(2024) 29:409–423
T ans o ma ion G oups
h ps://doi.o g/10.1007/s00031-022-09744-6
Ha d Le sche z P ope y o Isome ic Flows
Jos´
e Ignacio Royo P ie o1·Ma in xo Sa alegi-A angu en2·Robe Wolak3
Recei ed: 6 Ap il 2021 / Accep ed: 18 May 2022
©The Au ho (s) 2022
Abs ac
The ha d Le sche z p ope y (HLP) is an impo an p ope y which has been s udied
in se e al ca ego ies o he symplec ic wo ld. Fo Sasakian mani olds, his duali y is
sa is ied by he basic cohomology (so, i is a ans e se p ope y), bu a new e sion
o he HLP has been ecen ly gi en in e ms o duali y o he cohomology o he
mani old i sel in [1]. Bo h p ope ies we e p o ed o be equi alen (see [2]) in he
case o K-con ac lows. In his pape , we ex end bo h e sions o he HLP ( ans e se
and no ) o he mo e gene al ca ego y o isome ic lows, and show ha hey a e
equi alen . We also gi e some explici examples which illus a e he ca ego ies whe e
he HLP could be conside ed.
Keywo ds Le sche z ha d p ope y ·Con ac mani olds ·Isome ic low
Ma hema ics Subjec Classifica ion (2010) 53C12 ·53D10 ·53C25
In oduc ion
The o igins o he ha d Le sche z p ope y (HLP in he sequel) go back o Le sche z’s
s udy o opological p ope ies o algeb aic eal p ojec i e a ie ies [3], whe e he
Jos´
e Ignacio Royo P ie o
[email p o ec ed]
Ma in xo Sa alegi-A angu en
ma in.sa aleguia angu en@uni -a ois.
Robe Wolak
[email p o ec ed]
1Ma ema ika Saila, Zien zia e a Teknologia Fakul a ea, Uni e si y o he Basque Coun y
UPV/EHU, Ba io Sa iena s/n 48940 Leioa, Spain
2UR 2462, Labo a oi e de Ma h´
ema iques de Lens (LML), Uni . A ois,
F-62300 Lens, F ance
3Ins y u Ma ema yki, Uniwe sy e Jagiellonski, ul. p o . S anisława Łojasiewicza 6, 30-348
K ak´
ow, Poland
/Published online: 8 July 2022
J.I. Royo P ie o e al.
p o ed ha he epea ed cup p oduc by he cohomology class o a hype plane gi es
an isomo phism in he cohomology o he a ie y. La e , a e sion o ha heo em
was p o ed by Hodge (see [4]) o gene al compac K¨
ahle mani olds, s a ing iso-
mo phisms be ween de Rham cohomology g oups o complemen a y deg ees gi en
by mul iplica ion by a powe o he symplec ic o m. This p ope y was conside ed
o be one o he mos impo an o his class o mani olds. Compac K¨
ahle mani olds
ha e e y s ong and pa icula cohomological p ope ies. A lo o e o was pu in o
dis inguishing such p ope ies which could cha ac e ize K¨
ahle mani olds wi hin he
ca ego y o compac symplec ic mani olds. Now we know among o he hings ha
•The e a e compac symplec ic mani olds which a e no K¨
ahle , c . [5] o he i s
such an example;
•The o us is he only nilmani old which is K¨
ahle , c . [6];
•The e a e compac symplec ic mani olds whose cohomology ing is o mal
which a e no K¨
ahle , c . [7];
•The e a e compac He mi ian mani olds wi h collapsing F ¨
oliche spec al
sequence which a e no K¨
ahle , c . [8];
•The e a e compac symplec ic mani olds sa is ying he HLP which a e no
K¨
ahle , c . [9]. Howe e , a nilmani old ha ing he HLP is di eomo phic o a
o us, c . [6], hus a K¨
ahle mani old.
O e he yea s, many examples ha e been discussed and published. Among o he
publica ions, le us men ion he pape s [10–15] and he book [16].
