Deformations of astheno-K\"ahler metrics

The property of admitting an astheno-K\"ahler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this paper, we prove necessary cohomological conditions for the existence of curves of astheno-K\"ahler metrics along curves of deformations starting from an initial compact complex manifold endowed with an astheno-K\"ahler metric. Furthermore, we apply our results providing obstructions to the existence of curves of astheno-K\"ahler metrics on two different families of real $8$-dimensional nilmanifolds endowed with invariant nilpotent complex structures.


Introduction
The study of small deformations of the complex structure of a compact complex manifold yields many interesting insights on its homotopy and complex invariants.A remarkable result by Kodaira and Spencer in [19] is the proof of the stability under small deformations of the Kähler condition, i.e., the property of admitting a Hermitian metric whose fundamental form is closed.As a matter of fact, the Kähler condition is strongly related to the topology of a compact even dimensional manifold, e.g., it forces many costraints on the Betti numbers, that is, the dimensions of the de Rham cohomology spaces, and it implies the formality according to Sullivan, that is, the algebra of differential forms is equivalent to the algebra of its de Rham cohomology ( [30,31,8]).Stability under deformation has been proved also for important invariants directly related to the complex structure of a manifold, e.g., the property of satisfying the ∂∂-lemma, i.e., every d-closed d-exact form is also ∂∂-exact, and different versions that have been recently introduced and studied (see [5,24,29]).
Concerning the stability of special Hermitian structures, i.e., Hermitian metrics whose fundamental forms belong to the kernel of differential operators depending on the complex structure of the manifold, the property of admitting a standard metric in the sense of Gauduchon has been proved to be stable.In fact, by the celebrated Gauduchon theorem (see [15]), given any Hermitian metric on a complex manifolds, its fundamental form is conformal to a standard metric in the sense of Gauduchon.However, this class of metrics represents a vary special case, since for many other notions of special metrics on a n-dimensional complex manifold, e.g., • strong Kähler with torsion metrics, i.e., metrics whose fundamental form ω satisfies ∂∂ω = 0, • astheno-Kähler metrics, i.e., metrics whose fundamental form ω satisfies ∂∂ω n−2 = 0, • astheno-Kähler metrics, i.e, metrics whose fundamental form ω satisfies dω n−1 = 0, stability does not hold in general (see, respectively, [12,2]).On the other hand, sufficient conditions for the stability of balanced and SKT metrics have been proved in [5,21,25].Hence, it is natural to investigate whether there exist assumptions on either the initial manifold or the performed deformation of the complex structure of a compact complex manifold so that the astheno-Kähler condition is stable.
The notion of astheno-Kähler metrics has been first introduced by Jost and Yau in the study of solutions to certain non linear elliptic systems [17] and has been later used to prove rigidity theorems regarding projectively flat manifolds in [20].By definition, it is clear that on a complex manifold of complex dimension lesser than 2 every Hermitian metric is an astheno-Kähler metric.In complex dimension 3, the notion of astheno-Kähler coincides with the notion of strong Kähler with torsion and, hence, the results valid for the latter class of metrics hold also for the former, e.g., on 6-dimensional real nilmanifold endowed with an invariant complex structure, the existence of invariant SKT (and hence, astheno-Kähler) metrics depends only on the complex structure (see [10]).Nonetheless, for n > 3, this notions are not related, i.e., there exists examples of manifolds with astheno-Kähler metrics which are not strong Kähler with torsion, and viceversa (see [13,26]).
Furthermore, whereas for SKT metrics it has been conjectured that on the same non-Kähler compact complex manifold there cannot exist both SKT metrics and balanced metrics with respect to the same complex structure (see [14]), in [22] it has been proved that an astheno-Kähler metric is balanced if and only if the metric is also Kähler.Nonetheless, in [11] the authors show the existence of a compact complex non-Kähler manifold which admits both a balanced and astheno-Kähler metric.Moreover, the existence of SKT or balanced metrics is equivalent to the existence of, respectively, 1-pluriclosed and (n − 1)-Kähler structures on a manifold (see [1]), so that their existence on a compact manifolds can be intrinsically characterized in terms of currents (see [9]), analogously to the Kähler case (see [16]).Even though for astheno-Kähler metrics such a characterization does not hold, the existence of an astheno-Kähler metric yet implies certain restraint on a manifold, i.e., the existence of a (n − 2)-pluriclosed form and the closedness with respect to the exterior differentential of any holomorphic 1-form.In [13], the authors study the behaviour of astheno-Kähler metrics under complex blowup and identify a sufficient differential condition for stability; in [28] it is shown that under weaker differential conditions, stability is not assured.
