Invariant complex structures on 6-nilmanifolds classification, Frölicher spectral sequence and special Hermitian metrics

arXiv:1111.5873v5 [math.DG] 10 Oct 2014 INVARIANT COMPLEX STRUCTURES ON 6-NILMANIFOLDS: CLASSIFICATION, FR ¨ OLICHER SPECTRAL SEQUENCE AND SPECIAL HERMITIAN METRICS M. CEBALLOS, A. OTAL, L. UGARTE, AND R. VILLACAMPA Abstract. We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Fr¨olicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation. 1. Introduction Let gbe a Lie algebra endowed with an endomorphism J:g−→ gsuch that J2=−Id. The endomorphism Jis a complex structure if the integrability condition [JX, JY ] = J[JX, Y ] + J[X, JY ] + [X, Y ] is satisfied for any X, Y ∈g; equivalently, the i-eigenspace g1,0of Jin gC=g⊗RC is a complex subalgebra of gC. Nilpotent Lie algebras gadmitting a complex structure were classified by Salamon [28] up to dimension 6. More recently, Andrada, Barberis and Dotti classified in [2] the 6-dimensional Lie algebras ghaving a complex structure Jof abelian type, that is, the complex subalgebra g1,0is abelian, or equivalently [JX, JY ] = [X, Y ] for any X, Y ∈g. A related question is to determine the complex structures on a given Lie algebra gup to isomorphism in the following sense. Two complex structures Jand J′on g are equivalent if there exists an automorphism F:g−→ gof the Lie algebra such that J=F−1◦J′◦F. The latter condition is equivalent to say that F, extended to gC, satisfies F(gJ 1,0)⊂gJ′ 1,0. If C(g) denotes the space of complex structures on g then C(g)/Aut(g) parametrizes the equivalence classes of complex structures on g. A classification of abelian complex structures in dimension 6 is given in [2]. Some partial results on nilpotent Lie algebras can be found in several papers [6, 19, 30, 31], although to our knowledge there is no complete classification of complex structures on 6-dimensional nilpotent Lie algebras. This is our first goal here. The classification of complex structures on nilpotent Lie algebras provides a classification of invariant complex structures on nilmanifolds. Let M= Γ\Gbe a nilmanifold, i.e. a compact quotient of a simply-connected nilpotent Lie group G by a lattice Γ of maximal rank. If Jis a complex structure on the Lie algebra gof G, then it gives rise to a left-invariant complex structure on Gwhich descends to a complex structure on the quotient Min a natural way. Several interesting aspects of this complex geometry have been investigated, as for instance the Dolbeault cohomology [7, 13, 26], complex deformations [6, 8, 20, 27] or the existence of special Hermitian metrics [16, 30]. Recently, it is proved in [4] that the canonical 1 2 bundle of any complex nilmanifold is holomorphically trivial and some applications to hypercomplex geometry are given. As a first application of the classification of complex structures we study the behaviour of the Fr¨olicher sequence [17]. Recall that the Fr¨olicher sequence Er(M, J) of a complex manifold (M, J) is the spectral sequence associated to the double complex (Ωp,q(M, J), ∂, ¯ ∂), where ∂+¯ ∂=dis the decomposition, with respect to J, of the exterior differential d. The first term E1(M, J) is precisely the Dolbeault cohomology of (M, J) and after a finite number of steps the sequence converges to the de Rham cohomology of M. The first examples of compact complex manifolds for which E26∼ =E∞were independently found in [9] and [21]. The examples in [9] are complex nilmanifolds of complex dimension 3, which is the lowest possible dimension for which the Fr¨olicher sequence can be non-degenerate at E2. More recently, Rollenske has constructed in [25] complex nilmanifolds for which the sequence {Er} can be arbitrarily non-degenerate. The behaviour of the Fr¨olicher sequence has been studied for some other complex manifolds [14, 29], but as far as we know its general behaviour for complex nilmanifolds has not been studied, although some partial results can be found in [10, 11, 12]. Here we study the Fr¨olicher spectral sequence for general invariant complex structures on a 6-dimensional nilmanifold. A remarkable consequence of this study is the existence of a compact complex manifold on which the ∂¯ ∂-lemma fails but E1∼ =E∞and the Hodge diamond is symmetric. As a second application of the classification of complex structures we consider strongly Gauduchon (sG for short) metrics in the sense of Popovici [22, 23]. Any balanced Hermitian metric is sG and any sG metric is a Gauduchon metric [18]. In [24] the relation between the degeneration of the Fr¨olicher sequence at E1and the existence of sG metrics is studied, showing that these two notions are unrelated. We study the existence of sG or balanced metrics on 6-nilmanifolds in relation to the general behaviour of the Fr¨olicher sequence. Moreover, Popovici proved in [23] that the sG property of compact complex manifolds is open under holomorphic deformations, and conjectured in [24] that the sG property and the balanced property of compact complex manifolds are closed under holomorphic deformations. We construct a counterexample to both closedness conjectures. The paper is structured as follows. In Section 2 we first review some general facts about complex structures on a 6-dimensional nilpotent Lie algebra g. By [28] such gmust be isomorphic to h1,...,h16,h− 19 or h+ 26 (see Theorem 2.1 for a description of the Lie algebras). Of special interest is h5because it corresponds to the real Lie algebra underlying the Iwasawa manifold, whose complex geometry is studied in [19]. For the first sixteen classes the complex structure is necessarily of nilpotent type in the sense of [13]. We classify the non-abelian nilpotent complex structures on 2-step and 3-step nilpotent Lie algebras in Sections 2.1 and 2.2, respectively. Then, using the classification of non-nilpotent complex structures obtained in [31] as well as the classification of abelian structures given in [2], we present in Tables 1 and 2 of Section 3 the complete classification