Wi hin he ealm o olia ions, he ounda ions o he heo y o ans e sely K¨
ahle
olia ions we e p esen ed by El Kacimi in [17].Co de oandWolakin wopape s
p esen ed a se ies o examples showing ha he co esponding ans e se p ope ies
o he basic cohomology do no cha ac e ize ans e sely K¨
ahle olia ions, c . [18,
19].
These esul s p o ed o be o pa icula impo ance in he s udy o he odd-
dimensional coun e pa o K¨
ahle mani olds, i.e., Sasakian mani olds. In pa icula ,
by [17, pa . 3.4.7], he basic cohomology o a compac Sasakian mani old sa is ies
he HLP. In ecen yea s, a lo o esea ch has been done o dis inguish Sasakian
mani olds wi hin he class o con ac me ic mani olds and K-con ac mani olds in
pa icula , e.g., c . [1,20,21].
One o he p ope ies used in hese conside a ions was a new e sion o he HLP
o Sasakian mani olds demons a ed in [22] which s a ed Le sche z- ype isomo -
phisms no o he basic cohomology g oups, bu o he de Rham g oups o he
mani old i sel . The au ho s o ha pape ex ended he scope o he p ope y by gi -
ing a de ini ion o Le sche z con ac mani old in he same global e ms. Examples
o non-Sasakian Le sche z con ac mani olds ha e been gi en in [22]and[23], all
o hem wi hin he ca ego y o isome ic lows (i.e., he Reeb ield associa ed o he
con ac s uc u e is a Killing ec o ield).
So, a p io i he e a e wo di e en p ope ies (global and basic) ha a con ac man-
i old may o may no sa is y, bo h o hem gene alizing he HLP sa is ied by Sasakian
mani olds. In [2], he au ho p o es ha bo h p ope ies a e equi alen o compac
K-con ac mani olds. To de ine he Le sche z map, he au ho uses he symplec ic
Hodge heo y. We do no know whe he ha equi alence is held o all con ac lows.
410
Ha d Le sche z P ope y o Isome ic Flows
Le sche z- ype isomo phisms also exis in he ealm o isome ic lows, whe e he
ole o he class o he symplec ic o m is played by he Eule class. In his wo k, we
de ine wo duali y p ope ies o isome ic lows which esemble he HLP: a ans e -
sal one THLand a global one HL. Al hough ou de ini ion is essen ially opological
and no symplec ic s uc u e is needed, in he case o K-con ac lows, ou new de -
ini ions ag ee wi h he p e ious e sions o he HLP in oduced abo e. In Sec ion
1, we p o e ha bo h p ope ies a e equi alen o isome ic lows. So, we can call
Le sche z isome ic lows he isome ic lows sa is ying THLo HL.
In Fig. 1, we show he ca ego ies whe e he HLP has been de ined. The HLP is sa -
is ied in he ec angula egion. We do no know whe he he shaded a ea is nonemp y
( ha is, whe he he e exis Le sche z con ac lows which a e no K-con ac ), bu all
o he egions a e, as we illus a e wi h some examples in Sec ion 2. In Example 2.2,
we p o ide a Le sche z isome ic low which does no admi a con ac s uc u e. In
o de o ind an example o a low which is con ac , Le sche z bu no isome ic, we
ha e o look o a low which is no Riemannian as he Le sche z condi ion ensu es
au ness in he Riemannian ealm; hus, ou low would be isome ic. In he case
o ans e sely symplec ic bu no Riemannian olia ions, we can encoun e in ini e
dimensional basic cohomology which makes he Le sche z condi ion p oblema ic, as
happens in Example 2.9. We do no know whe he he ans e sal and he global de -
ini ions o he HLP a e equi alen i he con ac low is no isome ic. In he e e ed
example, nei he o hem is sa is ied, bu he p oblems appea a non-co esponding
deg ees, di e en ly as in he isome ic case.
