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

a Xi :1111.5873 5 [ma h.DG] 10 Oc 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 Abs ac . We classi y in a ian complex s uc u es on 6-dimensional nilman- i olds up o equi alence. As an applica ion, he beha iou o he associa ed F ¨oliche sequence is s udied as well as i s ela ion o he exis ence o s ongly Gauduchon me ics. We also show ha he s ongly Gauduchon p ope y and he balanced p ope y a e no closed unde holomo phic de o ma ion. 1. In oduc ion Le gbe a Lie algeb a endowed wi h an endomo phism J:g−→ gsuch ha J2=−Id. The endomo phism Jis a complex s uc u e i he in eg abili y condi ion [JX, JY ] = J[JX, Y ] + J[X, JY ] + [X, Y ] is sa is ied o any X, Y ∈g; equi alen ly, he i-eigenspace g1,0o Jin gC=g⊗RC is a complex subalgeb a o gC. Nilpo en Lie algeb as gadmi ing a complex s uc- u e we e classi ied by Salamon [28] up o dimension 6. Mo e ecen ly, And ada, Ba be is and Do i classi ied in [2] he 6-dimensional Lie algeb as gha ing a com- plex s uc u e Jo abelian ype, ha is, he complex subalgeb a g1,0is abelian, o equi alen ly [JX, JY ] = [X, Y ] o any X, Y ∈g. A ela ed ques ion is o de e mine he complex s uc u es on a gi en Lie algeb a gup o isomo phism in he ollowing sense. Two complex s uc u es Jand J′on g a e equi alen i he e exis s an au omo phism F:g−→ go he Lie algeb a such ha J=F−1◦J′◦F. The la e condi ion is equi alen o say ha F, ex ended o gC, sa is ies F(gJ 1,0)⊂gJ′ 1,0. I C(g) deno es he space o complex s uc u es on g hen C(g)/Au (g) pa ame izes he equi alence classes o complex s uc u es on g. A classi ica ion o abelian complex s uc u es in dimension 6 is gi en in [2]. Some pa ial esul s on nilpo en Lie algeb as can be ound in se e al pape s [6, 19, 30, 31], al hough o ou knowledge he e is no comple e classi ica ion o complex s uc u es on 6-dimensional nilpo en Lie algeb as. This is ou i s goal he e. The classi ica ion o complex s uc u es on nilpo en Lie algeb as p o ides a classi ica ion o in a ian complex s uc u es on nilmani olds. Le M= Γ Gbe a nilmani old, i.e. a compac quo ien o a simply-connec ed nilpo en Lie g oup G by a la ice Γ o maximal ank. I Jis a complex s uc u e on he Lie algeb a go G, hen i gi es ise o a le -in a ian complex s uc u e on Gwhich descends o a complex s uc u e on he quo ien Min a na u al way. Se e al in e es ing aspec s o his complex geome y ha e been in es iga ed, as o ins ance he Dolbeaul cohomology [7, 13, 26], complex de o ma ions [6, 8, 20, 27] o he exis ence o special He mi ian me ics [16, 30]. Recen ly, i is p o ed in [4] ha he canonical 1 2 bundle o any complex nilmani old is holomo phically i ial and some applica ions o hype complex geome y a e gi en. As a i s applica ion o he classi ica ion o complex s uc u es we s udy he be- ha iou o he F ¨oliche sequence [17]. Recall ha he F ¨oliche sequence E (M, J) o a complex mani old (M, J) is he spec al sequence associa ed o he double com- plex (Ωp,q(M, J), ∂, ¯ ∂), whe e ∂+¯ ∂=dis he decomposi ion, wi h espec o J, o he ex e io di e en ial d. The i s e m E1(M, J) is p ecisely he Dolbeaul coho- mology o (M, J) and a e a ini e numbe o s eps he sequence con e ges o he de Rham cohomology o M. The i s examples o compac complex mani olds o which E26∼ =E∞we e independen ly ound in [9] and [21]. The examples in [9] a e complex nilmani olds o complex dimension 3, which is he lowes possible dimen- sion o which he F ¨oliche sequence can be non-degene a e a E2. Mo e ecen ly, Rollenske has cons uc ed in [25] complex nilmani olds o which he sequence {E } can be a bi a ily non-degene a e. The beha iou o he F ¨oliche sequence has been s udied o some o he complex mani olds [14, 29], bu as a as we know i s gene al beha iou o complex nilmani olds has no been s udied, al hough some pa ial e- sul s can be ound in [10, 11, 12]. He e we s udy he F ¨oliche spec al sequence o gene al in a ian complex s uc u es on a 6-dimensional nilmani old. A ema kable consequence o his s udy is he exis ence o a compac complex mani old on which he ∂¯ ∂-lemma ails bu E1∼ =E∞and he Hodge diamond is symme ic. As a second applica ion o he classi ica ion o complex s uc u es we conside s ongly Gauduchon (sG o sho ) me ics in he sense o Popo ici [22, 23]. Any balanced He mi ian me ic is sG and any sG me ic is a Gauduchon me ic [18]. In [24] he ela ion be ween he degene a ion o he F ¨oliche sequence a E1and he exis ence o sG me ics is s udied, showing ha hese wo no ions a e un ela ed. We s udy he exis ence o sG o balanced me ics on 6-nilmani olds in ela ion o he gene al beha iou o he F ¨oliche sequence. Mo eo e , Popo ici p o ed in [23] ha he sG p ope y o compac complex mani olds is open unde holomo phic de o ma ions, and conjec u ed in [24] ha he sG p ope y and he balanced p op- e y o compac complex mani olds a e closed unde holomo phic de o ma ions. We cons uc a coun e example o bo h closedness conjec u es. The pape is s uc u ed as ollows. In Sec ion 2 we i s e iew some gene al ac s abou complex s uc u es on a 6-dimensional nilpo en Lie algeb a g. By [28] such gmus be isomo phic o h1,...,h16,h− 19 o h+ 26 (see Theo em 2.1 o a desc ip ion o he Lie algeb as). O special in e es is h5because i co esponds o he eal Lie algeb a unde lying he Iwasawa mani old, whose complex geome y is s udied in [19]. Fo he i s six een classes he complex s uc u e is necessa ily o nilpo en ype in he sense o [13]. We classi y he non-abelian