On generalized gauduchon nilmanifolds

ON GENERALIZED GAUDUCHON NILMANIFOLDS
A. LATORRE, L. UGARTE, AND R. VILLACAMPA
Abs ac . We cons uc in a ian gene alized Gauduchon me ics on he p oduc o wo com-
plex nilmani olds ha do no necessa ily admi his kind o me ics. In pa icula , we p o e
ha he p oduc o a locally con o mal K¨ahle nilmani old and a balanced nilmani old admi s
a gene alized Gauduchon me ic. In complex dimension 4, gene alized Gauduchon nilmani olds
wi h ( he highes possible) nilpo ency s ep s= 5 a e gi en, as well as 3-s ep and 4-s ep examples
o which he cen e o hei unde lying Lie algeb as does no con ain any non- i ial J-in a ian
ideal. These examples show s ong di e ences be ween he SKT and he gene alized Gauduchon
geome ies o nilmani olds.
1. In oduc ion
Le Xbe a compac complex mani old o complex dimension n, and le Fbe a He mi ian me ic
on X. When he Lee o m is co-closed, o equi alen ly Fn−1is ∂∂-closed, he me ic Fis called
s anda d o Gauduchon. By [13] he e is a s anda d me ic in he con o mal class o e e y
He mi ian me ic on X. Fu, Wang, and Wu in es iga e in [11] he ollowing gene aliza ion o
Gauduchon me ics. Fo 1 ≤k≤n−1, a He mi ian me ic Fon Xis called k- h Gauduchon i
∂∂Fk∧Fn−k−1= 0. In [11] a unique cons an γk(F) is associa ed o any Fon X. This cons an
is in a ian by biholomo phisms and depends smoo hly on he me ic F. Mo eo e , i is p o ed
ha γk(F) = 0 i and only i he e exis s a k- h Gauduchon me ic in he con o mal class o F.
Fo k=n−1, (n−1)- h Gauduchon me ics a e by de ini ion he usual Gauduchon me ics, and
i is showed in [11] ha γn−1(F) = 0, acco dingly o [13].
In his pape we a e mainly conce ned wi h gene alized Gauduchon me ics F o k= 1, i.e.
hose sa is ying ∂∂F ∧Fn−2= 0. Some compac complex mani olds wi h 1-s Gauduchon me ics
a e cons uc ed in [9, 11, 15] by di e en me hods. A pa icula ly in e es ing subclass o 1-s
Gauduchon me ics is ha cons i u ed by he plu iclosed o s ong K¨ahle wi h o sion (SKT
o sho ) me ics. They a e de ined by he condi ion ∂¯
∂F = 0 and ha e been s udied by many
au ho s on non-K¨ahle compac complex mani olds (see o ins ance [4, 6, 7, 8, 10, 20] and he
e e ences he ein).
The class o complex nilmani olds has p o ed o be an impo an sou ce o compac complex
mani olds admi ing hese ypes o special He mi ian me ics. He e by a complex nilmani old we
mean a compac complex mani old o he o m X= (Γ G, J), whe e Γ Gis a compac quo ien o
a simply-connec ed nilpo en Lie g oup Gby a uni o m disc e e subg oup Γ, and Jis an in a ian
complex s uc u e. Fo ins ance, Fino, Pa on, and Salamon ind in [7] he complex nilmani olds
o complex dimension 3 wi h in a ian SKT me ics, and hey show ha he exis ence o such
me ics only depends on he complex s uc u e J. Mo eo e , i is p o ed in [9] ha any in a ian
1-s Gauduchon me ic on a complex nilmani old o complex dimension 3 is necessa ily SKT.
Ou i s goal in his pape is o cons uc in a ian 1-s Gauduchon me ics on he p oduc o
wo complex nilmani olds ha do no necessa ily admi his ype o me ics. Fo his pu pose, in
P oposi ion 2.3 we s udy he cons an c1(F+F0) gi en by (1) ha measu es he 1-s Gauduchon
condi ion o he p oduc o wo He mi ian me ics Fand F0. As a consequence, we conclude
Key wo ds and ph ases. Nilmani old; complex s uc u e; He mi ian me ics.
