Abstract
We provide families of compact astheno-Kähler nilmanifolds and we study the behaviour of the complex blowup of such manifolds. We prove that the existence of an astheno-Kähler metric satisfying an extra differential condition is not preserved by blowup. We also study the interplay between Strong Kähler with torsion metrics and geometrically Bott–Chern metrics. We show that Fino–Parton–Salamon nilmanifolds are geometrically-Bott–Chern-formal, whereas we obtain negative results on the product of two copies of primary Kodaira surface, Inoue surface of type \(\mathcal {S}_M\) and on the product of a Kodaira surface with an Inoue surface.
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1 Introduction
Compact Kähler manifolds share many remarkable cohomological and metric properties, e.g., they satisfy the \({\partial }{\overline{\partial }}\)-Lemma, all the complex cohomologies, Dolbeault, Aeppli and Bott-Chern, are isomorphic, their underlying structure of smooth manifold is formal in the sense of Sullivan, they satisfy the Hard Lefschetz Condition. As a consequence, the existence of a Kähler structure on a compact smooth manifold M forces many topological constraints, e.g., the Betti numbers of even degree are positive, the Betti numbers of odd index are even and Massey products vanish. On the other hand, a large and natural class of compact complex manifolds arises as a quotient of simply connected nilpotent Lie groups by a discrete uniform subgroup; nevertheless, in view of Benson and Gordon [11], the underlying structure of the smooth manifold of such compact quotients admits a Kähler structure if and only if the group is Abelian, that is the manifold is diffeomorphic to a torus, and, more in general, by Hasegawa (see [27, 28]) a compact quotient of a simply-connected solvable Lie group by a closed subgroup carries a Kähler structure if and only if it is a finite quotient of a complex torus. Yet, this class of compact manifolds admits special Hermitian metrics, e.g., Strong Kähler with torsion metrics or pluriclosed (see [12, 23]), astheno-Kähler metrics in the sense of Jost and Yau (see [31]). More precisely, a Hermitian metric g with fundamental form F on an n-dimensional complex manifold (M, J) is said to be
-
strong Kähler with torsion, or shortly SKT, if \({\partial }{\overline{\partial }} F=0\) or, equivalently \(dd^cF=0\);
-
astheno-Kähler, if \({\partial }{\overline{\partial }} F^{n-2}=0\) or, equivalently \(dd^c F^{n-2}=0\),
where \(d^c=J^{-1}dJ\). The SKT metrics have been studied by many authors and they have also applications in type II string theory and in 2-dimensional supersymmetric \(\sigma \)-models [23, 30, 50]. Moreover, they have also relations with generalized Kähler geometry (see for instance [10, 15, 23, 26, 29]).
Some rigidity theorems concerning compact astheno-Kähler manifolds have been showed in [31, Theorem 6] and in [34], where, in particular, a generalization to higher dimension of the Bogomolov’s theorem on \(VII_0\) surfaces is proved (see [34, Corollary 3]). Astheno-Kähler structures on Calabi–Eckmann manifolds have been constructed in [36]. For other results on SKT and astheno-Kähler metrics and special metrics on complex manifolds see also [14, 17,18,19,20,21, 30, 37, 40,41,42, 44, 53] and the references therein.
In the first part of the present paper we are interested in providing astheno-Kähler metrics on compact nilmanifolds, endowed with left-invariant complex structures and in studying the behaviour of blowup of compact complex manifolds endowed with astheno-Kähler metrics, satisfying certain extra differential conditions. First of all, we construct a family of simply-connected 2-step nilpotent Lie groups G, admitting discrete uniform subgroups \(\Gamma \) and endowed with a left-invariant complex structure J, such that \((\Gamma \backslash G,J)\) carries an astheno-Kähler metric (see Theorem 4.1 for the precise statement). Such a construction will be applied in the study of the behaviour of blowups.
In [21] respectively [20, Proposition 2.4] it is proved that the existence of an SKT metric respectively a Hermitian metric g with fundamental form F on an n-dimensional compact complex manifold M, satisfying \({\partial }{\overline{\partial }} F=0\), \({\partial }{\overline{\partial }} F^2=0\), is stable under blowups of M. In contrast, we prove the following
Theorem
(See Theorem 5.3) On a compact complex manifold of dimension n, the existence of a Hermitian metric F such that
is not preserved by blowup.
Note that metrics whose fundamental form satisfies \({\partial }{\overline{\partial }} F^{k}=0\) belong, in particular, to a class of Hermitian metrics introduced Fu, Wang, and Wu in [22] as a generalization of the notion of Kähler metrics.
In the second part of the paper we focus on the interplay between SKT metrics and geometrically-Bott-Chern-formal metrics. From the cohomological point of view, besides the Dolbeault cohomology groups, a useful tool in the study of compact complex (non-Kähler) manifolds is provided by Bott-Chern and Aeppli cohomology groups. Indeed, the property of satisfying the \({\partial }{\overline{\partial }}\)-Lemma can be characterized in terms of their dimensions (see [8]). By adapting the construction of Massey triple products respectively Dolbeault Massey triple products as in [16, 51], respectively in Neisendorfer and Taylor (see [39]), in [9] the notion of triple Aeppli–Bott-Chern–Massey products are introduced with the aim to give obstructions to the existence of geometrically-Bott-Chern-formal metrics, that is Hermitian metrics whose space of Bott-Chern harmonic forms has the structure of algebra. The notion of formal Riemannian metrics has been previously introduced and studied intensively by Kotschick in [32]. Very recently Milivojević and Stelzig in [38] introduced the n-fold Massey products in the spectral sequence sense, for a commutative bigraded differential algebra. In particular, for \(n=3\), they recover the definition of triple Aeppli–Bott-Chern–Massey products as in [9]. Furthermore, they consider the notions of weak formality, respectively strong formality which have applications in the study of cohomological properties of complex non Kähler manifolds. For other results on these topics we refer to [47,48,49, 55].
Concerning the relation between SKT metrics and geometrically-Bott-Chern-formal metrics, we study the 6-dimensional nilmanifolds with a left-invariant complex structure admitting a left-invariant SKT metric, which have been characterized by Fino, Parton and Salamon in [19, Theorem 1.2]. In particular, for such manifolds they provide explicit structure equations depending on 5 complex parameters satisfying a real algebraic equation and they prove that the SKT condition is satisfied by every left-invariant Hermitian metric. Denoting by FPS-nilmanifold any such a manifold, we prove the following result.
Theorem
(See Theorem 7.2) Let (M, J) be a FPS-nilmanifold. Then, any left-invariant metric is geometrically-Bott-Chern-formal.
Moreover, we extend this result to a class of nilmanifolds which are a generalization of FPS-manifolds in a arbitrary higher dimension (see Theorem 7.4).
In contrast to the mentioned positive results, on a compact complex manifold the existence of a SKT metric does not imply the existence of geometrically-Bott-Chern-formal metrics. More precisely, we prove this for the product of a pair of certain compact complex surfaces by providing a non vanishing Aeppli–Bott-Chern–Massey product on each manifold.
Theorem
(See Theorem 7.5) Let (M, J) be the product of either two Kodaira surfaces, two Inoue surfaces, or a Kodaira surface and a Inoue surface. Then (M, J) admits SKT metrics but does not admit geometrically-Bott-Chern-formal metrics.
Furthermore, a similar result holds also for manifolds which are not a product of manifolds, as it is shown for a family of nilmanifolds of complex dimension 4 in Theorem 7.6.
The paper is organized as follows. In Sect. 2, we briefly recall the notions of Dolbeault, Bott-Chern, and Aeppli cohomologies on compact complex manifolds, and we recollect the basis facts about the complex geometry of nilmanifolds. In Sect. 3, following [2, 52], we recall the notions of p-pluriclosed forms on an almost complex manifold of real dimension 2n (in particular, when J is integrable, for \(p=1\), respectively \(p=n-2\), we recover the notion of SKT metrics, respectively of astheno-Kähler metrics). Moreover, we obtain a useful obstruction for the existence of such structures (see Lemma 3.5), which will be used in both Proposition 4.3 and Theorem 5.3. Sections 4 and 5 are devoted to the proofs of Theorems 4.1 and 5.3. In Sect. 6, we briefly recall the notions of Aeppli–Bott-Chern–Massey products and geometrically-Bott-Chern-formal metrics. Finally, Sect. 7 is mainly devoted to the proof of Theorems 7.2, 7.4, 7.5, 7.6.