1 Le sche z Duali y and T ans e se Duali y o Isome ic Flows
1.1 P elimina ies
Th oughou his sec ion, (M, g) deno es a closed Riemannian mani old endowed
wi h an isome ic low F, ha is, a 1-dimensional olia ion de ined by he o bi s
o a locally ee R-ac ion by isome ies. Le Xbe he uni ec o ield de ining he
Fig. 1 Some ca ego ies whe e he HLP has been conside ed
411
J.I. Royo P ie o e al.
low. H∗
Mand H∗
Bs and o he de Rham cohomology o Mand he basic cohomol-
ogy o he low, espec i ely. The la e is he cohomology o he complex o basic
o ms {ω∈(M)|iXω=iXdω =0}. The closu e o Rin he g oup o isome-
ies Iso(M, g) is an abelian compac and connec ed g oup, and hence, a o us G.
We ha e an isomo phism ∗:H∗
M→H∗((M)G)be ween he de Rham and he
G-in a ian cohomology g oups (see [24, Th. 1 in p. 151]).
We ha e he Gysin exac sequence (see [25, Th. 6.13]):
(1.1)
whe e ιkis induced by he na u al inclusion o he basic complex in o he de Rham
complex, εkis he mul iplica ion by he Eule Class [e]=[dχ]∈H2
Bo F,being
χ=iXg he cha ac e is ic o m o F,andρk=iX◦∗,beingiX he con ac ion
ope a o (no ice ha ρk([ω])=[iXω]when ωis X-in a ian ). In he li e a u e, he
mul iplica ion by he Eule class is also known as he Le sche z ope a o ,andis
deno ed by
1.2 Ha d Le sche z Duali y P ope ies
De ini ion 1.1 Le Fbe an isome ic low on he closed mani old M,whe e
dim M=2n+1, and le [e]∈H2
Bdeno e i s Eule class. We will say ha Fsa is-
ies he ans e sal ha d Le sche z p ope y a deg ee k∈Zi he ollowing p ope y
holds:
whe e Ln−k([β])=[β∧en−k]. We also de ine he ollowing p ope ies:
(T HL)≤k:(T HL)jholds o e e y j≤k
(T HL) :(T H L)jholds o e e y j∈Z.
In his las case, we will say ha Fsa is ies he ans e sal ha d Le sche z p ope y.
Rema k 1.2 (T H L)kholds i ially i k<0o k>2n.Fo k=0, on one hand,
i Fis ans e sally symplec ic, hen (T HL)0is sa is ied. On he o he hand, one
can easily cons uc an S1-p incipal bundle o e B=T4=R4/Z4wi h a non i ial
Eule class (say, [e]=[dx1∧dx2]∈H2
B). We ha e ha [e] = 0, bu [e2]=0, and
hus (T HL)0does no hold.
De ini ion 1.3 We de ine he k- h basic p imi i e cohomology g oup as he ke nel
o he map , ha is,PH k
B=[β]∈Hk
B|[β∧en−k+1]=0
in H2n−k+2
B.
De ini ion 1.4 We will say ha Fsa is ies he k- h p imi i e condi ion (and deno e
i by Pk) i he inclusion o o ms induces he ollowing wo isomo phisms:
412
Ha d Le sche z P ope y o Isome ic Flows
(P1)k:
(P2)k:Hk
B=PH k
B⊕LHk−2
B.
Rema k 1.5 No ice ha as PH 0
B=H0
B=H0
M, henP0is always ue. (P1)1is
no always ue, bu o e e y β∈1
Bwe ha e β∧en∈2n+1
B=0, so we ha e
PH 1
B=H1
B,and hen(P2)1always holds.
Lemma 1.6 Fo e e y k≤n,
P oo By Rema k 1.5, he esul holds i ially o k=0. I k≥1, by (T H L)k−1,
Ln−k+1=L◦L◦···◦Lis an isomo phism, and hus, a monomo phism. So, he i s
map in ha composi ion L=εk:Hk−1
B−→ Hk+1
Bmus be a monomo phism, oo.
The exac ness o he Gysin sequence (1.1) implies ha ιkis an epimo phism.
Rema k 1.7 By deg ee easons ι1:H1
B→H1
Mis always a monomo phism ( his
holds o any olia ion). So, by Rema k 1.5 and Lemma 1.6, we ha e ha (T H L)0
implies P1.
