Table Of ContentFamilies of monotone symplectic manifolds
constructed via symplectic cut and their
0
1
0 Lagrangian submanifolds
2
n
Agnès GADBLED ∗
a
J
3 January 2010
1
]
G
Abstract
S
. Wedescribefamiliesofmonotonesymplecticmanifoldsconstructedvia
h
thesymplecticcuttingprocedureofLerman[15]from thecotangentbun-
t
a dle of manifolds endowed with a free circle action. We also give obstruc-
m
tionstothemonotoneLagrangian embeddingofsomecompactmanifolds
[ in these symplecticmanifolds.
1
v Introduction
5
2
Wedescribefamiliesofexamplesofmonotonesymplecticmanifoldsconstructed
3
2 by the symplectic cutting procedure of Lerman [15] from the cotangent bundle
. of manifolds endowed with a free circle action.
1
0 If V is a closed manifold endowed with a free circle action, this action in-
0 duces a Hamiltonian action of the circle on the cotangent bundle T∗V of V. If
1 H denotes the Hamiltonian associated to this action, W the symplectic man-
ξ
v: ifold obtained by symplectic cut of T∗V at the level ξ and Qξ the reduced
i manifold H−1(ξ)/S1, we prove that W has the following properties:
X ξ
r 1) (Lemma 2.7) The manifold W is a complex line bundle over Q .
ξ ξ
a
2) (Proposition2.5) The quotientQ is asymplectic submanifoldofW diffeo-
ξ ξ
morphic to the cotangent bundle of B = V/S1. If e is the Euler class of
the circle bundle V B and q is the projection Q B, then the class
ξ
→ →
of the symplectic form over Q is 2πξq⋆e.
ξ
−
∗MSCclassification: 53D05,53D12,53D20,53D40.
Keywords: Monotone symplectic manifold, monotone Lagrangian submanifold, symplectic
cut,Floerhomology,Maslovindex.
Partially supported by ANR project "Floer Power" ANR-08-BLAN-0291-03, SNF project
"Complexity and recurrence in Hamiltonian systems" and MSRI post-doc fellowship NSF
grantDMS-0441170
1
3) (Proposition 2.6) The class in H2(W ;R) of the symplectic form of W is
ξ ξ
related to the first Chern class of W by the formula:
ξ
[ω ]= 2πξc (W ).
ξ 1 ξ
−
In particular, W is monotone for negative ξ.
ξ
4) (Proposition 4.5) The manifold W is endowed with a Hamiltonian circle
ξ
action and the sum of the weights of this action in a fixed point is equal
to 1.
5) (Theorem3.3)ThezerosectionV ofthecotangentbundlecanbeembedded
as a monotone Lagrangiansubmanifold of W for any negative level ξ.
ξ
6) (Corollary 4.10) Any exact compact Lagrangian submanifold of T∗V with
vanishing Maslov class can be embedded as a monotone Lagrangian sub-
manifold of W for some negative level ξ.
ξ
In the general case of a monotone Lagrangian submanifold in a symplectic
manifold, Seidel [21] has proved, under some assumptions on the symplectic
manifold and the Lagrangiansubmanifold, that the Floer homology can be ab-
solutelygraded. Healsoprovedthatifthesymplecticmanifoldisendowedwith
a Hamiltoniancircle action,the Floer homologyof the Lagrangiansubmanifold
is periodic with respect to the absolute grading with period the sum of the
weights of the action in a fixed point. Applying these results to the monotone
Lagrangiansubmanifolds of the symplectic cut W , we get:
ξ
1) (Theorem4.8)Anobstructionresultontheembeddingofacompactsimply-
connectedmanifoldasamonotoneLagrangiansubmanifoldinasymplectic
cut under conditions on the Euler class e.
2) (Theorems 4.11 and 4.12) Results on the index of the fundamental groupof
an exactcompactLagrangiansubmanifold with vanishing Maslovclass in
the cotangent bundle of a manifold endowed with a free circle action.