1
2
ha he p oduc o wo He mi ian nilmani olds wi h cons an s c1(F) and c1(F0) o opposi e signs
admi s a 1-s Gauduchon me ic (see Co olla y 2.4).
As a i s applica ion, we show in Theo em 2.5 ha he p oduc o a locally con o mal K¨ahle
nilmani old and a balanced nilmani old always admi s a 1-s Gauduchon me ic. Recall ha a
locally con o mal K¨ahle (LCK o sho ) me ic is a He mi ian me ic ha is con o mal o some
local K¨ahle me ic in a neighbo hood o each poin o he mani old, and a He mi ian me ic Fis
balanced i he Lee o m anishes, o equi alen ly Fn−1is a closed o m. Balanced nilmani olds
a e s udied in [1, 22], whe eas he complex nilmani olds admi ing LCK me ics a e classi ied
in [19].
A second applica ion is gi en in Theo em 2.11, whe e we conside he p oduc o wo He mi ian
nilmani olds Xand X0o complex dimension 3. In his case, he sign o he cons an c1(F)
o an in a ian J-He mi ian me ic Fon X( espec i ely, X0) only depends on he complex
s uc u e J, as seen in [9] (see also P oposi ion 2.7 and Table 1). Thanks o i , we can p o ide
classi ica ions o hose complex s uc u es o which c1(F) is nega i e, ze o (SKT case), o posi i e
(see P oposi ions 2.9 and 2.10). Using hese classi ica ions, one can easily choose Xand X0wi h
cons an s c1’s o opposi e signs in o de o cons uc in a ian 1-s Gauduchon me ics on he
p oduc nilmani old Y=X×X0.
En ie i, Fino, and Vezzoni p o ed in [4, P oposi ion 3.1] ha he exis ence o an SKT me ic
on a complex nilmani old implies ha he cen e o i s unde lying Lie algeb a is J-in a ian .
Fu he mo e, [4, Theo em 1.2] asse s ha an SKT nilmani old (no a o us) is necessa ily 2-s ep.
The aim o Sec ion 3 is o show ha he gene alized Gauduchon geome y o nilmani olds is much
mo e lexible ha he SKT geome y. Since in complex dimension 3 he in a ian 1-s Gauduchon
me ics coincide wi h he in a ian SKT me ics [9], we a e led o s udy complex nilmani olds
o complex dimension 4. Al hough he e a e some examples o 1-s Gauduchon nilmani olds o
complex dimension 4 in he li e a u e [9, 16], we no ice ha all o hem a e 2-s ep.
In P oposi ion 3.1 we cons uc o s= 3,4 an s-s ep nilmani old o complex dimension 4 wi h
in a ian 1-s Gauduchon me ics such ha he cen e Z(g) o he unde lying Lie algeb a gis no
J-in a ian ; mo eo e , Z(g) does no con ain any non- i ial J-in a ian ideal. In P oposi ion 3.2
we gi e 5-s ep nilmani olds o complex dimension 4 ha ing in a ian 1-s Gauduchon me ics. No e
ha his esul is op imal, since o s≥6 he e do no exis s-s ep 8-dimensional nilmani olds
admi ing in a ian complex s uc u es by [2, 12, 14] (see Rema k 3.3).
The examples cons uc ed in P oposi ions 3.1 and 3.2 a e i educible. By aking app op ia e
p oduc s, in Theo em 3.5 we conclude ha o 3 ≤s≤5 and o any n≥4, he e exis s an s-s ep
nilmani old o complex dimension nadmi ing in a ian 1-s Gauduchon me ics. Fu he mo e,
by [16, P oposi ion 2.2] all he 1-s Gauduchon me ics gi en in his pape also sa is y he k- h
Gauduchon p ope y o e e y 2 ≤k≤n−1 (see Rema k 3.4).
2. Gene alized Gauduchon me ics on p oduc nilmani olds
In his sec ion we s udy he exis ence o gene alized Gauduchon me ics on a p oduc o complex
nilmani olds endowed wi h in a ian He mi ian me ics. As a consequence, we ob ain many
examples o gene alized Gauduchon nilmani olds.
Le us s a e iewing he de ini ion and some o he main p ope ies o he gene alized Gaudu-
chon me ics ob ained by Fu, Wang, and Wu in [11].