P oposi ion 1.8 Fo e e y k≤n,
P oo By Rema ks 1.5 and 1.7, we ha e P0and P1. Conside k≥2. On one hand,
he Gysin sequence (1.1)gi es
Hk
B∼
=[imιk⊕ke ιk∼
=Hk
M⊕[imεk−1∼
=Hk
M⊕LHk−2
B,(1.2)
whe eweha eused ha ιkis an epimo phism, which holds by (T HL)k−1and
Lemma 1.6. On he o he hand, we conside he sum PH k
B+LHk−2
B≤Hk
B.We
now show ha he sum is a di ec one: ake [β]∈PH k
B∩LHk−2
B. Then, he e
exis s [γ]∈Hk−2
Bsuch ha [β]=[γ∧e]∈PH k
B, which implies
0=β∧en−k+1=γ∧en−k+2=Ln−k+2([γ]),
and by (T HL)k−2we ha e [γ]=0. Thus, [β]=0, and he sum is di ec . F om
Eq. 1.2,wege
PH k
B⊕LHk−2
B≤Hk
B=Hk
M⊕LHk−2
B,(1.3)
413
J.I. Royo P ie o e al.
which implies ha dim PH k
B≤dim Hk
M. Hence, i we p o e ha ik:
is an epimo phism, i would be an isomo phism, yielding (P1)kand, by Eq. 1.3,
(P2)k.
We comple e he p oo by showing ha ikis an epimo phism. Le [α]∈Hk
M.By
(T HL)k−1and Lemma 1.6, he e exis s [β]∈Hk
Bsuch ha ιk([β])=[α]. F om
(T HL)k−2, he e exis s [γ]∈Hk−2
Bsuch ha
β∧en−k+1=Ln−k+2([γ])=γ∧en−k+2∈H2n−k+2
B,
which leads o [(β−γ∧e) ∧en−k+1]=0∈H2n−k+2
Band hus, [β−γ∧e]∈PH k
B.
Finally, we ha e
ik([β−γ∧e])=[β−d(χ ∧γ)]=[β]∈Hk
M.
De ini ion 1.9 Le Fbe an isome ic low on he closed mani old M,whe e
dim(M) =2n+1, and le [e]∈H2
Bdeno e i s Eule class. We say ha Fsa -
is ies he ha d Le sche z p ope y a deg ee k(and deno e i by (H L)k)i he e
exis s an isomo phism Ln−k:Hk
M−→ H2n−k+1
Mmaking he ollowing diag am
commu a i e:
(1.4)
We shall also use he ollowing no a ions:
(HL)≤k:(HL)jholds o e e y j≤k
(HL) :(HL)jholds o e e y j∈Z.
In his las case, we will say ha Fsa is ies he ha d Le sche z p ope y.
Theo em 1.10 Le Fbe an isome ic low on a closed o ien ed mani old Mo
dimension 2n+1. Then, o e e y k≤n, we ha e (T HL)≤k⇐⇒ (H L)≤k.
P oo Assume (T HL)≤k. By P oposi ion 1.8, Pkis ue, and so ikis an isomo -
phism. To de ine Ln−k,we ixabasis{[βi]}io PH k
B.As[βi∧en−k+1]=0∈
H2n−k+2
B, he e exis s a basic o m γi∈2n−k+1
Bsuch ha βi∧en−k+1=dγi,and
so, χ∧βi∧en−k−γi∈2n−k+1
Mis a closed o m. Thus, we can de ine
Ln−k([βi])=[χ∧βi∧en−k−γi](1.5)
and ex end i by linea i y. As γiis basic and χ∧βi∧en−k−γiis X-in a ian , we ha e
ρ2n−k+1([χ∧βi∧en−k−γi])=[iX(χ ∧βi∧en−k−γi)]=[βi∧en−k],
and he diag am (1.4) is commu a i e. Finally, om Eq. 1.4,weha e
ke Ln−k≤ke (ρ ◦Ln−k)≤ke (Ln−k|PH k
B◦(ik)−1)={0},
414
Ha d Le sche z P ope y o Isome ic Flows
because ikand Ln−ka e isomo phisms. So, Ln−kis a monomo phism be ween
Hk
Mand H2n−k+1
M, who ha e he same dimension by Poinca ´
e duali y. Hence, an
isomo phism.
