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 =e2β
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¯
2i=|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−λ2d
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 −is2zω123¯
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.