2 Preliminaries
Let M be a compact 2n-dimensional differentiable manifold endowed with an integrable almost complex structure J, i.e., \(J\in \text {End}(TM)\) such that \(J^2=-{{\,\textrm{id}\,}}_{TM}\) and the Nijenhuis tensor associated to J
vanishes for every \(X,Y\in TM\). By Newlander–Nirenberg, J endows M with the structure of a compact complex manifold of complex dimension n. We will denote such manifold by (M, J).
The \(\mathbb {C}\)-linear extension of the endorphism J to the complexified tangent bundle \(T_{\mathbb {C}}M:=TM\otimes \mathbb {C}\) gives rise to following decomposition in terms of the \(\pm i\)-eigenspaces of J
where \(T^{1,0}M:=\{X\in T_{\mathbb {C}}M:\, JX=iX \}\) and \(T^{0,1}M:=\{X \in T_{\mathbb {C}}M\,:\, JX=-iX\}\). Such decomposition of \(T_{\mathbb {C}} M\) extends to the exterior powers bundles \(\bigwedge _\mathbb {C}^kM:=\bigwedge ^k T_\mathbb {C}M\) of complex k-forms, that is
where \(\bigwedge ^{p,q}M:=\bigwedge ^p T^{1,0}M\otimes \bigwedge ^q T^{0,1}M\) is the bundle of (p, q)-forms. We will denote the global sections \(\Gamma (M,\bigwedge _\mathbb {C}^kM)\) and \(\Gamma (M,\bigwedge ^{p,q}M)\) of the mentioned bundles by, respectively, \(\mathcal {A}_\mathbb {C}^kM\) and \(\mathcal {A}^{p,q}M\).
At the level of (p, q)-forms, the exterior differential d acts as
therefore, by setting \(\partial :=\pi ^{p+1,q}\circ d\) and \(\overline{\partial }:=\pi ^{p,q+1}\circ d\), we obtain that d splits as \(d=\partial +\overline{\partial }\). Since \(d^2=0\), it immediately follows that \(\partial ^2=\overline{\partial }^2=0\).
Associated to a compact complex manifold (M, J), one may define the de Rham, Dolbeault, Bott-Chern, and Aeppli cohomologies,
For the sake of simplicity, we will denote \(H_{\sharp }^{\bullet ,\bullet }(M,J)=:H_{\sharp }^{\bullet ,\bullet }(M)\), for \(\sharp \in \{\overline{\partial }, BC, A\}\). We will denote the Betti numbers by \(b_k=\dim _{\mathbb {C}}H_{dR}^k(M;\mathbb {C})\) and the Hodge numbers and the dimensions for the Bott-Chern and Aeppli cohomologies by \(h_{\sharp }^{p,q}:=\dim _{\mathbb {C}}H_{\sharp }(M)\), for \(\sharp \in \{\overline{\partial },BC,A\}\).
From now on, (M, J) will denote a compact complex manifold of complex dimension n.
A Hermitian metric g on (M, J) is a Riemannian metric on M such that J is an isometry with respect to g, i.e., \(g(JX,JY)=g(X,Y)\), for every \(X,Y\in \Gamma (TM)\). For any given Hermitian metric g, we will denote by F the fundamental form of g, defined as \(F(X,Y)= g(JX,Y)\) forn any \(X,Y\in \Gamma (TM)\).
We will consider the \(\mathbb {C}\)-antilinear extension of g to \(\Gamma (T_\mathbb {C}M)\) given by\( g(X\otimes \lambda ,Y\otimes \mu ):=\lambda \overline{\mu }\,g(X,Y), \) for every \(X\otimes \lambda \), \(Y\otimes \mu \in \Gamma (T_\mathbb {C}M)\). Once fixed a Hermitian metric g on (M, J), then the complex cohomology groups recalled above, are isomorphic to the kernel of suitable elliptic self-adjoint operators. More precisely, setting
where as usual \(*:\mathcal {A}^{p,q}\rightarrow \mathcal {A}^{n-q,n-p}\) is the \(\mathbb {C}\)-linear Hodge operator defined as
and \(\overline{\partial }^*=-*\partial *\), \(\partial ^*=-*\overline{\partial }*\), and denoting the spaces of Dolbeault harmonic, respectively Bott-Chern harmonic and Aeppli harmonic forms by
we have the following complex vector spaces isomorphisms
see, e.g., [25], and
see [46]. It turns out that
We recall now some basic facts of complex geometry of nilmanifolds. Let M be a nilmanifold, that is \(M=\Gamma \backslash G\), where G is a simply connected nilpotent Lie group and \(\Gamma \) is a lattice in G, with \(\dim _\mathbb {R}M=2n\). Let \(\mathfrak {g}\) be the Lie algebra of G. Then any given left-invariant almost complex structure J on G gives rise to an almost complex structure J on M; therefore, any almost complex structure J on \(\mathfrak {g}\) gives rise to an almost complex structure on M, denoted with the same symbol J. According to the Newlander and Nirenberg theorem, J is a complex structure if and only
Equivalently, an almost complex structure on \(\mathfrak {g}\) can be defined by assigning an n-dimensional complex subspace \(\mathfrak {g}^{1,0}\) of \(\mathfrak {g}^*_\mathbb {C}\), such that \(\mathfrak {g}^{1,0}\cap \overline{\mathfrak {g}^{1,0}}=\{0\}\). We set, as usual,
Then it turns out that a nilpotent Lie algebra \(\mathfrak {g}\) has a complex structure if and only if there exists a basis \(\{\eta ^1,\ldots ,\eta ^n\}\) of \(\mathfrak {g}^{1,0}\) such that
where \(\mathcal {I}(\eta ^{1},\ldots ,\eta ^k)\) denotes the ideal generated by \(\{\eta ^{1},\ldots ,\eta ^k\}\) in \(\bigwedge ^*\mathfrak {g}^*_\mathbb {C}\) (see [45, Theorem 1.3]) and d denotes the extension of the Chevalley–Eilenberg exterior differential on the Lie algebra \(\mathfrak {g}\) to \(\bigwedge ^*\mathfrak {g}_\mathbb {C}^*\).
3 p-pluriclosed structures
We review the notion of positive forms on almost complex manifolds. We start by recalling some preliminary linear algebra notions (see, e.g., [5]). Let V be a real 2n-dimensional vector space endowed with a complex structure J, that is an endomorphism J of V satisfying \(J^2=-{\textrm{id}}_V\). Denote by \(V^*\) the dual space of V and denote by the symbol J the complex structure on \(V^*\) naturally induced by J on V. Then the complexified \(V^{*\mathbb {C}}\) splits as the direct sum of the \(\pm \,i\)-eigenspaces, \(V^{1,0}\), \(V^{0,1}\) of the extension of J to \(V^{*\mathbb {C}}\), given by
that is
According to the above decomposition, the space \(\bigwedge ^r(V^{*\mathbb {C}})\) of complex r-covectors on \(V^\mathbb {C}\) decomposes as
where
If \(\{\eta ^1,\ldots ,\eta ^{n}\}\) is a basis of \(V^{1,0}\), then
is a basis of \(\bigwedge ^{p,q}(V^{*\mathbb {C}})\). Set \(\sigma _p=i^{p^2}2^{-p}\). Then, given any \(\eta \in \bigwedge ^{p,0}(V^{*\mathbb {C}})\) we have that
that is \(\sigma _p\eta \wedge \overline{\eta }\) is a (p, p)-real form. Consequently, denoting by
we get that
is a basis of \(\bigwedge _{\mathbb {R}}^{p,p}(V^{*\mathbb {C}})\). By definition, \(\psi \in \bigwedge ^{p,0}(V^{*\mathbb {C}})\) is said to be simple or decomposable if
for suitable \(\eta ^1,\ldots ,\eta ^p\in V^{1,0}\).