In this paper, following a similar approach to [23] and [27], we prove necessary conditions of cohomological type for the existence of a curve of astheno-Kähler metrics starting from a fixed astheno Kähler metric on a compact complex manifold.More specifically, if e ιϕ ι ϕ denotes the extension map introduced in [25] and recalled in section 2, and ι ψ is the contraction by a (0, 1)vector form with values in a holomorphic vector bundle, as we recall in section 3, we have obtained the following results.
Theorem.Let (M, J) be a n-dimensional compact complex manifold endowed with an astheno-Kähler metric g and associated fundamental form ω. Let {M t } t∈I be a differentiable family of compact complex manifolds with M 0 = M and parametrized by ϕ(t) ∈ A 0,1 (T 1,0 (M )), for t ∈ I ∶= (− , ), > 0. Let {ω t } t∈I be a smooth family of Hermitian metrics along {M t } t∈I , written as Then, if every metric ω t is astheno-Kähler, for t ∈ I, it must hold that As a direct consequence, we immediately have the following corollary.
Corollary.Let (M, J) be a compact Hermitian manifold endowed with an asteno-Kähler metric g and associated fundamental form ω. If there exists a smooth family of astheno-Kähler metrics which coincides with ω in t = 0, along the family of deformations {M t } t with M 0 = M and parametrized by the (0, 1)-vector form ϕ(t) on M , then the following equation must hold

BC
(M ) = 0. We note that the proof of the main theorem makes use of the formulas proved in [25] for the differential operators ∂ t and ∂ t acting on (p, q)-forms on each M t , t ∈ I, combined with the Taylor series expansion centered in t = 0.
As an application of our results, we characterize the obstructions to the existence of curves of astheno-Kähler metrics along certain curves of deformations, starting from two families of 4dimensional nilmanifolds endowed with invariant nilpotent complex structures.Thanks to general results (see e.g., [6,4,3]), nilmanifolds with invariant complex structures (see, e.g., [7]) are a natural source of examples of compact complex non-Kähler manifolds on which both cohomological and metric properties can be explicitly verified at the level of invariant tensors.
The paper is organized as follows.In section 2, we fix the notations that will be used throughout the paper.In section 3, we briefly recall the necessary tools of deformation theory, among which the extension map and the formulas for the differential operators ∂ t and ∂ t acting on differential forms on a family of deformations.In section 4, we will state and prove the main theorem.In section 5, we recall the main facts about the geometry of nilmanifolds and special complex structures and in section 6 we characterize the obstructions yielded by Theorem 4.1 and Corollary 4.2 starting from two families of 4-dimensional nilmanifolds with invariant nilpotent complex structure.
Acknowledgements.The author would like to kindly thank Adriano Tomassini for many useful suggestions and comments.

Notation
Let (M, J) be a compact complex manifold of complex dimension n, where M is a 2n-dimensional compact differentiable manifold and J is a integrable almost complex structure on M , i.e., an endomorphism of T M such that J 2 = − id T M and the Nijenhuis tensor N J identically vanishes, that is, for every X, Y ∈ T M .Once the complex structure J is extended to the complexified version of the bundles T M , we obtain the decomposition in terms of the ±i-eigenspaces of J.A similar decomposition holds when we consider the induced action of J on complex forms, i.e., where ⋀ p,q M ∶= ⋀ p (T 1,0 M ) * ⊗ ⋀ q (T 0,1 M ) * is the bundle of (p, q)-forms on M .Let A k C (M ) and A p,q (M ) denote the global sections of the bundles of, respectively, complex k-forms and (p, q)-forms on (M, J).Then, since J is integrable, the differential d acts on (p, q)-forms on M as d∶ A p,q (M ) → A p+1,q (M ) ⊕ A p,q+1 (M ), so that d = ∂ + ∂, where we have set ∂ ∶= π p+1,q and ∂ ∶= π p,q+1 .Clearly, ∂ 2 = ∂ 2 = 0 and ∂∂ = −∂∂.Let g be a Hermitian metric on (M, J), i.e., a Riemannian metric on M such that g(JX, JY ) = g(X, Y ) for every X, Y ∈ T M .Let ω(X, Y ) ∶= g(JX, Y ) for every X, Y ∈ T M , be the fundamental form associated to g.The extension of ω to a form on A 2 C M is a positive (1, 1)-form on M .A astheno-Kähler metric on a n-dimensional complex manifold (M, J) is a Hermitian metric g such that its fundamental form satisfies ∂∂ω n−2 = 0.A strong Kähler with torsion metric on (M, J) is a Hermitian metric g such that its fundamental form satisfies ∂∂ω = 0.