nilpo en complex s uc u es on 2-s ep and 3-s ep nilpo en Lie algeb as in Sec ions 2.1 and 2.2, espec i ely. Then, using he classi ica ion o non-nilpo en complex s uc u es ob ained in [31] as well as he classi ica ion o abelian s uc u es gi en in [2], we p esen in Tables 1 and 2 o Sec ion 3 he comple e classi ica ion o complex s uc u es on 6-dimensional nilpo en Lie algeb as up o equi alence. Since Jequi alen o J′implies ha he e ms in he associa ed F ¨oliche se- quences a e isomo phic, as an applica ion we s udy he gene al beha iou o he F ¨oliche sequence E (Γ G, J) in Sec ion 4 (see Theo em 4.1 o de ails). We ind ha E26∼ =E∞i and only i he unde lying Lie algeb a g∼ =h13,h14 o h15. Mo e- o e , E1∼ =E26∼ =E3∼ =E∞ o any Jwhen g∼ =h13 o h14. In con as , h15 has a ich complex geome y wi h espec o F ¨oliche sequence because i admi s 3 complex s uc u es o which E16∼ =E2∼ =E∞,E1∼ =E26∼ =E3∼ =E∞o e en E16∼ =E26∼ =E3∼ =E∞. In Example 4.8 we gi e a amily J o non-equi alen complex s uc u es on h15 along which he F ¨oliche sequence has hese h ee be- ha iou s. We also show ha a nilmani old wi h unde lying Lie algeb a h6has a complex s uc u e wi h degene a e F ¨oliche sequence and sa is ying hp,q ¯ ∂=hq,p ¯ ∂ o e e y p, q ∈N, which p o ides an answe o a ques ion ecen ly posed in [3] (see P oposi ion 4.3). In sec ion 5 we s udy he exis ence o sG me ics on 6-dimensional nilmani olds endowed wi h an in a ian complex s uc u e and show ha he unde lying Lie algeb a mus be isomo phic o h1,...,h6o h− 19. I is also p o ed ha he exis ence o sG me ic implies he degene a ion o he F ¨oliche sequence a E2. Using [32] we gi e in P oposi ion 5.5 a classi ica ion o complex s uc u es ha ing sG me ics bu no admi ing any balanced me ic, as well as a classi ica ion o nilpo en complex s uc u es admi ing balanced me ic (see Table 3). Based on he complex geome y o he Lie algeb a h4, in Theo em 5.9 we show ha nei he he sG p ope y no he balanced p ope y o compac complex mani olds a e closed unde holomo phic de o ma ion. 2. Nilpo en complex s uc u es on 6-dimensional nilpo en Lie algeb as Gi en a Lie algeb a g, le g∗ Cbe he dual o he complexi ica ion gCo g. I J:g−→ g is an endomo phism such ha J2=−Id, hen he e is a na u al big adua ion induced on V∗g∗ C=⊕p,q Vp,q(g∗), whe e he spaces V1,0(g∗) and V0,1(g∗), which we shall also deno e by g1,0and g0,1, a e he eigenspaces o he eigen alues ±i o Jas an endomo phism o g∗ C, espec i ely. Now, i d:V∗g∗ C−→ V∗+1 g∗ Cis he ex ension o he complexi ied ex e io algeb a o he usual Che alley-Eilenbe g di e en ial, hen i is well known ha Jis a complex s uc u e i and only i π0,2◦d|g1,0≡0, whe e π0,2:V2g∗ C−→ V0,2(g∗) deno es he canonical p ojec ion. We shall ocus on nilpo en Lie algeb as (NLA o sho ). Salamon has p o ed in [28] he ollowing equi alen condi ion o he in eg abili y o Jon a 2n-dimensional NLA g:Jis a complex s uc u e on gi and only i g1,0has a basis {ωj}n j=1 such ha dω1= 0 and dωj∈ I(ω1,...,ωj−1), o j= 2,...,n, whe e I(ω1,...,ωj−1) is he ideal in V∗g∗ Cgene a ed by {ω1,...,ωj−1}. Recall ha a complex s uc u e Jon a 2n-dimensional NLA gis nilpo en [13] i he e exis s a basis {ωj}n j=1 o g1,0sa is ying dω1= 0 and (1) dωj∈^2hω1,...,ωj−1, ω1,...,ωj−1i, o j= 2,...,n. An impo an special class o nilpo en complex s uc u es is he abelian class con- sis ing o hose s uc u es Jsa is ying [JX, JY ] = [X, Y ], o all X, Y ∈g, o equi alen ly d(g1,0)⊂V1,1(g∗). They a e also cha ac e ized by he ac ha he subalgeb a g1,0is abelian. In six dimensions, he classi ica ion o NLAs in e ms o he di e en ypes o complex s uc u es ha hey admi is as ollows. 4 Theo em 2.1. [28, 30] Le gbe an NLA o dimension 6. Then, ghas a complex s uc u e i and only i i is isomo phic o one o he ollowing Lie algeb as: 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). Mo eo e : (a)Any complex s uc u e on h− 19 and h+ 26 is non-nilpo en ; (b)Fo 1≤k≤16, any complex s uc u e on hkis nilpo en ; (c)Any complex s uc u e on h1,h3,h8and h9is abelian; (d)The e exis bo h abelian and non-abelian nilpo en complex s uc u es on h2,h4,h5and h15; (e)Any complex s uc u e on h6,h7,h10,h11,h12,h13,h14 and h16 is no abelian. Rema k 2.2. He e we use he usual no a ion, i.e. o ins ance h2= (0,0,0,0,12,34) means ha he e is a basis {ej}6 j=1 sa is ying de1=de2=de3=de4= 0, de5=e1∧e2,de6=e3∧e4; equi alen ly, he Lie b acke is gi en in e ms o i s dual basis {ej}6 j=1 by [e1, e2] = −e5, [e3, e4] = −e6. Le gbe a Lie algeb a endowed wi h wo complex s uc u es Jand J′. We ecall ha Jand J′a e said o be equi alen i he e is an au omo phism F:g−→ go he Lie algeb a such ha J′=F−1◦J◦F, ha is, Fis a linea au omo phism such ha F∗:g∗−→ g∗commu es wi h he Che alley-Eilenbe g di e en ial d and Fcommu es wi h he complex s uc u es Jand J′. The la e condi ion is equi alen o say ha F∗, ex ended o he complexi ied ex e io algeb a, p ese es he big adua ions induced by Jand J′. No ice ha i g1,0 Jand g1,0 J′deno e he (1,0)-subspaces o g∗ Cassocia ed o Jand J′, hen he complex s uc u es Jand J′a e equi alen i and only i he e is a C-linea isomo phism F∗:g1,0 J−→ g1,0 J′such ha d◦F∗=F∗◦d. In dimension 6, by Theo em 2.1, i he NLA gadmi s complex s uc u es hen all o hem a e ei he nilpo en o non-nilpo en . The classi ica ion o abelian complex s uc u es up o equi alence is ob ained in [2], whe eas he non-nilpo en complex s uc u es a e classi ied in [31] (see Sec ion 3 o de ails). The e o e, i emains o s udy he equi alence classes o non-abelian nilpo en complex s uc u es. In o de o p o ide such classi ica ion, ou s a ing poin is he ollowing educ ion o he nilpo en condi ion (1). P oposi ion 2.3. [30] Le Jbe a nilpo en complex s uc u e on an NLA go dimension 6. The e is a basis {ωj}3 j=1 o g1,0sa is ying (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, whe e A, B, C, D ∈Cand ǫ, ρ ∈ {0,1}. 