of complex structures on 6-dimensional nilpotent Lie algebras up to equivalence. Since Jequivalent to J′implies that the terms in the associated Fr¨olicher sequences are isomorphic, as an application we study the general behaviour of the Fr¨olicher sequence Er(Γ\G, J) in Section 4 (see Theorem 4.1 for details). We find that E26∼ =E∞if and only if the underlying Lie algebra g∼ =h13,h14 or h15. Moreover, E1∼ =E26∼ =E3∼ =E∞for any Jwhen g∼ =h13 or h14. In contrast, h15 has a rich complex geometry with respect to Fr¨olicher sequence because it admits 3 complex structures for which E16∼ =E2∼ =E∞,E1∼ =E26∼ =E3∼ =E∞or even E16∼ =E26∼ =E3∼ =E∞. In Example 4.8 we give a family Jtof non-equivalent complex structures on h15 along which the Fr¨olicher sequence has these three behaviours. We also show that a nilmanifold with underlying Lie algebra h6has a complex structure with degenerate Fr¨olicher sequence and satisfying hp,q ¯ ∂=hq,p ¯ ∂for every p, q ∈N, which provides an answer to a question recently posed in [3] (see Proposition 4.3). In section 5 we study the existence of sG metrics on 6-dimensional nilmanifolds endowed with an invariant complex structure and show that the underlying Lie algebra must be isomorphic to h1,...,h6or h− 19. It is also proved that the existence of sG metric implies the degeneration of the Fr¨olicher sequence at E2. Using [32] we give in Proposition 5.5 a classification of complex structures having sG metrics but not admitting any balanced metric, as well as a classification of nilpotent complex structures admitting balanced metric (see Table 3). Based on the complex geometry of the Lie algebra h4, in Theorem 5.9 we show that neither the sG property nor the balanced property of compact complex manifolds are closed under holomorphic deformation. 2. Nilpotent complex structures on 6-dimensional nilpotent Lie algebras Given a Lie algebra g, let g∗ Cbe the dual of the complexification gCof g. If J:g−→ g is an endomorphism such that J2=−Id, then there is a natural bigraduation induced on V∗g∗ C=⊕p,q Vp,q(g∗), where the spaces V1,0(g∗) and V0,1(g∗), which we shall also denote by g1,0and g0,1, are the eigenspaces of the eigenvalues ±i of Jas an endomorphism of g∗ C, respectively. Now, if d:V∗g∗ C−→ V∗+1 g∗ Cis the extension to the complexified exterior algebra of the usual Chevalley-Eilenberg differential, then it is well known that Jis a complex structure if and only if π0,2◦d|g1,0≡0, where π0,2:V2g∗ C−→ V0,2(g∗) denotes the canonical projection. We shall focus on nilpotent Lie algebras (NLA for short). Salamon has proved in [28] the following equivalent condition for the integrability of Jon a 2n-dimensional NLA g:Jis a complex structure on gif and only if g1,0has a basis {ωj}n j=1 such that dω1= 0 and dωj∈ I(ω1,...,ωj−1),for j= 2,...,n, where I(ω1,...,ωj−1) is the ideal in V∗g∗ Cgenerated by {ω1,...,ωj−1}. Recall that a complex structure Jon a 2n-dimensional NLA gis nilpotent [13] if there exists a basis {ωj}n j=1 for g1,0satisfying dω1= 0 and (1) dωj∈^2hω1,...,ωj−1, ω1,...,ωj−1i,for j= 2,...,n. An important special class of nilpotent complex structures is the abelian class consisting of those structures Jsatisfying [JX, JY ] = [X, Y ], for all X, Y ∈g, or equivalently d(g1,0)⊂V1,1(g∗). They are also characterized by the fact that the subalgebra g1,0is abelian. In six dimensions, the classification of NLAs in terms of the different types of complex structures that they admit is as follows. 4 Theorem 2.1. [28, 30] Let gbe an NLA of dimension 6. Then, ghas a complex structure if and only if it is isomorphic to one of the following Lie algebras: h1= (0,0,0,0,0,0), h2= (0,0,0,0,12,34), h3= (0,0,0,0,0,12 + 34), h4= (0,0,0,0,12,14 + 23), h5= (0,0,0,0,13 + 42,14 + 23), h6= (0,0,0,0,12,13), h7= (0,0,0,12,13,23), h8= (0,0,0,0,0,12), h9= (0,0,0,0,12,14 + 25), h10 = (0,0,0,12,13,14), h11 = (0,0,0,12,13,14 + 23), h12 = (0,0,0,12,13,24), h13 = (0,0,0,12,13 + 14,24), h14 = (0,0,0,12,14,13 + 42), h15 = (0,0,0,12,13 + 42,14 + 23), h16 = (0,0,0,12,14,24), h− 19 = (0,0,0,12,23,14 −35), h+ 26 = (0,0,12,13,23,14 + 25). Moreover: (a)Any complex structure on h− 19 and h+ 26 is non-nilpotent; (b)For 1≤k≤16, any complex structure on hkis nilpotent; (c)Any complex structure on h1,h3,h8and h9is abelian; (d)There exist both abelian and non-abelian nilpotent complex structures on h2,h4,h5and h15; (e)Any complex structure on h6,h7,h10,h11,h12,h13,h14 and h16 is not abelian. Remark 2.2. Here we use the usual notation, i.e. for instance h2= (0,0,0,0,12,34) means that there is a basis {ej}6 j=1 satisfying de1=de2=de3=de4= 0, de5=e1∧e2,de6=e3∧e4; equivalently, the Lie bracket is given in terms of its dual basis {ej}6 j=1 by [e1, e2] = −e5, [e3, e4] = −e6. Let gbe a Lie algebra endowed with two complex structures Jand J′. We recall that Jand J′are said to be equivalent if there is an automorphism F:g−→ gof the Lie algebra such that J′=F−1◦J◦F, that is, Fis a linear automorphism such that F∗:g∗−→ g∗commutes with the Chevalley-Eilenberg differential d and Fcommutes with the complex structures Jand J′. The latter condition is equivalent to say that F∗, extended to the complexified exterior algebra, preserves the bigraduations induced by Jand J′. Notice that if g1,0 Jand g1,0 J′denote the (1,0)-subspaces of g∗ Cassociated to Jand J′, then the complex structures Jand J′are equivalent if and only if there is a C-linear isomorphism F∗:g1,0 J−→ g1,0 J′such that d◦F∗=F∗◦d. In dimension 6, by Theorem 2.1, if the NLA gadmits complex structures then all of them are either nilpotent or non-nilpotent. The classification of abelian complex structures up to equivalence is obtained in [2], whereas the non-nilpotent complex structures are classified in [31] (see Section 3 for details). Therefore, it remains to study the equivalence classes of non-abelian nilpotent complex structures. In order to provide such classification, our starting point is the following reduction of the nilpotent condition (1). Proposition 2.3. [30] Let Jbe a nilpotent complex structure on an NLA gof dimension 6. There is a basis {ωj}3 j=1 for g1,0satisfying (2)    dω1= 0, dω2=ǫ ω1¯ 1, dω3=ρ ω12 + (1 −ǫ)A ω1¯ 1+B ω1¯ 2+C ω2¯ 1+ (1 −ǫ)D ω2¯ 2, where A, B, C, D ∈Cand ǫ, ρ ∈ {0,1}. 