and Hodge
theory from a “complex point of view” on
$(A^{\bullet,0},\partial,\partial_{J})$ or from a “symplectic point of view”
on $(A^{\bullet,0},\partial,\partial^{\Lambda})$.
Concerning the latter case, one can contextualize everything in the more
general setting of Lefschetz spaces and with this approach one can then prove
that all the natural Laplacian operators that arise in Section 2 and at the
beginning of Section 3 are related. From the complex point of view, naturally,
the quaternionic Dolbeault, Bott-Chern and Aeppli cohomology groups can be
defined:

$H^{p,0}_{\partial}(M):=\frac{\mathrm{Ker}(\partial|_{A^{p,0}(M)})}{\partial
A^{p-1,0}(M)}\,,\qquad
H^{p,0}_{\partial_{J}}(M):=\frac{\mathrm{Ker}(\partial_{J}|_{A^{p,0}(M)})}{\partial_{J}A^{p-1,0}(M)}\,,$
$H^{p,0}_{\mathrm{BC}}(M):=\frac{\mathrm{Ker}(\partial|_{A^{p,0}(M)})\cap\mathrm{Ker}(\partial_{J}|_{A^{p,0}(M)})}{\partial\partial_{J}A^{p-2,0}(M)}\,,$
$H^{p,0}_{\mathrm{A}}(M):=\frac{\mathrm{Ker}(\partial\partial_{J}|_{A^{p,0}(M)})}{\partial
A^{p-1,0}(M)+\partial_{J}A^{p-1,0}(M)}\,,$

when $M$ is compact all these groups are finite-dimensional [20], indeed, as
usual, once fixed an hyperhermitian metric, one can show that each of these
cohomology group is isomorphic to the kernel of the following Laplacians
acting on $(p,0)$-forms

$\Delta_{\partial}:=\partial\partial^{*}+\partial^{*}\partial\,,\qquad\Delta_{\partial_{J}}:=\partial_{J}\partial^{*}_{J}+\partial^{*}_{J}\partial_{J}\,,$
$\Delta_{\mathrm{BC}}:=\partial^{*}\partial+\partial^{*}_{J}\partial_{J}+\partial\partial_{J}\partial_{J}^{*}\partial^{*}+\partial^{*}_{J}\partial^{*}\partial\partial_{J}+\partial^{*}_{J}\partial\partial^{*}\partial_{J}+\partial^{*}\partial_{J}\partial_{J}^{*}\partial\,,$
$\Delta_{\mathrm{A}}:=\partial\partial^{*}+\partial_{J}\partial_{J}^{*}+\partial\partial_{J}\partial_{J}^{*}\partial^{*}+\partial^{*}_{J}\partial^{*}\partial\partial_{J}+\partial\partial_{J}^{*}\partial_{J}\partial^{*}+\partial_{J}\partial^{*}\partial\partial_{J}^{*}\,.$

For each of these we denote with a calligraphic letter the corresponding space
of harmonic forms, thus, for instance,
$\mathcal{H}_{\partial}^{p,0}(M):=\mathrm{Ker}(\Delta_{\partial}|_{A^{p,0}(M)})$.
It is well known that on a compact Kähler manifold the spaces of Dolbeault,
Bott-Chern and Aeppli-harmonic forms all coincide. We prove that the analogue
result is also true for balanced HKT manifolds:

######  Theorem 1.1 (Propositions 3.4, 3.8, 3.13).

On a compact balanced HKT manifold $M$ the spaces of harmonic forms all
coincide:

$\mathcal{H}^{p,0}_{\partial}(M)=\mathcal{H}^{p,0}_{\partial_{J}}(M)=\mathcal{H}^{p,0}_{\mathrm{BC}}(M)=\mathcal{H}^{p,0}_{\mathrm{A}}(M)\,.$

In particular, there are isomorphisms

$H^{p,0}_{\partial}(M)\cong H^{p,0}_{\partial_{J}}(M)\cong
H^{p,0}_{\mathrm{BC}}(M)\cong H^{p,0}_{\mathrm{A}}(M)$

for every $p$.

We remark that the equality of $\Delta_{\partial_{J}}$ and $\Delta_{\partial}$
on balanced HKT manifolds is already implicitly proved in [33, Theorem 10.2].
Along the way we shall also study the Hard Lefschetz condition on these spaces
(see Theorems 3.1 and 3.10).

Another interesting notion to be investigated is formality since it provides
an obstruction to the $\partial\partial_{J}$-lemma and so to the existence of
HKT $\mathrm{SL}(n,\mathbb{H})$-structures. More precisely, we prove

######  Theorem 1.2 (Theorem 4.1).

Let $(M,I,J,K)$ be a compact hypercomplex manifold satisfying the
$\partial\partial_{J}$-lemma, then the differential graded algebra
$(A^{\bullet,0}(M),\partial)$ is formal.

As a consequence, we obtain that triple $\partial$-Massey products vanish,
which allows us to derive the following interesting obstruction for a compact
complex $4n$-dimensional manifold to allow HKT
$\mathrm{SL}(n,\mathbb{H})$-structures.

######  Corollary 1.3 (cf. Corollary 4.9).

Let $(M,I)$ be a $4n$-dimensional compact complex manifold with
holomorphically trivial canonical bundle and such that there exists a non
trivial $\partial$-Massey product. Then $(M,I)$ does not admit any complex
structures $J,K$ such that $(M,I,J,K)$ is hypercomplex and admits a HKT
metric.

Notice that, as a consequence of Theorems 4.10, 4.11, on nilmanifolds with
invariant HKT structures the $\partial$-Massey products vanish. For this
reason, we use such obstruction to give some examples of complex solvmanifolds
without invariant HKT $\mathrm{SL}(n,\mathbb{H})$-structures.

The organization of the paper is the following. In Section 2 we briefly study
a class of differential graded algebras which we call “Lefschetz”. Building on
the work of Tomassini and Wang [31] we define a generalization of the Hodge
star operator, which allows us to take into account formal adjoints and
discuss some relations between Laplacians. This algebraic picture is then
applied to HKT manifolds in Section 3, leading us to the proof of Theorem 1.1.
Section 4 is then devoted to investigate the notion of formality. Here we
prove Theorem 1.2 and Corollary 1.3. Finally, in Section 5 we briefly study
hypercomplex structures on solvmanifolds and their relation with the balanced
condition.

Acknowledgements. Both authors are deeply grateful to Luigi Vezzoni for
proposing this research argument and for many helpful discussions. This paper
was partly written during the first author’s visits to IMPA (Rio de Janeiro)
and IMECC/UNICAMP (Campinas). He wishes to thank both institutions for the
warm hospitality. In particular, he expresses his most sincere gratitude to
Misha Verbitsky for countless discussions, remarkable helpfulness and guidance
during the visit experience at IMPA and he is profoundly thankful to Henrique
Sá Earp for many instructive conversations. The second author would like to
thank Riccardo Piovani, Tommaso Sferruzza and Adriano Tomassini for
interesting discussions on formality and Massey products. Both authors would
like to thank Adrián Andrada and Alejandro Tolcachier for sharing a
preliminary version of [4] and for pointing out an inaccuracy in the first
version of Theorem 5.4. We also thank the anonymous referee for their comments
that improved the presentation of the paper. The second author has financially
been supported by the Programme “FIL-Quota Incentivante” of University of
Parma and co-sponsored by Fondazione Cariparma. This work was partially
supported by GNSAGA of INdAM.

##  2\. Lefschetz spaces

In this section, inspired by the algebraic treatment of Tomassini and Wang
[31], we wish to push a little further their work, proving some identities
between Laplacians defined in a fairly general context. We start by recalling
the main definitions and results from [31] (see also [36]).

######  Definition 2.1.

Let $A=\bigoplus_{p=0}^{2n}A^{p}$ be a direct sum of complex vector spaces.
Let $L$ be a ${\mathbb{C}}$-linear endomorphism of $A$ such that
$L(A^{p})\subseteq A^{p+2}$ for $p=0,\dots,2n-2$ and
$L(A^{2n-1})=L(A^{2n})=0$. We say that $(A,L)$ is a _Lefschetz space_ if $L$
satisfies the Hard Lefschetz Condition (HLC), i.e.

$L^{n-p}\colon A^{p}\to A^{2n-p}$

is an isomorphism for all $p=0,\dots,n$.

If a Lefschetz space $(A,L)$ is equipped with a ${\mathbb{C}}$-linear
endomorphism $d$ such that $d(A^{p})\subseteq A^{p+1}$ for $p=0,\dots,2n-1$,
while $d(A^{2n})=0$ we call the triple $(A,L,d)$ a _differential Lefschetz
space_.
If moreover $d^{2}=0$ then the triple $(A,L,d)$ is called a _Lefschetz
complex_.

On a Lefschetz space we say that $\alpha\in A^{p}$ is a _primitive form_ if
$p\leq n$ and $L^{n-p+1}\alpha=0$. By the HLC immediately follows the
decomposition into primitive forms (see [36]), more precisely, for every
$\alpha\in A^{p}$ there exist unique primitive $\alpha^{k}\in A^{p-2k}$ such
that

(1) $\alpha=\sum_{k=0}^{\lfloor p/2\rfloor}\frac{1}{k!}L^{k}\alpha^{k}\,.$

As a generalization of the symplectic star operator Tomassini and Wang [31]
introduced the _Lefschetz star operator_ $*_{L}\colon A\to A$, acting on a
primitive form $\beta\in A^{p}$ as follows:

$*_{L}\frac{1}{k!}L^{k}\beta:=(-1)^{1+2+\dots+p}\frac{1}{(n-p-k)!}L^{n-p-k}\beta\,.$

Clearly the definition is then extended by linearity to any $\alpha\in A^{p}$
via the Lefschetz decomposition (1). Notice that $*_{L}^{2}=1$.