Remark 3.1
The complex structure J acts on the space of real k-covectors \(\bigwedge ^k(V^*)\) by setting, for any given \(\alpha \in \bigwedge ^k(V^*)\),
Then it is immediate to check that if \(\psi \in \bigwedge _{\mathbb {R}}^{p,p}(V^{*\mathbb {C}})\) then \(J\psi =\psi \). For \(k=2\), the converse holds.
Set
then
that is \(\textrm{Vol}\) is a volume form on V. A real (n, n)-form \(\psi \) is said to be positive respectively strictly positive if
where \(a\ge 0\), respectively \(a>0\).
Let \(\Omega \in \bigwedge _{\mathbb {R}}^{p,p}(V^{*\mathbb {C}})\). Then \(\Omega \) is said to be weakly positive if given any non-zero simple \((n-p)\)-covector \(\eta \), the real (n, n)-form
is positive. The real (p, p)-form \(\Omega \) is said to be transverse if, given any non-zero simple \((n-p)\)-covector \(\eta \), the real (n, n)-form
is strictly positive.
The notion of positivity on complex vector spaces can be transferred pointwise to almost complex manifolds. Let (M, J) be an almost complex manifold of real dimension 2n; let \(\bigwedge ^{p,q}(M)\) be the bundle of (p, q)-forms on (M, J). Denote by \(A^{p,q}(M):=\Gamma (M, \Lambda _J^{p, q} M)\) the space of (p, q)-forms on (M, J) and by
the space of real (p, p)-forms. Then the exterior differential d satisfies
and, consequently, d decomposes as
where \(\mu _J = \pi ^{p + 2, q - 1} \circ d\), \(\overline{\partial }_J = \pi ^{p, q + 1} \circ d\). Set
Definition 3.2
Let (M, J) be an almost complex manifold of real dimension 2n and let \(1\le p\le n\). A p-pluriclosed form on (M, J) is a real \(dd^c\)-closed transverse (p, p)-form \(\Omega \), that is \(\Omega \) is \(dd^c\)-closed and, at every \(x\in M\), \(\Omega _x\in \bigwedge ^{p,p}_{\mathbb {R}}(T_x^*M)\) is transverse. The triple \((M,J,\Omega )\) is said to be an almost p-pluriclosed manifold.
Let (M, J) be an n-dimensional complex manifold and g be a Hermitian metric with fundamental form F. Then \(d^c=i(\overline{\partial }-\partial )\) and consequently \(dd^c=2i{\partial }{\overline{\partial }}\).
Definition 3.3
The Hermitian metric g is said to be astheno-Kähler in the terminology by Jost and Yau [31] if
g is said to be strong Kähler with torsion, shortly SKT, if
Therefore, if g is an astheno-Kähler metric respectively SKT metric on (M, J), then \((M,J,F^{n-2})\) respectively (M, J, F) is an \((n-2)\)-pluriclosed respectively 1-pluriclosed manifold.
Following Gauduchon [24], a Hermitian metric g on (M, J) is said to be standard if \(F^{n -1}\) is \(\partial {\overline{\partial }}\)-closed. As observed in [20], if a Hermitian metric on a 4-dimensional compact complex manifold is at the same time SKT and astheno-Kähler, then it must be also standard. Furthermore, in [31, Lemma 6] a necessary condition for the existence of astheno-Kähler metrics on compact complex manifolds was provided, showing that any given holomorphic 1-form must be d-closed.
In order to recall the characterization theorem of compact complex manifolds admitting a p-pluriclosed structure, we review some known facts on positive currents. Let M be an n-dimensional complex manifold and let \(A^{p,q}(\Omega )\) respectively \({\mathcal D}^{p,q}(\Omega )\)) be the space of (p, q)-forms respectively (p, q)-forms with compact support on M. Consider the \({{\mathcal {C}}}^\infty \)-topology on \({\mathcal D}^{p,q}(M)\). The space of currents of bi-dimension (p, q) or of bi-degree \((n-p,n-q)\) is the topological dual \({{\mathcal {D}}}'_{p,q}(M)\) of \({{\mathcal {D}}}^{p,q}(M)\). A current of bi-dimension (p, q) on M can be identified with a \((n-p,n-q)\)-form on M with coefficients distributions. A current T of bi-dimension (p, p) is said to be real if \(T(\eta ) =T(\overline{\eta })\), for any \(\eta \in {\mathcal D}^{p,q}(M)\). A real current \(T\in {{\mathcal {D}}}'_{p,p}(M)\) is said to be strongly positive if,
for every weakly positive (p, p)-form \(\Omega \). We have the following (see [4, Theorem 2.4,(4)])
Theorem 3.4
A compact n-dimensional complex manifold N has a strictly weakly positive (p, p)-form \(\Omega \) with \({\partial }{\overline{\partial }}\Omega =0\) if and only if N has no strongly positive currents \(T\ne 0\) of bidimension (p, p), such that \(T=i{\partial }{\overline{\partial }} A\) for some current A of bidimension \((p+1,p+1)\).
We end this section by proving a simple yet useful lemma, which yields an obstruction to the existence of p-pluriclosed forms on a closed almost complex manifold.
Lemma 3.5
Let (M, J) be a closed almost complex manifold of real dimension 2n. Let \(\alpha \) be a \((2n-2p-2)\)-form which is not \(dd^c\)-closed and such that
with \(\psi ^k\) simple \((n-p,0)\)-covectors and \(c_k\ne 0\) constants having the same sign. Then (M, J) does not admit a \(p-\)pluriclosed form.
In particular,
-
for \(p=1\), (M, J) does not admit SKT metrics;
-
for \(p=n-2\), (M, J) does not admit astheno-Kähler metrics.
Proof
We prove this lemma by contradiction. Suppose there exists a p-pluriclosed form \(\Omega \) on (M, J), i.e., \(\Omega \) is a (p, p)-real form which is \(dd^c\)-closed and, for every \(x\in M\), \(\Omega _x\in \bigwedge ^{p,p}(T_xM^*)\) is transverse. Then, let \(\alpha \) be a \((2n-2p-2)\)-form on (M, J) as above and let us assume, for example, that each \(c_k>0\). Since M is closed, by Stokes theorem we have that
which is a contradiction. To end the proof, notice that if F is an astheno-Kähler metric on (M, J), the \((n-2,n-2)\)-form \(F^{n-2}\) is a \((n-2)\)-pluriclosed form on (M, J). Analogously, if F is a SKT metric on (M, J), F is 1-pluriclosed form on (M, J). \(\square \)
Remark 3.6
In Lemma 3.5 the thesis on the non existence of Hermitian metrics satisfying \(dd^cF=0\), for \(p=1\), respectively \(dd^cF^{n-2}=0\), for \(p=n-2\), is still valid, without assuming the integrability of J.
4 Astheno-Kähler metrics on 5-dimensional nilmanifolds
We now proceed to construct a family of nilmanifolds of complex dimension 5 endowed with a left-invariant complex structure admitting an astheno-Kähler metric.