5 Here ωjk (resp. ωjk) means the wedge product ωj∧ωk(resp. ωj∧ωk), where ωkindicates the complex conjugated of ωk. From now on, we shall use a similar abbreviated notation for “basic” forms of arbitrary bidegree. Notice that in the equations (2) the complex structure is not abelian if and only if ρ= 1. Next we study the 2-step and 3-step cases in Sections 2.1 and 2.2, respectively. 2.1. Non-abelian complex structures in the 2-step case. Any 6-dimensional 2-step NLA ghas first Betti number at least 3, and if it is equal to 3 then necessarily the coefficient ǫin (2) is non-zero. We consider firstly ǫ= 0, i.e. the Lie algebra has first Betti number ≥4, and we will finish the section by considering the remaining case ǫ= 1. The following proposition provides a further reduction of the equations (2) when ǫ= 0 and the structure is not complex-parallelizable. Recall that Jis complexparallelizable if [JX, Y ] = J[X, Y ], for all X, Y ∈g, or equivalently d(g1,0)⊂ V2,0(g∗). These structures are the natural complex structures of complex Lie algebras, and in six dimensions they correspond to ǫ=A=B=C=D= 0 and the possible Lie algebras are h1(for ρ= 0) and h5(for ρ= 1). Proposition 2.4. Let Jbe a complex structure on a 2-step NLA gof dimension 6 with first Betti number ≥4. If Jis not complex-parallelizable, then there is a basis {ωj}3 j=1 for g1,0such that (3) dω1=dω2= 0, dω3=ρ ω12 +ω1¯ 1+λ ω1¯ 2+D ω2¯ 2, where ρ∈ {0,1},λ∈Rsuch that λ≥0, and D∈Cwith Im D≥0. Moreover, if we denote x=Re Dand y=Im D, then: (i) If λ=ρ, then the Lie algebra gis isomorphic to (i.1) h2, for y > 0; (i.2) h3, for ρ=y= 0 and x6= 0; (i.3) h4, for ρ= 1,y= 0 and x6= 0; (i.4) h6, for ρ= 1 and x=y= 0; (i.5) h8, for ρ=x=y= 0. (ii) If λ6=ρ, then the Lie algebra gis isomorphic to (ii.1) h2, for 4y2>(ρ−λ2)(4x+ρ−λ2); (ii.2) h4, for 4y2= (ρ−λ2)(4x+ρ−λ2); (ii.3) h5, for 4y2<(ρ−λ2)(4x+ρ−λ2). Proof. In [30, Lemma 11] it is proved that under these conditions there is a basis {σj}3 j=1 for g1,0such that (4) dσ1=dσ2= 0, dσ3=ρ σ12 +σ1¯ 1+B σ1¯ 2+D σ2¯ 2, where B, D ∈Cand ρ∈ {0,1}. If B6= 0 then we can take any non-zero solution zof ¯zB |B|=z, and the equations (4) reduce to (3) with λ=|B|with respect to the new basis {ω1=z σ1, ω2= ¯z σ2, ω3=|z|2σ3}. Consider now B=λwith λ∈R≥0in (4). If D6= 0, then with respect to the new basis {ω1=−¯ D σ2, ω2=σ1+λ σ2, ω3=¯ D σ3}we get (3) with ¯ Dinstead of D. Finally, the second part of the proposition follows directly from [30, Proposition 13].  6 From now on we consider ρ= 1. By Proposition 2.4 any two complex structures on the Lie algebra h6are equivalent. Thus, it remains to classify up to equivalence the non-abelian structures Jon h2,h4and h5. Any such Jis identified with a triple (1, λ, D) through equations (3) with ρ= 1, λ≥0 and Im D≥0. We will say that two triples (1, λ, D) and (1, λ′, D′) are equivalent, denoted by (1, λ, D)∼(1, λ′, D′), if the corresponding structures Jand J′are equivalent. So, the problem reduces to classify triples (1, λ, D) up to equivalence. Lemma 2.5. Let us consider two triples (1, λ, D)and (1, t, E)as above. (i) If D= 0 then, (1, t, E)∼(1, λ, 0) if and only if t=λand E= 0. (ii) If D6= 0 then, (1, t, E)∼(1, λ, D)if and only if there exist non-zero complex numbers e, f such that E=De/¯eand (5) |f|2 ¯e−1(¯ D¯e−De)2= (λ¯ f−tf)(λ¯ D¯ef −tDe ¯ f). Proof. The structure equations corresponding to the triples (1, λ, D) and (1, t, E) are dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λω1¯ 2+Dω2¯ 2, dσ1=dσ2= 0, dσ3=σ12 +σ1¯ 1+tσ1¯ 2+Eσ2¯ 2, where λ, t ≥0 and Im D, Im E≥0. Then (1, t, E)∼(1, λ, D) if and only if there exists an automorphism of the Lie algebra preserving the complex equations, i.e. there is (mij )∈GL(3,C) such that σi=P3 j=1 mij ωjand dσi= 3 X j=1 mij dωj, i = 1,2,3. These conditions are equivalent to σ1=a ω1+b ω2, σ2=c ω1+f ω2, σ3=m31 ω1+m32 ω2+e ω3, and (6)                (I) e=af −bc, (II) e=|a|2+t a¯c+E|c|2, (III) λe =a¯ b+t a ¯ f+Ec ¯ f, (IV) 0 = ¯ab +t b¯c+E¯cf, (V) De =|b|2+t b ¯ f+E|f|2. Notice that m13 =m23 = 0, e6= 0 and the coefficients m31 and m32 are not relevant. It is straightforward to see that coefficient fmust be non-zero (otherwise λ=t and D=E) and so we can express aas a=e+bc f. First of all, let us suppose that D= 0. Replacing ain (IV) and using (V) we obtain that b= 0 and therefore E= 0 by equation (V). Combining (I) and (III) we get that λf =t¯ f. Since λand tare real non-negative numbers, we conclude that λ=t, i.e. (1, λ, 0) defines an equivalence class for every λ≥0. This completes the proof of (i). 7 We suppose next that D6= 0. In order to solve (6) we transform it into an equivalent system by doing the following substitutions. Replacing ain equation (IV) and using (V) we can express ¯c=−b¯e De. Next, in (II) we can substitute aand cand use again (V) to obtain that De =E¯e, which implies in particular |D|=|E|. Notice that since D6= 0 we can assume E6=¯ Dby Proposition 2.4. Now, ¯c=−b/E. Proceeding in a similar way in equation (III) we get ¯ b=λf −t¯ f 1−D/ ¯ E. Finally, using the expressions of a,b,cabove, equation (V) is equivalent to (5). Therefore, given e, f ∈C−{0}satisfying De =E¯eand (5), it is always possible to find a, b, c ∈Csuch that system (6) is satisfied.  