The starting point of the discussion by Tomassini and Wang is the following
general Demailly-Griffiths-Kähler identity [31, Theorem A].

######  Theorem 2.2.

Let $(A,L,d)$ be a differential Lefschetz space and $\Lambda=*_{L}^{-1}L*_{L}$
the dual Lefschetz operator. Define $d^{\Lambda}\in\mathrm{End}(A)$ by

$d^{\Lambda}|_{A^{p}}:=(-1)^{p+1}*_{L}d*_{L}\,,$

and assume that $[L,[d,L]]=0$, then

$[d^{\Lambda},L]=d+[\Lambda,[d,L]]\,,\qquad[d,\Lambda]=d^{\Lambda}+[[\Lambda,d^{\Lambda}],L]\,.$

Notice that if $(A,L,d)$ is a Lefschetz complex then $d^{2}=0$ implies
$(d^{\Lambda})^{2}=0$. In case $[d,L]=0$, one also obtains that

$[d,d^{\Lambda}]=0.$

Therefore, on a Lefschetz complex with $[d,L]=0$ one has that the triple
$(A,d,d^{\Lambda})$ is a double complex.

We summarize here the main consequences which we are interested in (cf. [31,
Theorems 3.3, 3.5]).

######  Theorem 2.3.

Let $(A,L,d)$ be a Lefschetz complex. Suppose $[d,L]=0$ and denote with
$\mathcal{H}^{p}_{L}$ the space of _Lefschetz harmonic_ $p$-forms, i.e.
elements $\alpha\in A^{p}$ such that

$d\alpha=0=d^{\Lambda}\alpha.$

Then $(\mathcal{H}^{\bullet}_{L},L)$ and $(\mathcal{H}^{\bullet}_{L},\Lambda)$
satisfy the HLC. Furthermore the following are equivalent:

  * •

$(A^{\bullet},L)$ satisfies the $dd^{\Lambda}$-lemma, i.e.,

$\mathrm{Ker}\,d\cap\mathrm{Ker}\,d^{\Lambda}\cap(\mathrm{Im}\,d+\mathrm{Im}\,d^{\Lambda})=\mathrm{Im}\,dd^{\Lambda}\,$

  * •

There is a Lefschetz harmonic representative in each cohomology class of
$H^{\bullet}_{d}$;

  * •

$(H_{d}^{\bullet},L)$ satisfies the HLC;

  * •

$(H_{d^{\Lambda}}^{\bullet},\Lambda)$ satisfies the HLC.

Now we wish to introduce in the picture (a generalization of) the Hodge star
operator, in order to do so, we need a complex structure on our Lefschetz
space.

######  Definition 2.4.

A _Lefschetz (differential) graded algebra_ is a (differential) Lefschetz
space $A=\bigoplus_{p=0}^{2n}A^{p}$ which is also a graded algebra that is
generated by $A^{1}$ over ${\mathbb{C}}$.

Let $A$ be a Lefschetz graded algebra and assume that $A^{1}$ is equipped with
an endomorphism $\mathcal{J}$ such that $\mathcal{J}^{2}=-I$. We extend the
action of $\mathcal{J}$ on $A$ by setting on homogeneous elements

$\mathcal{J}(\alpha_{1}\cdots\alpha_{k})=\mathcal{J}\alpha_{1}\cdots\mathcal{J}\alpha_{k}\,,\qquad\text{for
every }\alpha_{1},\dots,\alpha_{k}\in A^{1}\,,$

and then extending by ${\mathbb{C}}$-linearity.

We make the assumption that $\mathcal{J}L=L\mathcal{J}$ and consequently
introduce a generalization of the Hodge star operator by setting

(2) $*:=\mathcal{J}*_{L}=*_{L}\mathcal{J}$

or equivalently

(3)
$*\frac{1}{k!}L^{k}\beta:=(-1)^{1+2+\dots+p}\frac{1}{(n-p-k)!}L^{n-p-k}\mathcal{J}\beta\,,$

for a primitive $\beta\in A^{p}$ and then extend the definition on all $A$ by
bilinearity via the Lefschetz decomposition (1). It follows that

$*^{2}|_{A^{p}}=\mathcal{J}^{2}|_{A^{p}}=(-1)^{p}\,.$

######  Remark 2.5.

Let $(M,J,\omega)$ be an almost Kähler manifold, namely $\omega$ is a
symplectic structure on a smooth manifold $M$ and $J$ is a compatible almost
complex structure. Clearly when $J$ is integrable (and so $(M,J)$ is a complex
manifold) then $(M,J,\omega)$ is a Kähler manifold. Set $L=\omega\wedge-$ for
the usual Lefschetz operator and let $A=\bigoplus_{p=0}^{2n}A^{p}$ be the
Lefschetz graded algebra of differential forms on $M$. The almost complex
structure $J\colon TM\to TM$ naturally induces a complex structure
$\mathcal{J}$ on $A^{1}$. Since $\omega$ is a $(1,1)$-form we have
$\mathcal{J}L=L\mathcal{J}$ and the description above is coherent with the
well-known almost Kähler case. Indeed, formula (3), where $*$ is the usual
Hodge operator, is sometimes referred to as the Weil relation [37].

Now, suppose $A$ is equipped with a differential $d$. Consider the dual
Lefschetz operator

$\Lambda=*_{L}^{-1}L*_{L}=*^{-1}\mathcal{J}L\mathcal{J}^{-1}*=*^{-1}L*$

and define as before the “Lefschetz adjoint” of $d$, i.e.
$d^{\Lambda}\in\mathrm{End}(A)$ given by
$d^{\Lambda}|_{A^{p}}:=(-1)^{p+1}*_{L}d*_{L}$. Then, by Theorem 2.2, if
$[d,L]=0$ we have $d^{\Lambda}=[d,\Lambda]$ and $d=[d^{\Lambda},L]$.

We may take the “Hodge adjoints”

$d^{*}=-*d*\,,\qquad d^{\Lambda*}=-*d^{\Lambda}*\,,$

and obtain also $d^{*}=[\Lambda,d^{\Lambda*}]$.

Now, we consider the following operators and we aim to study the relations
between them:

$\Delta_{d}=dd^{*}+d^{*}d\,,\qquad\Delta_{d^{\Lambda}}=d^{\Lambda}d^{\Lambda*}+d^{\Lambda*}d^{\Lambda}\,,$
$\Delta^{\mathrm{BC}}_{d^{\Lambda}}=d^{*}d+d^{\Lambda*}d^{\Lambda}+dd^{\Lambda}d^{\Lambda*}d^{*}+d^{\Lambda*}dd^{*}d^{\Lambda}+d^{*}d^{\Lambda}d^{\Lambda*}d+d^{\Lambda*}d^{*}dd^{\Lambda}\,,$
$\Delta^{\mathrm{BC}}_{d^{\Lambda*}}=d^{*}d+d^{\Lambda}d^{\Lambda*}+dd^{\Lambda*}d^{\Lambda}d^{*}+d^{\Lambda}dd^{*}d^{\Lambda*}+d^{*}d^{\Lambda*}d^{\Lambda}d+d^{\Lambda}d^{*}dd^{\Lambda*}\,.$

We will denote with $\mathcal{H}^{\bullet}_{d}$ and
$\mathcal{H}^{\bullet}_{d^{\Lambda}}$ the kernels of $\Delta_{d}$ and
$\Delta_{d^{\Lambda}}$ respectively. All these operators where originally
introduced for symplectic manifolds in [32].

Since it will be useful in the following we recall that for the graded bracket
$[A,B]=AB-(-1)^{\text{deg}\,A\,\text{deg}\,B}BA$, the graded Jacobi identity
holds

$[A,[B,C]]=[[A,B],C]+(-1)^{\text{deg}\,A\,\text{deg}\,B}[B,[A,C]]\,.$

######  Proposition 2.6.

In the previous assumptions it holds

$\Delta_{d}=\Delta_{d^{\Lambda}}-[\Lambda,[d,d^{\Lambda*}]]\,.$

In particular, if $[d,d^{\Lambda*}]=0$ the kernels of $\Delta_{d}$ and
$\Delta_{d^{\Lambda}}$ coincide, namely for every $p$ we have

$\mathcal{H}^{p}_{d}=\mathcal{H}^{p}_{d^{\Lambda}}\,.$

###### Proof.

Using $[\Lambda,d^{\Lambda*}]=d^{*}$ and $[d,\Lambda]=d^{\Lambda}$ we obtain

$\Delta_{d}=[d,d^{*}]=[d,[\Lambda,d^{\Lambda*}]]=[[d,\Lambda],d^{\Lambda*}]-[\Lambda,[d,d^{\Lambda*}]]=[d^{\Lambda},d^{\Lambda*}]-[\Lambda,[d,d^{\Lambda*}]]=\Delta_{d^{\Lambda}}-[\Lambda,[d,d^{\Lambda*}]]\,.\qed$

######  Proposition 2.7.

If $[d,d^{\Lambda*}]=0=[d,L]$, then

$\displaystyle\Delta^{\mathrm{BC}}_{d^{\Lambda}}$
$\displaystyle=\Delta_{d^{\Lambda}}\Delta_{d^{\Lambda}}+d^{*}d+d^{\Lambda*}d^{\Lambda}$
$\displaystyle=\Delta^{\mathrm{BC}}_{d^{\Lambda*}}+d^{\Lambda*}d^{\Lambda}-d^{\Lambda}d^{\Lambda*}\,.$