Let \(\{\eta ^1,\ldots ,\eta ^5\}\) be the set of complex forms of type (1, 0), such that
where \(a_h,b_k,c_r,d_s \in \mathbb {C}\), \(h = 1, \ldots , 7\), \(k = 1, \ldots , 6\), \(r = 1, \ldots , 5\), \(s = 1, \ldots , 4\) and we set as usual \(\eta ^{AB}=\eta ^A\wedge \eta ^B\). Then, setting \(\mathfrak {g}^{1,0}={{\,\textrm{Span}\,}}\langle \eta ^1,\ldots ,\eta ^5\rangle \), we obtain that \(\mathfrak {g}_\mathbb {C}^*=\mathfrak {g}^{1,0}\oplus \overline{\mathfrak {g}^{1,0}}\) gives rise to an integrable almost complex structure J on the real 2-step nilpotent Lie algebra \(\mathfrak {g}\). Let G be the simply-connected and connected Lie group with Lie algebra \(\mathfrak {g}\). Then, for any given choice of parameters \(a_h,b_k,c_r,d_s \in \mathbb {Q}[i]\) as a consequence of Malcev’s theorem [35], there exist lattices \(\Gamma \subset G\), so that \((M=\Gamma \backslash G,J)\) is a nilmanifold endowed with a complex structure J with \(\dim _\mathbb {C}M=5\). We have the following
Theorem 4.1
Let \(M=\Gamma \backslash G\) and J be the complex structure on M defined by (4.1). Then
-
I)
The diagonal metric g on (M, J) whose fundamental form is
$$\begin{aligned} F=\frac{i}{2}\sum _{h=1}^5\eta ^{h\bar{h}} \end{aligned}$$is astheno-Kähler if and only if the following condition holds
$$\begin{aligned}{} & {} 2{\mathfrak {R}}{\mathfrak {e}}\,\,(d_4\bar{a}_4 + d_4\bar{b}_4+d_4\bar{c}_4+c_4\bar{a}_4+c_4\bar{b}_4+b_4\bar{a}_4 )\nonumber \\{} & {} \quad = \vert a_1\vert ^2+\vert a_2\vert ^2+\vert a_3\vert ^2+\vert a_5\vert ^2+\vert a_6\vert ^2+\vert a_7\vert ^2 +\nonumber \\{} & {} \qquad +\vert b_1\vert ^2+\vert b_2\vert ^2+\vert b_3\vert ^2+\vert b_5\vert ^2+\vert b_6\vert ^2+\vert c_1\vert ^2+\nonumber \\{} & {} \qquad +\vert c_2\vert ^2+\vert c_3\vert ^2+\vert c_5\vert ^2+\vert d_1\vert ^2+\vert d_2\vert ^2+\vert d_3\vert ^2. \end{aligned}$$(4.2) -
II)
Let
$$\begin{aligned}{} & {} a_2=a_3=a_5=a_6=a_7=b_1=b_2=b_3=b_5=b_6=c_2 =c_3=c_5\\{} & {} =d_1=d_2=d_3=0. \end{aligned}$$Then the metric g satisfies \(dd^cF^3=0\) and \(dd^cF^2=0\) if and only if the following conditions hold
$$\begin{aligned} \left\{ \begin{array}{l} 2\,{\mathfrak {R}}{\mathfrak {e}}\,\,(d_4\bar{a}_4+d_4\bar{b}_4+d_4\bar{c}_4+c_4\bar{a}_4 + c_4\bar{b}_4+b_4\bar{a}_4 )=\vert a_1\vert ^2+\vert c_1\vert ^2\\ 2\,{\mathfrak {R}}{\mathfrak {e}}\,\,(c_4\bar{a}_4 + c_4\bar{b}_4+b_4\bar{a}_4 )=\vert a_1\vert ^2\\ {\mathfrak {R}}{\mathfrak {e}}\,\,(c_4\bar{b}_4 - d_4\bar{a}_4)= 0\\ {\mathfrak {R}}{\mathfrak {e}}\,\,(b_4\bar{d}_4 - c_4\bar{a}_4)= 0. \end{array} \right. \end{aligned}$$(4.3)
Proof
As for I), with the aid of Sagemath and structure equations (4.1), it is easy to the see that
i.e., the metric F is astheno-Kähler on (M, J) if and only if (4.2) holds.
II) Under the assumption
taking into account (4.4) and by a straightforward computation, we obtain that
if and only if
The last system is equivalent to (4.3). \(\square \)
Remark 4.2
Recall that an Hermitian metric g on a n-dimensional complex manifold (M, J) is said to be balanced if its fundamental form \(\omega \) satisfies \(d\omega ^{n-1}=0\). In [54, p. 185] the authors asked for an example of a non-Kähler compact complex manifold which admits both balanced and astheno-Kähler metrics. In [18], and independently in [33], the authors constructed explicit examples of such manifolds in any dimension. As a direct application of Theorem 4.1, we obtain families of 5-dimensional complex nilmanifolds carrying both astheno-Kähler and balanced metrics. We apply a similar construction as in [33, Remark 2.6]. Let
where A is a positive real number. Then \(d\hat{F}^4=0\) if and only if
where \(a_4,b_4,c_4,d_4\) are the parameters as in (4.1) Let g be the diagonal metric whose fundamental form is
Then, according to I) of Theorem 4.1, g is astheno-Kähler if and only if condition (4.2) holds.
Take
Then, with this choice of parameters, we obtain
that is (4.5) is satified and so, for such a choice of parameters, \(\hat{F}\) gives rise to a balanced metric on \(M=\Gamma \backslash G\). A straightforward calculation yields
Therefore, the Hermitian metric g is astheno-Kähler if and only if condition (4.2) reads as
One can check that there exist solutions in \(\mathbb {Q}[i]\) of Eq. (4.6), so that, for any given solution, the associated complex nilmanifold defined as in (4.1) admits both a balanced metric and an astheno-Kähler metric.
As an application of Lemma 3.5, we provide a family of compact almost complex nilmanifolds without 2-pluriclosed forms.
Proposition 4.3
Let \(\{\psi ^1,\ldots ,\psi ^4\}\) be the set of complex forms of type (1, 0), such that
where \(a_1,\ldots ,a_7 \in \mathbb {Q}[i]\). Let G be the corresponding simply-connected and connected nilpotent Lie group and \(\Gamma \subset G\) be a lattice such that \(N=\Gamma \backslash G\) is a compact nilmanifold. Assume that
and set \(a=(a_1,\ldots ,a_7)\). Then \((N,J_a)\) does not admit any 2-pluriclosed form.
Proof
A straightforward calculation using (4.7) yields to
The thesis follows immediately from Lemma 3.5. \(\square \)
Remark 4.4
For any given a such that \((a_6,a_7)\ne (0,0)\), \(J_a\) is a non integrable almost complex structure on N. Consequently, for such an a, \((N,J_a)\) is an almost complex manifold with no 2-pluriclosed forms.
5 Blow-ups of astheno-Kähler metrics
By classical results and more recent ones, (see [4, 13, 21, 56]), we know that, for compact complex manifolds, the property of admitting, respectively, Kähler, balanced, or SKT metrics, is stable under blowups either in a point or along a compact complex submanifold. Regarding astheno-Kähler metrics, in [20], it is proved the following result.
Proposition 5.1
[20, Proposition 2.4] Let (M, J, g) be an astheno-Kähler manifold of complex dimension n such that its fundamental 2-form F satisfies
Then both the blow-up \(\tilde{M}_p\) of M at a point \(p\in M\) and the blow-up \(\tilde{M}_Y\) of M along a compact complex submanifold Y admit an astheno-Kähler metric satisfying (5.1), too.
In this section, we will show that blowups of astheno-Kähler metrics do not preserve additional differential properties of the metric, namely we construct an example of a 5-dimensional manifold M admitting a metric F satisfying
and we will consider the blowup of such manifold along a submanifold. We will prove that such blowup does not admit any Hermitian metric \(\tilde{F}\) which satisfies \(dd^c\tilde{F}^2=0\) and \(dd^c\tilde{F}^3=0\).
We note that if \(dd^cF=0\), conditions (5.1) of [20] would be verified, thus yielding stability. Therefore, when we consider a Hermitian metric F which satisfies weaker conditions than (5.1), e.g., the astheno-Kähler condition and the differential condition \(dd^cF^{n-3}=0\), in general such conditions are not stable under blowups
Now, we construct a family of 5-dimensional compact complex nilmanifolds endowed with a Hermitian metric whose fundamental form F satisfies (5.2) and such that the blowup of M along a suitable 3-dimensional complex submanifold Y has no Hermitian metrics satisfying (5.2). To this purpose, we start by considering the following nilpotent Lie group \(G:=(\mathbb {C}^5,*)\), where the operation \(*\) is defined for every \(w=(w_1,w_2,w_3,w_4,w_5),z=(z_1,z_2,z_3,z_4,z_5)\in \mathbb {C}^5\) by
with \(a_1,a_4,b_4,c_1,c_4,d_4\in \mathbb {Q}[i]\). We can then consider the following forms on G
It can be easily seen that \(\{\eta ^1,\dots ,\eta ^5\}\) are left invariant global forms on G with structure equations
The dual left invariant complex vectors fields \(\{Z_1,Z_2,Z_3,Z_4,Z_5\}\) on G are given by
We note that \(T_{\mathbb {C}}G=\langle Z_1,\dots ,Z_5,\overline{Z}_1,\dots ,\overline{Z}_5\rangle \) and the distribution \(D=\langle Z_1,\dots ,Z_5\rangle \) is integrable. Therefore, if we denote by J the almost complex structure on G for which \(\{Z_1,\dots ,Z_5\}\) is a frame of (1, 0)-vector fields and \(\{\eta ^1,\dots ,\eta ^5\}\) is a coframe of (1, 0)-forms, then J is an integrable left invariant almost complex structure on G.