3) (Theorem 4.20) The monotone Lagrangian tori in a simply connected sym-
plectic cut have a Maslov number equal to 2.
In the first section, we explain the symplectic cutting construction for our
specific circle action and describe it in some examples. In the second section,
westudy the structureofthe spaceswhichappearinthe constructionandtheir
monotonicity. The third part is essentially devoted to the monotonicity of the
canonicalembedding of V in the symplectic cut. After briefly recalling Seidel’s
periodicityresults,weapply themtothe monotoneLagrangiansubmanifoldsof
the symplectic cut in the fourth part and prove the results stated above. An
appendix contains complements to some proofs and a discussion of the mono-
tonicity in the case of a semi-free circle action.
2
Acknowledgements. This is a part of the author’s PhD thesis at the Univer-
sity of Strasbourg. I would like to thank Michèle Audin and Mihai Damian for
theirhelpfulsuggestionsandcommentsandfortheircarefulreadingofaearlier
version of this article. I am greatly indebted to them for the time they spend
discussing this subject with me.
1 Symplectic cut of cotangent bundles
Let π : V B be a principal circle bundle, the base B being path connected.
→
DenotebyX thefundamentalvectorfieldassociatedwiththecircleactiononV.
As this S1-action is free, the fundamental vector field cannot vanish.
Remark 1.1. We refer to appendix B for the study of the case of a semi-free
circle action.
Denote bye H2(B;Z)the Euler classofthe principalbundle π andby N
e
the nonnegative∈generator of the subgroup e,π (B) of Z.
2
h i
Example 0: In the case of the trivial bundle V = B S1 B, π is the
× →
projection on the first factor, the circle action is the complex multiplication on
thefactorS1 (seenasthegroupofcomplexnumbersofmodule1)andtheEuler
class is zero.
A subcaseis the caseofS1 acting onitselfby multiplication, the quotientB
being reduced to a point.
Example 1: If V =S2n+1 B =CPn (n 1) is the Hopf bundle, the Eu-
ler class is equal to the opposit→e of the preferre≥d generator of H2(CPn;Z) (this
can also be considered as a choice for the preferred generator of H2(CPn;Z)).
Example 2: Let V be the lens space L2n+1, quotient of the sphere S2n+1
p
by the action of the subgroup Z/p of S1 consisting of the p-roots of unity.
S2n+1
HHHHHHHHH$$
L2n+1
p
(cid:15)(cid:15) zzvvvvvvvvv
CPn
The free action of S1 on S2n+1 enables to define a free action of S1 on L2n+1,
p
with quotient CPn, such that the Euler class of the principal circle bundle
L2n+1 CPn is p times the preferred generator of H2(CPn;Z) (chosen in
p → −
Example 1).
3
Example 3: We can also consider the Stiefel manifold V (Rn+2) consisting
2
of the pairs of orthonormal vectors of Rn+2. This manifold V (Rn+2) can also
2
be described as the quotient SO(n +2)/SO(n). The subgroup SO(2) S1
of SO(n + 2) acts freely on V (Rn+2), the quotient of this action being≃the
2
Grassmannian B =G (Rn+2) of the (2-dimensional) oriented planes in Rn+2.
2
ThisGrassmanniancanbeidentifiedwiththequadricQn inCPn+1 ofequa-
e
n+1
tion z2 = 0. To see this, we can use the first description of the Stiefel
i
Xj=0
manifold V (Rn+2) as a subset of (Rn+2)2:
2
V (Rn+2)= (x,y) (Rn+2)2 x y =0, x = y =1 .
2
{ ∈ | · k k k k }
The action of SO(2) S1 can then be written:
≃
eiθ (x,y)=(cos(θ)x sin(θ)y,sin(θ)x+cos(θ)y).