###### Proof.

Notice that under our assumptions we also have $[d^{*},d^{\Lambda}]=0$ and
$[d^{*},d^{\Lambda*}]=0$. We start by considering
$\Delta_{d^{\Lambda}}\Delta_{d^{\Lambda}}$. By Proposition 2.6

$\Delta_{d^{\Lambda}}\Delta_{d^{\Lambda}}=\Delta_{d}\Delta_{d^{\Lambda}}=dd^{*}d^{\Lambda}d^{\Lambda*}+dd^{*}d^{\Lambda*}d^{\Lambda}+d^{*}dd^{\Lambda}d^{\Lambda*}+d^{*}dd^{\Lambda*}d^{\Lambda}=(I)+(II)+(III)+(IV)$

Now we will treat the four terms separately. Using that $[d,d^{\Lambda*}]=0$
and $[d,d^{\Lambda}]=0$

$\displaystyle(I)$
$\displaystyle=dd^{*}d^{\Lambda}d^{\Lambda*}=-dd^{\Lambda}d^{*}d^{\Lambda*}=dd^{\Lambda}d^{\Lambda*}d^{*}\,,$
$\displaystyle(II)$
$\displaystyle=dd^{*}d^{\Lambda*}d^{\Lambda}=-dd^{\Lambda*}d^{*}d^{\Lambda}=d^{\Lambda*}dd^{*}d^{\Lambda}\,,$
$\displaystyle(III)$
$\displaystyle=d^{*}dd^{\Lambda}d^{\Lambda*}=-d^{*}d^{\Lambda}dd^{\Lambda*}=d^{*}d^{\Lambda}d^{\Lambda*}d\,,$
$\displaystyle(IV)$
$\displaystyle=d^{*}dd^{\Lambda*}d^{\Lambda}=-d^{*}d^{\Lambda*}dd^{\Lambda}=d^{\Lambda*}d^{*}dd^{\Lambda}\,.$

Now, putting the four terms together we have

$\Delta_{d^{\Lambda}}\Delta_{d^{\Lambda}}=\Delta_{d}\Delta_{d^{\Lambda}}=dd^{\Lambda}d^{\Lambda*}d^{*}+d^{\Lambda*}dd^{*}d^{\Lambda}+d^{*}d^{\Lambda}d^{\Lambda*}d+d^{\Lambda*}d^{*}dd^{\Lambda}=\Delta^{\mathrm{BC}}_{d^{\Lambda}}-d^{*}d-d^{\Lambda*}d^{\Lambda}\,.$

Furthermore using again that $[d,d^{\Lambda*}]=0$ and $[d,d^{\Lambda}]=0$ we
obtain

$\displaystyle\Delta^{\mathrm{BC}}_{d^{\Lambda}}=$ $\displaystyle
d^{*}d+d^{\Lambda*}d^{\Lambda}+dd^{\Lambda}d^{\Lambda*}d^{*}+d^{\Lambda*}dd^{*}d^{\Lambda}+d^{*}d^{\Lambda}d^{\Lambda*}d+d^{\Lambda*}d^{*}dd^{\Lambda}$
$\displaystyle=$ $\displaystyle
d^{*}d+d^{\Lambda*}d^{\Lambda}+d^{\Lambda}dd^{*}d^{\Lambda*}+dd^{\Lambda*}d^{\Lambda}d^{*}+d^{\Lambda}d^{*}dd^{\Lambda*}+d^{*}d^{\Lambda*}d^{\Lambda}d$
$\displaystyle=$ $\displaystyle
d^{\Lambda*}d^{\Lambda}-d^{\Lambda}d^{\Lambda*}+\Delta^{\mathrm{BC}}_{d^{\Lambda*}}$

as desired. ∎

######  Remark 2.8.

Let $(M,\omega,J,g)$ be an almost-Kähler manifold, then $[d,d^{\Lambda*}]=0$
if and only if $[d,d^{c}]=0$ if and only if $J$ is integrable. In such a case
$(M,\omega,J,g)$ is a Kähler manifold and we recover the usual equalities for
the Laplacians in Propositions 2.6 and 2.7.

##  3\. Hodge theory on HKT manifolds

###  3.1. Lefschetz spaces on HKT manifolds

We specialize the results of the previous section to the cohomology of HKT
manifolds.

Let $(M,I,J,K)$ be a $4n$-dimensional compact hypercomplex manifold and
$\Omega\in A^{2,0}(M)$ a non-degenerate $(2,0)$-form on $(M,I)$. Then, as
usual we set

$L\colon A^{r,0}(M)\to A^{r+2,0}(M)\,,\qquad L:=\Omega\wedge-$

for the Lefschetz operator.
Then $(A^{\bullet,0}(M),L)$ is a Lefschetz space.
Moreover, if we consider as differential operator $\partial$ (always taken
with respect to $I$), since $I$ is integrable, $\partial^{2}=0$ and so
$(A^{\bullet,0}(M),L,\partial)$ defines a Lefschetz complex. If
$\partial\Omega=0$ then we have

$[\partial,L]=0.$

Denote with $\mathcal{H}^{p,0}_{L}(M)$ the space of _Lefschetz harmonic_
$(p,0)$-forms, i.e. forms $\alpha\in A^{p,0}(M)$ such that
$\partial\alpha=0=\partial^{\Lambda}\alpha$, where
$\partial^{\Lambda}=[\partial,\Lambda]$.

We can therefore apply the results of the previous section to infer

######  Theorem 3.1.

Let $(M,I,J,K,\Omega)$ be a compact $4n$-dimensional hypercomplex manifold and
$\Omega\in A^{2,0}(M)$ a non-degenerate $(2,0)$-form on $(M,I)$ such that
$\partial\Omega=0$. Then $(\mathcal{H}^{\bullet,0}_{L}(M),L)$ and
$(\mathcal{H}^{\bullet,0}_{L}(M),\Lambda)$ satisfy the HLC. Furthermore the
following are equivalent:

  * •

$(A^{\bullet,0}(M),L)$ satisfies the $\partial\partial^{\Lambda}$-lemma, i.e.,

$\mathrm{Ker}\,\partial\cap\mathrm{Ker}\,\partial^{\Lambda}\cap(\mathrm{Im}\,\partial+\mathrm{Im}\,\partial^{\Lambda})=\mathrm{Im}\,\partial\partial^{\Lambda}\,;$

  * •

There is a Lefschetz harmonic representative in each Dolbeault cohomology
class of $H^{\bullet,0}_{\partial}(M)$;

  * •

$(H_{\partial}^{\bullet,0}(M),L)$ satisfies the HLC;

  * •

$(H_{\partial^{\Lambda}}^{\bullet,0}(M),\Lambda)$ satisfies the HLC.

Moreover, by the general results in the previous section we obtain

######  Proposition 3.2.

Let $(M,I,J,K,\Omega)$ be a compact $4n$-dimensional hypercomplex manifold and
$\Omega\in A^{2,0}(M)$ a non-degenerate $(2,0)$-form on $(M,I)$ such that
$\partial\Omega=0$. Then,

$\Delta_{\partial}=\Delta_{\partial^{\Lambda}}-[\Lambda,[\partial,\partial^{\Lambda*}]]\,.$

In particular, if $[\partial,\partial^{\Lambda*}]=0$

$\Delta_{\partial}=\Delta_{\partial^{\Lambda}},$

and for every $p$ we have

$\mathcal{H}^{p,0}_{\partial}(M)=\mathcal{H}^{p,0}_{\partial^{\Lambda}}(M)\,.$

Moreover, if $[\partial,\partial^{\Lambda*}]=0$

$\Delta^{\mathrm{BC}}_{\partial^{\Lambda}}=\Delta^{\mathrm{BC}}_{\partial^{\Lambda*}}+\partial^{\Lambda*}\partial^{\Lambda}-\partial^{\Lambda}\partial^{\Lambda*}=\Delta_{\partial^{\Lambda}}\Delta_{\partial^{\Lambda}}+\partial^{*}\partial+\partial^{\Lambda*}\partial^{\Lambda}\,.$

###  3.2. Complex Hodge theory on HKT manifolds

If we further assume that $\Omega$ is q-positive, in the sense that
$J\Omega=\bar{\Omega}$ and $\Omega(Z,J\bar{Z})>0$ for every $Z\in
T^{1,0}_{I}M$, $Z\neq 0$, then it must be the HKT form corresponding to a HKT
metric $g$ on $(M,I,J,K)$. If $(M,I,J,K,g,\Omega)$ is HKT by [33] we have

$[\partial^{*},L]=-\partial_{J}+\theta_{J}\wedge-$

where $\theta_{J}$ is a $1$-form defined as follows. Since $\Omega$ is non-
degenerate, there exists a $(0,1)$-form $\bar{\theta}$ such that

$\overline{\partial}\Omega^{n}=\bar{\theta}\wedge\Omega^{n}\,$

then, by definition, $\theta_{J}:=J\bar{\theta}\in A^{1,0}(M)$. Notice that
$(M,I,J,K,g,\Omega)$ is balanced if and only if $\theta_{J}=0$ and so for
balanced HKT manifolds we have

$[\partial^{*},L]=-\partial_{J}.$

Moreover recall that a balanced HKT manifold has holonomy in
$\mathrm{SL}(n,\mathbb{H})$. Denoting with $\Lambda$ the adjoint of $L$, one
can easily get by duality or by applying $J$ the following (see [33])

######  Proposition 3.3.

Let $(M,I,J,K,g,\Omega)$ be a compact balanced HKT manifold. Then, the
following identities hold

  * •

$[\partial^{*},L]=-\partial_{J}\,,$

  * •

$[\partial,\Lambda]=\partial_{J}^{*}\,,$

  * •

$[L,\partial_{J}^{*}]=-\partial\,,$

  * •

$[\Lambda,\partial_{J}]=\partial^{*}\,.$

Now, we shall show that the framework of the previous section can be used to
study quaternionic cohomologies. First of all, we set
$\mathcal{J}\alpha=J\bar{\alpha}$ for every $\alpha\in A^{1,0}(M)$, thus
$\mathcal{J}$ is a complex structure on $A^{1,0}(M)$ and naturally extends to
$A^{p,0}(M)$ by imposing compatibility with the wedge product. Since $\Omega$
is q-real we have $\mathcal{J}L=L\mathcal{J}$ and we can use (2) to define a
Hodge-type operator.