The Bott-Chern cohomology of (M, J) is given by the spaces H p,q BC (M ) ∶= Ker d∶ A p,q (M ) → A p+1,q (M ) ⊕ A p,q+1 (M ) whereas we say that a (p, q)-form α on M is said Bott-Chern harmonic if ∆ BC α = 0, where the Bott-Chern Laplacian is the fourth-order self-adjoint elliptic operator.Since M is assumed to be compact, α is Bott-Chern harmonic if, and only if, dα = 0 and ∂∂ * α = 0, where * is the C-antilinear Hodge * -operator with respect to g.Let us denote H p,q ∆ BC (M ) the space of Bott-Chern harmonic (p, q)-forms.Hodge theory adapted to Bott-Chern cohomology by Schweitzer in [32] implies that there exist the following isomorphisms of vector spaces Let π∶ E → M be a holomorphic vector bundle of rank r on M .Then, we set ⋀ p,q (M, E) ∶= ⋀ p,q M ⊗ E for the bundle of (p, q)-forms with values in E, and A p,q (M, E) will denote the its global sections.
We can define a ∂ E operator on A p,q (M, E) in the following way.Let ϕ = ∑ i η i ⊗s i ∈ A p,q (M, E), where each η i ∈ A p,q (M ) and (s 1 , . . ., s r ) is the expression of s in a local triviallization on E. Then (2.1) Throughout this paper, we will consider essentially as E the holomorphic tangent bundle T 1,0 M and its conjugate T 0,1 M and we will refer the sections of A 0,q (M, T 1,0 M ) as (0, q)-vector forms on M ; we will omit the term "M " in the notations when understood.

Review of deformation theory and the operators ∂ t and ∂ t along curves of deformations
In this section we recall the formulas fo the action of the differential operators ∂ t and ∂ t on each element of a differentiable family of deformations {M t } of a compact complex manifold (M, J); we follow the approach in [25], in which such operators are expressed in terms of the operators ∂ and ∂ on (M, J) and on a (0, 1)-vector form which parametrizes the differential family {M t } t .
Let B be a differential manifold of real dimension m.

Definition 3.1 ([18]
).A differentiable family of compact complex manifold is a differentiable manifold M and a differentiable proper map π∶ M → B such that • for every t ∈ B, the fiber M t ∶= π −1 (t) has a structure of complex manifold; • the rank of the Jacobian of π is constantly equal to m, at every point p ∈ M.
If (M, J) is a compact complex manifold, then a differentiable family of compact complex manifolds is a differential family of deformations of (M, J) if there exists t 0 ∈ B such that M t 0 = (M, J).We refer to this fixed element of the family as the central fiber.Without loss of generality, we can assume that the space of parameters B to be a polydisc in R m , i.e., B ∶= {(t 1 , . . ., t m ) ∈ R m ∶ t j < , j ∈ {1, . . ., m}}, for > 0, and t 0 will always be 0 ∈ R m .
Viceversa, starting from a compact complex manifold (M, J), in [18] it is shown that one can construct families of deformations {M t } t∈B of (M, J) by setting where J t is an integrable complex structure on the differentiable manifold M parametrized by a (0, 1)-vector form ϕ(t) on the central fiber (M, J), i.e, ϕ(t) ∈ A 0,1 (T 1,0 M ).From M 0 = (M, J), it follows that ϕ(0) = 0. We remark that in order for each J t to define an integrable complex structure on M , the (0, 1)-vector form ϕ(t) must satisfy the Maurer-Cartan equation, i.e., For the sake of semplicity, we will set the dimension of the parameters space B to be m = 1, i.e., B = I = (− , ), for > 0.
Let then (M, J) be a compact n-dimensional complex manifold and let ϕ(t) ∈ A 0,1 (T 1,0 (M )), t ∈ I, parametrize a differentiable family of deformations {M t } t∈I of (M, J).We will refer to differential families over an interval I as curves of deformations.
We now recall a map which links the (p, q)-forms on the central fiber (M, J) and on every element of the family M t = (M, J t ).First, let us then define the maps where i k ψ is the contraction by the vector form ψ repeated k times.Note that since M is compact, each summation is finite.