· −
The map
V (Rn+2) CPn+1
2
−→
(x,y) [x+iy]
7−→
descendstothequotientanddefinesanembeddingoftheGrassmannianG (Rn+2)
2
in CPn+1, the image of which is the quadric Qn (see [4]).
e
For n 3, by the Lefschetz hyperplane theorem, the embedding of the
quadric ind≥uces an isomorphism between H2(G (Rn+2);Z) and H2(CPn+1;Z).
2
Therefore, H2(G (Rn+2);Z) is isomorphic to Z, a preferred generator being
2 e
given by the pull-back of the preferred generator of H2(CPn+1;Z). Besides,
e
V (Rn+2) is the total space of the pullback of the Hopf bundle S2n+3 CPn+1
2
→
by the embedding j of G (Rn+2) in CPn+1:
2
je⋆S2n+3 =V (Rn+2) // S2n+3
2
(cid:15)(cid:15) (cid:15)(cid:15)
G (Rn+2) j //CPn+1
2
e 1
Indeed, with the map V (Rn+2) S2n+3 givenby (x,y) (x+iy)and the
2
→ 7→ √2
projection V (Rn+2) G (Rn+2), the above diagram commutes. We deduce
2 2
→
by naturality of the Euler class that:
e
e(V (Rn+2) G (Rn+2)) = j⋆e(S2n+3 CPn+1)
2 2
→ →
and hence, for n 3, the Eeuler class of the bundle V (Rn+2) G (Rn+2) is
2 2
≥ →
equal to the opposite of the generator of G (Rn+2).
2 e
Forthelowdimensionsn=1andn=2,thedescriptionoftheStiefelmanifolds
e
and the corresponding Grassmannians are well known (see for instance [13]).
ThelongexactsequenceassociatedwiththeprincipalcirclebundleV (Rn+2)
2
→
G (Rn+2) gives the numbers: for n=1, N =2 and for n=2, N =1.
2 e e
e
4
Associated Hamiltonian action on the cotangent bundle
For u S1, denote by ρ(u) : V V (defined by ρ(u)(x) = u x) the action
of S1 o∈n V. With this circle actio→n on V is associated a S1-actio·n on T∗V:
S1 T∗V T∗V
× −→
(u,(x,ϕ)) (u x,u ϕ)
7−→ · ·
where
u ϕ= t(T ρ(u)−1)(ϕ),
x
·
namely, if v T V, then
u·x
∈
u ϕ=ϕ(T ρ(u)−1v).
x
·
IfT∗V is endowedwithits canonicalsymplectic structure,this actionis Hamil-
tonian with Hamiltonian function:
H : T∗V R
−→
(x,ϕ) ϕ,X(x)
7−→ h i
(see for instance [16, exercise 3.12]).
This action is free since the circle action is free on V and the canonical
projection p :T∗V V is an equivariant map.
V
→
In particular, this action has no fixed point, the associated Hamiltonian
vector field X does not vanish (because the projection of X on TV by Tp
H H V
is X, a vector field that never vanishes) and all the values of the Hamiltonian
are regular.
Symplectic cut of the cotangent bundle
A symplectic manifold endowed with a Hamiltonian circle action being given,
Lerman [15] has developped a construction which enables to embed the re-
ducedspacesascodimension2symplecticsubmanifoldsofasymplecticmanifold
(called symplectic cut)
WedescribethisconstructioninthecaseofthecotangentbundleT∗V. This
symplectic cut is naturally endowed with a Hamiltonian circle action (Propo-
sition 4.5). We are interested in the monotonicity of this symplectic manifold
(Section2) inorderto apply aresult ofSeidel[21](reminded inSection 4.1)on
the periodicity of the LagrangianFloer homology.
To define the symplectic cut construction, we consider the following Hamil-
tonian action of S1 on the product T∗V C :
×
S1 (T∗V C) T∗V C
× × −→ ×
(u,(x,ϕ,z)) (u x,u ϕ,u¯z)
7−→ · ·
with associated Hamiltonian:
H : T∗V C R
× −→
(x,ϕ,z) H(x,ϕ) 1 z 2.
e 7−→ − 2| |
5
As for H, this action is free and the levels of the Hamiltonian H are regular.