5 He e ωjk ( esp. ωjk) means he wedge p oduc ωj∧ωk( esp. ωj∧ωk), whe e ωkindica es he complex conjuga ed o ωk. F om now on, we shall use a simila abb e ia ed no a ion o “basic” o ms o a bi a y bideg ee. No ice ha in he equa ions (2) he complex s uc u e is no abelian i and only i ρ= 1. Nex we s udy he 2-s ep and 3-s ep cases in Sec ions 2.1 and 2.2, espec i ely. 2.1. Non-abelian complex s uc u es in he 2-s ep case. Any 6-dimensional 2-s ep NLA ghas i s Be i numbe a leas 3, and i i is equal o 3 hen necessa ily he coe icien ǫin (2) is non-ze o. We conside i s ly ǫ= 0, i.e. he Lie algeb a has i s Be i numbe ≥4, and we will inish he sec ion by conside ing he emaining case ǫ= 1. The ollowing p oposi ion p o ides a u he educ ion o he equa ions (2) when ǫ= 0 and he s uc u e is no complex-pa allelizable. Recall ha Jis complex- pa allelizable i [JX, Y ] = J[X, Y ], o all X, Y ∈g, o equi alen ly d(g1,0)⊂ V2,0(g∗). These s uc u es a e he na u al complex s uc u es o complex Lie alge- b as, and in six dimensions hey co espond o ǫ=A=B=C=D= 0 and he possible Lie algeb as a e h1( o ρ= 0) and h5( o ρ= 1). P oposi ion 2.4. Le Jbe a complex s uc u e on a 2-s ep NLA go dimension 6 wi h i s Be i numbe ≥4. I Jis no complex-pa allelizable, hen he e is a basis {ωj}3 j=1 o g1,0such ha (3) dω1=dω2= 0, dω3=ρ ω12 +ω1¯ 1+λ ω1¯ 2+D ω2¯ 2, whe e ρ∈ {0,1},λ∈Rsuch ha λ≥0, and D∈Cwi h Im D≥0. Mo eo e , i we deno e x=Re Dand y=Im D, hen: (i) I λ=ρ, hen he Lie algeb a gis isomo phic o (i.1) h2, o y > 0; (i.2) h3, o ρ=y= 0 and x6= 0; (i.3) h4, o ρ= 1,y= 0 and x6= 0; (i.4) h6, o ρ= 1 and x=y= 0; (i.5) h8, o ρ=x=y= 0. (ii) I λ6=ρ, hen he Lie algeb a gis isomo phic o (ii.1) h2, o 4y2>(ρ−λ2)(4x+ρ−λ2); (ii.2) h4, o 4y2= (ρ−λ2)(4x+ρ−λ2); (ii.3) h5, o 4y2<(ρ−λ2)(4x+ρ−λ2). P oo . In [30, Lemma 11] i is p o ed ha unde hese condi ions he e is a basis {σj}3 j=1 o g1,0such ha (4) dσ1=dσ2= 0, dσ3=ρ σ12 +σ1¯ 1+B σ1¯ 2+D σ2¯ 2, whe e B, D ∈Cand ρ∈ {0,1}. I B6= 0 hen we can ake any non-ze o solu ion zo ¯zB |B|=z, and he equa- ions (4) educe o (3) wi h λ=|B|wi h espec o he new basis {ω1=z σ1, ω2= ¯z σ2, ω3=|z|2σ3}. Conside now B=λwi h λ∈R≥0in (4). I D6= 0, hen wi h espec o he new basis {ω1=−¯ D σ2, ω2=σ1+λ σ2, ω3=¯ D σ3}we ge (3) wi h ¯ Dins ead o D. Finally, he second pa o he p oposi ion ollows di ec ly om [30, P oposi- ion 13].  6 F om now on we conside ρ= 1. By P oposi ion 2.4 any wo complex s uc u es on he Lie algeb a h6a e equi alen . Thus, i emains o classi y up o equi alence he non-abelian s uc u es Jon h2,h4and h5. Any such Jis iden i ied wi h a iple (1, λ, D) h ough equa ions (3) wi h ρ= 1, λ≥0 and Im D≥0. We will say ha wo iples (1, λ, D) and (1, λ′, D′) a e equi alen , deno ed by (1, λ, D)∼(1, λ′, D′), i he co esponding s uc u es Jand J′a e equi alen . So, he p oblem educes o classi y iples (1, λ, D) up o equi alence. Lemma 2.5. Le us conside wo iples (1, λ, D)and (1, , E)as abo e. (i) I D= 0 hen, (1, , E)∼(1, λ, 0) i and only i =λand E= 0. (ii) I D6= 0 hen, (1, , E)∼(1, λ, D)i and only i he e exis non-ze o complex numbe s e, such ha E=De/¯eand (5) | |2 ¯e−1(¯ D¯e−De)2= (λ¯ − )(λ¯ D¯e − De ¯ ). P oo . The s uc u e equa ions co esponding o he iples (1, λ, D) and (1, , E) a e dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λω1¯ 2+Dω2¯ 2, dσ1=dσ2= 0, dσ3=σ12 +σ1¯ 1+ σ1¯ 2+Eσ2¯ 2, whe e λ, ≥0 and Im D, Im E≥0. Then (1, , E)∼(1, λ, D) i and only i he e exis s an au omo phism o he Lie algeb a p ese ing he complex equa ions, i.e. he e is (mij )∈GL(3,C) such ha σi=P3 j=1 mij ωjand dσi= 3 X j=1 mij dωj, i = 1,2,3. These condi ions a e equi alen o σ1=a ω1+b ω2, σ2=c ω1+ ω2, σ3=m31 ω1+m32 ω2+e ω3, and (6)                (I) e=a −bc, (II) e=|a|2+ a¯c+E|c|2, (III) λe =a¯ b+ a ¯ +Ec ¯ , (IV) 0 = ¯ab + b¯c+E¯c , (V) De =|b|2+ b ¯ +E| |2. No ice ha m13 =m23 = 0, e6= 0 and he coe icien s m31 and m32 a e no ele an . I is s aigh o wa d o see ha coe icien mus be non-ze o (o he wise λ= and D=E) and so we can exp ess aas a=e+bc . Fi s o all, le us suppose ha D= 0. Replacing ain (IV) and using (V) we ob ain ha b= 0 and he e o e E= 0 by equa ion (V). Combining (I) and (III) we ge ha λ = ¯ . Since λand a e eal non-nega i e numbe s, we conclude ha λ= , i.e. (1, λ, 0) de ines an equi alence class o e e y λ≥0. This comple es he p oo o (i). 7 We suppose nex ha D6= 0. In o de o sol e (6) we ans o m i in o an equi alen sys em by doing he ollowing subs i u ions. Replacing ain equa ion (IV) and using (V) we can exp ess ¯c=−b¯e De. Nex , in (II) we can subs i u e aand cand use again (V) o ob ain ha De =E¯e, which implies in pa icula |D|=|E|. No ice ha since D6= 0 we can assume E6=¯ Dby P oposi ion 2.4. Now, ¯c=−b/E. P oceeding in a simila way in equa ion (III) we ge ¯ b=λ − ¯ 1−D/ ¯ E. Finally, using he exp essions o a,b,cabo e, equa ion (V) is equi alen o (5). The e o e, gi en e, ∈C−{0}sa is ying De =E¯eand (5), i is always possible o ind a, b, c ∈Csuch ha sys em (6) is sa is ied.  