Since the constant structures \(a_1,a_4,b_4,c_1,c_4,d_4\) are numbers in \(\mathbb {Q}[i]\), Malcev theorem assures the existence of a discrete uniform subgroup \(\Gamma \) such that \(M:=\Gamma \backslash G\) is a compact nilmanifold. In particular, since J is left invariant on G, it descends to M, i.e., (M, J) is a complex 5-dimensional nilmanifold. In particular \(\{Z_1,\dots ,Z_5\}\) and \(\{\eta ^1,\dots ,\eta ^5\}\) are a global left invariant frame of (1, 0)-vector fields, respectively (1, 0)-forms on M.
In particular, we point out that M is the nilmanifold associated to the Lie algebra \(\mathfrak {g}\) of Sect. 4, with structure constants
If we denote by
the natural quotient projection from G to \(\Gamma \backslash G\) and we set
then \(p(Y_0)=:Y\subset M\) is a compact complex 3-dimensional submanifold of M whose complexified tangent bundle \(T_{\mathbb {C}}Y\) is spanned by \(\{Z_1,Z_3,Z_5,\overline{Z}_1,\overline{Z}_3,\overline{Z}_5\}\).
It is immediate to check that Y is a 3-dimensional nilmanifold and \(\{\eta ^1,\eta ^3,\eta ^5\}\) is a global coframe of (1, 0)-forms on Y with complex structure equations given by
For the convenience of the reader, we set \(\alpha ^1:=\eta ^1\), \(\alpha ^2:=\eta ^3\), and \(\alpha ^3:=\eta ^5\), so that we can rewrite (5.3) as
Now fix the following constant structures
and consider the metric
For such choice of coefficients, by Theorem 4.1, we have that
Now, let us consider the blowup \(\pi :\tilde{M}_Y\rightarrow M\) of M along the compact complex submanifold Y, with E the exceptional divisor. We note that E has complex dimension 4, since each fiber \(\pi ^{-1}(y)\subset \tilde{M}_Y\) over a point \(y\in Y\) has dimension 1 and \(\dim _{\mathbb {C}}Y=3\).
By contradiction, now let us assume that the astheno-Kähler condition \(dd^cF^3=0\) and the condition \(dd^cF^2=0\) are stable, i.e., there exists a Hermitian metric on \(\tilde{M}_Y\) such that \(dd^c\tilde{F}^3=0\) and \(dd^c\tilde{F}^2=0\).
Then, the restriction of \(\tilde{F}\) on E gives rise to a astheno-Kähler metric on E, that is 
, i.e., E is a 2-pluriclosed manifold.
We now recall the following useful proposition by Alessandrini [3, Proposition 3.1].
Proposition 5.2
Let M and N be connected compact complex manifolds, with \(\dim N = n > m = \dim M \ge 1\), and let \(f:N\rightarrow M\) be a holomorphic submersion, where \(a:= n-m = \dim f^{-1}(x), x \in M\), is the dimension of the standard fibre F. If N is “ p-Kähler" for some p, \(a <p \le n - 1\), then M is “ \((p - a)\)-Kähler".
Note that in the above statement a “p-Kähler" manifold means a compact complex manifold admitting one of the following
-
(1)
a p-Kähler form, i.e., a closed transverse (p, p)-form;
-
(2)
a weakly p-Kähler form, i.e., a transverse (p, p)-form \(\Omega \) such that \(\partial \Omega ={\partial }{\overline{\partial }}\alpha \);
-
(3)
a p-symplectic form, i.e., a real closed 2p-form \(\Psi \) whose (p, p)-component is transverse;
-
(4)
a p-pluriclosed form, i.e., a \({\partial }{\overline{\partial }}\)-closed transverse (p, p)-form.
We refer to [1, Section 2] for further details.
Let us consider the map 
. We note that 
is a holomorphic submersion with 1-dimensional fibers, therefore by Proposition 5.2, we have that Y is 1-pluriclosed, i.e., it admits a SKT metric.
However, this is absurd by either the characterization of 3-dimensional SKT nilmanifolds by [19], or Lemma 3.5, observing that \(dd^c(-\alpha ^{3\overline{3}})=8\alpha ^{12\overline{12}}\).
Summing up, we have proved the following
Theorem 5.3
On a compact complex manifold of dimension n, the existence of a Hermitian metric F such that
is not preserved by blowup.
6 Aeppli–Bott-Chern–Massey products and geometrically-Bott-Chern-formal metrics
In this section, we review the definitions and basic facts of the notions of Aeppli–Bott-Chern–Massey products on a compact complex manifold and of geometrically-Bott-Chern-formal Hermitian metrics.
From rational homotopy theory, a differentiable manifold M is said to be Sullivan formal if the algebra of differential forms endowd with the exterior differential d, i.e., the pair \((\mathcal {A}_\mathbb {C}^{\bullet },d)\), is equivalent to a differential graded algebra \((B,d_{B})\) with zero differential, i.e., \(d_{B}\equiv 0\).
In the 50’s, Massey introduced certain cosets of the de Rham cohomology, the Massey triple products, which, if non trivial, yield an obstruction to Sullivan’s formality.
More recently, in [9] a notion of Massey triple products has been introducted as an adaptation of classical Massey products for the Bott-Chern cohomology of complex manifolds.
Let (M, J) be a compact complex manifold, and choose
such that
i.e., there exists \(f_{\alpha \beta }\in \mathcal {A}^{p+r-1,q+s-1}M\) and \(f_{\beta \gamma }\in \mathcal {A}^{r+u-1,s+v-1}M\) such that
Then, the triple Aeppli–Bott-Chern–Massey product of a, b, c is given by
We note that such a construction does not depend on the choice of representatives \(\alpha ,\beta ,\gamma \) or the choice of the primitives \(f_{\alpha \beta }, f_{\beta \gamma }\).
Furthermore, a notion of geometric formality analogous to the one in the sense of Kotschick (see [32]) has been defined in [9].
Let g be a Hermitian metric on a compact complex manifold (M, J) and let \(\mathcal {H}_{\Delta _{BC}}^{\bullet ,\bullet }(M)\) be the space of Bott-Chern harmonic forms on (M, J) with respect to g. The Hermitian metric g is said to be geometrically-Bott-Chern-formal if \(\mathcal {H}_{\Delta _{BC}}^{\bullet ,\bullet }(M)\) has a structure of algebra induced by the \(\wedge \) product, i.e., if for every two Bott-Chern harmonic forms \(\alpha \in \mathcal {H}_{\Delta _{BC}}^{p,q}(M)\), \(\beta \in \mathcal {H}_{\Delta _{BC}}^{r,s}(M)\), we have that
in particular, by the characterization (2.1),
It is clear that, since first two conditions are always satisfied by Leibniz rule, since \(\alpha \) and \(\beta \) are d-closed, the only condition that needs to be checked (and the only one involving the metric g) is \({\partial }{\overline{\partial }}*_g(\alpha \wedge \beta )=0\).
It turns out that Aeppli–Bott-Chern–Massey products are an obstruction to the existence of geometrically-Bott-Chern-formal metrics, as proved in [9].
Theorem
[9, Theorem 2.4] Triple Aeppli–Bott-Chern–Massey products vanish on compact complex geometrically-Bott-Chern-formal manifolds.