Fi s no ice ha Fis ans e sally o ien able, and by [26,Th.A]and
[27, Th. 4.10], H∗
Bsa is ies he Poinca ´
e duali y. In pa icula , Hk
Band H2n−k
Bha e
he same dimension. So, in o de o p o e ha Ln−kis an isomo phism, i su ices o
show ha Ln−kis an epimo phism.
As he s a emen is i ial o k<0, we shall p oceed by induc ion on ks a ing
a k=−2. Assume ha (HL)≤kholds and assume (HL)≤k−1⇒(T HL)≤k−1.By
P oposi ion 1.8, Pkholds, gi ing ha ikis an isomo phism. To show ha Ln−kis
on o, we now ake [ϕ]∈H2n−k
B.By(T HL)k−2,Ln−k+2is an isomo phism, and so,
he e exis s [γ]∈Hk−2
Bsuch ha
[ϕ∧e]=Ln−k+2[γ]=γ∧en−k+2∈H2n−k+2
B.
We ha e he ollowing commu a i e diag am,
(1.6)
whose igh column is pa o he Gysin sequence (1.1). We ha e
[ϕ−γ∧en−k+1]∈ke ε2n−k+1=imρ2n−k+1,
and so, by Eq. 1.6, he e exis s [β]∈Hk
Bsuch ha
[β∧en−k]=Ln−k[β]=[ϕ−γ∧en−k+1],
which implies
[ϕ]=[β∧en−k+γ∧en−k+1]=[(β+γ∧e) ∧en−k]=Ln−k([β+γ∧e]),
which concludes he p oo .
Now, he ollowing de ini ion makes sense.
De ini ion 1.11 Wesay ha anisome ic lowFon a closed mani old Mis an
isome ic Le sche z low i i sa is ies (T HL) o (HL).
Rema k 1.12 Gi en an isome ic low Fon a compac mani old, in [28, Sec ion 3.2],
i is p o ed ha he Eule classes associa ed o wo in a ian me ics a e he same
up o a mul iplica i e nonze o cons an . As a esul , whe he an isome ic low is
415
J.I. Royo P ie o e al.
Le sche z o no is a opological p ope y in he sense ha i does no depend on he
chosen in a ian me ic, bu only on he olia ion Fi sel .
In [1], he au ho s de ine a (2n+1)-dimensional con ac mani old (M, η) wi h
Reeb ec o ield ξ o be a Le sche z con ac mani old i o e e y k≤n, he ela ion
be ween Hk
Mand H2n+1−k
Mde ined by
Rk=[β],η∧(dη)n−kβ|β∈k
M,dβ=0,i
ξβ=0,(dη)
n−k+1∧β=0
(1.7)
is he g aph o an isomo phism Hk
M∼
=H2n−k+1
M.
Rema k 1.13 No ice ha R0is always he g aph o he isomo phism H0
M∼
=H2n+1
M
because (dη)n+1=0.
We now see ha i ξis Killing (i.e., we ha e a K-con ac low), hen his no ion is
equi alen o (HL).
P oposi ion 1.14 Le (M, η) be a (2n+1)-dimensional K-con ac mani old. Then,
he isome ic low de ined by i s Reeb ec o ield is an isome ic Le sche z low i
and only i (M, η) is a Le sche z con ac mani old.