Rema k 2.6. As a consequence o Lemma 2.5 (ii), when D6= 0 a necessa y condi ion o (1, , E) o be equi alen o (1, λ, D) is ha |D|=|E|. Mo eo e , o ind an equi alen complex s uc u e (1, , E) i su ices o ind ≥0 and e, ∈ C−{0}sa is ying (5), because Eis necessa ily gi en by E=De/¯e. Co olla y 2.7. Le E6=¯ D. I (1, , E)∼(1, λ, D) hen, =λi and only i E=D. P oo . By hypo hesis Dcanno be ze o, so we a e in case (ii) o Lemma 2.5. Suppose i s ha λ= in (5), i.e. (¯ D¯e−De)2| |2 ¯e−1=λ2(¯ − )( ¯ D¯e −De ¯ ). The igh hand side o he p e ious equali y is a eal numbe . I i is ze o hen e=| |2(o he wise De =¯ D¯ewould imply E=¯ D); hus, eis a eal numbe and since E=De/¯ewe conclude ha D=E. On he o he hand, i i is a non-ze o eal numbe , hen | |2 ¯e−1 mus be a eal numbe and hen e∈Rand again D=E. Con e sely, le us suppose ha E=D6= 0. In his case e∈Rand by (5) we can exp ess i as e=| |2−(λ¯ − )(λ¯ D − D ¯ ) (¯ D−D)2. No ice ha by hypo hesis D6=¯ E=¯ D. To ensu e ha e∈Ri mus happen ha (λ¯ − )(λ¯ D − D ¯ )∈Ro equi alen ly, | |2(λ2− 2)( ¯ D−D) = 0. As (¯ D−D)6= 0 he only possibili y o sol e he p e ious equa ion is λ= . F om he p e ious esul s i ollows ha i emains o conside he case when D6= 0 and λ6= . The nex lemma p o ides a simpli ica ion o equa ion (5). Lemma 2.8. Le us suppose ha λ6= ,D=x+iy 6= 0 and e∈C−{0}. Then, (1, λ, D)∼(1, , De/¯e)i and only i (7) 4y2−( 2−λ2)(4x+ 2−λ2)≥0. 8 P oo . By Lemma 2.5 (ii), we know ha (1, λ, D)∼(1, , De/¯e) i and only i (5) is sa is ied. This condi ion eads, wi h espec o H=De, as (¯ H−H)2¯ D| |2−¯ H=¯ H(λ¯ − )(λ ¯ H− ¯ H). Taking eal and imagina y pa s in he exp ession abo e we ob ain (8)          4H2 2(H1−x| |2) = | |2( 2−λ2)H2 2+| |2( 2+λ2)H2 1 −2λ ( 2 1− 2 2)H2 1−4λ H1H2 1 2, 4H2 2(y| |2−H2) = 2λH2 H1( 2 1− 2 2) + 2 H2 1 2−λ| |2H1, whe e H=H1+iH2and = 1+i 2. Obse e ha H26= 0, o he wise we ge a con adic ion using he i s equa ion o (8). Subs i u ing he second equa ion o (8) in he i s one and eplacing Hby De, we can exp ess he sys em (8) as    e2 1( 2−λ2) + 4ye1e2+e2 2( 2−λ2+ 4x) = 0, 2H2(y| |2−H2) = λ H1( 2 1− 2 2) + 2 H2 1 2−λ| |2H1, (9) whe e e=e1+ie2. To sol e he i s equa ion in (9) as a second deg ee equa ion in e1we need he disc iminan o be g ea e han o equal o 0, i.e. 4y2−( 2−λ2)(4x+ 2−λ2)≥0, which is p ecisely condi ion (7). Now, suppose ha (7) holds. Then we ob ain ha e1=e2β λ2− 2, e =e2β λ2− 2+i, whe e β= 2y+p4y2−( 2−λ2)(4x+ 2−λ2) and e2is de e mined by he second equa ion in (9).  Co olla y 2.9. Le us suppose ha λ6= and D=x+iy 6= 0. I (7) holds hen (1, λ, D)∼1, , D β2−(λ2− 2)2 β2+ (λ2− 2)2+2β(λ2− 2) β2+ (λ2− 2)2i, whe e β= 2y+p4y2−( 2−λ2)(4x+ 2−λ2). Compa ing he inequali ies (ii.1) and (ii.2) in P oposi ion 2.4 wi h he condi- ion (7), we obse e ha o h2and h4i is possible o ake = 1 in he p e ious co olla y in o de o ge equi alences wi h he complex s uc u es (i.1) and (i.3), espec i ely. The e o e, using Co olla y 2.7, we conclude: P oposi ion 2.10. Le us conside he amily o complex s uc u es (10) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+ω1¯ 2+D ω2¯ 2,Im D≥0. Then: (i) Any non-abelian complex s uc u e on h2is equi alen o one and only one s uc u e in (10) wi h Im D > 0; (ii) Any non-abelian complex s uc u e on h4is equi alen o one and only one s uc u e in (10) wi h D∈R−{0}. The classi ica ion o complex s uc u es on h5 equi es a mo e sub le s udy. 9 Lemma 2.11. Any non-abelian complex s uc u e on h5which is no complex- pa allelizable belongs o one o he ollowing amilies: (I) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+iy ω2¯ 2,whe e 0≤2y < |1−λ2|; (II) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+ (x+iy)ω2¯ 2,whe e 4y2<1 + 4x. Mo eo e , (i) he s uc u es in amily (I) a e non-equi alen ; (ii) he s uc u es in amily (II) a e non-equi alen ; (iii) a s uc u e (1, λ, iy)in amily (I) is equi alen o a s uc u e in amily (II) i and only i 2λ2∈[0,1) and 2y∈[λ2,1−λ2). P oo . Le us conside a complex s uc u e gi en by (1, λ, D =x+i y) on h5, i.e. 4y2<(1 −λ2)(4x+ 1 −λ2), acco ding o P oposi ion 2.4 (ii.3). I λ2≥2x, hen (1, λ, D)∼(1,√λ2−2x, i|D|) because (7) exp esses simply as 4|D|2≥0 and i i ially holds. On he o he hand, i λ2<2x, hen (1, λ, D)∼(1,0, E), whe e Eis gi en in Co olla y 2.9, because in his case 4y2+λ2(4x−λ2)≥0, ha is, condi ion (7) is sa is ied. To s udy u he equi alences, i is clea ha s uc u es in amily (I) a e non- equi alen and he same holds o s uc u es in amily (II). Now le us conside he iples (1, λ, iy) and (1,0, E). Then, (7) exp esses simply as (11) 4y2≥λ4. Condi ion o amily (I) implies ha 4y2<(1 −λ2)2, which is equi alen o 4y2− λ4<1−2λ2, so i 2λ2≥1 hen (11) does no hold. Now, i 0 ≤λ2<1 2 hen he condi ion o amily (I) is equi alen o y < 1 2−λ2 2, and he e o e when 2y∈[λ2,1−λ2) he iple (1, λ, iy) in amily (I) is equi alen o he iple (1,0, E = −1 2(λ2−p4y2−λ4i)) in amily (II).  P oposi ion 2.12. Any non-abelian complex s uc u e on h5which is no complex- pa allelizable is equi alen o one and only one s uc u e in he ollowing amilies: (I) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+D ω2¯ 2, whe e Re D= 0 and    0≤2Im D < λ2,0< λ2<1 2;o 0≤2Im D < |1−λ2|,1 2≤λ2. (II) dω1=dω2= 0, dω3=ω12+ω1¯ 1+D ω2¯ 2,whe e 4(Im D)2<1+4 Re D. To inish his sec ion, i emains o s udy he case o 2-s ep NLAs gwi h i s Be i numbe equal o 3, which co esponds o ǫ= 1 in (2). P oposi ion 2.13. Le Jbe a nilpo en complex s uc u e on an NLA ggi en by (2) wi h ǫ= 1, i.e. dω1= 0, dω2=ω1¯ 1, dω3=ρ ω12 +B ω1¯ 2+C ω2¯ 1, wi h ρ∈ {0,1}and B, C ∈Csuch ha (ρ, B, C)6= (0,0,0). Then gis 2-s ep nilpo en i and only i B=ρ= 1 and C= 0. In such case gis isomo phic o h7 and all he complex s uc u es a e equi alen . 