If α = α i 1 ...ipj 1 ...jq dz i 1 ∧ ⋅ ⋅ ⋅ ∧ dz ip ∧ dz j 1 ∧ ⋅ ⋅ ⋅ ∧ dz jq is the local expression of a (p, q)-form α on (M, J), then the extension map e ι ϕ(t) ι ϕ(t) is defined as ).Note that such a map defines a global object, since ϕ(t) is a global (0, 1)-vector form on (M, J).At the level of each space of (p, q)-forms, we have the following result, see [25, Lemma 2.9] Lemma 3.2.For a fixed t ∈ I, the exponential map defines the following isomorphisms Hence, once we fix t ∈ I, we can see any (p, q)-form α t on M t as the image e ι ϕ(t) ι ϕ(t) (α) of a certain α ∈ A p,q (M ).
Exploiting Lemma 3.2, the Maurer-Cartan equation (3.1) for the integrability of J t can be restated in the following versatile terms.Let {η 1 , . . ., η n } be a global frame of (1, 0)-forms on (M, J), and {η j t ∶= e ι ϕ(t) ι ϕ(t) (η j )} n j=1 the corresponding frame of (1, 0)-forms on M t .Then J t defines an integrable complex structure on M if (dη j t ) 0,2 = 0, for every j ∈ {1, . . ., n}, where (dη j t ) 0,2 denotes the (0, 2)-component of dη j t with respect to the decomposition given by Note that this condition is equivalent to (3.1).We end this section by recalling the formulas for the operators ∂ t and ∂ t acting on (p, q) forms on each M t .We will usually omit the dependence on t of the (0, 1)-vector form ϕ(t).
Let {M t } t∈I be a curve of deformations of (M, J) such that each M t = (M, J t ) and each complex structure J t is parametrized by a (0, 1)-vector form ϕ(t), t ∈ I, on (M, J).Then, from the expression of d at that level of (p, q)-forms on each M t , i.e., the operators ∂ t and ∂ t are, by definition, ∂ t ∶= π p+1,q t ○ d and ∂ t ∶= π p,q+1 t ○ d.In [25], the authors provides the formulas for the action of such operators on both functions and (p, q)-forms.Let then f ∶ M → C be a smooth function on M .Then , where we use the notations ϕϕ = ϕ ⌟ ϕ, ϕ ⌟ ϕ = ϕ ⌟ ϕ, and I is the identity map (see [25, Formula 2.13]).In particular, a map f ∶ M → C is holomorphic with respect to the complex structure J t if, and only if, ∂f − (ϕ(t) ⌟ ∂)f = 0. We recall that the simultaneous contraction by a (0, 1)-vector form ϕ (or any operator) acts on a (p, q) form α as Such a contraction is well defined and can be used to rewrite the exponential map as Then, from [25, Prop 2.13], we obtain that the action of ∂ t and ∂ t on any e ιϕ ι ϕ α ∈ A p,q (M t ), for α ∈ A p,q (M ), is defined as
We can then apply formulas (3.4) and (3.5) to (4.1), and by making use of Taylor series expansion and differentiating with respect to t in t = 0, we are able to prove the main theorem.
Theorem 4.1.Let (M, J) be a n-dimensional compact complex manifold endowed with an astheno-Kähler metric g and associated fundamental form ω. Let {M t } t∈I be a differentiable family of compact complex manifolds with M 0 = M and parametrized by ϕ(t) ∈ A 0,1 (T 1,0 (M )), for t ∈ I ∶= (− , ), > 0. Let {ω t } t∈I be a smooth family of Hermitian metrics along {M t } t∈I , written as where, locally, Then, if every metric ω t is astheno-Kähler, for t ∈ I, it must hold that As a direct consequence, we immediately have the following corollary.
Corollary 4.2.Let (M, J) be a compact Hermitian manifold endowed with an asteno-Kähler metric g and associated fundamental form ω. If there exists a smooth family of astheno-Kähler metrics which coincides with ω in t = 0, along the family of deformations {M t } t with M 0 = M and parametrized by the (0, 1)-vector form ϕ(t) on M , then the following equation must hold Proof of Theorem 4.1.The metrics ω t are astheno-Kähler for every t ∈ I, i.e., ∂ t ∂ t ω n−2 t = 0.This implies Let us then compute the right hand side of (4.5) through formulas (3.4) and (3.5) for the operators ∂ t and ∂ t .By the extension map we have that and then, by (3.4) and (3.5), we have . Now, we expand in Taylor series centered in t = 0 the terms Combining with (3.3) and usig the notation ⌟, we obtain that which is equivalent to hence, concluding the proof.