De ini ion 2.1. [11] Le Xbe a compac complex mani old o complex dimension n, and le
1≤k≤n−1be an in ege . A He mi ian me ic Fon Xis called k- h Gauduchon i i sa is ies
he condi ion
∂∂Fk∧Fn−k−1= 0.
3
F om he de ini ion, one can see ha he alue k=n−1 eco e s he classical s anda d
(Gauduchon) me ics. Mo eo e , i is clea ha any SKT me ic is a 1-s Gauduchon me ic.
Ex ending he esul p o ed by Gauduchon in [13] o s anda d me ics, in [11] i is shown ha ,
o any 1 ≤k≤n−1, he e exis s a unique cons an γk(F) and a (unique up o a cons an )
unc ion ∈ C∞(X) such ha
i
2∂∂(e Fk)∧Fn−k−1=γk(F)e Fn.
I is seen ha o k=n−1 one always has γn−1(F) = 0. Mo eo e , i Xadmi s a K¨ahle
me ic F, hen γk(F) = 0 and is a cons an unc ion o any 1 ≤k≤n−1.
Fu he mo e, he cons an γk(F) is in a ian unde biholomo phisms, and by [11, P oposi-
ion 11] he sign o γk(F) is in a ian in he con o mal class o F.
To compu e he sign o he cons an γk(F) one can use he ollowing esul :
P oposi ion 2.2. [11] Fo a He mi ian me ic Fon a compac complex mani old Xo complex
dimension n, he numbe γk(F)is >0 (= 0, o <0) i and only i he e exis s a me ic ˜
Fin he
con o mal class o Fsuch ha
i
2∂∂ ˜
Fk∧˜
Fn−k−1>0 (= 0,o <0).
In his pape we ocus ou a en ion on hese He mi ian me ics in he special class o nil-
mani olds. We ecall ha a complex nilmani old is a compac complex mani old o he o m
X= (Γ G, J), whe e Γ Gis a compac quo ien o a simply-connec ed nilpo en Lie g oup Gby
a uni o m disc e e subg oup Γ, and Jis an in a ian complex s uc u e. This means ha Jis an
endomo phism J:g−→ go he Lie algeb a go Gsuch ha J2=−Id and i is in eg able, ha
is, he i-eigenspace g1,0o Jin gC=g⊗RCis a complex subalgeb a o gC. We will be mainly
conce ned wi h He mi ian me ics Fon Xwhich a e also in a ian .
Fo any in a ian He mi ian me ic Fon a complex nilmani old o complex dimension n≥2,
he eal (n, n)- o m i
2∂¯
∂F ∧Fn−2is p opo ional o he olume o m Fn. The e o e,
(1) i
2∂¯
∂F ∧Fn−2=c1(F)Fn,
o some cons an c1(F)∈R. Obse e ha Fis 1-s Gauduchon i and only i c1(F) = 0.
Le us no ice ha by P oposi ion 2.2 he sign o c1(F) coincides wi h he sign o he con-
s an γ1(F). In ou s udy o gene alized Gauduchon me ics we will conside c1(F) ins ead
o γ1(F), because i s p ecise alue on nilmani olds can be de e mined easily, and hus i s sign.
P oposi ion 2.3. Le Xand X0be complex nilmani olds endowed wi h in a ian He mi ian
me ics Fand F0, espec i ely.
(i) Fo any eal λ > 0, we ha e
(2) c1(λ F) = c1(F)
λ.
(ii) Le Y=X×X0be he p oduc nilmani old endowed wi h he p oduc He mi ian me ic
F+F0. Then,
(3) c1(F+F0) = n(n−1)
(n+n0)(n+n0−1) c1(F) + n0(n0−1)
(n+n0)(n+n0−1) c1(F0),
whe e n= dimCXand n0= dimCX0.
P oo . Le us s a wi h pa (i). A he sigh o (1), one has he ollowing exp ession o he
He mi ian me ic λ F:i
2∂¯
∂(λ F)∧(λ F)n−2=c1(λ F) (λ F)n.
4
I we expand he le -hand side o his equali y, we ob ain
i
2∂¯
∂(λ F)∧(λ F)n−2=λn−1i
2(∂¯
∂F ∧Fn−2) = λn−1c1(F)Fn=λ−1c1(F) (λ F)n,
and he esul comes s aigh o wa d.