16 (c) I g∼ =h5and Jis a complex s uc u e on h5gi en in Table 1, hen: (c.1) E16∼ =E2∼ =E∞when Jis complex-pa allelizable; (c.2) E1∼ =E∞i and only i Jis no complex-pa allelizable and ρD 6= 0; mo eo e , E16∼ =E2∼ =E∞when ρD = 0. (d) I g∼ =h16 o h+ 26, hen E16∼ =E2∼ =E∞ o any J. (e) I g∼ =h13 o h14, hen E1∼ =E26∼ =E3∼ =E∞ o any J. ( ) I g∼ =h15 and Jis a complex s uc u e on h15 gi en in Table 1, hen: ( .1) E16∼ =E2∼ =E∞, when c= 0 and |B−ρ| 6= 0; ( .2) E1∼ =E26∼ =E3∼ =E∞, when ρ= 1 and |B−1| 6=c6= 0; ( .3) E16∼ =E26∼ =E3∼ =E∞, when ρ= 0 and |B| 6=c6= 0. P oo . The p oo is s aigh o wa d and we only gi e i explici ly o he case ( ), ha is, g∼ =h15, because i is he mos in iguing case whe e di e en non- i ial beha iou s can be p oduced. We will use he no a ion E|k| =⊕p+q=kEp,q . Since E|k| ∞∼ =Hk dR, i is clea ha dim E|k| ≥bk= dim Hk dR o all k, and he equali ies hold i and only i E ∼ =E∞. Recall ha b1(h15) = 3, b2(h15) = 5 and b3(h15) = 6 (see [28]). Fo he calcula ion o he i s e m E1, ha is, he Dolbeaul cohomology, by he Se e duali y i su ices o s udy he spaces Ep,q 1=Hp,q ¯ ∂ o (p, q) = (1,0),(0,1),(2,0),(1,1),(0,2),(3,0) and (2,1). Le Jbe a complex s uc u e on h15 gi en in Table 1. I Jis abelian hen (B, c) = (0,1) o (1, c) wi h c6= 1, he e o e (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, whe e δc 0is equal o 0 i c6= 0, and equals 1 i c= 0. Since dim E|1| 1= 4 >3 = b1(h15) we ge ha E16∼ =E∞ o any abelian J. When Jis no abelian, i.e. ρ= 1, he Dolbeaul cohomology g oups a e (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, whe e δB 0has a simila de ini ion as o δc 0abo e. No ice ha he coe icien Bc+δB 0 is non-ze o excep o B6= 0 and c= 0. Thus, dim E|2| 1≥6>5 = b2(h15) and so E16∼ =E∞also o any non-abelian J. In o de o p o e ( .1) we need o s udy independen ly he abelian and he non- abelian complex s uc u es wi h c= 0 and B6=ρon h15. We s a wi h he abelian ones. In his case, by Table 1 we can suppose B= 1 and om (18) i ollows ha he dimensions o E|2| 1and E|3| 1a e dim E|2| 1= 9 >5 = b2(h15),dim E|3| 1= 12 >6 = b3(h15). 17 Fo he ollowing d1-homomo phisms E0,1 1 d1 −→ E1,1 1 d1 −→ E2,1 1 d1 −→ E3,1 1, he classes [ω¯ 3], [ω1¯ 3], [ω3¯ 2], [ω13¯ 3] ha e linea ly independen images. On he o he hand, o E0,2 1 d1 −→ E1,2 1 d1 −→ E2,2 1 d1 −→ E3,2 1, he images o he classes [ω¯ 2¯ 3], [ω3¯ 2¯ 3], [ω2¯ 2¯ 3+ω3¯ 1¯ 3] and [ω13¯ 2¯ 3] a e also independen . Coun ing dimensions o E|k| 2we ge ha 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 ha E2∼ =E∞because necessa ily dim E|k| 2=bk(h15) o all k. I ρ= 1 and c= 0, hen B6= 1 and by (19) we ha e dim E|1| 1=b1(h15) + δB 0. So E|1| 1∼ =E|1| ∞when B6= 0. Fo B= 0, since d1([ω3]) 6= 0 and d1([ω3¯ 1¯ 2¯ 3]) 6= 0, we conclude ha 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 he e o e, E|k| 2∼ =E|k| ∞i k= 1 o k= 5. Now, o B6= 1 we ha e ha 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 o de o conclude ha E2∼ =E∞ i su ices o obse e ha o he ollowing homomo phisms E1,1 1 d1 −→ E2,1 1 d1 −→ E3,1 1, E0,2 1 d1 −→ E1,2 1 d1 −→ E2,2 1 he classes [ω1¯ 3+ω2¯ 2], [ω3¯ 2], [ω13¯ 3+ω23¯ 2], [ω¯ 2¯ 3], [ω3¯ 2¯ 3] and [Bω2¯ 2¯ 3+ω3¯ 1¯ 3] ha e linea ly independen images. Fo case ( .2), we conside ρ= 1 and |B−1| 6=c6= 0. As dim E|1| 1= 3 = b1(h15), we ge ha E|1| 1∼ =E|1| ∞. Now, o he map E0,2 2 d2 −→ E2,1 2we ha e 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+∂γ1,1 o 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 ha E|2| ∞∼ =E|2| 36∼ =E|2| 2∼ =E|2| 1. Simila ly, d2:E1,2 2−→ E3,1 2is non-ze o ( o ins ance, 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 ha E|3| ∞∼ =E|3| 36∼ =E|3| 2∼ =E|3| 1. By he same a gumen b4(h15)≤dim E|4| 3≤dim E|4| 2−1≤dim E|4| 1−1 = 6 −1 = 5 = b4(h15) and he e o e E|4| ∞∼ =E|4| 36∼ =E|4| 2∼ =E|4| 1. Summing up all he in o ma ion, we conclude ha E1∼ =E26∼ =E3∼ =E∞in case ( .2). Fo he las case ( .3), we i s obse e ha d1([ω¯ 3]) = −c[ω1¯ 2]−¯ B[ω2¯ 1]. Since his class is ze o i and only i c ω1¯ 2+¯ Bω2¯ 1∈¯ ∂(V1,0) = hω1¯ 1, Bω1¯ 2+c ω2¯ 1i, i.e. |B|=c, he map d1:E0,1 1−→ E1,1 1is non-ze o. The e o e, dim E|1| 2≤dim E|1| 1−1 = 3, i.e. E|1| 16∼ =E|1| 2∼ =E|1| ∞. Mo eo e , since d2([ω¯ 2¯ 3]) 6= 0, we deduce ha 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 ha E|3| ∞∼ =E|3| 36∼ =E|3| 2∼ =E|3| 1. We also ha e b4(h15)≤dim E|4| 3≤dim E|4| 2−1≤dim E|4| 1−2 = 7 −2 = 5 = b4(h15), and he e o e E|4| ∞∼ =E|4| 36∼ =E|4| 26∼ =E|4| 1. Consequen ly, E16∼ =E26∼ =E3∼ =E∞in case ( .3).  Rema k 4.2. Le (M= Γ G, J) be a 6-dimensional nilmani old endowed wi h an in a ian complex s uc u e Jand suppose ha g=h7. In [27, Theo em 4.4] i is p o ed ha he e is a dense subse o he space o all in a ian complex s uc u es o which he complex nilmani old admi s he s uc u e o p incipal holomo phic bundle o ellip ic cu es o e a Kodai a su ace, bu his is no ue o all complex s uc u es. In ac , he in a ian complex s uc u e Jmay no be compa ible wi h he la ice Γ (see [27, Example 1.14]), so one canno ensu e he exis ence o he isomo phism (17), and hence o a canonical isomo phism be ween Ep,q (g, J) and Ep,q (M, J), o any in a ian Jon he nilmani old M. Howe e , no ice ha up o equi alence he e is only one complex s uc u e on h7and i can be p o ed ha i sa is ies ha he sequence degene a es a he i s s ep, i.e. E1(h7)∼ =E∞(h7). In [3] he au ho s posed he ollowing p oblem: o cons uc a compac complex mani old such