7 Geometric Bott-Chern formality and Strong Kähler with Torsion metrics
In this section we investigate the relation between the notions of SKT metrics and geometrically-Bott-Chern-formal metrics in the setting of nilmanifolds endowed with a left-invariant complex structure J and a Hermitian metric g.
In complex dimension 3, the existence of SKT metrics is fully characterized by Fino, Parton, and Salamon, in terms of the complex structure equation of the manifold, as we recall in the following.
Theorem 7.1
[19, Theorem 1.2] Let \(M = \Gamma \backslash G\) be a 6-dimensional nilmanifold with an invariant complex structure J. Then the SKT condition is satisfied by either all invariant Hermitian metrics g or by none. Indeed, it is satisfied if and only if J has a basis \((\alpha ^i)\) of (1, 0)-forms such that
where A, B, C, D, E are complex numbers such that
We will refer to 6-dimensional nilmanifolds satisfying (7.1) and (7.2) as Fino–Parton–Salamon-nilmanifolds, shortly FPS-nilmanifolds and we will denote the Lie algebra of the group G by the symbol \(\mathfrak {g}\). As a consequence of Theorem 7.1, any left-invariant Hermitian metric on a FPS-nilmanifold is SKT.
By this classification result, we are able to prove the following theorem.
Theorem 7.2
Let (M, J) be an FPS-nilmanifold. Then, any left-invariant metric is geometrically-Bott-Chern-formal.
Before proving Theorem 7.2, we will need the following lemma for the \({\partial }{\overline{\partial }}\) operator on this class of manifolds.
Lemma 7.3
Let (M, J) be a FPS-nilmanifold. Then,
Proof of Lemma 7.3
We begin by observing that it suffices to prove that \({\partial }{\overline{\partial }}\alpha ^{3\overline{3}}=0.\) In fact, let us consider the left invariant (p, q)-form on M
We note that if \(\sigma \) does not contain \(\alpha ^{3\overline{3}}\), then \({\partial }{\overline{\partial }}\sigma =0\). In fact, let us consider the two cases:
- (1):
-
\(i_k\ne 3,\overline{j}_l\ne \overline{3}\) for every \(k\in \{1,\dots ,p\}\), \(l\in \{1,\dots ,q\}\).
- (2):
-
\(i_k=3\) for some \(k\in \{1,\dots ,p\}\) and \(\overline{j}_l\ne \overline{3}\) for every \(l\in \{1,\dots ,q\}\), or \(i_k\ne 3\) for every \(k\in \{1,\dots ,p\}\) and \(\overline{j}_l=\overline{3}\) for some \(l\in \{1,\dots ,q\}\).
In case (1), by structure equations (7.1) we immediately have that \(\overline{\partial }\alpha ^{i_{k}}=\overline{\partial }\alpha ^{\overline{j}_l}=0\). Hence, by Leibnitz rule, \({\partial }{\overline{\partial }}\sigma =\partial (\overline{\partial }\sigma )=0\).
For case (2), we first assume that \(i_k=3\) for \(k\in \{1,\dots ,p\}\) and \(\overline{j}_l\ne \overline{3}\), for \(j\in \{1,\dots , q\}\). Then, up to a sign change, by Leibnitz rule we have that \(\overline{\partial }\sigma =\overline{\partial }\alpha ^{3}\wedge \hat{\sigma }\), where \(\hat{\sigma }\) is \(\sigma \) from which we remove \(\alpha ^{3}\). Since \(\overline{\partial }{\alpha ^{3}}=A\alpha ^{\overline{1}2}+B\alpha ^{2\overline{2}}+C\alpha ^{1\overline{1}}+D\alpha ^{1\overline{2}}\), we can write that
Since \(\overline{\partial }\sigma \) does not contain \(\alpha ^3\) nor \(\alpha ^{\overline{3}}\), once again by (7.1) and Leibnitz rule, we obtain \({\partial }{\overline{\partial }}\sigma =\partial (\overline{\partial }\sigma )=0.\) Analogous computations can be carried out for the other case, i.e., when \(i_k\ne 3\) for every \(k\in \{1,\dots ,p\}\) and \(\overline{j}_l=\overline{3}\) for some \(l\in \{1,\dots ,q\}\).
Let us then compute \({\partial }{\overline{\partial }}\alpha ^{3\overline{3}}\). If g is any left-invariant metric on (M, J) with fundamental form
then, by the above argument \({\partial }{\overline{\partial }} F=\frac{i}{2}F_{3\overline{3}}{\partial }{\overline{\partial }}\alpha ^{3\overline{3}}\). By Theorem 7.1, any left-invariant Hermitian metric on (M, J) is SKT, therefore, g is SKT, i.e., \({\partial }{\overline{\partial }} F={\partial }{\overline{\partial }} \alpha ^{3\overline{3}}=0\).
Therefore, up to swapping the forms and changing the sign accordingly, for a left-invariant form \(\sigma =\alpha ^{3\overline{3}}\wedge \hat{\sigma }\) with \(\hat{\sigma }\) not containing \(\alpha ^3\) nor \(\alpha ^{\overline{3}}\), as a a consequence of the above argument we obtain that
Then, by linearity of the \({\partial }{\overline{\partial }}\) operator, we can conclude. \(\square \)
Proof of Theorem 7.2
First of all, we observe that the complex structure J on the nilmanifold \(M:=\Gamma \backslash G\) is nilpotent, i.e., there exists a basis of (1, 0)-forms \(\{\alpha _i\}_{i=1}^3\), such that \(d\alpha \in {{\,\textrm{Span}\,}}_{\mathbb {C}}\langle \alpha ^{ij},\alpha ^{i\overline{j}}\rangle _{i,j=1}^2\). Hence, [6, Theorem 3.7] (see also [43, Corollary 3.12]) yields the isomorphisms
i.e., the Bott-Chern cohomology of (M, J) can be computed via the subcomplex of left-invariant forms.
Now, let g be a left-invariant metric on (M, J) with fundamental form F. We will show that g is geometrically-Bott-Chern-formal. Let us then fix two Bott-Chern harmonic forms \(\beta \in \mathcal {H}_{BC}^{p,q}(M,g)\), \(\gamma \in \mathcal {H}_{BC}^{r,s}(M,g)\). Then, the product \(\beta \wedge \gamma \) is Bott-Chern harmonic with respect to g if, and only if,
By Leibnitz rule, \(d(\beta \wedge \gamma )=0\) since both \(\beta \) and \(\gamma \) are Bott-Chern harmonic. Moreover, by Lemma 7.3, \({\partial }{\overline{\partial }}(*_g\beta \wedge \gamma )=0\), i.e., \(\beta \wedge \gamma \in \mathcal {H}_{BC}^{p+r,q+s}(M,g)\). Hence, g is a geometrically-Bott-Chern-formal metric on (M, J). \(\square \)
A similar result also holds for a class of manifolds which generalizes the FPS manifolds in higher dimensions.
Theorem 7.4
Let M be any 2n-dimensional nilmanifold endowed with an invariant integrable almost complex structure J induced by a coframe \(\{\eta ^1,\dots , \eta ^n\}\) of left invariant (1, 0)-forms on (M, J) with structure equations given by
Then, any invariant SKT metric is geometrically-Bott-Chern-formal.
Proof
In a similar fashion to proof of Theorem 7.2, it can be shown that, if there exists a left invariant SKT metric on (M, J), then \({\partial }{\overline{\partial }}\eta ^{n\overline{n}}=0\), and in particular the \({\partial }{\overline{\partial }}\) operator vanishes on any left-invariant form on (M, J). Notice that the Bott-Chern cohomology of (M, J) can be computed via the subcomplex of left-invariant forms and its Bott-Chern harmonic representatives are invariant since the complex structure J is nilpotent and [6, Theorem 3.8] applies. Hence, if g is a SKT metric on (M, J) and we take two Bott-Chern harmonic forms \(\alpha \) and \(\beta \) of bedegree (p, q), respectively (r, s), then \(\alpha \) and \(\beta \) are left-invariant and \(\alpha \wedge \beta \in \bigwedge ^{p+r,q+s}\mathfrak {g}\) satisfies \(d(\alpha \wedge \beta )=0\) and by, structure equations, \({\partial }{\overline{\partial }}(*_{g}\alpha \wedge \beta )=0\). Therefore \(\alpha \wedge \beta \) is Bott-Chern harmonic, i.e., the product of two left-invariant Bott-Chern harmonic forms with respect to g is Bott-Chern harmonic. This implies that every left-invariant SKT metric is geometrically-Bott-Chern-formal. \(\square \)
In higher dimension and under more general conditions on the complex structure of the nilmanifold, however, similar results do not hold. Certain products of compact complex surfaces, e.g., are SKT but do not admit geometrically-Bott-Chern-formal metrics, as proved in the following theorem.