We nex p o e (ii). On he one hand, he equa ion (1) o he He mi ian me ic F+F0on he
(n+n0)-dimensional complex nilmani old Y=X×X0 eads as
i
2∂¯
∂(F+F0)∧(F+F0)n+n0−2=c1(F+F0) (F+F0)n+n0.
On he o he hand, we conside
i
2∂¯
∂(F+F0)∧(F+F0)n+n0−2=i
2(∂¯
∂F +∂¯
∂F0)∧(F+F0)n+n0−2
=i
2(∂¯
∂F +∂¯
∂F0)∧(α Fn−2∧F0n0+ζ Fn−1∧F0n0−1+β Fn∧F0n0−2)
=i
2∂¯
∂F ∧(α Fn−2∧F0n0) + i
2∂¯
∂F0∧(β Fn∧F0n0−2)
=α(i
2∂¯
∂F ∧Fn−2)∧F0n0+β Fn∧(i
2∂¯
∂F0∧F0n0−2)
=α c1(F) + β c1(F0)Fn∧F0n0
=α
νc1(F) + β
νc1(F0)(F+F0)n+n0,
whe e α=n+n0−2
n0,ζ=n+n0−2
n−1,β=n+n0−2
n, and ν=n+n0
n. The e o e,
α
ν=n(n−1)
(n+n0)(n+n0−1) and β
ν=n0(n0−1)
(n+n0)(n+n0−1). This gi es us (3). 
As a di ec consequence o P oposi ion 2.3, he p oduc o 1-s Gauduchon nilmani olds is again
a 1-s Gauduchon nilmani old. None heless, we can also conside He mi ian nilmani olds wi h
opposi e signs o hei cons an s c1in o de o p oduce examples o 1-s Gauduchon nilmani olds.
Indeed:
Co olla y 2.4. Le Xand X0be complex nilmani olds endowed wi h in a ian He mi ian me ics
Fand F0, espec i ely, such ha c1(F)>0and c1(F0)<0. Then, he p oduc nilmani old X×X0
has a 1-s Gauduchon me ic.
P oo . Le n,n0be he complex dimensions o Xand X0, espec i ely. Le us conside
λ=−n0(n0−1) c1(F0)
n(n−1) c1(F)>0,
which is in ac a posi i e eal numbe , since c1(F)>0 and c1(F0)<0 by hypo hesis. Now, we
can ake he He mi ian me ic ˜
F0=λ F0on he complex nilmani old X0. A di ec calcula ion
using (2) and (3) shows ha he He mi ian me ic F+˜
F0on he complex p oduc nilmani old
X×X0sa is ies
(n+n0)(n+n0−1) c1(F+˜
F0) = n(n−1) c1(F) + n0(n0−1) c1(F0)
λ
=n(n−1) c1(F)−n(n−1) c1(F)
= 0.
Hence, F+˜
F0is a 1-s Gauduchon me ic. 
5
Le us ecall ha a He mi ian mani old (X, F) o complex dimension nis called locally con o mal
K¨ahle (LCK o sho ) i Fis con o mal o some local K¨ahle me ic in a neighbo hood o
each poin o X. Ano he impo an class o He mi ian mani olds is ha o balanced mani olds,
cha ac e ized by he condi ion dFn−1= 0. Nilmani olds wi h balanced me ics a e gi en in [1, 22]
and LCK nilmani olds a e s udied in [19].
In he ollowing esul , he complex s uc u es on he nilmani olds a e in a ian , bu he LCK
me ic and he balanced me ic a e no necessa ily o in a ian ype.
Theo em 2.5. The p oduc o a locally con o mal K¨ahle nilmani old by a balanced nilmani old
admi s a 1-s Gauduchon me ic.
P oo . Le Xbe a complex nilmani old o complex dimension nadmi ing a balanced me ic. By
[5, Theo em 4.1], he exis ence o a balanced me ic on Ximplies he exis ence o an in a ian
one. Le us deno e Fan in a ian balanced me ic on X. By [15, Lemma 3.7], he cons an
c1(F)>0.