We warn the reader that in this framework the operator defined by (2) slightly
differs from the usual Hodge operator. To distinguish them, let us denote here
$*:A^{p,0}(M)\to A^{2n-p,0}(M)$ the operator defined in (2) and
$\hat{*}:A^{p,q}(M)\to A^{2n-p,2n-q}(M)$ the usual Hodge star operator, then
one can easily show that for every $\alpha,\beta\in A^{p,0}(M)$

$\alpha\wedge*\beta=g(\alpha,\beta)\frac{\Omega^{n}}{n!}\,,$

where $g$ here is the Hermitian product induced by the Riemannian metric on
$A^{p,0}(M)$, while, by definition,

$\alpha\wedge\hat{*}\beta=g(\alpha,\beta)\frac{\Omega^{n}\wedge\bar{\Omega}^{n}}{(n!)^{2}}\,.$

However, we can identify the formal adjoints of $\partial$ and $\partial_{J}$
with respect to $*$ and $\hat{*}$ in the following way.

Suppose $M$ is a $\mathrm{SL}(n,\mathbb{H})$-manifold and fix a q-positive
q-real holomorphic $(2n,0)$-form $\Theta$. Define the following
$L^{2}$-products:

$(\alpha,\beta)_{1}:=\int_{M}g(\alpha,\beta)\frac{\Omega^{n}\wedge\bar{\Omega}^{n}}{(n!)^{2}}=\int_{M}\alpha\wedge\hat{*}\beta\,,\qquad(\alpha,\beta)_{2}:=\int_{M}g(\alpha,\beta)\frac{\Omega^{n}}{n!}\wedge\bar{\Theta}=\int_{M}\alpha\wedge*\beta\wedge\bar{\Theta}\,.$

Then the adjoint of $\partial$ and $\partial_{J}$ with respect to
$(\cdot,\cdot)_{1}$ are $\partial^{\hat{*}}=-\hat{*}\partial\hat{*}$ and
$\partial_{J}^{\hat{*}}=-\hat{*}\partial_{J}\hat{*}$, while those with respect
to $(\cdot,\cdot)_{2}$ are $\partial^{*}=-*\partial*$ and
$\partial_{J}^{*}=-*\partial_{J}*$ (cf. [24]). Since $\Theta$ is q-positive,
there exists a real-valued function $f>0$ such that
$\Theta=f\frac{\Omega^{n}}{n!}$, moreover, the holomorphicity of $\Theta$
translates into the condition $\partial f+f\theta=0$. Now, observe that
$(\cdot,\cdot)_{2}=(f\cdot,\cdot)_{1}$ thus

$(\alpha,\partial^{*}\beta)_{2}=(\partial\alpha,\beta)_{2}=(f\partial\alpha,\beta)_{1}=(\partial(f\alpha)-\partial
f\wedge\alpha,\beta)_{1}=(\alpha,\partial^{\hat{*}}\beta)_{2}+(\theta\wedge\alpha,\beta)_{2}$

and similarly, working with $\partial_{J}^{*}$ and $\partial_{J}^{\hat{*}}$
one obtains

$(\alpha,\partial_{J}^{*}\beta)_{2}=(\alpha,\partial^{\hat{*}}_{J}\beta)_{2}-(\theta_{J}\wedge\alpha,\beta)_{2}\,.$

In particular if $M$ is balanced then $\theta=\theta_{J}=0$ and $f$ is
constant, so that the two $L^{2}$-products coincide up to a constant and
$\partial^{*}=\partial^{\hat{*}}$ and
$\partial^{*}_{J}=\partial^{\hat{*}}_{J}$. In particular the usual Laplacians
obtained by means of the Riemannian Hodge star operator coincide with those
Laplacians considered in Section 2 and the related results can be applied.

In particular, if $M$ is compact and if $\alpha\in A^{p,0}(M)$ one immediately
obtains

$\begin{cases}\alpha\in\mathcal{H}^{p,0}_{\partial}(M)\ &\iff\
\partial\alpha=0,\quad\partial^{*}\alpha=0;\\\
\alpha\in\mathcal{H}^{p,0}_{\partial_{J}}(M)\ &\iff\
\partial_{J}\alpha=0,\quad\partial_{J}^{*}\alpha=0;\\\
\alpha\in\mathcal{H}^{p,0}_{BC}(M)\ &\iff\
\partial\alpha=0,\quad\partial_{J}\alpha=0,\quad\partial_{J}^{*}\partial^{*}\alpha=0;\\\
\alpha\in\mathcal{H}^{p,0}_{A}(M)\ &\iff\
\partial^{*}\alpha=0,\quad\partial_{J}^{*}\alpha=0,\quad\partial\partial_{J}\alpha=0.\end{cases}$

Proposition 3.3 shows that $\partial^{\Lambda}=\partial_{J}^{*}$ and we
readily obtain from Proposition 2.6.

######  Proposition 3.4.

Let $(M,I,J,K,\Omega)$ be a compact balanced HKT manifold, then

$\Delta_{\partial_{J}}=\Delta_{\partial}\,.$

In particular, the spaces of harmonic forms coincide, namely for every $p$ we
have

$\mathcal{H}^{p,0}_{\partial_{J}}(M)=\mathcal{H}^{p,0}_{\partial}(M)\,.$

######  Remark 3.5.

If we do not assume the compact HKT manifold $(M,I,J,K,\Omega)$ to be balanced
we would have, in general

$[\partial^{*},L]=-\partial_{J}+\theta_{J}\wedge-\,.$

Setting $\tau(\alpha):=\theta_{J}\wedge\alpha$ and
$\psi(\alpha):=\theta\wedge\alpha$, then, we would get

$\partial^{\Lambda}=[\partial,\Lambda]=\partial_{J}^{*}-\tau^{*}\,.$

In particular $[\partial,\partial^{\Lambda*}]=-\partial\theta_{J}\wedge-$ and
in such a case the Laplacians $\Delta_{\partial}$,
$\Delta_{\partial^{\Lambda}}$ and $\Delta_{\partial_{J}}$ do not coincide, and
in fact by a direct computation one gets

$\Delta_{\partial_{J}}=\Delta_{\partial}+[\psi^{*},\partial]+[\partial_{J},\tau^{*}].$

Notice that $\psi^{*}=\iota_{\theta^{\sharp}}$ and
$\tau^{*}=\iota_{\theta_{J}^{\sharp}}$.

We also observe that the condition $[\partial,\partial^{\Lambda*}]=0$, i.e.
$\partial\theta_{J}=0$ is only satisfied when the manifold is balanced, indeed
$\partial\theta_{J}=0$ is equivalent to $\partial_{J}\theta=0$, from which we
obtain

$\displaystyle\partial_{J}\partial\bar{\Omega}^{n}=$
$\displaystyle\,\partial_{J}(\theta\wedge\bar{\Omega}^{n})=\partial_{J}\theta\wedge\bar{\Omega}^{n}-\theta\wedge\partial_{J}\bar{\Omega}^{n}=-\theta\wedge
J^{-1}\bar{\partial}\Omega^{n}$ $\displaystyle=$ $\displaystyle\,-\theta\wedge
J^{-1}(\bar{\theta}\wedge\Omega^{n})=\theta\wedge
J\bar{\theta}\wedge\bar{\Omega}^{n}\,.$

Therefore by integrating we infer

$0=\int_{M}\partial_{J}\bar{\Omega}^{n}\wedge\partial\Omega^{n-1}=\int_{M}\partial_{J}\partial\bar{\Omega}^{n}\wedge\Omega^{n-1}=\int_{M}\theta\wedge
J\bar{\theta}\wedge\bar{\Omega}^{n}\wedge\Omega^{n-1}=\frac{1}{n}\int_{M}\|\theta\|^{2}\,\Omega^{n}\wedge\bar{\Omega}^{n}$

and the claim follows.

######  Remark 3.6.

We can also reinterpret Proposition 3.4 as follows. We first notice that if
$(M,I,J,K,\Omega)$ is a compact balanced HKT manifold. Then,
$[\partial,\partial^{\Lambda*}]=-[\partial,\partial_{J}]=0$.
Hence, by Propositions 2.6 and 2.7 we obtain that

$\Delta_{\partial}=\Delta_{\partial^{\Lambda}}$

and for every $p$ we have

$\mathcal{H}^{p}_{\partial}(M)=\mathcal{H}^{p}_{\partial^{\Lambda}}(M)\,.$

Moreover,

$\Delta^{\mathrm{BC}}_{\partial^{\Lambda}}=\Delta_{\partial^{\Lambda}}\Delta_{\partial^{\Lambda}}+\partial^{*}\partial+\partial^{\Lambda*}\partial^{\Lambda}\,=\Delta^{\mathrm{BC}}_{\partial^{\Lambda*}}+\partial^{\Lambda*}\partial^{\Lambda}-\partial^{\Lambda}\partial^{\Lambda*}.$

As a consequence of Proposition 3.4 we obtain isomorphisms for the associated
cohomology groups (cf. [24, Proposition 2.3] where it is noticed that an
isomorphism, induced by $J$ and conjugation with respect to $I$, holds in
general for hypercomplex manifolds).