Let ξ R be a (regular) value of H. The level H−1(ξ) is the disjoint union
∈ e
of
e e
(x,ϕ,0)H(x,ϕ)=ξ ,
{ | }
diffeomorphic to H−1(ξ), and
(x,ϕ,z)H(x,ϕ)>ξ and z = 2(H(x,ϕ) ξ) ,
{ | | | − }
p
diffeomorphic to H−1(ξ,+ ) S1, the S1-action preserving both sets.
Notice that S1 acts also∞fre×ely on the level H−1(ξ) since
q : H−1(ξ) V
ξ
−→
(x,ϕ) x
e 7−→
is an equivariant fibration (see Lemma 2.1) and the action is free on the base
space.
As the action of S1 on the regular level H−1(ξ) is free, we can carry out a
symplecticreduction. Equippedwiththereducedsymplecticform,thequotient
e
W =H−1(ξ)/S1 is a symplectic manifold. The decomposition above descends
ξ
tothequotient,theimageofthepartH−1(ξ)isasymplecticsubmanifoldQ of
e ξ
W diffeomorphictoH−1(ξ)/S1. Itscomplementissymplecticallydiffeomorphic
ξ
to the open set H−1(ξ,+ ) (see Figure 1.1).
∞
H
T∗V
W
ξ
V
0 V
ξ Q
ξ
Figure 1.1: The cotangent bundle (left) and its symplectic cut (right)
Remark 1.2. If ξ <0, then H−1(ξ,+ ) is a neighbourhood of the zero section
∞
in the cotangent bundle T∗V and this neighbourhood is symplectically embedded
as an open submanifold of W . In particular, V is canonically embedded as a
ξ
Lagrangian submanifold of the symplectic cut W .
ξ
6
Because of Remark 1.2 and of the monotonicity condition on W (Proposi-
ξ
tion 2.6), we will deal in the following only with the case ξ <0.
Example 0: Let us first study the case of the circle V =S1 acting on itself
by multiplication with quotient B reducedto a point. The fundamental field of
theactionatthe pointxisthenix(weconsiderS1 asthe unitcircleinC). The
cotangent space T∗S1 is diffeomorphic to S1 R: if x is a point of S1 and ϕ is
an element of T∗S1, there exists a unique rea×l number λ such that ϕ= λiu,
(denoting by ,x the scalar product on C = R2). An explicit diffeomohrphism·i
fromT∗S1 tohS·1·i R canbe givenby mapping the pair (x,ϕ) onthe pair (x,λ).
The circle act×ion on T∗S1 can then be written on S1 R:
×
u (x,λ)=(ux,λ)
·
and the Hamiltonians become:
H(x,λ)=H(x,ϕ)=ϕ(ix)=λ
and
1
H(x,λ,z)=H(x,ϕ,z)=λ z 2.
− 2| |
Thus, the level H−1(ξe) = (x,ξ) x S1 is a circle and the quotient Q =
ξ
H−1(ξ)/S1 is a point. { | ∈ }
For H, the level H−1(ξ) is the set (x,ξ+ 1 z 2,z) x S1,z C and the
{ 2| | | ∈ ∈ }
quotient W = H−1(ξ)/S1 is symplectomorphic to C. Indeed, the one-to-one
e ξ e
map
e iC : C Wξ
−→
z 1,ξ+ 1 z 2,z
7−→ 2| |
(cid:2) (cid:3)
is also onto as each element
1 1
x,ξ+ z 2,z = 1,ξ+ z 2,x−1z
(cid:20) 2| | (cid:21) (cid:20) 2| | (cid:21)
ofWξ canbewrittenastheimagebyiC ofacomplexnumber(x−1z =x¯z here).