Nilmanifolds with nilpotent complex structure
In this section, we briefly recall the basic notions on the geometry of nilmanifolds endowed with invariant complex structure.
A nilmanifold M is the datum of a quotient M = Γ G of a simply connected connected Lie group G by a discrete uniform subgroup Γ, such that the Lie algebra g associated to the Lie group G is nilpotent, i.e., the series {g (k) } ends in {0}, where ∶= g, g (1)  ∶= [g, g], g (j)  ∶= [g, g (j−1) ].An invariant complex structure J on a nilmanifold M is a left-invariant integrable almost complex structure on G, which descendens to the quotient M .Such complex structures on M can be characterized by a set of linearly independent left-invariant forms {η j } j on the Lie group G such that C is the space of left-invariant complex forms on G.By left-invariance, the coframe {η j } j then descends to the quotient M .
Therefore, one can construct nilmanifolds by setting C a set of linearly independent complex covectors with structure equations , it holds that g (k 0 ) = {0}, for some k 0 .Indeed, under these hypotesis, the Lie algebra g underlying g * C is nilpotent, and its corresponding Lie group G admits a discrete uniform subgroup Γ by Malcev theorem, so that (M = Γ G, J) is a nilmanifold endowed with an invariant complex structure J and ⊕ g * (0,1) where In particular, we say that the complex structure J on M is • nilpotent if there exists a coframe of left-invariant (1, 0)-forms {η j } on G such that, for every j, • abelian if dg * (1,0) ⊂ g * (1,1) .

Applications
We now apply Corollary 4.2, providing examples of obstructions to the existence of curve of astheno-Kähler metrisc on two families of 4-dimensional nilmanifolds.6.1.Example 1.Let (M = Γ G, J) be the 4-dimensional nilmanifold with left-invariant complex structure J induced by the covectors {η 1 , η 2 , η 3 , η 4 } on the complexified dual g * C of the Lie algebra g of G, which satisfy structure equations Remark 6.1.If the complex manifold (M, J) is holomorphically parallelizable, i.e., a 3 = a 4 = a 5 = a 7 = a 8 = a 9 = a 9 = a 10 = a 11 = a 12 = 0, then metric g is astheno Kähler on (M, J) if, and only if, also a 3 = a 8 = a 12 = 0, i.e., (M, J) is a complex torus.This is in line with the more general argument that on a compact holomorphically parallelizable manifold there exists a global coframe of holomorphic (1, 0)-form; however, if a manifold admits an astheno-Kähler metric, every holomorphic 1-form is d-closed.Therefore, on a holomorphically parallelizable manifold endowed with an astheno-Kähler metric, each form of the global holomorphic coframe is d-closed, hence the manifold is a torus.
Proof.(1 ).By simple computations, it holds that Let us rewrite structure equations (6.1) as For the sake of completeness, we write Then, it is easy to see that Note that the (1, 1)-forms and do not contain η 4 nor η 4 .Then, i.e, the form ∂ ○ ι ϕ ′ (0) ○ ∂ω 2 is a (3, 3)-form on (M, J) which does not contain η 4 nor η 4 .Hence, , where C J,ϕ ′ (0) ∈ C is a constant depending on the the structure equations defining the complex structure J and the derivative of ϕ(t) in t = 0.
For computations in higher dimensions, it is rather easy to see that, following Remark 6.1, Lemma 6.2, and Remark 6.3, similar arguments are valid also for any n-dimensional nilmanifold (M, J) with left-invariant complex structure J characterized by analogous structure equations, i.e., it holds the following.Lemma 6.4.Let (M = Γ G, J) be a nilmanifold whose complex structure is determined by a base {η 1 , . . ., η n } of the complexified dual of the Lie algebra g of G such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ and whose structure constants are elements of Q[i].More specifically, if ω = i 2 ∑ n j=1 η jj is the fundamental form associated to the diagonal metric g and ϕ(t As in the 4-dimensional case, the existence of curves of astheno-Kähler metrics along the family of deformations parametrized by ϕ could be obstructed by Corollary 4.2 if the canonical diagonal metric g is also SKT and the complex structure J is neither abelian nor holomorphically parallelizable. In the 4-dimensional example, let us consider an element (M, J) of the family of 4-dimensional nilmanifolds, with a 2 = a 5 = a 6 = a 9 = a 10 = a 11 = a 12 = 0.In this case, structure equations become Note that the manifold (M, J) can be realized as the direct product of a complex 1-dimensional torus and a 3-dimensionl nilmanifold.