######  Corollary 3.7.

Let $(M,I,J,K,\Omega)$ be a compact HKT balanced manifold, then

$H^{p,0}_{\partial_{J}}(M)\simeq H^{p,0}_{\partial}(M).$

In particular, we have the equalities

$h^{p,0}_{\partial_{J}}(M)=h^{p,0}_{\partial}(M).$

Invoking Proposition 2.7 we obtain that, similarly to the Kähler case, the
Laplacians $\Delta_{\mathrm{BC}}$ and
$\Delta_{\partial_{J}}=\Delta_{\partial}$ are related.

######  Proposition 3.8.

Let $(M,I,J,K,\Omega)$ be a compact HKT balanced manifold, then

$\displaystyle\Delta_{\mathrm{BC}}$
$\displaystyle=\Delta_{\partial_{J}}\Delta_{\partial_{J}}+\partial^{*}\partial+\partial_{J}^{*}\partial_{J}$
$\displaystyle=\Delta_{\partial}\Delta_{\partial}+\partial^{*}\partial+\partial_{J}^{*}\partial_{J}\,.$

In particular, the spaces of harmonic forms coincide, namely for every $p$ we
have

$\mathcal{H}^{p,0}_{\mathrm{BC}}(M)=\mathcal{H}^{p,0}_{\partial_{J}}(M).$

Consequently we obtain isomorphisms for the associated cohomology groups.

######  Corollary 3.9.

Let $(M,I,J,K,\Omega)$ be a compact balanced HKT manifold, then for every $p$,

$H^{p,0}_{\mathrm{BC}}(M)\simeq H^{p,0}_{\partial_{J}}(M)\simeq
H^{p,0}_{\partial}(M).$

In particular, we have the equalities

$h^{p,0}_{\mathrm{BC}}(M)=h^{p,0}_{\partial_{J}}(M)=h^{p,0}_{\partial}(M).$

Notice that these results are the analogue of the ones proved in [27] for
compact Kähler manifolds.

As a consequence of the previous results we prove that, under the same
hypothesis, the Hard Lefschetz condition holds for the cohomologies
$H^{\bullet,0}_{\partial}(M)$, $H^{\bullet,0}_{\partial_{J}}(M)$,
$H^{\bullet,0}_{\mathrm{BC}}(M)$, thus generalizing [6, Proposition 4.7].

######  Theorem 3.10.

Let $(M,I,J,K,\Omega)$ be a compact $4n$-dimensional balanced HKT manifold,
then for every $i$,

$L^{n-i}:\mathcal{H}^{i,0}_{\partial}(M)\to\mathcal{H}^{2n-i,0}_{\partial}(M),$
$L^{n-i}:\mathcal{H}^{i,0}_{\partial_{J}}(M)\to\mathcal{H}^{2n-i,0}_{\partial_{J}}(M)\,,$
$L^{n-i}:\mathcal{H}^{i,0}_{\mathrm{BC}}(M)\to\mathcal{H}^{2n-i,0}_{\mathrm{BC}}(M)\,\quad$

are isomorphisms.
In particular,

$h^{i,0}_{\mathrm{BC}}=h^{i,0}_{\partial}=h^{i,0}_{\partial_{J}}=h^{2n-i,0}_{BC}=h^{2n-i,0}_{\partial}=h^{2n-i,0}_{\partial_{J}}.$

###### Proof.

In view of Propositions 3.4, 3.8 it is sufficient to prove that

$L^{n-i}:\mathcal{H}^{i,0}_{\partial}(M)\to\mathcal{H}^{2n-i,0}_{\partial}(M)$

are isomorphisms. Notice that by hypothesis
$\partial\Omega=\partial_{J}\Omega=0$, hence

$[\partial,L]=0\,,\qquad[\partial_{J},L]=0\,.$

Let
$\alpha\in\mathcal{H}^{i,0}_{\partial}(M)=\mathcal{H}^{i,0}_{\partial_{J}}(M)$.
Then,

$\partial\alpha=0\,,\quad\partial_{J}\alpha=0\,,\quad\partial^{*}\alpha=0\,,\quad\partial^{*}_{J}\alpha=0.$

As a consequence

$\partial(L^{n-i}\alpha)=L^{n-i}\partial\alpha=0,$

and, using $[\partial^{*},L]=-\partial_{J}$,

$\partial^{*}(L^{n-i}\alpha)=L^{n-i}\partial^{*}\alpha-(n-i)L^{n-i-1}\partial_{J}\alpha=0.$

Hence, $L^{n-i}\alpha\in\mathcal{H}^{2n-i,0}_{\partial}(M)$. The result
follows from $\Omega$ being non-degenerate. ∎

Notice that combining this result with Theorem 3.1 we have

######  Corollary 3.11.

Let $(M,I,J,K,\Omega)$ be a compact $4n$-dimensional balanced HKT manifold,
then it satisfies the $\partial\partial^{\Lambda}$-lemma and there exists a
Lefschetz harmonic representative in each Dolbeault cohomology class of
$H^{\bullet,0}_{\partial}(M)$.

######  Remark 3.12.

Notice that, in general, on a compact $4n$-dimensional HKT manifold
$(M,I,J,K,\Omega)$ we cannot expect the HLC for $H^{i,0}_{\partial}(M)$.
Indeed, there are examples of HKT manifolds with $K_{(M,I)}$ holomorphically
non-trivial and $H^{2n,0}_{\partial}\simeq
H^{0,2n}_{\overline{\partial}}\simeq
H^{2n,0}_{\overline{\partial}}=\left\\{0\right\\}$ (e.g. quaternionic Hopf
surfaces).
But HLC would imply that

$L^{n}:\mathcal{H}^{0,0}_{\partial}(M)\to\mathcal{H}^{2n,0}_{\partial}(M),$

is an isomorphism, which is absurd.

######  Proposition 3.13.

Let $(M,I,J,K,\Omega)$ be a compact $4n$-dimensional balanced HKT manifold,
then for every $p$ we have

$\mathcal{H}^{p,0}_{\mathrm{BC}}(M)=\mathcal{H}^{p,0}_{\mathrm{A}}(M).$

###### Proof.

We first show the inclusion
$\mathcal{H}^{p,0}_{\mathrm{BC}}(M)\subseteq\mathcal{H}^{p,0}_{\mathrm{A}}(M)$.
Let $\alpha\in\mathcal{H}^{p,0}_{\mathrm{B}C}(M)$. By Propositions 3.4, 3.8
$\alpha\in\mathcal{H}^{p,0}_{\partial}(M)=\mathcal{H}^{p,0}_{\partial_{J}}(M)$,
namely

$\partial\alpha=0\,,\quad\partial_{J}\alpha=0\,,\quad\partial^{*}\alpha=0\,,\quad\partial^{*}_{J}\alpha=0.$

Hence, $\alpha\in\mathcal{H}^{p,0}_{\mathrm{A}}(M)$. The opposite inclusion
$\mathcal{H}^{p,0}_{\mathrm{A}}(M)\subseteq\mathcal{H}^{p,0}_{\mathrm{BC}}(M)$
follows from Theorem 3.10 and [20, Remark 21], indeed for every $p$,

$h^{p,0}_{\mathrm{BC}}(M)=h^{2n-p,0}_{\mathrm{BC}}(M)=h^{p,0}_{\mathrm{A}}(M).\qed$

As a corollary we have

######  Corollary 3.14.

Let $(M,I,J,K,\Omega)$ be a compact balanced HKT manifold, then for every $p$,

$H^{p,0}_{\mathrm{BC}}(M)\simeq H^{p,0}_{\mathrm{A}}(M).$

##  4\. Formality of HKT manifolds

It is well known that formality in the sense of Sullivan is an obstruction to
Kählerianity, more precisely compact complex manifolds satisfying the
$\partial\overline{\partial}$-lemma are formal (see [12]). However, notice
that the HKT condition does not imply formality, indeed there are examples of
non tori nilmanifolds that are HKT but it is well know that non tori
nilmanifolds are not formal in the sense of Sullivan [22].

In this section we study formality for compact hypercomplex manifolds. We
first recall some definitions.
Let $(\mathcal{A},d_{\mathcal{A}})$ and $(\mathcal{B},d_{\mathcal{B}})$ be two
differential graded algebras (DGA for short) over a field $\mathbb{K}$. A
_DGA-homomorphism_ between $\mathcal{A}$ and $\mathcal{B}$ is a
$\mathbb{K}$-linear map $f:\mathcal{A}\longrightarrow\mathcal{B}$ such that

  * i)

$f(\mathcal{A}^{i})\subset\mathcal{B}^{i}$;

  * ii)

$f(\alpha\cdot\beta)=f(\alpha)\cdot f(\beta)$;

  * iii)

$d_{\mathcal{B}}\circ f=f\circ d_{\mathcal{A}}$.

Any DGA-homomorphism
$f:(\mathcal{A},d_{\mathcal{A}})\longrightarrow(\mathcal{B},d_{\mathcal{B}})$
induces a DGA-homomorphism in cohomology

$H(f):(H^{\bullet}(\mathcal{A},d_{\mathcal{A}}),0)\longrightarrow(H^{\bullet}(\mathcal{B},d_{\mathcal{B}}),0)\,.$

A DGA-homomorphism
$f:(\mathcal{A},d_{\mathcal{A}})\longrightarrow(\mathcal{B},d_{\mathcal{B}})$
is called _quasi-isomorphism_ if $H(f)$ is an isomorphism.
Two DGA $(\mathcal{A},d_{\mathcal{A}})$ and $(\mathcal{B},d_{\mathcal{B}})$
are said to be _equivalent_ if there exists a sequence of quasi-isomorphisms
of the following form:

$\begin{array}[]{lclclclclclcl}&&(\mathcal{C}_{1},d_{\mathcal{C}_{1}})&&&&\cdots&&&&(\mathcal{C}_{n},d_{\mathcal{C}_{n}})&&\\\
&\swarrow&&\searrow&&\swarrow&&\searrow&&\swarrow&&\searrow&\\\
(\mathcal{A},d_{\mathcal{A}})&&&&(\mathcal{C}_{2},d_{\mathcal{C}_{2}})&&&&\cdots&&&&(\mathcal{B},d_{\mathcal{B}}).\end{array}$

A DGA $(\mathcal{A},d_{\mathcal{A}})$ is called _formal_ if
$(\mathcal{A},d_{\mathcal{A}})$ is equivalent to a DGA
$(\mathcal{B},d_{\mathcal{B}}=0)$.