Le X0be a complex nilmani old o complex dimension n0admi ing an LCK me ic. As a
consequence o [21, P oposi ion 34], he e mus exis an in a ian LCK me ic F0on X0. Using
[15, P oposi ion 3.8] we ha e ha he cons an c1(F0)<0.
Now, i su ices o apply Co olla y 2.4 o he pai (X, F) and (X0, F0) o ensu e he exis ence
o a 1-s Gauduchon me ic on he p oduc nilmani old X×X0.
Nex , we will apply he p e ious esul s o he p oduc o low dimensional complex nilmani olds.
In complex dimension 2, all he in a ian He mi ian me ics Fsa is y c1(F) = 0; in ac , o he
complex o us e e y me ic Fis K¨ahle , and on he Kodai a-Thu s on mani old any in a ian
He mi ian me ic Fsa is ies ∂¯
∂F = 0, so c1(F) = 0. The e o e, in o de o cons uc gene alized
Gauduchon me ics on nilmani olds making use o Co olla y 2.4 we need o conside wo complex
nilmani olds o complex dimension a leas 3.
Recall ha in complex dimension 3, he nilpo en Lie algeb as unde lying he nilmani olds ha
admi an in a ian complex s uc u e a e classi ied by Salamon in [18], whe eas he classi ica ion
o in a ian complex s uc u es is ca ied ou in [3]. Fo he desc ip ion o he complex s uc u es
we use a complex basis o (in a ian ) o ms {ωj}3
j=1 o bideg ee (1,0) wi h espec o he complex
s uc u e. Remembe ha he e exis wo complex-pa allelizable nilmani olds, de ined by he
equa ions
(4) dω1=dω2= 0, dω3=ρ ω12,
whe e ρ∈ {0,1}. One is he o us (ρ= 0) and he o he one is he Iwasawa mani old (ρ= 1).
We nex use he desc ip ion o he emaining in a ian complex s uc u es ob ained in [3], whe e
hey a e di ided in o h ee amilies:
Family (I): dω1=dω2= 0, dω3=ρ ω12 +ω1¯
1+λ ω1¯
2+D ω2¯
2,
whe e ρ∈ {0,1},λ∈R≥0, and D∈Cwi h Im D≥0;
Family (II): dω1= 0, dω2=ω1¯
1, dω3=ρ ω12 +B ω1¯
2+c ω2¯
1,
whe e ρ∈ {0,1},B∈C,c∈R≥0, wi h (ρ, B, c)6= (0,0,0);
Family (III): dω1= 0, dω2=ω13 +ω1¯
3, dω3=ε i ω1¯
1+i δ(ω1¯
2−ω2¯
1),
whe e ε∈ {0,1}and δ=±1.
Any in a ian He mi ian me ic Fon Xis gi en in e ms o a (1,0)-basis {ω1, ω2, ω3}by
F=P3
j,k=1 xj¯
kωj¯
k, whe e xj¯
k∈Cand xk¯
j=−xj¯
k. Tha is, Fcan be w i en as
(5) F=x1¯
1ω1¯
1+x2¯
2ω2¯
2+x3¯
3ω3¯
3+x1¯
2ω1¯
2−x1¯
2ω2¯
1+x1¯
3ω1¯
3−x1¯
3ω3¯
1+x2¯
3ω2¯
3−x2¯
3ω3¯
2.

6
No ice ha he posi i e de ini eness o he me ic Fimplies ha in pa icula
−i xj¯
j∈R+, i de (xj¯
k)>0.
Rema k 2.6. When Jis a complex s uc u e gi en by (4), i.e. Jis complex-pa allelizable, he
sign o he cons an c1(F) is well known o any J-He mi ian me ic F. Indeed, i is clea ha
i ρ= 0 hen Fis closed, so c1(F) = 0. I ρ= 1, i.e. he complex nilmani old is he Iwasawa
mani old, hen Fis balanced and hus c1(F)>0.
The ollowing esul shows he sign o any in a ian He mi ian me ic o he amilies o complex
s uc u es (I), (II), and (III).
P oposi ion 2.7. Le Xbe a complex nilmani old o complex dimension 3, and le Fbe any
in a ian He mi ian me ic on X. Suppose ha he complex s uc u e Jon Xis no complex-
pa allelizable. We ha e:
(i) I Jis a complex s uc u e in Family (I), hen
i
2∂¯
∂F ∧F=i x2
3¯
3
12 de (xj¯
k)ρ+λ2−2Re DF3,
and hus he sign o c1(F)only depends on he complex s uc u e. Indeed,
c1(F)is >0 (= 0, o <0) i and only i 2Re D−ρ−λ2<0 (= 0, o >0).