Remark 2.6. As a consequence of Lemma 2.5 (ii), when D6= 0 a necessary condition for (1, t, E) to be equivalent to (1, λ, D) is that |D|=|E|. Moreover, to find an equivalent complex structure (1, t, E) it suffices to find t≥0 and e, f ∈ C−{0}satisfying (5), because Eis necessarily given by E=De/¯e. Corollary 2.7. Let E6=¯ D. If (1, t, E)∼(1, λ, D)then, t=λif and only if E=D. Proof. By hypothesis Dcannot be zero, so we are in case (ii) of Lemma 2.5. Suppose first that λ=tin (5), i.e. (¯ D¯e−De)2|f|2 ¯e−1=λ2(¯ f−f)( ¯ D¯ef −De ¯ f). The right hand side of the previous equality is a real number. If it is zero then e=|f|2(otherwise De =¯ D¯ewould imply E=¯ D); thus, eis a real number and since E=De/¯ewe conclude that D=E. On the other hand, if it is a non-zero real number, then |f|2 ¯e−1 must be a real number and then e∈Rand again D=E. Conversely, let us suppose that E=D6= 0. In this case e∈Rand by (5) we can express it as e=|f|2−(λ¯ f−tf)(λ¯ Df −tD ¯ f) (¯ D−D)2. Notice that by hypothesis D6=¯ E=¯ D. To ensure that e∈Rit must happen that (λ¯ f−tf)(λ¯ Df −tD ¯ f)∈Ror equivalently, |f|2(λ2−t2)( ¯ D−D) = 0. As f(¯ D−D)6= 0 the only possibility to solve the previous equation is λ=t. From the previous results it follows that it remains to consider the case when D6= 0 and λ6=t. The next lemma provides a simplification of equation (5). Lemma 2.8. Let us suppose that λ6=t,D=x+iy 6= 0 and e∈C−{0}. Then, (1, λ, D)∼(1, t, De/¯e)if and only if (7) 4y2−(t2−λ2)(4x+t2−λ2)≥0. 8 Proof. By Lemma 2.5 (ii), we know that (1, λ, D)∼(1, t, De/¯e) if and only if (5) is satisfied. This condition reads, with respect to H=De, as (¯ H−H)2¯ D|f|2−¯ H=¯ H(λ¯ f−tf)(λf ¯ H−t¯ fH). Taking real and imaginary parts in the expression above we obtain (8)          4H2 2(H1−x|f|2) = |f|2(t2−λ2)H2 2+|f|2(t2+λ2)H2 1 −2λt(f2 1−f2 2)H2 1−4λtH1H2f1f2, 4H2 2(y|f|2−H2) = 2λH2tH1(f2 1−f2 2) + 2tH2f1f2−λ|f|2H1, where H=H1+iH2and f=f1+if2. Observe that H26= 0, otherwise we get a contradiction using the first equation of (8). Substituting the second equation of (8) in the first one and replacing Hby De, we can express the system (8) as    e2 1(t2−λ2) + 4ye1e2+e2 2(t2−λ2+ 4x) = 0, 2H2(y|f|2−H2) = λtH1(f2 1−f2 2) + 2tH2f1f2−λ|f|2H1, (9) where e=e1+ie2. To solve the first equation in (9) as a second degree equation in e1we need the discriminant to be greater than or equal to 0, i.e. 4y2−(t2−λ2)(4x+t2−λ2)≥0, which is precisely condition (7). Now, suppose that (7) holds. Then we obtain that e1=e2β λ2−t2, e =e2β λ2−t2+i, where β= 2y+p4y2−(t2−λ2)(4x+t2−λ2) and e2is determined by the second equation in (9).  Corollary 2.9. Let us suppose that λ6=tand D=x+iy 6= 0. If (7) holds then (1, λ, D)∼1, t, D β2−(λ2−t2)2 β2+ (λ2−t2)2+2β(λ2−t2) β2+ (λ2−t2)2i, where β= 2y+p4y2−(t2−λ2)(4x+t2−λ2). Comparing the inequalities (ii.1) and (ii.2) in Proposition 2.4 with the condition (7), we observe that for h2and h4it is possible to take t= 1 in the previous corollary in order to get equivalences with the complex structures (i.1) and (i.3), respectively. Therefore, using Corollary 2.7, we conclude: Proposition 2.10. Let us consider the family of complex structures (10) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+ω1¯ 2+D ω2¯ 2,Im D≥0. Then: (i) Any non-abelian complex structure on h2is equivalent to one and only one structure in (10) with Im D > 0; (ii) Any non-abelian complex structure on h4is equivalent to one and only one structure in (10) with D∈R−{0}. The classification of complex structures on h5requires a more subtle study. 9 Lemma 2.11. Any non-abelian complex structure on h5which is not complexparallelizable belongs to one of the following families: (I) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+iy ω2¯ 2,where 0≤2y < |1−λ2|; (II) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+ (x+iy)ω2¯ 2,where 4y2<1 + 4x. Moreover, (i) the structures in family (I) are non-equivalent; (ii) the structures in family (II) are non-equivalent; (iii) a structure (1, λ, iy)in family (I) is equivalent to a structure in family (II) if and only if 2λ2∈[0,1) and 2y∈[λ2,1−λ2). Proof. Let us consider a complex structure given by (1, λ, D =x+i y) on h5, i.e. 4y2<(1 −λ2)(4x+ 1 −λ2), according to Proposition 2.4 (ii.3). If λ2≥2x, then (1, λ, D)∼(1,√λ2−2x, i|D|) because (7) expresses simply as 4|D|2≥0 and it trivially holds. On the other hand, if λ2<2x, then (1, λ, D)∼(1,0, E), where Eis given in Corollary 2.9, because in this case 4y2+λ2(4x−λ2)≥0, that is, condition (7) is satisfied. To study further equivalences, it is clear that structures in family (I) are nonequivalent and the same holds for structures in family (II). Now let us consider the triples (1, λ, iy) and (1,0, E). Then, (7) expresses simply as (11) 4y2≥λ4. Condition for family (I) implies that 4y2<(1 −λ2)2, which is equivalent to 4y2− λ4<1−2λ2, so if 2λ2≥1 then (11) does not hold. Now, if 0 ≤λ2<1 2 then the condition for family (I) is equivalent to y < 1 2−λ2 2, and therefore when 2y∈[λ2,1−λ2) the triple (1, λ, iy) in family (I) is equivalent to the triple (1,0, E = −1 2(λ2−p4y2−λ4i)) in family (II).  Proposition 2.12. Any non-abelian complex structure on h5which is not complexparallelizable is equivalent to one and only one structure in the following families: (I) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+D ω2¯ 2, where Re D= 0 and    0≤2Im D < λ2,0< λ2<1 2;or 0≤2Im D < |1−λ2|,1 2≤λ2. (II) dω1=dω2= 0, dω3=ω12+ω1¯ 1+D ω2¯ 2,where 4(Im D)2<1+4 Re D. To finish this section, it remains to study the case of 2-step NLAs gwith first Betti number equal to 3, which corresponds to ǫ= 1 in (2). Proposition 2.13. Let Jbe a nilpotent complex structure on an NLA ggiven by (2) with ǫ= 1, i.e. dω1= 0, dω2=ω1¯ 1, dω3=ρ ω12 +B ω1¯ 2+C ω2¯ 1, with ρ∈ {0,1}and B, C ∈Csuch that (ρ, B, C)6= (0,0,0). Then gis 2-step nilpotent if and only if B=ρ= 1 and C= 0. In such case gis isomorphic to h7 and all the complex structures are equivalent. 