Spherical Lagrangians via ball packings and symplectic cutting

0.49 MB·English

by  Matthew Strom Borman

Spherical Lagrangians via ball packings and
symplectic cutting
3
1 Matthew Strom Borman∗, Tian-Jun Li†, and Weiwei Wu†
0
2 January 29, 2013

n
a
J

7
Abstract
2
In this paper we prove the connectedness of symplectic ball packings in

] the complement of a spherical Lagrangian, S2 or RP2, in symplectic mani-
G
foldsthatarerationalorruled.Viaasymplecticcuttingconstructionthisis
S anaturalextensionofMcDuff’sconnectednessofballpackingsinotherset-
. tings and this result has applications to several different questions: smooth
h
t knotting and unknottedness results for spherical Lagrangians, the transi-
a tivity of the action of the symplectic Torelli group, classifying Lagrangian
m
isotopyclassesinthepresenceofknotting,anddetectingFloer-theoretically
[ essential Lagrangian tori in the del Pezzo surfaces.

2
v 1 Introduction
2
5
9 In[28]thesecondandthirdnamedauthorsinvestigatedtheexistenceandunique-
5 ness (unknottedness) problems of Lagrangian S2 in rational manifolds via tools
. from symplectic field theory and the study of symplectic ball-packings [37, 34]. In
1
1 this paper we continue to explore the connections between symplectic ball pack-
2 ing, symplectic cutting, and Lagrangian unknottedness, while answering several
1 questions from [28] and extending the results to Lagrangian RP2s. See [9] for
:
v an early survey of the problem of Lagrangian knots and more recent results in
i [5, 8, 11, 12, 16, 40, 41].
X
Our first result, conjectured in [28, Remark 5.2], is on the connectedness of
ar symplectic ball packings in the complement of a Lagrangian S2 or RP2:
Theorem 1.1. Let (M4,ω) be a closed 4-dimensional symplectic manifold that is
rational or ruled and let L ⊂ M be a Lagrangian S2 or RP2, then the space of
symplectic ball packings in M\L is connected.
∗PartiallysupportedbyNSF-grantDMS1006610.
†SupportedbyNSF-grantDMS0244663.
1
Via symplectic cutting, Theorem 1.1 follows from the connectedness of the space
of ball packings in the complement of a symplectic sphere. This is established
for certain symplectic surfaces in Proposition 2.1 by work of McDuff on relative
inflation [34, 36].
Recallthatasymplecticrationalmanifold(M4,ω)iswhereM iseitherCP2,
a symplectic blow-up of CP2, or S2×S2. In [28], building on work of Evans [11],
Hamiltonian unknottedness for Lagrangian S2’s in symplectic rational manifolds
was established when the Euler characteristic χ(M) ≤ 7, except for the case of
a characteristic homology class. We complete the picture here in Theorem A.2 in
theappendix.ThisresultissharpduetoSeidel’s[40]constructionofHamiltonian
knotted Lagrangian spheres in CP2#5CP2.
As noted in [28, Section 6.4.2], Theorems 1.1 and A.2 have the following
consequences, which we also prove in the appendix. Recall that the symplectic
Torelli group Symp (M,ω) is the subgroup of Symp(M,ω) that acts trivially on
h
homology H (M;Z).
∗
Corollary 1.2. Suppose (M4,ω) is a symplectic rational manifold.
(1) (Symplectic unknottedness of Lagrangian S2) The symplectic Torelli group
Symp (M,ω) acts transitively on homologous Lagrangian spheres.
h
(2) (SmoothunknottednessofLagrangianS2)HomologousLagrangianspheresare
smoothly isotopic to each other.
The smooth unknottedness for Lagrangian spheres was first noticed by Evans [11]
in the case of del Pezzo surfaces, i.e. monotone rational symplectic manifolds.
Let (X ,ω ) be a monotone CP2#5CP2. By [28, Theorem 1.4] one can ex-
5 0
plicitly classify the homology classes ξ ∈ H (X ;Z) that can be represented by
2 5
a Lagrangian sphere in (X ,ω ). Furthermore Evans in [12, Theorem 1.3] com-
5 0
putes the weak homotopy type of Symp (X ,ω ) and in [12, Section 6.1] shows
h 5 0
π (Symp (X ,ω )) is a Z -quotient of the pure braid group PBr(S2,5) on S2
0 h 5 0 2
with5strands.ThesetworesultsabovetogetherwithCorollary1.2showthatLa-
grangian spheres in (X ,ω ) are unique up to Hamiltonian isotopy and a certain
5 0
(explicit) braid group action.
Corollary1.2thereforeallowsforthefirstexplicitdescriptionofHamiltonian
isotopy classes of Lagrangian spheres in a closed symplectic manifold where there
is Lagrangian knotting. Hind [17] has done this in the non-compact setting with
theplumbingoftwoT∗S2.Inforthcomingworkofthethirdauthor[44]thisbraid
group action will be connected with Lagrangian Dehn twists along a finite set of
Lagrangian spheres and the Hamiltonian isotopy classes of Lagrangian spheres in
an A -singularity will be studied.
n