Theorem 7.5
Let (M, J) be the product of either two Kodaira surfaces, two Inoue surfaces, or a Kodaira surface and a Inoue surface. Then (M, J) admits SKT metrics but does not admit geometrically-Bott-Chern-formal metrics.
Proof
We begin by noticing that given the product of any two of the above compact complex surfaces \((M,J)=(M',J')\times (M'',J'')\), such manifold admits an SKT metric.
Let us consider the product metric \(g:=g'+g''\), given by the sum of the diagonal constant metrics \(g'\) and \(g''\) with respect to certain coframes \(\{\eta ^1,\eta ^2\}\) and \(\{\eta ^3,\eta ^4\}\) on, respectively, \((M',J')\) and \((M'',J'')\) and let
be the fundamental forms associated to, respectively, \(g'\) and \(g''\). By a dimension argument, we have that on each factor
Therefore, if \(F:=F'+F''\), it is clear that
i.e., the product metric g is SKT on (M, J). (We will refer to such metric by g.)
We will show that none of the above product manifolds admits geometrically-Bott-Chern-formal metrics by exhibiting a non vanishing Aeppli–Bott-Chern–Massey product on each manifold.
Note that on each product, Bott-Chern and Aeppli cohomologies can be computed via the subcomplex of invariant complex forms, as follows. First of all, the de Rham cohomology and the Dolbeault cohomology of compact surfaces diffeomorphic to solvmanifolds can be computed in terms of invariant forms, see, e.g., [7]. Therefore, applying Kunneth formula, it follows that the de Rham and the Dolbeault cohomologies of the product of any two such surfaces can be computed in terms of the invariant forms. By [6, Theorem 3.7], also the Bott-Chern and Aeppli cohomologies can be computed in terms of the invariant forms.
(i) The product of two Kodaira surfaces of primary type.
Let \((M,J)=(KT,J_{KT})\times (KT,J_{KT})\) be the product of two Kodaira surfaces. The complex structure J is determined by the coframe \(\{\eta ^1,\eta ^2,\eta ^3,\eta ^4\}\) of left-invariant (1, 0)-forms such that its structure equations read
for \(A,B\in \mathbb {C}\setminus \{0\}\).
From (7.4), it is easy to see that the following Bott-Chern cohomology classes
are non zero. Also, we have that
Then, it is well defined the following Aeppli–Bott-Chern–Massey product
Since \(d*\eta ^{23\overline{4}}=-d(\eta ^{123\overline{14}})=0\), the form \(\eta ^{23\overline{4}}\) is Aeppli harmonic, hence, as a cohomology class in \(H_{A}^{2,1}(M)\), we have that
It remains to show that \(\left[ -\frac{1}{A\overline{B}}\eta ^{23\overline{4}}\right] _A\notin H_A^{1,0}(M)\cup \,[\eta ^{1\overline{1}}]_{BC}+H_A^{1,1}(M)\cup \,[\eta ^{3}]_{BC}\).
Let us then suppose, by contradiction, the opposite, i.e.,
where \(h_A^{p,q}:=\dim \mathcal {H}_{A}^{p,q}(M,g)\), \(r_i,s_j\in \mathbb {C}\), \(R\in \mathcal {A}^{1,1}(M)\), \(S\in \mathcal {A}^{2,0}(M)\), and \(\{\xi ^i\}\) and \(\{\psi ^j\}\) are the left-invariant harmonic representatives of, respectively, \(H_A^{1,0}(M)\), and \(H_A^{1,1}(M)\), with respect to g.
It is immediate to compute the invariant Aeppli cohomology of (M, J) of bi-degree (1, 0) and (1, 1), resulting in
Then, Eq. (7.6) can be rewritten as
We note that the form \(\eta ^{12\overline{134}}\) is d-closed. Therefore, if we multiply (7.7) by \(\eta ^{12\overline{134}}\), we obtain
i.e.,
By integrating (7.8) and applying Stokes theorem on a manifold with empty boundary, we obtain that \(s_{13}=0.\)
If we repeat the same argument, multiplying now (7.7) by the d-closed form \(\eta ^{14\overline{123}}\), we obtain
which, by integrating and Stokes theorem, leads to a contradiction.
To summarize,
i.e., we obtained a non vanishing Aeppli–Bott-Chern–Massey product, which, by [9, Theorem 2.4], implies that (M, J) does not admit geometrically-Bott-Chern-formal metrics.
(ii) The product of two Inoue surfaces of type \(\mathcal {S}_{M}\).
Let \((M,J)=(\mathcal {S}_M,J_{\mathcal {S}_M})\times (\mathcal {S}_M,J_{\mathcal {S}_M})\) be product of two Inoue surfaces of type \(\mathcal {S}_M\). The complex structure J is determined by the left invariant (1, 0)-coframe \(\{\eta ^1,\eta ^2, \eta ^3,\eta ^4\}\) with structure equations
for \(\alpha ,\gamma \in \mathbb {R}{\setminus }\{0\}\), \(\beta ,\delta \in \mathbb {R}\).
From (7.9), it is clear that the following Bott-Chern cohomology classes
are well defined and non zero. Moreover,
hence the following Aeppli–Bott-Chern–Massey product
is well defined.
Note that since \(d(*_g\eta ^{234\overline{34}})=-d(\eta ^{12\overline{1}})=0\), the form \(\eta ^{234\overline{34}}\) is Aeppli-harmonic and, as a Aeppli cohomology class,
It remains to show that \(\left[ \frac{1}{\alpha \gamma }\eta ^{234\overline{34}} \right] _A\notin H_A^{2,1}\cup [\eta ^{2\overline{2}}]_{BC}+H_A^{2,1}(M)\cup [\eta ^{4\overline{4}}]_{BC}\). In order to do, we prove that \(H_A^{2,1}(M)=\{0\}\), yielding that \(H_A^{2,1}\cup [\eta ^{2\overline{2}}]_{BC}+H_A^{2,1}(M)\cup [\eta ^{4\overline{4}}]_{BC}=0\).
By definition, we observe that
With the aid of structure equations (7.9) and Sagemath, we can compute
so that
Therefore, \(H_{A}^{2,1}(M)=\{0\}\) and
which, by [9, Theorem 2.4] implies that (M, J) does not admit any geometrically-Bott-Chern-formal metric.
(iii) The product of a Inoue surface of type \(\mathcal {S}_M\) and a primary Kodaira surface
Let \((M,J)=(\mathcal {S}_M,J_{\mathcal {S}_M})\times (KT,J_{KT})\) be the product of a Inoue surface of type \(\mathcal {S}_M\) and a primary Kodaira surfaces. The complex structure J is determined by the coframe of left-invariant (1, 0)-form \(\{\eta ^1,\eta ^2,\eta ^3,\eta ^4\}\) with structure equations
with \(\alpha \in \mathbb {R}{\setminus }\{0\}\), \(\beta \in \mathbb {R}\), \(B\in \mathbb {C}{\setminus } \{0\}.\)
We consider the following Bott-Chern cohomology classes
They are clearly well defined and they are not zero. Moreover,
Therefore, the Aeppli–Bott-Chern–Massey product
is well defined.
Note that \(d*(\frac{1}{i\alpha \overline{A}}\eta ^{23\overline{4}})=-\frac{1}{i\alpha \overline{A}}d(\eta ^{123\overline{14}})=0\), i.e., the form \(\frac{1}{i\alpha \overline{A}}\eta ^{23\overline{4}}\) is Aeppli-harmonic and
as a Aeppli cohomology class.