ha E1∼ =E∞and hp,q ¯ ∂=hq,p ¯ ∂ o e e y p, q ∈Nbu o which he ∂¯ ∂-lemma does no hold. Since nilmani olds do no sa is y he ∂¯ ∂-lemma, unless hey a e complex o i, he ollowing esul p o ides a solu ion. P oposi ion 4.3. Le Jbe any in a ian complex s uc u e on a nilmani old M wi h unde lying Lie algeb a isomo phic o h6. Then E1(M)∼ =E∞(M)and he Hodge numbe s sa is y 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. P oo . Any complex s uc u e Jon h6is equi alen o he complex s uc u e gi en in Table 1, ha is, ρ=λ= 1 and D= 0. I s Dolbeaul cohomology g oups Hp,q ¯ ∂ o (p, q) = (1,0),(0,1),(2,0),(1,1),(0,2),(3,0) and (2,1) a e 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 Se e duali y we ge he abo e Hodge diamond which is symme ic. Mo eo e , dim E|1| 1= 4 = b1(h6),dim E|2| 1= 9 = b2(h6),dim E|3| 1= 12 = b3(h6), so he F ¨oliche spec al sequence degene a es a he i s s ep.  19 The ollowing esul shows ha he e a e many complex nilmani olds o which he F ¨oliche spec al sequence is s able unde small de o ma ions o he complex s uc u e. P oposi ion 4.4. Le M= Γ Gbe a 6-dimensional nilmani old endowed wi h an in a ian complex s uc u e J, and le gbe he Lie algeb a o G. I g∼ = h1,h3,h6,h8,h9,h10,h11,h12,h13,h14,h16,h− 19 o h+ 26, hen dim Ep,q (M, J)is s able unde small de o ma ions o J o any p, q and any ≥1. P oo . By [26, Theo em 2.6], all small de o ma ions o he complex s uc u e Ja e again in a ian complex s uc u es. P oceeding as in he p oo o Theo em 4.1, i can be p o ed ha i g6∼ =h2,h4,h5o h15, hen dim Ep,q (M) does no depend on he in a ian complex s uc u e on M o any p, q and any ≥1, so i is s able unde small de o ma ions o J. Rema k 4.5. The 6-dimensional nilmani olds wi h unde lying Lie algeb a isomo - phic o h2,h4,h5o h15 a e he only ones ha ha e bo h abelian and non-abelian complex s uc u es (see Table 1). Mo e gene ally, le Mbe a 2n-dimensional nil- mani old, g he unde lying Lie algeb a, Jan abelian complex s uc u e and J′a non-abelian in a ian complex s uc u e on M. I is well known ha Jis abelian i and only i he e is a basis {ω1,...,ωn}o in a ian o ms o ype (1,0) sa is ying ∂ωj= 0 o 1 ≤j≤n; he e o e, by [7] one has ha h0,1 ¯ ∂(M, J) = nbecause (17) holds o abelian s uc u es. Howe e , o J′we ha e dim H0,1 ¯ ∂(g, J′)< n and, i an isomo phism like (17) holds, hen he Hodge numbe sa is ies h0,1 ¯ ∂(M, J′)< n. Thus, he exis ence o Jand J′on a nilmani old Mmigh lead o he non-s abili y o dim E0,1 1unde small de o ma ions. A na u al ques ion a ises in his con ex : is he F ¨oliche spec al sequence s able unde small de o ma ions i and only i he nilmani old does no admi bo h abelian and non-abelian complex s uc u es? P oposi ion 4.4 abo e gi es an a i ma i e answe o n= 3. Nex we p o ide some examples o explici amilies o complex s uc u es on nil- mani olds co esponding o h5and h15 along which he F ¨oliche sequence a ies. In Co olla ies 5.11 and 5.12 below, u he p ope ies o he F ¨oliche spec al sequence on nilmani olds a e shown. Example 4.6. Le Jbe a non complex-pa allelizable and non-abelian complex s uc u e on h5gi en in Table 1 wi h non-degene a e F ¨oliche sequence, i.e. E1≇ E∞ o J. We will cons uc a amily o complex s uc u es J by de o ming he p e ious one, i.e. J0=J, such ha he F ¨oliche spec al sequence degene a es a he i s s ep o any 6= 0. Acco ding o Theo em 4.1, Jhas complex s uc u e equa ions o he o m dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2, o some non-nega i e λ6= 1, whe e {ω1, ω2, ω3}is a (1,0)-basis o J. Wi h espec o he eal basis {e1,...,e6}gi en by e1+i e2=ω1,1 1 + λ(e3−e1) + i 1−λ(e2+e4) = ω2, e5+ie6=ω3, he complex s uc u e Jexp esses 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 Fo any ∈[0,1 2), conside he complex s uc u e J gi en by J e1=4d(1−λ) α2e1−1−λ2 αe2−2d(1−λ)2 α2e3+8d2(1−λ) α3e4, J e2=1−λ2 αe1+2d(1−λ2) α2e4, J e3=−2d (1−λ)2e1−2α (1−λ2)(1−λ)e2−(1+λ)2 αe4, J e4=−2(1−λ) αe1+2d 1−λ2e2+(1−λ)2 αe3−4d(1−λ) α2e4, J e5=2d 1−λ2e5−4d2+(1−λ2)2 α(1−λ2)e6, J e6=α 1−λ2e5−2d 1−λ2e6, whe e α=p(1 −λ2)2−4d2, and d( , λ) =          , i λ= 0, λ2/4,i λ2∈(0,1/2), (1 −λ2)/4,i λ2∈[1/2,1), − (1 −λ2)/4,i λ2>1. No ice ha J0=J. Now, he o ms ω1 =1−λ2 αe1+2d(1−λ2) α2e4+i e2, ω2 =1−λ α(e3−e1)−2d(1−λ) α2e4+i 1−λ2d αe1+e2+(1−λ2)2 α2e4, ω3 =e5−2d αe6+i1−λ2 αe6, sa is y J ωk =i ωk o k= 1,2,3, i.e. {ω1 , ω2 , ω3 }is a basis o ype (1,0) o J . Fu he mo e, wi h espec o his basis he complex s uc u e equa ions a e dω1 =dω2 = 0, dω3 =ω12 +ω1¯ 1 +λ ω1¯ 2 +D ω2¯ 2 , wi h D=i d( , λ). Acco ding o Theo em 4.1, he F ¨oliche spec al sequence degene a es i and only i D6= 0, i.e. i and only i > 0. In conclusion, Jcan be de o med in o a non-abelian complex s uc u e wi h E1∼ =E∞. Co olla y 4.7. Le M= Γ Gbe he nilmani old unde lying he Iwasawa mani old, i.e. g∼ =h5. Le Jbe a non complex-pa allelizable and non-abelian complex s uc u e on Mgi en in Table 1 wi h E16∼ =E∞. Then, Jcan be de o med in o an in a ian complex s uc u e wi h degene a e F ¨oliche spec al sequence. The Lie algeb a h15 has a ich complex geome y wi h espec o he F ¨oliche sequence and in he nex example we cons uc a amily J along which he h ee cases in ( ) o Theo em 4.1 a e ealized. 