P oo We ha e ha X=ξis a Killing ec o ield, χ=ηand e=dη.I (M, η) is
a Le sche z con ac mani old, hen o e e y k≤n, he isomo phism Ln−kwhose
g aph is he ela ion (1.7) clea ly makes he diag am (1.4) commu a i e and so, X
de ines an isome ic Le sche z low. Con e sely, i Xis a Le sche z isome ic low,
i sa is ies (T HL) and we can cons uc an isomo phism Ln−k:Hk
M→H2n−k+1
M
as in he ⇒pa o Theo em 1.10. As (M, η) is con ac , by [29, Th. 11(1)], each
basic cohomology class has a ha monic ep esen a i e, and hus, each p imi i e basic
cohomology class admi s a p imi i e basic ep esen a i e1.So,asHk
M∼
=PH k
B,we
can ind a basis {[βi]}io Hk
M,whe eβia e p imi i e closed basic o ms. In he p oo
o Theo em 1.10, we ha e o add he o ms γi o ge closed o ms, bu as he βia e
p imi i e, we can choose γi=0andsoLn−k([β])=χ∧β∧en−kde ines an
isomo phism whose g aph is he ela ion (1.7). Thus, (M, η) is a Le sche z con ac
mani old.
In [1], he au ho s p o e ha he small odd Be i numbe s (up o he middle dimen-
sion) o a Le sche z Con ac low a e e en. As we show now, he same algeb aic
p oo wo ks o p o e he co esponding esul o isome ic Le sche z lows.
Theo em 1.15 Le Fbe an isome ic Le sche z low on he compac mani old M,
being dim(M) =2n+1. Then, he Be i numbe bk(M) is e en o e e y odd k≤n.
1This is [2, Lemma 2.11], which can also be p o ed by he las pa ag aph o he p oo o Theo em 0.1 o
[30], which applies e ba im o he complex o basic o ms o he con ac mani old.
416
Ha d Le sche z P ope y o Isome ic Flows
P oo Conside a basis o p imi i e basic classes {[βi]}io Hk
M, and cons uc an
isomo phism Ln−k:Hk
M→H2n−k+1
Mas in he p oo o Theo em 1.10. Conside
he non-degene a e bilinea o m Bon Hk
Mde ined as he composi ion
being P([ω1],[ω2])=Mω1∧ω2 he usual non-degene a e pai ing. Now, we ha e
B([βi],[βj])=P([βi],Ln−k[βj])
=M
(βi∧χ∧βj∧en−k−βi∧γi)
=M
βi∧χ∧βj∧en−k
whe eweha eused ha βi∧γi=0 because i is a basic o m o deg ee 2n+1. As
βiχβj=(−1)kβjχβi, i ollows ha B([βi],[βj])=(−1)kB([βj],[βi]).So,Bis a
non-degene a e skew-symme ical bilinea o m, and he dimension o Hk
Mmus be
e en.
2 Examples
Example 2.1 (Sasakian mani olds) Conside a Sasakian mani old Mo dimension
2n+1 ( o he essen ials o Sasakian geome y, we e e he eade o [31]). Recall
ha i s associa ed Reeb ec o ield Xde ines an isome ic low wi h espec o he
me ic go he Sasakian s uc u e, being he associa ed con ac o m χ=iXg he
cha ac e is ic o m o he isome ic low. As any Sasakian mani old is ans e sally
K¨
ahle , i sa is ies (T HL) (c .[17, 3.4.7]), and by Theo em 1.10, i sa is ies (H L),
which has been p o ed in [1, Sec ion 4].
In [32], Boo hby and Wang use he cons uc ion desc ibed by Kobayashi in
[33, Th. 2]) o ge examples o con ac mani olds ou o in eg al symplec ic o ms.
The same cons uc ion can also be applied o ge isome ic lows wi h a p esc ibed
in eg al Eule o m as ollows: gi en an in eg al closed o m ω∈2(B), Kobayashi’s
cons uc ion gi es an S1-p incipal bundle π:M→Bwhose connec ion o m
χ∈1(M)S1sa is ies dχ =π∗ω.Le Fbe he olia ion on Mde ined by he
o bi s o he p incipal S1-ac ion and conside on TM =TF⊕ke χ he Rieman-
nian me ic g=χ⊗χ+π∗
bgB,beinggBany me ic on Band πb he es ic ion o
π∗:TM →TB o ke χ, which is an isomo phism by deg ee easons. Then, Fis an
isome ic low on Mwhose Eule o m is dχ =π∗ω. We shall use his cons uc ion
in he ollowing examples. Fi s , we show ha Theo em 1.10 applies o new cases
ou side he con ac ca ego y:
417