(ii) I Jis a complex s uc u e in Family (II), hen
i
2∂¯
∂F ∧F=i x2
3¯
3
12 de (xj¯
k)ρ+|B|2+c2F3,
and c1(F)>0, o any F.
(iii) I Jis a complex s uc u e in Family (III), hen
i
2∂¯
∂F ∧F=i(x2
2¯
2+x2
3¯
3)
6 de (xj¯
k)F3,
and hus c1(F)>0, o any F.
P oo . The esul is a di ec applica ion o [9, Lemma 3.2] aking in o accoun he educed complex
s uc u e equa ions ob ained in [3] and gi en in he amilies (I), (II), and (III) abo e. 
As a consequence o he p e ious p oposi ion, he sign o c1(F) only depends on he complex
s uc u e. Fu he mo e, as i was no iced in [9, P oposi ion 3.3], an in a ian He mi ian me ic F
on a complex nilmani old Xo complex dimension 3 sa is ies c1(F) = 0, i.e. i is 1-s Gauduchon,
i and only i Fis SKT. Recall ha he classi ica ion o nilmani olds admi ing an in a ian SKT
me ic was gi en in [7].
In Table 1 we analyze he sign o c1 o in a ian He mi ian me ics on complex nilmani olds o
complex dimension 3, i.e. on 6-dimensional nilmani olds endowed wi h in a ian complex s uc-
u es J. The algeb as in he i s column co espond o hose nilpo en Lie algeb as unde lying
such nilmani olds. He e, we ollow he no a ion gi en in he pape [18] o name and desc ibe
he di e en Lie algeb as. Fo ins ance, h2= (0,0,0,0,12,34) means ha he e is a basis o eal
1- o ms {ej}6
j=1 sa is ying de1=de2=de3=de4= 0, de5=e1∧e2, and de6=e3∧e4.
No e ha he nilpo en Lie algeb as admi ing a complex-pa allelizable s uc u e (4) a e h1( o
ρ= 0) and h5( o ρ= 1). Mo eo e , a Lie algeb a admi ing a complex s uc u e in Family (I)
is isomo phic o h2,...,h6, o h8, and i is always 2-s ep nilpo en . The nilpo en Lie algeb as
ha ing complex s uc u es in Family (II) a e h7and h9,...,h16. Finally, he nilpo en Lie algeb as
co esponding o Family (III) a e h−
19 ( o ε= 0) and h+
26 ( o ε= 1).
7
In he second column o Table 1, we indica e he nilpo ency s ep o he nilmani olds. Fo he
o he columns, we use he ollowing con en ion. The symbol “X” means ha o any in a ian
complex s uc u e Jon he co esponding nilmani old and o any in a ian J-He mi ian me -
ic F, he sign o c1(F) is always as indica ed in he able ( emembe ha he sign o c1(F) only
depends on he complex s uc u e, and c1(F) = 0 i and only i Fis SKT). The symbol “X
(J)”
means ha he e exis in a ian complex s uc u es Jon he co esponding nilmani old admi ing
in a ian J-He mi ian me ics Fwi h he gi en sign o c1(F), bu he e a e also o he complex
s uc u es wi h in a ian He mi ian me ics o di e en sign. In con as , he symbol “−” means
ha none o he in a ian complex s uc u es admi s in a ian He mi ian me ics o he gi en
sign.