21 Example 4.8. On h15, le us conside he eal basis {e1,...,e6}gi en in Theo- em 2.1 and he ollowing amily o complex s uc u es J e1=−s3(3 −sin )(7 + 3 sin ) (5 + sin )(11 −sin )e2, J e3=s3(3 −sin )(11 −sin ) (5 + sin )(7 + 3 sin )e4, J e5=−s(11 −sin )(7 + 3 sin ) 3(3 −sin )(5 + sin )e6, whe e ∈R. Le 4ω1 =p(11 −sin )(5 + sin )e1+ip3(3 −sin )(7 + 3 sin )e2, 8ω2 = (5 + sin )(7 + 3 sin )e3−ip3(5 + sin )(3 −sin )(11 −sin )(7 + 3 sin )e4, and 128 ω3 = (5 + sin )(7 + 3 sin )h3(3 −sin )p(11 −sin )(5 + sin )e5 +i(11 −sin )p3(3 −sin )(7 + 3 sin )e6i. Then, {ω1 , ω2 , ω3 }is a (1,0)-basis o J sa is ying dω1 = 0, dω2 =ω1¯ 1 , dω3 =1−sin 2ω12 + 2 ω1¯ 2 +1 + sin 4ω2¯ 1 . I sin = 1, he coe icien o ω12 anishes and he e o e J is an abelian com- plex s uc u e ha is equi alen o he one gi en by (ρ, B , c ) = (0,1,1 4) (see Lemma 2.14). I sin 6= 1, hen we can no malize he coe icien o ω12 and he complex s uc u e equa ions can be w i en in o m (14) as dω1 = 0, dω2 =ω1¯ 1, dω3 =ω12 +4 1−sin ω1¯ 2 +1 + sin 2(1 −sin )ω2¯ 1 , i.e. hey a e de e mined by he iple (ρ, B , c ) = 1,4 1−sin ,1+sin 2(1−sin ). Now, con- ce ning he F ¨oliche spec al sequence o he amily {J } ∈R, by Theo em 4.1 ( ) we ge •I sin = 1, hen (ρ , B , c ) = (0,1,1 4) and he e o e E16∼ =E26∼ =E3∼ =E∞. •I sin =−1, hen (ρ , B , c ) = (1,2,0) and E16∼ =E2∼ =E∞. •I |sin | 6= 1, E1∼ =E26∼ =E3∼ =E∞. As a consequence o his example, in he ollowing esul we show ha o ≥2 he dimension o he e m Ep,q (J ) in gene al is nei he uppe no lowe semi- con inuous unc ion o . This is in deep con as wi h he case = 1, as i is well known he uppe semicon inui y o he Hodge numbe s dim Hp,q ¯ ∂(J ) wi h espec o along a de o ma ion. Co olla y 4.9. Le Mbe a nilmani old wi h unde lying Lie algeb a h15 endowed wi h he in a ian complex s uc u es J gi en in Example 4.8. Then, dim E0,2 2(Jπ 2) = 3 >2 = dim E0,2 2(J ),dim E1,1 2(Jπ 2) = 2 <3 = dim E1,1 2(J ), 22 and dim E0,2 3(Jπ 2) = 2 >1 = dim E0,2 3(J ),dim E1,1 3(Jπ 2) = 2 <3 = dim E1,1 3(J ), o any ∈(π 2,3π 2). The e o e, he dimensions o he e ms E1,1 2(J )and E1,1 3(J ) a e no uppe semi-con inuous unc ions o , and he dimensions o he e ms E0,2 2(J )and E0,2 3(J )a e no lowe semi-con inuous unc ions o . P oo . I ollows di ec ly om he p oo o Theo em 4.1 aking in o accoun ha o =π 2 he complex s uc u e lies in case ( .3) and o any ∈(π 2,3π 2) he s uc u es J lie in case ( .2).  5. S ongly Gauduchon and balanced He mi ian me ics Le (M, J) be a complex mani old o complex dimension n. A He mi ian me ic g on (M, J) can be desc ibed by means o a posi i e de ini e smoo h o m Ω on M o bideg ee (1,1) wi h espec o J. We will use his app oach in wha ollows and we will e e o Ω as a He mi ian s uc u e o as a He mi ian me ic indis inc ly. A He mi ian s uc u e Ω is s ongly Gauduchon (sG o sho ) i ∂Ωn−1is ¯ ∂- exac [22, 23]. In pa icula , any balanced He mi ian s uc u e (i.e. dΩn−1= 0) is sG, and any sG me ic is a Gauduchon me ic [18], ha is, Ωn−1is ∂¯ ∂-closed o equi alen ly he Lee o m is co-closed. Nex we suppose ha (M= Γ G, J) is a nilmani old endowed wi h an in a ian complex s uc u e. I is p o ed in [15] ha (M= Γ G, J) has a balanced me ic i and only i i has an in a ian one. Mo eo e , by using he symme iza ion p ocess gi en in [5] (see also [15], [30] and [32, P oposi ion 3.2]) one easily a i es a : P oposi ion 5.1. (M= Γ G, J)has an sG me ic i and only i i has an in a ian one. The e o e, he exis ence o sG me ics on (M= Γ G, J) is educed o he exis- ence a he Lie algeb a le el go G. Co olla y 5.2. Le Ωbe an in a ian He mi ian s uc u e on (M= Γ G, J). I Jis abelian, hen Ωis sG i and only i i is balanced. P oo . Le gbe he Lie algeb a o G. Fi s we p o e ha ¯ ∂(Vn,k(g∗)) = 0 o e e y 1≤k≤n. Le us conside a decomposable o m α∈Vn,k(g∗) gi en by α=β∧γ, whe e β∈Vn,0(g∗) and γ∈V0,k(g∗). Since gis nilpo en and Jis abelian, one has ha dβ = 0 and dγ ∈V1,k(g∗), so in pa icula βand γa e ¯ ∂-closed. Hence, ¯ ∂α = (¯ ∂β)∧γ+ (−1)nβ∧(¯ ∂γ) = 0. Now, he s a emen in he co olla y ollows di ec ly om P oposi ion 5.1 and om he p e ious p ope y o k=n−2, i.e. ¯ ∂(Vn,n−2(g∗)) = 0.  F om now on we conside n= 3. P oposi ion 5.3. Le M= Γ Gbe a 6-dimensional nilmani old endowed wi h an in a ian complex s uc u e J. The e exis s an sG me ic on (M= Γ G, J)i and only i he Lie algeb a go Gis isomo phic o h1,...,h6o h− 19. P oo . By P oposi ion 5.1 i su ices o s udy he in a ian case. By [30], he undamen al 2- o m o any J-He mi ian me ic is gi en by (20) 2 Ω = i( 2ω1¯ 1+s2ω2¯ 2+ 2ω3¯ 3) + uω1¯ 2−¯uω2¯ 1+ ω2¯ 3−¯ ω3¯ 2+zω1¯ 3−¯zω3¯ 1, 23 whe e coe icien s 2, s2, 2a e non-ze o eal numbe s and u, , z ∈Csa is y 2s2> |u|2,s2 2>| |2, 2 2>|z|2and 2s2 2+ 2Re (i¯u¯ z)> 2|u|2+ 2| |2+s2|z|2. Le us s a wi h he non-nilpo en case. F om (16) 2∂Ω = (iǫ ∓iz)ω12¯ 1∓i ω12¯ 2+ (u−¯u−ǫ 2)ω13¯ 1+ (is2± 2)ω13¯ 2+ ω13¯ 3+ (is2∓ 2)ω23¯ 1 and he e o e 4∂Ω∧Ω = iǫ(s2 2−| |2)±( 2u+ 2¯u+i ¯z−i¯ z)ω123¯ 1¯ 2+u −is2zω123¯ 1¯ 3. Di ec compu a ions show ha ¯ ∂(V3,1(g∗)) = hω123¯ 1¯ 3i. I he He mi ian s uc u e (J, Ω) is sG hen ∓iǫ(s2 2−| |2) = 2(u+ ¯u) + i ¯z−i¯ z. Since he le -hand side is pu ely imagina y and he igh -hand side is eal, we ge ha ǫ= 0 and he e o e g∼ =h− 19. Fo he nilpo en case, le us conside he gene al complex equa ions (2). Now, he undamen al 2- o m o any J-He mi ian me ic is gi en also by (20). Using [30, Lemma 17 and P oposi ion 25], we ge 4∂Ω∧Ω=(1 −ǫ)¯ A(s2 2−| |2) + ¯ B(i 2u+ ¯ z)−¯ C(i 2¯u− ¯z) + (1 −ǫ)¯ D( 2 2−|z|2)ω123¯ 1¯ 2−ǫ(s2 2−| |2)ω123¯ 1¯ 3. I is s aigh o wa d o e i y ha ¯ ∂(V3,1(g∗)) = hρ ω123¯ 1¯ 2i, and he e o e, i he He mi ian s uc u e (J, Ω) is sG hen ǫ= 0, i.e. g∼ =hi o i= 1,...,6. Mo eo e , i in addi ion ρ= 1, hen any J-He mi ian s uc u e is sG. In conclusion, i he e exis s an sG me ic hen g∼ =h1,...,h6o h− 19. The con e se ollows di ec ly om [30, Theo em 26] because hese Lie algeb as admi balanced He mi ian me ics.  