We will assume a 1 ≠ 0.
Then, structure equations (6.7) and the astheno-Kähler (and SKT) condition for ω (6.8) still holds.Then solution set of (6.9) is then for δ > 0 sufficiently small.Hence the (0, 1)-vector form ϕ(t) parametrizing the integrable deformations of (M, J) is so that can consider the curve of deformations for > 0 sufficiently small, so that By direct computations, we obtain that Therefore, since the Bott-Chern cohomology class [η 123123 ] ≠ 0 by Lemma 6.2, condition (4.4) holds if and only if Let us consider the case a 7 ≠ 0. We have that the solution set of (6.9) is for δ > 0 sufficiently small.We can then pick as a curve of deformations for > 0 sufficiently small, so that Then, by computations similar to the previous case we obtain that By applying Corollary 4.2 to each case, we obtain the following theorem.
Theorem 6.5.Let (M = Γ G, J) be an element of the family of 4-dimensional nilmanifolds with complex structure J determined by the covectors {η 1 , η 2 , η 3 , η 4 } on the complexified dual of the Lie algebra g of G, with structure equations = 2Re(a 3 a 8 ).Then, • if a 4 ≠ 0 and (u 2 , u 3 ) ∈ C 2 , there exists no curve of astheno-Kähler metrics ω t such that ω 0 = ω along the curve of deformations t ↦ ϕ(t) • if a 7 ≠ 0 and (u 1 , u 3 ) ∈ C 2 , there exists no curve of astheno-Kähler metrics ω t such that 6.2.Example 2. We now show an application of Corollary 4.2 to a different family of 4dimensional nilmanifolds with invariant complex structure.Let (M = Γ G, J) be the nilmanifold with complex structure J determined by the base {η 1 , η 2 , η 3 , η 4 } of the complexified dual g * C of the Lie algebra g of G, with structure equations (6.11) As in Example 6.1, the complex structure J is 2-step nilpotent.However, the manifold (M, J) does not have the structure of a direct product of a 1-dimensional complex torus and a 3-dimensional nilmanifold.
Let ω = ∑ 4 j=1 η jj be the fundamental form associated to the diagonal metric g.By direct computations, it turns out that g is astheno-Kähler if and only if (6.12) = 0, if the metric g is astheno-Kähler, it is also SKT.
Let us consider the family of deformations {(M, J t )} t∈B of (M, J) parametrized by the (0, 1)vector form t = η 4 so that, by inverting the system, we obtain , for j ∈ {1, 2}.Since the form ϕ(t) defines an family of complex manifolds if and only if d(η j t ) 0,2 = 0, for every j ∈ {1, 2, 3, 4}, such a integrability condition is satisfied if and only if (dη 3 t ) 0,2 = 0, which yields Under the assumption that a 1 = b 1 , a 3 = b 3 , and a 4 = b 4 , we have that the condition of integrability is valid for t ∈ S, where S is the solution set of equation We now proceed as in the previous example, by considering the map and discussing the cases in which either ∇F and the diagonal metric g is astheno-Kähler if, and only if, the following condition holds We assume that b 1 ≠ 0. The solution set S for equation (6.13) is then Hence as a curve of deformation ϕ(t) with t ∈ S we can choose for > 0 sufficiently small.Then, ϕ ′ (0) = u 1 η 1 ⊗ Z 1 + u 2 η 2 ⊗ Z 2 .By computations, however, it turns out that ∂ ○ ι ϕ ′ (0) ○ ∂ω 2 = 0, hence Corollary 4.2 does not yield any obstruction.and the astheno-Kähler condition (6.12) on the diagonal metric g is still valid.The solution set S for (6.13) is then for δ > 0 sufficiently small.Once we fix u 2 ∈ C, we can pick the curve of deformations for > 0 sufficiently small, so that Then, we compute We summarize what we obtained in the following theorem.Then, • if b 3 ≠ 0 and u 2 ∈ C, there exists no curve of astheno-Kähler metrics ω t with ω 0 = ω along the curve of deformations ϕ(t) • if b 4 ≠ 0 and u 1 ∈ C, there exists no curve of astheno-Kähler metrics ω t with ω 0 = ω along the curve of deformations ϕ(t) = tu 1 η 1 ⊗ Z 1 + t tb