16 (c) If g∼ =h5and Jis a complex structure on h5given in Table 1, then: (c.1) E16∼ =E2∼ =E∞when Jis complex-parallelizable; (c.2) E1∼ =E∞if and only if Jis not complex-parallelizable and ρD 6= 0; moreover, E16∼ =E2∼ =E∞when ρD = 0. (d) If g∼ =h16 or h+ 26, then E16∼ =E2∼ =E∞for any J. (e) If g∼ =h13 or h14, then E1∼ =E26∼ =E3∼ =E∞for any J. (f) If g∼ =h15 and Jis a complex structure on h15 given in Table 1, then: (f.1) E16∼ =E2∼ =E∞, when c= 0 and |B−ρ| 6= 0; (f.2) E1∼ =E26∼ =E3∼ =E∞, when ρ= 1 and |B−1| 6=c6= 0; (f.3) E16∼ =E26∼ =E3∼ =E∞, when ρ= 0 and |B| 6=c6= 0. Proof. The proof is straightforward and we only give it explicitly for the case (f), that is, g∼ =h15, because it is the most intriguing case where different non-trivial behaviours can be produced. We will use the notation E|k| r=⊕p+q=kEp,q r. Since E|k| ∞∼ =Hk dR, it is clear that dim E|k| r≥bk= dim Hk dR for all k, and the equalities hold if and only if Er∼ =E∞. Recall that b1(h15) = 3, b2(h15) = 5 and b3(h15) = 6 (see [28]). For the calculation of the first term E1, that is, the Dolbeault cohomology, by the Serre duality it suffices to study the spaces Ep,q 1=Hp,q ¯ ∂for (p, q) = (1,0),(0,1),(2,0),(1,1),(0,2),(3,0) and (2,1). Let Jbe a complex structure on h15 given in Table 1. If Jis abelian then (B, c) = (0,1) or (1, c) with c6= 1, therefore (18) H1,0 ¯ ∂=h[ω1]i, H2,0 ¯ ∂=h[ω12], δc 0[ω13]i, H3,0 ¯ ∂=h[ω123]i, H0,1 ¯ ∂=h[ω¯ 1],[ω¯ 2],[ω¯ 3]i, H0,2 ¯ ∂=h[ω¯ 1¯ 2],[ω¯ 1¯ 3],[ω¯ 2¯ 3]i, H1,1 ¯ ∂=h(1 −δc 0)[ω1¯ 2],[ω1¯ 3], δc 0[ω2¯ 1],[Bω2¯ 2+ω3¯ 1], δc 0[ω3¯ 2]i, H2,1 ¯ ∂=hδc 0[ω12¯ 1],[ω12¯ 2],[ω12¯ 3],[Bω13¯ 2−cω23¯ 1], δc 0[ω13¯ 3]i, where δc 0is equal to 0 if c6= 0, and equals 1 if c= 0. Since dim E|1| 1= 4 >3 = b1(h15) we get that E16∼ =E∞for any abelian J. When Jis not abelian, i.e. ρ= 1, the Dolbeault cohomology groups are (19) H1,0 ¯ ∂=h[ω1], δB 0δc 0[ω3]i, H2,0 ¯ ∂=h[ω12], δc 0[ω13]i, H3,0 ¯ ∂=h[ω123]i, H0,1 ¯ ∂=h[ω¯ 1],[ω¯ 2]i, H0,2 ¯ ∂=h[ω¯ 1¯ 3],[ω¯ 2¯ 3]i, H1,1 ¯ ∂=h(Bc +δB 0)[ω1¯ 2],[ω1¯ 3+ω2¯ 2],[Bω1¯ 3−ω3¯ 1], δc 0[ω2¯ 1], δc 0[ω3¯ 2]i, H2,1 ¯ ∂=hδc 0[ω12¯ 1],[ω12¯ 2],[c ω12¯ 3+ω13¯ 2],[Bω12¯ 3+ω23¯ 1], δc 0[ω13¯ 3+ω23¯ 2]i, where δB 0has a similar definition as for δc 0above. Notice that the coefficient Bc+δB 0 is non-zero except for B6= 0 and c= 0. Thus, dim E|2| 1≥6>5 = b2(h15) and so E16∼ =E∞also for any non-abelian J. In order to prove (f.1) we need to study independently the abelian and the nonabelian complex structures with c= 0 and B6=ρon h15. We start with the abelian ones. In this case, by Table 1 we can suppose B= 1 and from (18) it follows that the dimensions of E|2| 1and E|3| 1are dim E|2| 1= 9 >5 = b2(h15),dim E|3| 1= 12 >6 = b3(h15). 17 For the following d1-homomorphisms E0,1 1 d1 −→ E1,1 1 d1 −→ E2,1 1 d1 −→ E3,1 1,the classes [ω¯ 3], [ω1¯ 3], [ω3¯ 2], [ω13¯ 3] have linearly independent images. On the other hand, for E0,2 1 d1 −→ E1,2 1 d1 −→ E2,2 1 d1 −→ E3,2 1,the images of the classes [ω¯ 2¯ 3], [ω3¯ 2¯ 3], [ω2¯ 2¯ 3+ω3¯ 1¯ 3] and [ω13¯ 2¯ 3] are also independent. Counting dimensions for E|k| 2we get that dim E|1| 2≤dim E|1| 1−1 = 3 = b1(h15), dim E|2| 2≤dim E|2| 1−4 = 5 = b2(h15), dim E|3| 2≤dim E|3| 1−6 = 6 = b3(h15), dim E|4| 2≤dim E|4| 1−4 = 5 = b4(h15), dim E|5| 2≤dim E|5| 1−1 = 3 = b5(h15). This implies that E2∼ =E∞because necessarily dim E|k| 2=bk(h15) for all k. If ρ= 1 and c= 0, then B6= 1 and by (19) we have dim E|1| 1=b1(h15) + δB 0. So E|1| 1∼ =E|1| ∞when B6= 0. For B= 0, since d1([ω3]) 6= 0 and d1([ω3¯ 1¯ 2¯ 3]) 6= 0, we conclude that dim E|1| 2≤dim E|1| 1−1 = 3 = b1(h15) and dim E|5| 2≤dim E|5| 1−1 = 3 = b1(h15), and therefore, E|k| 2∼ =E|k| ∞if k= 1 or k= 5. Now, for B6= 1 we have that dim E|2| 1= 8 + δB 0>5 = b2(h15), dim E|3| 1= 12 > 6 = b3(h15), dim E|4| 1= 8 + δB 0>5 = b4(h15).In order to conclude that E2∼ =E∞ it suffices to observe that for the following homomorphisms E1,1 1 d1 −→ E2,1 1 d1 −→ E3,1 1, E0,2 1 d1 −→ E1,2 1 d1 −→ E2,2 1 the classes [ω1¯ 3+ω2¯ 2], [ω3¯ 2], [ω13¯ 3+ω23¯ 2], [ω¯ 2¯ 3], [ω3¯ 2¯ 3] and [Bω2¯ 2¯ 3+ω3¯ 1¯ 3] have linearly independent images. For case (f.2), we consider ρ= 1 and |B−1| 6=c6= 0. As dim E|1| 1= 3 = b1(h15), we get that E|1| 1∼ =E|1| ∞. Now, for the map E0,2 2 d2 −→ E2,1 2we have d2([ω¯ 2¯ 3]) = h∂ω2¯ 3+1−¯ B cω3¯ 2i=|B−1|2−c2 c[ω12¯ 2]6= 0, because ω12¯ 26=¯ ∂β2,0+∂γ1,1for any β2,0and any ¯ ∂-closed γ1,1. Hence, b2(h15)≤dim E|2| 3≤dim E|2| 2−1≤dim E|2| 1−1 = 6 −1 = 5 = b2(h15) and we conclude that E|2| ∞∼ =E|2| 36∼ =E|2| 2∼ =E|2| 1. Similarly, d2:E1,2 2−→ E3,1 2is non-zero (for instance, d2([ω3¯ 1¯ 3+Bω2¯ 2¯ 3]) 6= 0). Thus, b3(h15)≤dim E|3| 3≤dim E|3| 2−2≤dim E|3| 1−2 = 8 −2 = 6 = b3(h15) and we conclude that E|3| ∞∼ =E|3| 36∼ =E|3| 2∼ =E|3| 1. By the same argument b4(h15)≤dim E|4| 3≤dim E|4| 2−1≤dim E|4| 1−1 = 6 −1 = 5 = b4(h15) and therefore E|4| ∞∼ =E|4| 36∼ =E|4| 2∼ =E|4| 1. Summing up all the information, we conclude that E1∼ =E26∼ =E3∼ =E∞in case (f.2). For the last case (f.3), we first observe that d1([ω¯ 3]) = −c[ω1¯ 2]−¯ B[ω2¯ 1]. Since this class is zero if and only if c ω1¯ 2+¯ Bω2¯ 1∈¯ ∂(V1,0) = hω1¯ 1, Bω1¯ 2+c ω2¯ 1i, i.e. |B|=c, the map d1:E0,1 1−→ E1,1 1is non-zero. Therefore, dim E|1| 2≤dim E|1| 1−1 = 3, i.e. E|1| 16∼ =E|1| 2∼ =E|1| ∞. Moreover, since d2([ω¯ 2¯ 3]) 6= 0, we deduce that b2(h15)≤dim E|2| 3≤dim E|2| 2−1≤dim E|2| 1−2 = 7 −2 = 5 = b2(h15), so E|2| ∞∼ =E|2| 36∼ =E|2| 26∼ =E|2| 1. Analogously, d2([ω3¯ 1¯ 3+Bω2¯ 2¯ 3]) 6= 0, which implies b3(h15)≤dim E|3| 3≤dim E|3| 2−2≤dim E|3| 1−2 = 8 −2 = 6 = b3(h15), 18 and we conclude that E|3| ∞∼ =E|3| 36∼ =E|3| 2∼ =E|3| 1. We also have b4(h15)≤dim E|4| 3≤dim E|4| 2−1≤dim E|4| 1−2 = 7 −2 = 5 = b4(h15), and therefore E|4| ∞∼ =E|4| 36∼ =E|4| 26∼ =E|4| 1. Consequently, E16∼ =E26∼ =E3∼ =E∞in case (f.3).  Remark 4.2. Let (M= Γ\G, J) be a 6-dimensional nilmanifold endowed with an invariant complex structure Jand suppose that g=h7. In [27, Theorem 4.4] it is proved that there is a dense subset of the space of all invariant complex structures for which the complex nilmanifold admits the structure of principal holomorphic bundle of elliptic curves over a Kodaira surface, but this is not true for all complex structures. In fact, the invariant complex structure Jmay not be compatible with the lattice Γ (see [27, Example 1.14]), so one cannot ensure the existence of the isomorphism (17), and hence of a canonical isomorphism between Ep,q r(g, J) and Ep,q r(M, J), for any invariant Jon the nilmanifold M. However, notice that up to equivalence there is only one complex structure on h7and it can be proved that it satisfies that the sequence degenerates at the first step, i.e. E1(h7)∼ =E∞(h7). In [3] the authors posed the following problem: to construct a compact complex manifold such that E1∼ =E∞and hp,q ¯ ∂=hq,p ¯ ∂for every p, q ∈Nbut for which the ∂¯ ∂-lemma does not hold. Since nilmanifolds do not satisfy the ∂¯ ∂-lemma, unless they are complex tori, the following result provides a solution. Proposition 4.3. Let Jbe any invariant complex structure on a nilmanifold M with underlying Lie algebra isomorphic to h6. Then E1(M)∼ =E∞(M)and the Hodge numbers satisfy h0,0 ¯ ∂(M) = 1, h1,0 ¯ ∂(M) = 2, h0,1 ¯ ∂(M) = 2, h2,0 ¯ ∂(M) = 2, h1,1 ¯ ∂(M) = 5, h0,2 ¯ ∂(M) = 2, h3,0 ¯ ∂(M) = 1, h2,1 ¯ ∂(M) = 5, h1,2 ¯ ∂(M) = 5, h0,3 ¯ ∂(M) = 1, h3,1 ¯ ∂(M) = 2, h2,2 ¯ ∂(M) = 5, h1,3 ¯ ∂(M) = 2, h3,2 ¯ ∂(M) = 2, h2,3 ¯ ∂(M) = 2, h3,3 ¯ ∂(M) = 1. Proof. Any complex structure Jon h6is equivalent to the complex structure given in Table 1, that is, ρ=λ= 1 and D= 0. Its Dolbeault cohomology groups Hp,q ¯ ∂ for (p, q) = (1,0),(0,1),(2,0),(1,1),(0,2),(3,0) and (2,1) are H1,0 ¯ ∂=h[ω1],[ω2]i, H2,0 ¯ ∂=h[ω12],[ω13]i, H3,0 ¯ ∂=h[ω123]i, H0,1 ¯ ∂=h[ω¯ 1],[ω¯ 2]i, H0,2 ¯ ∂=h[ω¯ 1¯ 3],[ω¯ 2¯ 3]i, H1,1 ¯ ∂=h[ω1¯ 2],[ω2¯ 1],[ω2¯ 2],[ω1¯ 3+ω3¯ 2],[ω3¯ 1+ω3¯ 2]i, H2,1 ¯ ∂=h[ω12¯ 2],[ω13¯ 1],[ω12¯ 3+ω23¯ 1],[ω12¯ 3−ω23¯ 2],[ω13¯ 2]i. By Serre duality we get the above Hodge diamond which is symmetric. Moreover, dim E|1| 1= 4 = b1(h6),dim E|2| 1= 9 = b2(h6),dim E|3| 1= 12 = b3(h6), so the Fr¨olicher spectral sequence degenerates at the first step.  19 The following result shows that there are many complex nilmanifolds for which the Fr¨olicher spectral sequence is stable under small deformations of the complex structure. Proposition 4.4. Let M= Γ\Gbe a 6-dimensional nilmanifold endowed with an invariant complex structure J, and let gbe the Lie algebra of G. If g∼ = h1,h3,h6,h8,h9,h10,h11,h12,h13,h14,h16,h− 19 or h+ 26, then dim Ep,q r(M, J)is stable under small deformations of Jfor any p, q and any r≥1. Proof. By [26, Theorem 2.6], all small deformations of the complex structure Jare again invariant complex structures. Proceeding as in the proof of Theorem 4.1, it can be proved that if g6∼ =h2,h4,h5or h15, then dim Ep,q r(M) does not depend on the invariant complex structure on Mfor any p, q and any r≥1, so it is stable under small deformations of J. Remark 4.5. The 6-dimensional nilmanifolds with underlying Lie algebra isomorphic to h2,h4,h5or h15 are the only ones that have both abelian and non-abelian complex structures (see Table 1). More generally, let Mbe a 2n-dimensional nilmanifold, gthe underlying Lie algebra, Jan abelian complex structure and J′a non-abelian invariant complex structure on M. It is well known that Jis abelian if and only if there is a basis {ω1,...,ωn}of invariant forms of type (1,0) satisfying ∂ωj= 0 for 1 ≤j≤n; therefore, by [7] one has that h0,1 ¯ ∂(M, J) = nbecause (17) holds for abelian structures. However, for J′we have dim H0,1 ¯ ∂(g, J′)< n and, if an isomorphism like (17) holds, then the Hodge number satisfies h0,1 ¯ ∂(M, J′)< n. Thus, the existence of Jand J′on a nilmanifold Mmight lead to the non-stability of dim E0,1 1under small deformations. A natural question arises in this context: is the Fr¨olicher spectral sequence stable under small deformations if and only if the nilmanifold does not admit both abelian and non-abelian complex structures? Proposition 4.4 above gives an affirmative answer for n= 3. Next we provide some examples of explicit families of complex structures on nilmanifolds corresponding to h5and h15 along which the Fr¨olicher sequence varies. In Corollaries 5.11 and 5.12 below, further properties of the Fr¨olicher spectral sequence on nilmanifolds are shown. Example 4.6. Let Jbe a non complex-parallelizable and non-abelian complex structure on h5given in Table 1 with non-degenerate Fr¨olicher sequence, i.e. E1≇ E∞for J. We will construct a family of complex structures Jtby deforming the previous one, i.e. J0=J, such that the Fr¨olicher spectral sequence degenerates at the first step for any t6= 0. According to Theorem 4.1, Jhas complex structure equations of the form dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2, for some non-negative λ6= 1, where {ω1, ω2, ω3}is a (1,0)-basis for J. With respect to the real basis {e1,...,e6}given by e1+i e2=ω1,1 1 + λ(e3−e1) + i 1−λ(e2+e4) = ω2, e5+ie6=ω3, the complex structure Jexpresses as Je1=−e2, Je3=−2 1−λe2−1+λ 1−λe4, Je5=−e6, Je2=e1, Je4=−2 1+λe1+1−λ 1+λe3, Je6=e5. 20 For any t∈[0,1 2), consider the complex structure Jtgiven by Jte1=4d(1−λ) α2e1−1−λ2 αe2−2d(1−λ)2 α2e3+8d2(1−λ) α3e4, Jte2=1−λ2 αe1+2d(1−λ2) α2e4, Jte3=−2d (1−λ)2e1−2α (1−λ2)(1−λ)e2−(1+λ)2 αe4, Jte4=−2(1−λ) αe1+2d 1−λ2e2+(1−λ)2 αe3−4d(1−λ) α2e4, Jte5=2d 1−λ2e5−4d2+(1−λ2)2 α(1−λ2)e6, Jte6=α 1−λ2e5−2d 1−λ2e6, where α=p(1 −λ2)2−4d2, and d(t, λ) =          t, if λ= 0, tλ2/4,if λ2∈(0,1/2), t(1 −λ2)/4,if λ2∈[1/2,1), −t(1 −λ2)/4,if λ2>1. Notice that J0=J. Now, the forms ω1 t=1−λ2 αe1+2d(1−λ2) α2e4+i e2, ω2 t=1−λ α(e3−e1)−2d(1−λ) α2e4+i 1−λ2d αe1+e2+(1−λ2)2 α2e4, ω3 t=e5−2d αe6+i1−λ2 αe6, satisfy Jtωk t=i ωk tfor k= 1,2,3, i.e. {ω1 t, ω2 t, ω3 t}is a basis of type (1,0) for Jt. Furthermore, with respect to this basis the complex structure equations are dω1 t=dω2 t= 0, dω3 t=ω12 t+ω1¯ 1 t+λ ω1¯ 2 t+D ω2¯ 2 t, with D=i d(t, λ). According to Theorem 4.1, the Fr¨olicher spectral sequence degenerates if and only if D6= 0, i.e. if and only if t > 0. In conclusion, Jcan be deformed into a non-abelian complex structure with E1∼ =E∞. Corollary 4.7. Let M= Γ\Gbe the nilmanifold underlying the Iwasawa manifold, i.e. g∼ =h5. Let Jbe a non complex-parallelizable and non-abelian complex structure on Mgiven in Table 1 with E16∼ =E∞. Then, Jcan be deformed into an invariant complex structure with degenerate Fr¨olicher spectral sequence. The Lie algebra h15 has a rich complex geometry with respect to the Fr¨olicher sequence and in the next example we construct a family Jtalong which the three cases in (f) of Theorem 4.1 are realized. 