s ep c1<0 SKT c1>0
h1= (0,0,0,0,0,0) 1 −X−
h2= (0,0,0,0,12,34) 2 X
(J)X
(J)X
(J)
h3= (0,0,0,0,0,12+34) 2 X
(J)−X
(J)
h4= (0,0,0,0,12,14+23) 2 X
(J)X
(J)X
(J)
h5= (0,0,0,0,13+42,14+23) 2 X
(J)X
(J)X
(J)
h6= (0,0,0,0,12,13) 2 − − X
h7= (0,0,0,12,13,23) 2 − − X
h8= (0,0,0,0,0,12) 2 −X−
h9= (0,0,0,0,12,14+25) 3 − − X
h10 = (0,0,0,12,13,14) 3 − − X
h11 = (0,0,0,12,13,14+23) 3 − − X
h12 = (0,0,0,12,13,24) 3 − − X
h13 = (0,0,0,12,13+14,24) 3 − − X
h14 = (0,0,0,12,14,13+42) 3 − − X
h15 = (0,0,0,12,13+42,14+23) 3 − − X
h16 = (0,0,0,12,14,24) 3 − − X
h−
19 = (0,0,0,12,23,14−35) 3 − − X
h+
26 = (0,0,12,13,23,14+25) 4 − − X
Table 1. Sign o c1 o in a ian He mi ian me ics on 6-nilmani olds
The exis ence o locally con o mal K¨ahle and balanced me ics on 6-dimensional nilmani olds
is s udied in [21]. On he one hand, i is seen ha , apa om he o us, he nilmani olds
admi ing balanced me ics ha e unde lying Lie algeb as h2,...,h6, o h−
19. On he o he hand, i
a 6-dimensional nilmani old has an LCK me ic hen i s unde lying Lie algeb a is h3. The e o e,
he only 6-nilmani old ha ing in a ian complex s uc u es wi h LCK and balanced me ics is he
one wi h h3as unde lying Lie algeb a. In he ollowing example we apply Theo em 2.5 o his
nilmani old.
Example 2.8. Le us s a ecalling ha he nilpo en Lie algeb a h3= (0,0,0,0,0,12+34)
admi s, up o equi alence, only wo complex s uc u es J±, which co espond o ρ=λ= 0 and
D=±1 in Family (I). The s uc u e J+admi s LCK me ics, whe eas J−admi s balanced ones.
8
Le us deno e by X± he complex nilmani olds espec i ely associa ed o (h3, J±). Conside he
J±-He mi ian me ics F+and F−
, wi h > 0, gi en as ollows:
(X+, F+) : (dω1=dω2= 0, dω3=ω1¯
1+ω2¯
2,
F+=i
2(ω1¯
1+ω2¯
2+ω3¯
3),
(X−, F−
) : (dσ1=dσ2= 0, dσ3=σ1¯
1−σ2¯
2,
F−
=i
2(σ1¯
1+σ2¯
2+ σ3¯
3), > 0.
We obse e ha dF+=θ∧F+wi h θ=ω3+ω¯
3(i.e. F+is an LCK me ic on X+) and
d(F−
)2= 0 (i.e. F−
is a balanced me ic on X−, o any > 0). Now, we can apply Theo em 2.5
o ensu e ha he p oduc mani old Y=X+×X−admi s a 1-s Gauduchon me ic.
Mo e conc e ely, by P oposi ion 2.7 one has
c1(F+) = −1
3, c1(F−
) =
3,
so he me ics
F =1
F++F−
de ined on Ya e 1-s Gauduchon o each > 0.
Nex we desc ibe he complex nilmani olds o complex dimension 3 o which he in a ian
He mi ian me ics Fsa is y c1(F)≤0. As he h3case is explained in Example 2.8, i emains o
s udy he Lie algeb as h2,h4, and h5:
Case h2:In [3] i is p o ed ha any complex s uc u e Jon h2is isomo phic o one and only
one in he ollowing amilies:
(h2, J1): dω1=dω2= 0, dω3=ω1¯
1+D ω2¯
2, D ∈Cwi h Im D= 1;
(h2, J2): dω1=dω2= 0, dω3=ω12 +ω1¯
1+ω1¯
2+D ω2¯
2, D ∈Cwi h Im D > 0.
By P oposi ion 2.7 (i), any J-He mi ian me ic Fon h2sa is ies
c1(F)<0 (= 0) i and only i Jis gi en by 




J1wi h Re D > 0 (= 0),
o
J2wi h Re D > 1 (= 1).
Case h4:As i is seen in [3], any complex s uc u e Jon h4is isomo phic o one and only one o
he ollowing ones:
(h4, J3): dω1=dω2= 0, dω3=ω1¯
1+ω1¯
2+1
4ω2¯
2;
(h4, J4): dω1=dω2= 0, dω3=ω12 +ω1¯
1+ω1¯
2+D ω2¯
2, D ∈R {0}.