We show now that for a compact hypercomplex manifold $M$ instead of
$(A^{\bullet}(M),d)$, the appropriate DGA to consider in this context is
$(A^{\bullet,0}(M),\partial)$ by proving the following

######  Theorem 4.1.

Let $(M,I,J,K)$ be a compact hypercomplex manifold satisfying the
$\partial\partial_{J}$-lemma, then the DGA $(A^{\bullet,0}(M),\partial)$ is
formal.

In order to prove this Theorem, we will need three lemmas.

######  Lemma 4.2.

Let $(M,I,J,K)$ be a compact hypercomplex manifold satisfying the
$\partial\partial_{J}$-lemma, then the natural inclusion

$i:\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)\to\left(A^{\bullet,0}(M),\,\partial\right)$

is a DGA quasi-isomorphism.

###### Proof.

Notice that
$\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)$ is a
DGA and the inclusion

$i:\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)\to\left(A^{\bullet,0}(M),\,\partial\right)$

is a morphism of DGAs. We are left to prove that the map induced in cohomology

$H_{\partial}(i):H_{\partial}\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)\to
H^{\bullet,0}_{\partial}(M)$

is an isomorphism.
We first prove that $H_{\partial}(i)$ is injective. Fix $k$, and let
$[\alpha]\in
H_{\partial}\left(A^{k,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)$
such that $H_{\partial}(i)([\alpha])=[\alpha]_{\partial}=0$, hence

$\alpha\in\mathrm{Ker}\,\partial_{J}\cap\mathrm{Im}\,\partial=\mathrm{Im}\,\partial\partial_{J}$

i.e., $\alpha=\partial\left(\partial_{J}\beta\right)$ for some form $\beta\in
A^{k-2,0}(M)$ and clearly $\partial_{J}\beta\in
A^{k-1,0}(M)\cap\mathrm{Ker}\,\partial_{J}$, hence

$[\alpha]=0\in
H_{\partial}^{k,0}\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)$

and so $H_{\partial}(i)$ is injective.
We now prove that $H_{\partial}(i)$ is is surjective. Let $a\in
H^{k,0}_{\partial}(M)$, $a=[\alpha]$ with $\partial\alpha=0$. Consider,

$\partial_{J}\alpha\in\mathrm{Im}\,\partial_{J}\cap\mathrm{Ker}\,\partial=\mathrm{Im}\,\partial_{J}\partial$

hence $\partial_{J}\alpha=\partial_{J}\partial\beta$ for some $\beta$.
Therefore, $\partial_{J}(\alpha-\partial\beta)=0$ and
$\partial(\alpha-\partial\beta)=\partial\alpha=0$. This means that
$\alpha-\partial\beta$ defines a class in
$H_{\partial}^{k,0}\left(A^{\bullet,0}M\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)$
and

$H_{\partial}(i)([\alpha-\partial\beta])=[\alpha-\partial\beta]_{\partial}=[\alpha]=a\,,$

concluding the proof. ∎

######  Lemma 4.3.

Let $(M,I,J,K)$ be a compact hypercomplex manifold satisfying the
$\partial\partial_{J}$-lemma, then the natural projection

$p:\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)\to\left(H_{\partial_{J}}^{\bullet,0}(M),\,\partial\right)$

is a DGA quasi-isomorphism.

###### Proof.

Notice that the projection

$p:\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)\to\left(H_{\partial_{J}}^{\bullet,0}(M),\,\partial\right)$

is a morphism of DGAs. We are left to prove that the map induced in cohomology

$H_{\partial}(p):H_{\partial}^{\bullet,0}\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)\to
H^{\bullet,0}_{\partial}\left(H_{\partial_{J}}^{\bullet,0}(M),\,\partial\right)$

is an isomorphism.
We first prove that $H_{\partial}(p)$ is injective. Fix $k$, and let
$[\alpha]\in
H_{\partial}^{k,0}\left(A^{\bullet,0}(M)\cap\mathrm{Ker}\,\partial_{J},\,\partial\right)$
such that $H_{\partial}(p)([\alpha])=0$. Hence,

$\alpha\in\mathrm{Im}\,\partial\cap\mathrm{Ker}\,\partial_{J}=\mathrm{Im}\,\partial\partial_{J}$

i.e., $\alpha=\partial\left(\partial_{J}\beta\right)$ for some form $\beta\in
A^{k-2,0}(M)$ and clearly $\partial_{J}\beta\in
A^{k-1,0}(M)\cap\mathrm{Ker}\,\partial_{J}$, hence

$[\alpha]=0\in
H^{\bullet,0}_{\partial}\left(H_{\partial_{J}}^{\bullet,0}(M),\,\partial\right)$

and so $H_{\partial}(p)$ is injective.
The surjectivity of $H_{\partial}(p)$ is immediate. ∎

######  Lemma 4.4.

Let $(M,I,J,K)$ be a compact hypercomplex manifold satisfying the
$\partial\partial_{J}$-lemma, then $\partial$ is the trivial operator on
$H_{\partial_{J}}^{\bullet,0}(M)$.

###### Proof.

Fix $k$ and let $a=[\alpha]_{\partial_{J}}\in H^{k,0}_{\partial_{J}}(M)$,
namely $\partial_{J}\alpha=0$. Now

$\partial a=[\partial\alpha]_{\partial_{J}}$

and

$\partial\alpha\in\mathrm{Im}\,\partial\cap\mathrm{Ker}\,\partial_{J}=\mathrm{Im}\,\partial_{J}\partial$

so $\partial\alpha=\partial_{J}\partial\beta$ for some $\beta$, giving
$\partial a=[\partial_{J}\partial\beta]_{\partial_{J}}=0\in
H^{k+1,0}_{\partial_{J}}(M)$, concluding the proof. ∎

Now we are able to prove Theorem 4.1.

###### Proof.

Under the assumptions and as a consequence of Lemmas 4.2, 4.3, 4.4 we have the
following diagram of quasi-isomorphisms of DGAs,

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3.0pt}}}}}}\ignorespaces{\hbox{\kern
28.87845pt\raise-29.76889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
151.18666pt\raise-15.51582pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
143.92113pt\raise-26.72137pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.65695pt\hbox{$\scriptstyle{\text{qis}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
176.49602pt\raise-29.76889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
202.31764pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-30.04475pt\raise-41.40665pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left(A^{\bullet,0}(M),\,\partial\right)}$}}}}}}}{\hbox{\kern
99.74649pt\raise-41.40665pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
175.44823pt\raise-41.40665pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left(H^{\bullet,0}_{\partial_{J}}(M),\,0\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

hence, by definition, $\left(A^{\bullet,0}(M),\,\partial\right)$ is a formal
DGA. ∎

As a consequence of [20, Theorem 6] and Theorem 4.1 we obtain

######  Corollary 4.5.

Let $(M,I,J,K,\Omega)$ be a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold,
then the DGA $(A^{\bullet,0}(M),\partial)$ is formal.

We recall now the definition of triple Massey products of a DGA in our
setting.

######  Definition 4.6.

Let $\mathfrak{a}=\left[\alpha\right]\in H^{p,0}_{\partial}(M)$,
$\mathfrak{b}=\left[\beta\right]\in H^{q,0}_{\partial}(M)$ and
$\mathfrak{c}=\left[\gamma\right]\in H^{r,0}_{\partial}(M)$ such that
$\mathfrak{a}\cup\mathfrak{b}=0\in H^{p+q,0}_{\partial}(M)$ and
$\mathfrak{b}\cup\mathfrak{c}=0\in H^{q+r,0}_{\partial}(M)$; more precisely
suppose that $\alpha\wedge\beta=\partial\lambda$ and
$\beta\wedge\gamma=\partial\mu$ for some $\lambda\in A^{p+q-1,0}$, $\mu\in
A^{q+r-1,0}$. The _triple $\partial$-Massey product_ of
$\mathfrak{a},\mathfrak{b},\mathfrak{c}$ is defined as

$\left\langle\mathfrak{a},\mathfrak{b},\mathfrak{c}\right\rangle:=\left[\lambda\wedge\gamma-(-1)^{p}\alpha\wedge\mu\right]\in\frac{H^{p+q+r-1,0}_{\partial}(M)}{H^{p+q-1,0}_{\partial}(M)\cup
H^{r,0}_{\partial}(M)+H^{p,0}_{\partial}(M)\cup H^{q+r-1,0}_{\partial}(M)}.$

Then, since for a formal DGA the associated Massey products vanish we have the
following

######  Corollary 4.7.

Let $(M,I,J,K)$ be a compact hypercomplex manifold satisfying the
$\partial\partial_{J}$-lemma, then the triple $\partial$-Massey products
vanish.

Hence, we have

######  Theorem 4.8.

Let $(M,I,J,K,\Omega)$ be a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold,
then the triple $\partial$-Massey products vanish.