It remains to show that \(\left[ \frac{1}{i\alpha \overline{A}}\eta ^{23\overline{4}}\right] _A\notin H_A^{1,0}(M)\cup [\eta ^{2\overline{2}}]_{BC}+H_A^{1,1}(M)\cup [\eta ^{3}]_{BC}\). Let us suppose by contradiction that this is the case, i.e.,
with \(\lambda _i,\mu _j\in \mathbb {C}\), \(R\in \mathcal {A}^{1,1}(M)\), \(S\in \mathcal {A}^{2,0}(M)\), and \(\{\xi ^i\}\) and \(\{\mu ^j\}\) are, respectively, a basis for \(\mathcal {H}_{A}^{1,0}(M,g)\) and \(\mathcal {H}_A^{1,1}(M,g)\). By structure equations (7.10), we can compute the spaces of Aeppli-harmonic forms with respect to g
Then, Eq. (7.11) becomes
Since the form \(\eta ^{14\overline{123}}\) is d-closed, if we multiply (7.12) by \(\eta ^{14\overline{123}}\), we obtain
which, by integrating over M and applying Stokes theorem, leads to contradiction.
Hence, we showed that
i.e., (M, J) admits a non vanishing Aeppli–Bott-Chern–Massey product. By [9, Theorem 2.4], this implies that (M, J) does not admit any geometrically-Bott-Chern-formal metric. \(\square \)
We prove one more result in this direction, showing that the existence of Aeppli–Bott-Chern–Massey products obstructs the existence of geometrically-Bott-Chern-formal metrics on a family of 4-dimensional complex nilmanifolds which cannot be constructed as a product of two or more manifolds.
We start by considering the set of complex forms \(\{\eta ^1,\eta ^2,\eta ^3,\eta ^4\}\) of type (1, 0) satisfying the following structure equations
with \(A,B_1,B_2,B_3\in \mathbb {Q}[i]\). Let \(\mathfrak {g}^*={{\,\textrm{Span}\,}}_\mathbb {R}\langle \mathfrak {Re}(\eta ^i)\mathfrak {Im}(\eta ^i)\rangle _{i=1,\dots ,4}\). Then, setting \(\mathfrak {g}^{1,0}={{\,\textrm{Span}\,}}_\mathbb {C}\langle \eta ^1,\eta ^2,\eta ^3,\eta ^4\rangle \), we obtain that \(\mathfrak {g}_\mathbb {C}^*=\mathfrak {g}^{1,0}\oplus \overline{\mathfrak {g}^{1,0}}\) gives rise to an integrable almost complex structure J on the real nilpotent Lie algebra \(\mathfrak {g}\). We will consider the natural complex structure J on \(\mathfrak {g}\) which arises by choosing \(\{\eta ^1,\eta ^2,\eta ^3,\eta ^4\}\) as a coframe of (1, 0)-form on \(\mathfrak {g}_{\mathbb {C}}^*\). Let G be the simply-connected and connected Lie group with Lie algebra \(\mathfrak {g}\). By Malcev’s theorem we have that the 8-dimensional real Lie group G associated to \(\mathfrak {g}\) admits a discrete uniform subgroup \(\Gamma \) such that \(M:=\Gamma \backslash G\) is compact and, in particular, (M, J) is an 8-dimensional nilmanifold with an invariant complex structure.
Moreover, since J is nilpotent complex structure on (M, J), by [6, Theorem 3.8] (see also, [43, Corollary 3.12]), we have the following isomorphisms
i.e., the Bott-Chern cohomology of (M, J) can be computed by means of the complex of left-invariant forms on \(\mathfrak {g}\).
Theorem 7.6
Let \(M=\Gamma \backslash G\) be a complex 4-dimensional nilmanifold endowed with a left-invariant complex structure J determined by a coframe of (1, 0)-forms \(\{\eta ^1,\eta ^2,\eta ^3,\eta ^4\}\) with structure equations
with \(A\in \mathbb {C}{\setminus }\{0\}\), \(B_j\in \mathbb {Q}[i]\), such that
Then (M, J) admits a SKT metric but does not admit any geometrically-Bott-Chern-formal metric.
Proof
Let now consider the diagonal metric g with fundamental associated form
With the aid of (7.13) and (7.14), we can see clearly that g is SKT, i.e., \({\partial }{\overline{\partial }} F=0\).
We will show that (M, J) admits a non vanishing ABC-Massey product, which suffices to prove that there exists no geometrically-Bott-Chern-formal metric on (M, J).
Let us consider the following Bott-Chern cohomology classes
Since
the Aeppli–Bott-Chern–Massey product
is well defined.
We notice that the \(d*_g(\frac{1}{|A|^2}\eta ^{23\overline{3}})=\frac{1}{|A|^2}d(\eta ^{124\overline{14}})=0\), i.e., the form \(\eta ^{23\overline{3}}\) is Aeppli harmonic and, as a Aeppli cohomology class, we have that \([-\frac{1}{|A|^2}\eta ^{23\overline{3}}]_A\ne 0\). It remains to show that \([-\frac{1}{|A|^2}\eta ^{23\overline{3}}]_A\notin H_A^{1,0}(M)\cup [\eta ^{1\overline{1}}]_{BC}+H_A^{1,1}(M)\cup [\eta ^{2}]_{BC}\).
Let us now suppose by contradiction that \([-\frac{1}{|A|^2}\eta ^{23\overline{3}}]_A\in H_A^{1,0}(M)\cup [\eta ^{1\overline{1}}]_{BC}+H_A^{1,1}(M)\cup [\eta ^{2}]_{BC}\). By straightforward computations, it is easy to check that the spaces \(\mathcal {H}_A^{1,0}(M,g)\) and \(\mathcal {H}_{A}^{1,1}(M,g)\) are generated, respectively, by \(\langle \psi ^j\rangle _{j=1}^2\) and \(\langle \xi ^i\rangle _{i=1}^{11}\), where
and
Then, \([-\frac{1}{|A|^2}\eta ^{23\overline{3}}]_A\in H_A^{1,0}(M)\cup [\eta ^{1\overline{1}}]_{BC}+H_A^{1,1}(M)\cup [\eta ^{2}]_{BC}\) implies that
for \(r_i,s_j\in \mathbb {C}\), \(R\in \mathcal {A}^{1,2}(M)\), \(S\in \mathcal {A}^{2,1}(M)\), so that
We note that the form \(\eta ^{13\overline{123}}\) is d-closed, therefore, if we multiply (7.16) by \(\eta ^{13\overline{123}}\), we obtain
which, by integrating and applying Stokes theorem, forces \(s_{11}=0\). Equation (7.16) reduces to
Now, the form \(\eta ^{14\overline{124}}\) is d-closed, so if we multiply (7.17) by \(\eta ^{14\overline{124}}\), we obtain
which, by integration and applying Stokes theorem, leads to contradiction.
Therefore, we showed that
i.e., (M, J) admits a non vanishing Aeppli–Bott-Chern–Massey product, which implies that (M, J) does not admit any geometrically-Bott-Chern-formal metric. \(\square \)
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Acknowledgements
The authors kindly thank Daniele Angella and Nicoletta Tardini for useful discussions and comments. Many thanks are also due to Anna Fino for reading carefully the manuscript and for useful remarks and to Jonas Stelzig for suggesting some possible further developments. They also would like to thank the anonymous referee for their useful comments and suggestions which improved the presentation of this paper.
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Tommaso Sferruzza has been supported by GNSAGA of INdAM. Adriano Tomassini has been supported by the project PRIN2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (code 2017JZ2SW5), and by GNSAGA of INdAM.
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Sferruzza, T., Tomassini, A. On cohomological and formal properties of strong Kähler with torsion and astheno-Kähler metrics. Math. Z. 304, 55 (2023). https://doi.org/10.1007/s00209-023-03303-9
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DOI: https://doi.org/10.1007/s00209-023-03303-9
Keywords
- Astheno–Kähler metric
- Strong Kähler with torsion metric
- Geometrically-Bott–Chern-formal metric
- Triple Aeppli–Bott-Chern–Massey product