Rema k 5.4. F om he p oo o he p e ious p oposi ion i ollows ha on h2,h4,h5 and h6, i Jis a non-abelian nilpo en complex s uc u e hen any in a ian J- He mi ian me ic is sG. This is in con as wi h h− 19, whe e o any complex s uc u e he space o balanced me ics is s ic ly con ained in he space o sG me ics, and mo eo e he e a e He mi ian me ics which a e no sG. Fo ins ance, conside a He mi ian me ic on h− 19 gi en 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), ha is, in (20) we ake = 1, = 0, uand z eal and s2= 2= 2(u2+z2+ 1): •i u=z= 0 hen he me ic is balanced; •i u= 0 and z6= 0 hen he me ic is sG bu no balanced; •i u6= 0 hen he me ic is no sG. No ice ha his indica es a con as be ween he sG and SKT geome ies, since by [16] he exis ence o an SKT s uc u e on a 6-dimensional nilpo en Lie algeb a depends only on he complex s uc u e. The e exis compac complex mani olds ha ing sG me ics bu no admi ing any balanced me ic [24, Theo em 1.8]. Nex we show he gene al si ua ion o nilmani olds in dimension 6. 24 P oposi ion 5.5. Le M= Γ Gbe a 6-dimensional nilmani old wi h an in a ian complex s uc u e Jsuch ha (M= Γ G, J)does no admi balanced me ics. I (M= Γ G, J)has sG me ic, hen Jis non-abelian nilpo en and gis isomo phic o h2,h4o h5. Mo eo e , acco ding o he classi ica ion in Table 1, such a Jis gi en by: Re D+ (Im D)2≥1 4on h2;Re D≥1 4on h4; and λ= 0,Im D6= 0 o λ=Im D= 0,Re D≥0on h5. P oo . Any complex s uc u e on h6o h− 19 admi s balanced me ics. F om [32] we ha e ha only h3and h5ha e abelian complex s uc u es Jadmi ing balanced me ic. In ac , any such Jon h5admi s balanced He mi ian me ics, whe eas o h3 he complex s uc u e mus be equi alen o he choice o (−)-sign in Table 1. F om Co olla y 5.2, i emains o s udy he non-abelian nilpo en complex s uc u es J on h2,h4and h5. Since any such Jadmi s sG me ics by Rema k 5.4, nex we show which o hem do no admi balanced me ics. In he h ee cases he complex equa ions a e o he o m (21) dω1=dω2= 0, dω3=ω12 +ω1¯ 1+λ ω1¯ 2+D ω2¯ 2. A simila a gumen as in he p oo o [32, P oposi ion 2.3] shows ha , up o equi - alence, he undamen al 2- o m o any J-He mi ian me ic is gi en by 2 Ω = i(ω1¯ 1+s2ω2¯ 2+ 2ω3¯ 3) + u ω1¯ 2−¯u ω2¯ 1, whe e s2>|u|2and 2>0. I D=x+iy and u=u1+iu2, he balanced condi ion is (22) s2+x+iy =u2λ+iu1λ. We dis inguish se e al cases depending on he alues o λ. I λ6= 0 hen Ω is balanced i and only i u1=y/λ and u2= (s2+x)/λ. The condi ion s2>|u|2is equi alen o s4+ (2x−λ2)s2+x2+y2<0 and i is easy o see ha a non-ze o ssa is ying his condi ion exis s i and only i he disc iminan o he p e ious equa ion as a second deg ee equa ion in s2is posi i e, i.e. (23) λ4−4xλ2−4y2>0. Acco ding o Table 1, non-abelian complex s uc u es on h2ha e λ= 1. In his case (23) eads as x+y2<1/4, which means ha any Jsuch ha x+y2≥1 4 has no balanced me ics. Simila ly, o h4any Jsuch ha x≥1 4does no admi balanced me ic. Fo h5and λ6= 0 we ha e ha x= 0 by Table 1. Thus, he e is no balanced me ics i and only i λ4≤4y2. Since y≥0, his is equi alen o λ2≤2y. Howe e , none o he h ee cases de ailed in Table 1 e i ies ha λ2≤2y, and he e o e any complex s uc u e on h5wi h λ6= 0 admi s balanced me ics. Finally, in he case λ= 0 on h5we ge ha he balanced condi ion (22) educes o y= 0 and s2=−x > 0. F om Table 1 we ha e ha 0 <1+4x, i.e. x∈(−1 4,∞). The e o e, i y6= 0 o y= 0, x ≥0 hen he e a e no balanced me ics.  As poin ed ou by Popo ici [24], he degene a ion o he F ¨oliche sequence a E1 and he exis ence o sG me ics a e un ela ed. F om he s udy o he sG geome y abo e and om Theo em 4.1 we ge : Theo em 5.6. Le M= Γ Gbe a 6-dimensional nilmani old endowed wi h an in a ian complex s uc u e J. I he e exis s an sG me ic hen he F ¨oliche 25 spec al sequence degene a es a he second le el, i.e. E2(M)∼ =E∞(M). Mo eo e , i he e exis s an sG me ic and g6∼ =h5, hen E1(M)∼ =E∞(M). P oo . By P oposi ion 5.3, he Lie algeb a gunde lying M= Γ Gmus be iso- mo phic o h1,...,h6o h− 19, so Theo em 4.1 implies ha he F ¨oliche sequence degene a es a he second le el. The las asse ion ollows di ec ly by aking in o accoun Co olla y 5.2 and Table 3 below.  I is in e es ing whe he his esul holds in gene al, ha is: Ques ion 5.7. Does he F ¨oliche spec al sequence degene a e a he second s ep o any compac complex mani old Mo complex dimension 3admi ing an sG me ic? In he ollowing able we show he complex s uc u es J, up o equi alence, on h1,...,h6 ha admi balanced He mi ian me ics. The classi ica ion ollows om he p oo o P oposi ion 5.5. Table 3: Classi ica ion o nilpo en complex s uc u es admi ing balanced me ics gAbelian s uc u es Non-Abelian Nilpo en s uc u es 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,wi h (λ, x, y) sa is ying one o : 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 Mo i a ed by [24, Theo em 1.9] nex we s udy he ela ion be ween he degene - a ion o he F ¨oliche spec al sequence and he exis ence o sG o balanced me ics. The possibili ies a e well illus a ed in he ollowing de o ma ions o he complex s uc u e co esponding o λ=x=y= 0 on a nilmani old wi h unde lying Lie algeb a h5. Example 5.8. Le us conside he Lie algeb a h5wi h he eal basis {e1,...,e6} desc ibed in Theo em 2.1. Le us conside he complex s uc u e J0,0gi en by J0,0e1=−e2, J0,0e3=−2e2−e4, J0,0e5=−e6, J0,0e2=e1, J0,0e4=−2e1+e3, J0,0e6=e5.