In particular, triple $\partial$-Massey products are an obstruction to the
existence of a HKT $\mathrm{SL}(n,\mathbb{H})$-structure on a compact
hypercomplex manifold. More precisely,

######  Corollary 4.9.

Let $(M,I)$ be a $4n$-dimensional compact complex manifold such that there
exists a non trivial $\partial$-Massey product, then $(M,I)$ does not admit
any complex structures $J,K$ such that $(M,I,J,K)$ is hypercomplex and admits
a HKT $\mathrm{SL}(n,\mathbb{H})$-structure.

Notice that, in fact, by [6] if a nilmanifold $N$ admits an invariant HKT
structure $(I,J,K,\Omega)$ then the complex structures $I,J,K$ are abelian and
in such a case the triple $\partial$-Massey products are trivial. Indeed, we
prove in general the following

######  Theorem 4.10.

Let $N=\Gamma\backslash G$ be a $2n$-dimensional nilmanifold and let $I$ be an
invariant abelian complex structure on $N$. Then, the triple $\partial$-Massey
products are all zero.

###### Proof.

Since $I$ is an invariant abelian complex structure on $N$, there exists a co-
frame of invariant $(1,0)$-forms $\left\\{\varphi^{i}\right\\}_{i=1,\ldots,n}$
on $(N,I)$ such that

$\partial\varphi^{i}=0,\quad\text{for }i=1,\ldots,n\,.$

Since $I$ is abelian, by [11] the Dolbeault cohomology of $N$ can be computed
using only invariant forms, hence

$H^{r,0}_{\partial}(N)\simeq
H^{r,0}_{\partial}(\mathfrak{g}^{\mathbb{C}})=\left\langle\varphi^{i_{1}}\wedge\ldots\wedge\varphi^{i_{r}}\right\rangle_{1\leq
i_{1}<\ldots<i_{r}\leq n}$

for $r=1,\ldots,n$, where, denoting with $\mathfrak{g}=\text{Lie}(G)$,
$H^{\bullet,\bullet}_{\partial}(\mathfrak{g}^{\mathbb{C}})$ denotes the
cohomology of the differential bigraded algebra
$\Lambda^{\bullet,\bullet}(\mathfrak{g}^{\mathbb{C}})^{*}$ with respect to the
operator $\partial$.
In order to construct a triple $\partial$-Massey product let
$\mathfrak{a}=\left[\alpha\right]\in H^{p,0}_{\partial}(N)$,
$\mathfrak{b}=\left[\beta\right]\in H^{q,0}_{\partial}(N)$ such that
$\mathfrak{a}\cup\mathfrak{b}=0\in H^{p+q,0}_{\partial}(N)$, hence
$\mathfrak{a}\cup\mathfrak{b}=0\in
H^{p+q,0}_{\partial}(\mathfrak{g}^{\mathbb{C}})$, namely there exists an
invariant $(p+q-1,0)$-form $\lambda$ such that

$\alpha\wedge\beta=\partial\lambda.$

But, on invariant $(r,0)$-forms the operator $\partial$ vanishes and so we can
take the primitive $\lambda=0$ itself. A similar conclusion is obtained taking
the third class in the definition of $\partial$-Massey products. This means
that we cannot construct non trivial $\partial$-Massey products since both
$\lambda$ and $\mu$ in the definition of $\partial$-Massey products would be
zero. ∎

An immediate consequence of this result combined with [6, Theorem 4.6] is the
following

######  Theorem 4.11.

Let $N=\Gamma\backslash G$ be a $4n$-dimensional nilmanifold and let
$(I,J,K,\Omega)$ be an invariant HKT structure on $N$. Then, the triple
$\partial$-Massey products are all zero.

Therefore, a relevant application of Corollary 4.9 can be given on
solvmanifolds.

######  Example 4.12.

Consider the $8$-dimensional almost abelian Lie algebra $\mathfrak{g}$ with
structure equations

$[e_{8},e_{2}]=e_{4},\quad[e_{8},e_{3}]=e_{5}.$

Let $G$ be the associated solvable simply connected Lie group. Then, by [10]
$G$ admits a lattice $\Gamma$ such that $S:=\Gamma\backslash G$ is a
solvmanifold. Define the complex structure setting as global co-frame of
$(1,0)$-forms

$\varphi^{1}=e^{1}+ie^{8},\quad\varphi^{2}=e^{2}+ie^{3},\quad\varphi^{3}=e^{4}+ie^{5},\quad\varphi^{4}=e^{6}+ie^{7}.$

The complex structure equations become

$d\varphi^{1}=d\varphi^{2}=d\varphi^{4}=0,\quad
d\varphi^{3}=\frac{i}{2}\varphi^{12}+\frac{i}{2}\varphi^{2\bar{1}}.$

Note that the form $\varphi^{1234}$ is closed, so $S$ has holomorphically
trivial canonical bundle. We now construct a non trivial triple
$\partial$-Massey product. Take $[\varphi^{1}]\in H^{1,0}_{\partial}(S)$,
$[\varphi^{2}]\in H^{1,0}_{\partial}(S)$ and $[\varphi^{2}]\in
H^{1,0}_{\partial}(S)$. Notice that
$\varphi^{1}\wedge\varphi^{2}=\partial(-2i\,\varphi^{3})$ and
$\varphi^{2}\wedge\varphi^{2}=0$. Hence, the $\partial$-Massey product is
given by

$[-2i\,\varphi^{3}\wedge\varphi^{2}]\in\frac{H^{2,0}_{\partial}(S)}{[\varphi^{1}]\cup
H^{1,0}_{\partial}(S)+H^{1,0}_{\partial}(S)\cup[\varphi^{2}]}.$

and this class is clearly non trivial. Therefore, by Corollary 4.9 the complex
manifold $(S,I)$ does not admit any complex structures $J,K$ such that the
solvmanifold $(S,I,J,K)$ is hypercomplex and admits a
$\mathrm{SL}(2,\mathbb{H})$ HKT structure.

The next two examples show that the converse of Corollary 4.7 (and hence
Theorem 4.8) does not hold in general. The first one is a compact HKT manifold
which is not $\mathrm{SL}(n,\mathbb{H})$, while the second one is
$\mathrm{SL}(n,\mathbb{H})$ but does not admit any HKT metric.

######  Example 4.13.

Consider $\mathrm{SU}(3)$ equipped with a homogeneous hypercomplex structure
$(I,J,K)$ as constructed in [23, 29]. There is an HKT metric on
$\mathrm{SU}(3)$ compatible with this hypercomplex structure [21, 26]. By [28]
the holonomy of the Obata connection on $\mathrm{SU}(3)$ is
$\mathrm{GL}(2,\mathbb{H})$ and, in fact, we claim that the
$\partial\partial_{J}$-lemma cannot hold on $\mathrm{SU}(3)$.

To see this, we observe that from [18], there exists a unitary co-frame
$\\{Z^{1},\dots,Z^{4}\\}$ of $(1,0)$-forms (with respect to $I$) on the Lie
algebra of $\mathrm{SU}(3)$ such that the HKT form is

$\Omega=Z^{12}+Z^{34}=\frac{1}{2}\partial Z^{2}.$

Now, if the $\partial\partial_{J}$-lemma hold we would have that
$\Omega=\partial\partial_{J}f$ for some function $f$, but since the HKT form
is q-positive, by E. Hopf’s maximum principle $f$ would be constant and thus
$\Omega=0$ which is a contradiction.

On the other hand the triple $\partial$-Massey products are all zero because
the same coframe satisfies

$\partial Z^{1}=0,\qquad\partial Z^{2}=2Z^{12}+2Z^{34},\qquad\partial
Z^{3}=(1+3i)Z^{13},\qquad\partial Z^{4}=(1-3i)Z^{14},$

which shows that $H^{1,0}_{\partial}(M)\simeq\langle Z^{1}\rangle$ and
$H^{i,0}_{\partial}(M)=0$ for $i>1$.

######  Example 4.14.

Consider the nilmanifold $M=\Gamma\backslash G$ whose structure equations of
the Lie algebra $\mathfrak{g}$ of $G$ are given by (see [24, Example 1])

$de^{1}=de^{2}=de^{3}=de^{4}=de^{5}=0,\quad de^{6}=e^{12}+e^{34},\quad
de^{7}=e^{13}-e^{24},\quad de^{8}=e^{14}+e^{23},$

where we use the standard notation $e^{ij}=e^{i}\wedge e^{j}$. Define the
following hypercomplex structure

$Ie^{1}=e^{2},\quad Ie^{3}=e^{4},\quad Ie^{5}=e^{6},\quad Ie^{7}=e^{8},$
$Je^{1}=e^{3},\quad Je^{2}=-e^{4},\quad Je^{5}=e^{7},\quad Je^{6}=-e^{8}.$

Then a co-frame for invariant $(1,0)$-forms with respect to $I$ on $M$ is
given by

$\varphi^{1}=e^{1}-ie^{2},\quad\varphi^{2}=e^{3}-ie^{4},\quad\varphi^{3}=e^{5}-ie^{6},\quad\varphi^{4}=e^{7}-ie^{8}$

and the complex structure equations become

$d\varphi^{1}=d\varphi^{2}=0,\quad
d\varphi^{3}=-\frac{1}{2}(\varphi^{1\bar{1}}+\varphi^{2\bar{2}}),\quad
d\varphi^{4}=\varphi^{12}.$

Since the hypercomplex structure is not abelian it does not admit any
compatible HKT metric (see also [24]). The conjugate Dolbeault cohomology in
bidegree $(p,0)$ is given by

$H^{1,0}_{\partial}(M)\simeq\left\langle\varphi^{1},\varphi^{2},\varphi^{3}\right\rangle\,,\qquad
H^{2,0}_{\partial}(M)\simeq\left\langle\varphi^{13},\varphi^{23},\varphi^{14},\varphi^{24}\right\rangle\,,$

We now construct a non trivial triple $\partial$-Massey product. Take
$[\varphi^{1}]\in H^{1,0}_{\partial}(M)$, $[\varphi^{2}]\in
H^{1,0}_{\partial}(M)$ and $[\varphi^{2}]\in H^{1,0}_{\partial}(M)$. Notice
that $\varphi^{1}\wedge\varphi^{2}=\partial\varphi^{4}$ and
$\varphi^{2}\wedge\varphi^{2}=0$. Hence, the $\partial$-Massey product is
given by

$[\varphi^{4}\wedge\varphi^{2}]\in\frac{H^{2,0}_{\partial}(M)}{[\varphi^{1}]\cup
H^{1,0}_{\partial}(M)+H^{1,0}_{\partial}(M)\cup[\varphi^{2}]}.$

and this class is clearly non trivial.