21 Example 4.8. On h15, let us consider the real basis {e1,...,e6}given in Theorem 2.1 and the following family of complex structures Jte1=−s3(3 −sin t)(7 + 3 sin t) (5 + sin t)(11 −sin t)e2, Jte3=s3(3 −sin t)(11 −sin t) (5 + sin t)(7 + 3 sin t)e4, Jte5=−s(11 −sin t)(7 + 3 sin t) 3(3 −sin t)(5 + sin t)e6, where t∈R. Let 4ω1 t=p(11 −sin t)(5 + sin t)e1+ip3(3 −sin t)(7 + 3 sin t)e2, 8ω2 t= (5 + sin t)(7 + 3 sin t)e3−ip3(5 + sin t)(3 −sin t)(11 −sin t)(7 + 3 sin t)e4, and 128 ω3 t= (5 + sin t)(7 + 3 sin t)h3(3 −sin t)p(11 −sin t)(5 + sin t)e5 +i(11 −sin t)p3(3 −sin t)(7 + 3 sin t)e6i. Then, {ω1 t, ω2 t, ω3 t}is a (1,0)-basis for Jtsatisfying dω1 t= 0, dω2 t=ω1¯ 1 t, dω3 t=1−sin t 2ω12 t+ 2 ω1¯ 2 t+1 + sin t 4ω2¯ 1 t. If sin t= 1, the coefficient of ω12 tvanishes and therefore Jtis an abelian complex structure that is equivalent to the one given by (ρ, Bt, ct) = (0,1,1 4) (see Lemma 2.14). If sin t6= 1, then we can normalize the coefficient of ω12 tand the complex structure equations can be written in form (14) as dω1 t= 0, dω2 t=ω1¯ 1, dω3 t=ω12 t+4 1−sin tω1¯ 2 t+1 + sin t 2(1 −sin t)ω2¯ 1 t, i.e. they are determined by the triple (ρ, Bt, ct) = 1,4 1−sin t,1+sin t 2(1−sin t). Now, concerning the Fr¨olicher spectral sequence for the family {Jt}t∈R, by Theorem 4.1 (f) we get •If sin t= 1, then (ρt, Bt, ct) = (0,1,1 4) and therefore E16∼ =E26∼ =E3∼ =E∞. •If sin t=−1, then (ρt, Bt, ct) = (1,2,0) and E16∼ =E2∼ =E∞. •If |sin t| 6= 1, E1∼ =E26∼ =E3∼ =E∞. As a consequence of this example, in the following result we show that for r≥2 the dimension of the term Ep,q r(Jt) in general is neither upper nor lower semicontinuous function of t. This is in deep contrast with the case r= 1, as it is well known the upper semicontinuity of the Hodge numbers dim Hp,q ¯ ∂(Jt) with respect to talong a deformation. Corollary 4.9. Let Mbe a nilmanifold with underlying Lie algebra h15 endowed with the invariant complex structures Jtgiven in Example 4.8. Then, dim E0,2 2(Jπ 2) = 3 >2 = dim E0,2 2(Jt),dim E1,1 2(Jπ 2) = 2 <3 = dim E1,1 2(Jt), 22 and dim E0,2 3(Jπ 2) = 2 >1 = dim E0,2 3(Jt),dim E1,1 3(Jπ 2) = 2 <3 = dim E1,1 3(Jt), for any t∈(π 2,3π 2). Therefore, the dimensions of the terms E1,1 2(Jt)and E1,1 3(Jt) are not upper semi-continuous functions of t, and the dimensions of the terms E0,2 2(Jt)and E0,2 3(Jt)are not lower semi-continuous functions of t. Proof. It follows directly from the proof of Theorem 4.1 taking into account that for t=π 2the complex structure lies in case (f.3) and for any t∈(π 2,3π 2) the structures Jtlie in case (f.2).  5. Strongly Gauduchon and balanced Hermitian metrics Let (M, J) be a complex manifold of complex dimension n. A Hermitian metric g on (M, J) can be described by means of a positive definite smooth form Ω on M of bidegree (1,1) with respect to J. We will use this approach in what follows and we will refer to Ω as a Hermitian structure or as a Hermitian metric indistinctly. A Hermitian structure Ω is strongly Gauduchon (sG for short) if ∂Ωn−1is ¯ ∂- exact [22, 23]. In particular, any balanced Hermitian structure (i.e. dΩn−1= 0) is sG, and any sG metric is a Gauduchon metric [18], that is, Ωn−1is ∂¯ ∂-closed or equivalently the Lee form is co-closed. Next we suppose that (M= Γ\G, J) is a nilmanifold endowed with an invariant complex structure. It is proved in [15] that (M= Γ\G, J) has a balanced metric if and only if it has an invariant one. Moreover, by using the symmetrization process given in [5] (see also [15], [30] and [32, Proposition 3.2]) one easily arrives at: Proposition 5.1. (M= Γ\G, J)has an sG metric if and only if it has an invariant one. Therefore, the existence of sG metrics on (M= Γ\G, J) is reduced to the existence at the Lie algebra level gof G. Corollary 5.2. Let Ωbe an invariant Hermitian structure on (M= Γ\G, J). If Jis abelian, then Ωis sG if and only if it is balanced. Proof. Let gbe the Lie algebra of G. First we prove that ¯ ∂(Vn,k(g∗)) = 0 for every 1≤k≤n. Let us consider a decomposable form α∈Vn,k(g∗) given by α=β∧γ, where β∈Vn,0(g∗) and γ∈V0,k(g∗). Since gis nilpotent and Jis abelian, one has that dβ = 0 and dγ ∈V1,k(g∗), so in particular βand γare ¯ ∂-closed. Hence, ¯ ∂α = (¯ ∂β)∧γ+ (−1)nβ∧(¯ ∂γ) = 0. Now, the statement in the corollary follows directly from Proposition 5.1 and from the previous property for k=n−2, i.e. ¯ ∂(Vn,n−2(g∗)) = 0.  From now on we consider n= 3. Proposition 5.3. Let M= Γ\Gbe a 6-dimensional nilmanifold endowed with an invariant complex structure J. There exists an sG metric on (M= Γ\G, J)if and only if the Lie algebra gof Gis isomorphic to h1,...,h6or h− 19. Proof. By Proposition 5.1 it suffices to study the invariant case. By [30], the fundamental 2-form of any J-Hermitian metric is given by (20) 2 Ω = i(r2ω1¯ 1+s2ω2¯ 2+t2ω3¯ 3) + uω1¯ 2−¯uω2¯ 1+vω2¯ 3−¯vω3¯ 2+zω1¯ 3−¯zω3¯ 1, 23 where coefficients r2, s2, t2are non-zero real numbers and u, v, z ∈Csatisfy r2s2> |u|2,s2t2>|v|2,r2t2>|z|2and r2s2t2+ 2Re (i¯u¯vz)> t2|u|2+r2|v|2+s2|z|2. Let us start with the non-nilpotent case. From (16) 2∂Ω = (iǫv ∓iz)ω12¯ 1∓iv ω12¯ 2+ (u−¯u−ǫ t2)ω13¯ 1+ (is2±t2)ω13¯ 2+ v ω13¯ 3+ (is2∓t2)ω23¯ 1 and therefore 4∂Ω∧Ω = iǫ(s2t2−|v|2)±(t2u+t2¯u+iv¯z−i¯vz)ω123¯ 1¯ 2+uv −is2zω123¯ 1¯ 3. Direct computations show that ¯ ∂(V3,1(g∗)) = hω123¯ 1¯ 3i. If the Hermitian structure (J, Ω) is sG then ∓iǫ(s2t2−|v|2) = t2(u+ ¯u) + iv¯z−i¯vz. Since the left-hand side is purely imaginary and the right-hand side is real, we get that ǫ= 0 and therefore g∼ =h− 19. For the nilpotent case, let us consider the general complex equations (2). Now, the fundamental 2-form of any J-Hermitian metric is given also by (20). Using [30, Lemma 17 and Proposition 25], we get 4∂Ω∧Ω=(1 −ǫ)¯ A(s2t2−|v|2) + ¯ B(it2u+ ¯vz)−¯ C(it2¯u−v¯z) + (1 −ǫ)¯ D(r2t2−|z|2)ω123¯ 1¯ 2−ǫ(s2t2−|v|2)ω123¯ 1¯ 3. It is straightforward to verify that ¯ ∂(V3,1(g∗)) = hρ ω123¯ 1¯ 2i, and therefore, if the Hermitian structure (J, Ω) is sG then ǫ= 0, i.e. g∼ =hifor i= 1,...,6. Moreover, if in addition ρ= 1, then any J-Hermitian structure is sG. In conclusion, if there exists an sG metric then g∼ =h1,...,h6or h− 19. The converse follows directly from [30, Theorem 26] because these Lie algebras admit balanced Hermitian metrics.  