Moreover,the symplectic formωW on W is the symplectic reductionof the
ξ ξ
2-formω ω whereω isthecanonicalsymplecticformonT∗S1 andω
can std can std
is the standa⊕rdsymplectic formon C. Butif ıis the embedding ofC inH−1(ξ)
defined as :
1 e e
ı(z)= 1,ξ+ z 2,z ,
(cid:18) 2| | (cid:19)
then e
(i⋆ωW) (ζ ,ζ ) = (ı⋆(ω ω )) (ζ ,ζ )
ξ z 1 2 can⊕ std z 1 2
= ω (ζ ,ζ )
estd 1 2
since z (1,ξ+ 1 z 2) takes value in a fibre of the projection T∗S1 S1 and
7→ 2| | →
the fibres are Lagrangiansubmanifolds.
7
The manifold V = S1 is embedded as the zero section in T∗V and as the
image of the map
x [x,0, 2ξ]=[1,0,x¯ 2ξ]
7→ − −
p p
in W if ξ <0. Its image in C by the symplectic diffeomorphism just described
ξ
is the circle centered at the origin and of radius √ 2ξ. This circle bounds a
−
disc of area 2πξ (which is also the area of the cylinder that lies between the
−
zero section and the level ξ) (see Figure 1.2).
R
W =C
ξ
V
H
0 V
ξ
Q
ξ
Figure 1.2: Symplectic cut of T∗S1
In the general case of a product V = B S1 B, there is a symplectic
diffeomorphismT∗V T∗B T∗S1. Thecircl×eacti→ononV beingconcentrated
on the factor S1, the≃action×on T∗V is concentrated on T∗S1 and the corre-
sponding Hamiltonian on T∗B T∗S1 can be written, for any ((b,ϕ),(x,λ))
T∗B T∗S1 T∗B (S1 R)×, ∈
× ≃ × ×
H((b,ϕ),(x,λ))=λ.
Thus, the symplectic cut of T∗V (at the level ξ) is the product of T∗B and of
the symplectic cut of T∗S1 (also at the level ξ) and hence is symplectomorphic
to T∗B C.
If ξ ×is negative, V = B S1 can be embedded as the product of the zero
×
section of T∗B and of the circle centered at the origin of radius √ 2ξ.
−
Letusnowanalysethestructureofthesemanifolds. Inparticular,wewould
liketoknowifW ismonotone(seeSection2)andinthiscase,ifthezerosection
ξ
is monotone in W (Section 3).
ξ
8
2 Monotonicity of the symplectic cut
Recall (see [18] and [19]) that a symplectic manifold (W,ω) is said to be mono-
toneifthereexistsapositiveconstantK suchthatforanyspherew:S2 W,
W
→
w⋆ω =K c ([w]), (1)
W 1
Z
where c is the first Chern class of the tangent bundle TW W.
1
→
IfW isasymplecticmanifold,denotebyN itsfirstChernnumber,namely
W
the nonnegative generator of the subgroup c (W),π (W) of Z.
1 2
h i
We will prove that the symplectic structure and the first Chern class of W
ξ
can be deduced of those of Q . To begin with, let us investigate the reduced
ξ
manifold Q =H−1(ξ)/S1.
ξ
2.1 The reduced symplectic manifold Q
ξ
Lemma 2.1. The projection q : H−1(ξ) V is an equivariant fibration and
ξ
→
descends to the quotients in a fibration
e
q : Q B
ξ ξ
−→
[x,ϕ] π(x).
7−→
Proof. Let us begin by looking at this projection for ξ =0. In this case,
q : (x,ϕ) ϕ,X(x) =0 V
0
{ |h i } −→
(x,ϕ) x
e 7−→
is a vector subbundle of the cotangent bundle T∗V V: it is the annihilator
→
of the vector field X in the cotangent bundle. Descending to the quotients, q
0
is also a vector bundle over B.