##  5\. Balanced HKT solvmanifolds

The following result can be seen as a generalization of [6, Proposition 4.11]
where it is proven that for a hyperhermitian nilmanifold with abelian
hypercomplex structure the metric is balanced. Notice that in the nilpotent
case the HKT assumption is automatic due to [14].

######  Theorem 5.1.

Let $(\Gamma\backslash G,I,J,K)$ be a $4n$-dimensional solvmanifold with an
invariant abelian hypercomplex structure. Then, every invariant hyperhermitian
metric $g$ is balanced.

###### Proof.

Let $g$ be an invariant hyperhermitian metric on $\Gamma\backslash G$, then by
[14] $g$ is HKT. We will denote with $(I,J,K,\Omega,g)$ the induced structure
on $G$. Since $\Omega$ is HKT the Bismut connections associated to $I,\,J,\,K$
coincide and we will denote them uniquely with $\nabla^{B}$. Since
$\Gamma\backslash G$ is a solvmanifold then $G$ is unimodular. Hence, by [6,
Lemma 2.4] the Lee form $\tau_{J}$ associated to $(J,g)$ is given by

$\tau_{J}(X)=\text{tr}\left(\frac{1}{2}J\nabla^{B}_{JX}\right)$

for any $X\in\mathfrak{g}$. Now we argue as in the proof of [6, Proposition
4.11] to show that $\tau_{J}=0$ and so $g$ is balanced with respect to $J$.
The argument is similar for $I$ and $K$.
Let
$X_{1},\,IX_{1},\,JX_{1},\,KX_{1},\,\cdots,\,X_{n},\,IX_{n},\,JX_{n},\,KX_{n}$
be an orthonormal basis of $\mathfrak{g}$. Now using that $\nabla^{B}$
preserves $I,\,J,\,K$ and that $g$ is quaternionic Hermitian we have

$\displaystyle\text{tr}\left(J\nabla^{B}_{JX}\right)$
$\displaystyle=\sum_{j=1}^{n}g(J\nabla^{B}_{JX}X_{j},X_{j})+\sum_{j=1}^{n}g(J\nabla^{B}_{JX}IX_{j},IX_{j})$
$\displaystyle\quad+\sum_{j=1}^{n}g(J\nabla^{B}_{JX}JX_{j},JX_{j})+\sum_{j=1}^{n}g(J\nabla^{B}_{JX}KX_{j},KX_{j})$
$\displaystyle=\sum_{j=1}^{n}g(J\nabla^{B}_{JX}X_{j},X_{j})+\sum_{j=1}^{n}g(JI\nabla^{B}_{JX}X_{j},IX_{j})$
$\displaystyle\quad+\sum_{j=1}^{n}g(\nabla^{B}_{JX}JX_{j},X_{j})+\sum_{j=1}^{n}g(JK\nabla^{B}_{JX}X_{j},KX_{j})$
$\displaystyle=\sum_{j=1}^{n}g(J\nabla^{B}_{JX}X_{j},X_{j})-\sum_{j=1}^{n}g(IJ\nabla^{B}_{JX}X_{j},IX_{j})$
$\displaystyle\quad+\sum_{j=1}^{n}g(\nabla^{B}_{JX}JX_{j},X_{j})-\sum_{j=1}^{n}g(KJ\nabla^{B}_{JX}X_{j},KX_{j})$
$\displaystyle=\sum_{j=1}^{n}g(\nabla^{B}_{JX}JX_{j},X_{j})-\sum_{j=1}^{n}g(\nabla^{B}_{JX}JX_{j},X_{j})$
$\displaystyle\quad+\sum_{j=1}^{n}g(\nabla^{B}_{JX}JX_{j},X_{j})-\sum_{j=1}^{n}g(\nabla^{B}_{JX}JX_{j},X_{j})=0.$

∎

######  Corollary 5.2.

Let $(\Gamma\backslash G,I,J,K)$ be a $4n$-dimensional solvmanifold with an
invariant abelian hypercomplex structure. Suppose that there exists an HKT
structure $\Omega$ on $(\Gamma\backslash G,I,J,K)$. Then, there exists a
balanced abelian HKT structure on $\Gamma\backslash G$.

###### Proof.

By [15] there exists a invariant HKT structure $\tilde{\Omega}$ on
$(\Gamma\backslash G,I,J,K)$. Now, by Theorem 5.1 we have that
$\tilde{\Omega}$ is balanced. ∎

######  Remark 5.3.

Notice that, differently from the nilpotent case (cf. [6]), the converse of
Theorem 5.1 is not true. Indeed, in [7] it is provided an example of a
balanced HKT solvmanifold with an hypercomplex structure that is not abelian.

In [34] Verbitsky showed that an $\mathrm{SL}(n,\mathbb{H})$-manifold has
holomorphically trivial canonical bundle. Andrada and Tolcachier [4] found a
counterexample to the converse exhibiting an hypercomplex solvmanifold that is
not $\mathrm{SL}(n,\mathbb{H})$ but admits a non-invariant holomorphic section
of the canonical bundle. We indeed now show that the
$\mathrm{SL}(n,\mathbb{H})$ condition is equivalent to the existence of an
invariant holomorphic section of the canonical bundle.

######  Theorem 5.4.

Let $(M:=\Gamma\backslash G,I,J,K,g)$ be a hypercomplex solvmanifold, then the
holonomy of the Obata connection $\nabla$ is contained in
$\mathrm{SL}(n,\mathbb{H})$ if and only if the canonical bundle admits an
invariant holomorphic section.

###### Proof.

Assume $M$ is a $\mathrm{SL}(n,\mathbb{H})$-manifold, then there exists a
q-positive holomorphic section $\eta$ of the canonical bundle. We claim that
$\eta$ must be invariant. To prove this we argue along the lines of [16,
Proposition 2.1]. Let $\Theta$ be an invariant q-positive section of the
canonical bundle (a priori not holomorphic). Since both $\eta$ and $\Theta$
are q-real and q-positive there exists a positive real-valued function $f$
such that $\eta=f\Theta$. Therefore
$0=\bar{\partial}\eta=\bar{\partial}f\wedge\Theta+f\bar{\partial}\Theta$, i.e.
$\bar{\partial}\Theta=-\bar{\partial}(\log f)\wedge\Theta$ because $f$ is
positive. Since $\Theta$ is invariant so is $\bar{\partial}\Theta$ and thus
there exists an invariant $(0,1)$-form $\alpha$ such that $\bar{\partial}(\log
f)=\alpha$. We now apply the well-known Belgun’s symmetrization process [9]
that for any $k$-form $\beta$ on $M$ returns an invariant $k$-form
$\mu(\beta)$. Since the hypercomplex structure is invariant $\mu$ preserves
$\bar{\partial}$ and we obtain

$\bar{\partial}(\log f)=\alpha=\mu(\alpha)=\mu(\bar{\partial}(\log
f))=\bar{\partial}\mu(\log f)=0$

because the symmetrization of a function is constant. In particular $f$ is
constant, showing that $\eta=f\Theta$ is invariant.

Conversely, let $\eta$ be an invariant nowhere vanishing holomorphic section
of the canonical bundle. The fact that
$\mathrm{Hol}(\nabla)\subseteq\mathrm{SL}(n,\mathbb{H})$, follows from the
fact that $\eta$ is parallel with respect to $\nabla$, which can be proved
along the lines of [6, Theorem 3.2]. ∎

Now we prove the following

######  Theorem 5.5.

Let $(M:=\Gamma\backslash G,I,J,K,g)$ be a solvmanifold with an invariant
holomorphic section of the canonical bundle and invariant HKT structure. Then
$g$ is balanced.

###### Proof.

Let $\bar{\eta}$ be an invariant non-vanishing $\partial$-closed section of
$A^{0,2n}(M)$, hence

$\bar{\Omega}^{n}=c\,\bar{\eta}$

with $c$ constant. Since $d\bar{\eta}=0$, then $d\bar{\Omega}^{n}=0$ and so
$\partial\bar{\Omega}^{n}=0$ proving that $g$ is balanced. ∎

As a consequence we confirm the conjecture by Alesker and Verbitsky on
solvmanifolds with invariant hypercomplex structure.

######  Theorem 5.6.

Let $(M:=\Gamma\backslash G,I,J,K)$ be a
$\mathrm{SL}(n,\mathbb{H})$-solvmanifold with invariant hypercomplex
structure. Suppose that there exists an HKT metric on $M$. Then there exists a
balanced HKT structure on $M$.

###### Proof.

Since $(M:=\Gamma\backslash G,I,J,K,g)$ is an HKT solvmanifold with a
$\mathrm{SL}(n,\mathbb{H})$ structure, then the canonical bundle of $M$ has an
invariant holomorphic section and, by [15], there exists an invariant HKT
structure on $M$. Hence, by the previous result the associated Hermitian
metric is balanced. ∎

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