Remark 5.4. From the proof of the previous proposition it follows that on h2,h4,h5 and h6, if Jis a non-abelian nilpotent complex structure then any invariant JHermitian metric is sG. This is in contrast with h− 19, where for any complex structure the space of balanced metrics is strictly contained in the space of sG metrics, and moreover there are Hermitian metrics which are not sG. For instance, consider a Hermitian metric on h− 19 given by Ω = i 2ω1¯ 1+ (u2+z2+ 1)i ω2¯ 2+ (u2+z2+ 1)i ω3¯ 3+u 2(ω1¯ 2−ω2¯ 1) + z 2(ω1¯ 3−ω3¯ 1), that is, in (20) we take r= 1, v= 0, uand zreal and s2=t2= 2(u2+z2+ 1): •if u=z= 0 then the metric is balanced; •if u= 0 and z6= 0 then the metric is sG but not balanced; •if u6= 0 then the metric is not sG. Notice that this indicates a contrast between the sG and SKT geometries, since by [16] the existence of an SKT structure on a 6-dimensional nilpotent Lie algebra depends only on the complex structure. There exist compact complex manifolds having sG metrics but not admitting any balanced metric [24, Theorem 1.8]. Next we show the general situation for nilmanifolds in dimension 6. 24 Proposition 5.5. Let M= Γ\Gbe a 6-dimensional nilmanifold with an invariant complex structure Jsuch that (M= Γ\G, J)does not admit balanced metrics. If (M= Γ\G, J)has sG metric, then Jis non-abelian nilpotent and gis isomorphic to h2,h4or h5. Moreover, according to the classification in Table 1, such a Jis given by: Re D+ (Im D)2≥1 4on h2;Re D≥1 4on h4; and λ= 0,Im D6= 0 or λ=Im D= 0,Re D≥0on h5. Proof. Any complex structure on h6or h− 19 admits balanced metrics. From [32] we have that only h3and h5have abelian complex structures Jadmitting balanced metric. In fact, any such Jon h5admits balanced Hermitian metrics, whereas for h3 the complex structure must be equivalent to the choice of (−)-sign in Table 1. From Corollary 5.2, it remains to study the non-abelian nilpotent complex structures J on h2,h4and h5. Since any such Jadmits sG metrics by Remark 5.4, next we show which of them do not admit balanced metrics. In the three cases the complex equations are of the form (21) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+D ω2¯ 2. A similar argument as in the proof of [32, Proposition 2.3] shows that, up to equivalence, the fundamental 2-form of any J-Hermitian metric is given by 2 Ω = i(ω1¯ 1+s2ω2¯ 2+t2ω3¯ 3) + u ω1¯ 2−¯u ω2¯ 1, where s2>|u|2and t2>0. If D=x+iy and u=u1+iu2, the balanced condition is (22) s2+x+iy =u2λ+iu1λ. We distinguish several cases depending on the values of λ. If λ6= 0 then Ω is balanced if and only if u1=y/λ and u2= (s2+x)/λ. The condition s2>|u|2is equivalent to s4+ (2x−λ2)s2+x2+y2<0 and it is easy to see that a non-zero ssatisfying this condition exists if and only if the discriminant of the previous equation as a second degree equation in s2is positive, i.e. (23) λ4−4xλ2−4y2>0. According to Table 1, non-abelian complex structures on h2have λ= 1. In this case (23) reads as x+y2<1/4, which means that any Jsuch that x+y2≥1 4 has no balanced metrics. Similarly, for h4any Jsuch that x≥1 4does not admit balanced metric. For h5and λ6= 0 we have that x= 0 by Table 1. Thus, there is no balanced metrics if and only if λ4≤4y2. Since y≥0, this is equivalent to λ2≤2y. However, none of the three cases detailed in Table 1 verifies that λ2≤2y, and therefore any complex structure on h5with λ6= 0 admits balanced metrics. Finally, in the case λ= 0 on h5we get that the balanced condition (22) reduces to y= 0 and s2=−x > 0. From Table 1 we have that 0 <1+4x, i.e. x∈(−1 4,∞). Therefore, if y6= 0 or y= 0, x ≥0 then there are no balanced metrics.  As pointed out by Popovici [24], the degeneration of the Fr¨olicher sequence at E1 and the existence of sG metrics are unrelated. From the study of the sG geometry above and from Theorem 4.1 we get: Theorem 5.6. Let M= Γ\Gbe a 6-dimensional nilmanifold endowed with an invariant complex structure J. If there exists an sG metric then the Fr¨olicher 25 spectral sequence degenerates at the second level, i.e. E2(M)∼ =E∞(M). Moreover, if there exists an sG metric and g6∼ =h5, then E1(M)∼ =E∞(M). Proof. By Proposition 5.3, the Lie algebra gunderlying M= Γ\Gmust be isomorphic to h1,...,h6or h− 19, so Theorem 4.1 implies that the Fr¨olicher sequence degenerates at the second level. The last assertion follows directly by taking into account Corollary 5.2 and Table 3 below.  It is interesting whether this result holds in general, that is: Question 5.7. Does the Fr¨olicher spectral sequence degenerate at the second step for any compact complex manifold Mof complex dimension 3admitting an sG metric? In the following table we show the complex structures J, up to equivalence, on h1,...,h6that admit balanced Hermitian metrics. The classification follows from the proof of Proposition 5.5. Table 3: Classification of nilpotent complex structures admitting balanced metrics gAbelian structures Non-Abelian Nilpotent structures h1dω2= 0, dω3= 0 — h2—dω2= 0, dω3=ω12 +ω1¯ 1+ω1¯ 2+ (x+iy)ω2¯ 2, y > 0, x +y2<1 4 h3dω2= 0, dω3=ω1¯ 1−ω2¯ 2— h4—dω2= 0, dω3=ω12 +ω1¯ 1+ω1¯ 2+x ω2¯ 2, x < 1 4, x 6= 0 dω2= 0, dω3=ω12 dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+ (x+iy)ω2¯ 2, dω2= 0,with (λ, x, y) satisfying one of: h5dω3=ω1¯ 1+ω1¯ 2+x ω2¯ 2,•λ=y= 0, x ∈−1 4,0; 0≤x < 1 4•0< λ2<1 2,0≤y < λ2 2, x = 0; •1 2≤λ2<1,0≤y < 1−λ2 2, x = 0; •λ2>1,0≤y < λ2−1 2, x = 0. h6—dω2= 0, dω3=ω12 +ω1¯ 1+ω1¯ 2 Motivated by [24, Theorem 1.9] next we study the relation between the degeneration of the Fr¨olicher spectral sequence and the existence of sG or balanced metrics. The possibilities are well illustrated in the following deformations of the complex structure corresponding to λ=x=y= 0 on a nilmanifold with underlying Lie algebra h5. Example 5.8. Let us consider the Lie algebra h5with the real basis {e1,...,e6} described in Theorem 2.1. Let us consider the complex structure J0,0given by J0,0e1=−e2, J0,0e3=−2e2−e4, J0,0e5=−e6, J0,0e2=e1, J0,0e4=−2e1+e3, J0,0e6=e5.