Let α be a family of linear forms α T∗V such that for all x V,
α ,X(x{)x=}x∈1Vand for all u S1, α =u αx.∈Forxinstance, it can be defi∈ned
x u·x x
husing an Si1-invariantRieman∈nian metric g ·on V and setting α =g ( ,X(x)).
x x
·
For any ξ, this allows to define an equivariant diffeomorphism
Φ : H−1(ξ) H−1(0)
ξ
−→
(x,ϕ) (x,ϕ ξα ).
e 7−→ − x
such that q =q Φ .
ξ 0 ξ
◦
Descendingtothequotients,thediffeomorphismΦ definesadiffeomorphism
e e e ξ
Φ :Q Q such that q =q Φ .
ξ ξ → 0 ξ 0◦ ξ e
Corollary2.2. ThelevelH−1(0)(andhenceH−1(ξ)aswell)hasthehomotopy
type of V and the quotient Q (and hence Q as well) has the homotopy type
0 ξ
of B, the homotopy equivalence being given by the projections.
9
Furthermore, the fibre bundle q is a particular fibre bundle on B:
0
Lemma 2.3. The space Q endowed with the symplectic form obtained by re-
0
duction of the canonical symplectic form on T∗V is symplectomorphic to the
cotangent bundle T∗B.
A proof of Lemma 2.3 can be found in [1, Section 5].
Corollary2.4. ThefirstChernclassc (TQ Q )ofQ iszeroinH2(Q ;R).
1 ξ ξ ξ ξ
→
Proof. Thanks to Lemma 2.3, Q is symplectomorphic to T∗B. As the first
0
Chernclassc (T(T∗B) T∗B)is a 2-torsionelementin H2(T∗B;Z) (because
1
→
therestrictiontoB ofthe tangentbundleofT∗B isisomorphictothe complex-
ified bundle of a real bundle, see [17]), c (TQ Q ) is 2-torsion element in
1 0 0
H2(Q ;Z). →
0
Moreover,Q variescontinuouslywithξ andthefirstChernclassofQ takes
ξ ξ
values in a discrete space so that c (TQ Q ) is also a 2-torsion element
1 ξ ξ
of H2(Q ;Z) and hence vanishes in H2(Q ;→R).
ξ ξ
Wedeterminetheclass[ω ]ofthesymplecticformonQ withtheDuistermaat-
ξ ξ
Heckman formula (see [10], we follow here the sign conventions of [2]):
Proposition2.5. TheclassinH2(Q ;R)ofthereducedsymplecticformonQ
ξ0 ξ
is equal to:
[ω ]= 2πξ q⋆e.
ξ − ξ
Proof. AsallthelevelsareregularforH,bytheDuistermaat-Heckmanformula,
for all ξ, ξ in R,
0
[ω ] [ω ]=(ξ ξ ) 2πe(H−1(ξ) Q )
ξ − ξ0 − 0 − → ξ
(cid:0) (cid:1)
in H2(Q ;R) where e(H−1(ξ) Q ) is the Euler class of the principal circle
ξ0 → ξ
bundle: π :H−1(ξ) Q .
Q ξ
→
We have now to compute this Euler class e(H−1(ξ) Q ). As the bundle
ξ
→
H−1(ξ) Q is the pullback of the fibre bundle π : V B by the projection
ξ
→ →
q :Q B:
ξ ξ
→
H−1(ξ) qeξ // V
πQ π
(cid:15)(cid:15) (cid:15)(cid:15)
Qξ qξ //B
we have by naturality:
e(H−1(ξ) Q )=q⋆(e(V B))=q⋆e,
→ ξ ξ → ξ
where q⋆ :H2(B;Z) H2(Q ;Z) is an isomorphism.
ξ → ξ
Eventually, the Duistermaat-Heckman formula gives:
[ω ] [ω ]=(ξ ξ )( 2πq⋆e).
ξ − ξ0 − 0 − ξ
10