No ice ha o J3we ha e 2 Re D= 1/2<1 = ρ+λ2. The e o e, by P oposi ion 2.7 (i) he
sign o c1o any J3-He mi ian me ic Fis posi i e, i.e. c1(F)>0. Simila ly, o any complex
s uc u e Jgi en by J4, P oposi ion 2.7 (i) implies ha c1(F)>0 i and only i D < 1.
The e o e, o any J-He mi ian me ic Fon h4we ha e:
c1(F)<0 (= 0) i and only i Jis gi en by J4wi h D > 1 (= 1).
Case h5:Using again [3], any complex s uc u e Jon h5is isomo phic o one and only one in
he ollowing lis :
(h5, J5): dω1=dω2= 0, dω3=ω12;
9
(h5, J6): dω3=ω1¯
1+ω1¯
2+D ω2¯
2,D∈[0,1
4);
(h5, J7): dω1=dω2= 0, dω3=ω12 +ω1¯
1+λ ω1¯
2+D ω2¯
2,wi h (λ, D)∈R≥0×Csuch ha :
•λ= 0 ≤Im D, 4(Im D)2<1+4Re D;
•0< λ2<1
2,0≤Im D < λ2
2,Re D= 0;
•1
2≤λ2<1,0≤Im D < 1−λ2
2,Re D= 0; o
•λ2>1,0≤Im D < λ2−1
2,Re D= 0.
We obse e ha J5is he complex-pa allelizable s uc u e on he Iwasawa mani old and any J5-
He mi ian me ic Fsa is ies c1(F)>0 (see Rema k 2.6). Fo any complex s uc u e Jgi en
by J6we ha e 0 ≤2Re D < 1
2<1 = ρ+λ2, hence by P oposi ion 2.7 (i) we also ge c1(F)>0
o any J6-He mi ian me ic F.
Using again P oposi ion 2.7 (i) one concludes ha o any J-He mi ian me ic Fon h5:
c1(F)<0 i and only i Jis gi en by J7wi h (λ= 0 ≤Im D,
4(Im D)2<1+4Re D, and Re D > 1
2,
and
c1(F) = 0 i and only i Jis gi en by J7wi h (λ= 0 ≤Im Dand
4(Im D)2<1+4Re D= 3.
As a di ec consequence o he p e ious discussion, we ge he ollowing classi ica ion o he
complex s uc u es J o which c1(F)<0 o any J-He mi ian me ic F.
P oposi ion 2.9. Le Xbe a complex nilmani old o complex dimension 3admi ing an in a ian
He mi ian me ic Fwi h c1(F)<0. Deno e (g, J) he Lie algeb a and he complex s uc u e
unde lying X. Then, he pai (g, J)is isomo phic o one (and only one) in he ollowing amilies:
(h2, J1)wi h Re D > 0(1-pa ame e amily o non-equi alen complex s uc u es),
(h2, J2)wi h Re D > 1(2-pa ame e amily),
(h3, J+)gi en in Example 2.8 ( his is a unique complex s uc u e),
(h4, J4)wi h D > 1(1-pa ame e amily),
(h5, J7)wi h λ= 0,Im D≥0and Re D > min 1
2,(Im D)2−1
4(2-pa ame e amily).
A second di ec consequence o he discussion abo e is he classi ica ion o complex s uc u es J
admi ing SKT me ics.
P oposi ion 2.10. Le Xbe a complex nilmani old o complex dimension 3admi ing an in-
a ian SKT me ic F(i.e. c1(F) = 0). Le (g, J)be he Lie algeb a and he complex s uc u e
unde lying X. Then, he pai (g, J)is isomo phic o one (and only one) in he ollowing amilies:
(h1, J)gi en by (4) wi h ρ= 0 (unique),
(h2, J1)wi h Re D= 0 (unique),
(h2, J2)wi h Re D= 1 (1-pa ame e amily),
(h4, J4)wi h D= 1 (unique),
(h5, J7)wi h λ= 0,Re D=1
2, and 0≤Im D < √3
2(1-pa ame e amily),
(h8, J)gi en by Family (I) wi h ρ=λ=D= 0 (unique).