Suchanexplicitdescriptionbringsupamoreintriguingquestion.Forasym-
plecticrationalmanifoldCorollary1.2(1)tellsuswehaveatransitivegroupaction
ofπ (Symp (M))=Symp (M)/Symp (M)onLag (M,S2),theHamiltonianiso-
0 h h 0 ξ
topy classes of Lagrangian spheres in the class ξ ∈ H (M;Z), and the stabilizer
2
2
of this action seems hard to understand. It may be possible to understand the
stabilizeroftheactionofπ (Symp (M))ontheFukayacategoryintermsofbraid
0 h
group elements [20].
The unknotting picture for Lagrangian RP2 is more intriguing. We first have
the following parallel results of Corollary 1.2 for small Betti numbers:
Theorem 1.3. Let (M4,ω) be a symplectic rational manifold and b−(M)≤8.
2
(1) Symp (M,ω) acts transitively on Z -homologous Lagrangian RP2’s.
h 2
(2) Z -homologous Lagrangian RP2’s are smoothly isotopic.
2
Note in Lemma 4.6 we prove there is an unique Z -homology class containing a
2
Lagrangian RP2 when b−(M)≤2.
2
While it is still possible that the uniqueness of Z -homologous RP2’s up to
2
smoothisotopyisvalidforanarbitrarysymplecticrationalmanifold,thefollowing
result gives some hints about the complication of the problem:
Proposition 1.4. Let M =CP2#kCP2.
(1) Fork ≥9thereexistsasymplecticformω onM withLaLagrangianRP2 and
S a symplectic (−1)-sphere, such that L and S have trivial Z -intersection,
2
but L and S cannot be made disjoint with a smooth isotopy.
(2) For k ≥10 there exists L and L that are Z -homologous smoothly embedded
0 1 2
RP2’s, which are not smoothly isotopic. Furthermore there are deformation
equivalent symplectic forms ω and ω on M so that L ⊂ (M,ω ) are La-
0 1 i i
grangians.
Our symplectic packing results also provide ways to construct disjoint La-
grangian S2’s and RP2’s, and this leads to Floer-theoretic statements about prop-
erties of Lagrangians, in particular showing that they are not superheavy with
respect to the fundamental class of quantum homology [10]. Let (X ,ω ) be a del
k 0
Pezzo surface, i.e. a monotone CP2#kCP2 with 0≤k ≤8.
Theorem 1.5. Any Lagrangian RP2 ⊂(X ,ω ) is not superheavy with respect to
k 0
the fundamental quantum homology class 1∈QH (X ,ω ) when k ≥2. Likewise
4 k 0
for Lagrangian spheres S2 ⊂X when k ≥3.
k
In Proposition 5.1 we build Lagrangian RP2’s in all X . We also note that this
k
has an interesting application to detecting non-displaceable toric fibers in certain
symplectic surfaces with “semi-toric type” structures (see Section 5.2).
Thestructureofthepapergoesasfollows.InSection2werecallthenecessary
tools from relative symplectic packing as well as the symplectic cutting procedure
that lets us switch between Lagrangians and divisors. This allows us to prove
Theorem 1.1. In Section 3 we study symplectic (−4)-spheres, which is related to
Lagrangian RP2’s via symplectic cuts. In Section 4 we provide proofs of Theorem
1.3andProposition1.4.InSection5weproveTheorem1.5anddiscusstherelation
between symplectic packing and Floer-theoretic properties.
3
Acknowledgements
TheauthorswarmlythankSelmanAkbulut,JosefDorfmeister,RonaldFintushel,
Robert Gompf, Dusa McDuff, and Leonid Polterovich for their interest in this
workandmanyhelpfulcorrespondences.ParticularthanksisduetoDusaMcDuff
for generously sharing early versions of her paper [36] with us, which plays a key
role in our arguments. We would also like to thank the anonymous referee for
various valuable comments, suggestions, clarifications, and pointing us to the fact
that Corollary 1.2(1) leads to a description of the Hamiltonian isotopy classes of
LagrangianspheresinacompactsymplecticmanifoldwherethereareHamiltonian
knotted Lagrangian spheres.
2 Connectedness of ball packings
Given a symplectic manifold (M2n,ω) the space of symplectic ball packings of M
of size λ¯ =(λ ,...,λ ) is the space of smooth embeddings
1 k
(cid:40) k (cid:41)
(cid:97)
Eλ¯(M,ω):= φ: B2n(λi)→M :φ∗ω =ωstd and φ is injective
i=1
where B2n(λ) = {z ∈ Cn : π|z|2 ≤ λ} and ω = dx∧dy. In [31, 32] McDuff es-
std
tablished the connection between symplectic packing and the symplectic blow-up.
Given a symplectic packing φ ∈ Eλ¯(M,ω) performing the symplectic blow-up re-
sultsinthesymplecticformω onM#kCPn where[ω ]=[π∗ω]−(cid:80)k λ PD(E )
φ φ i=1 i i
where π :M#kCPn →M is the blow-down map and PD(E ) is the Poincare dual
i
totheexceptionalclassE correspondingtotheblow-upofM atthei-thball.See
i
[32, 37] for more details.
Lalonde and McDuff in [21, 22, 34] developed a method known as inflation
which builds a symplectic deformation of (M,ω) by adding a two-form dual to a
symplectic submanifold C2 ⊂ (M4,ω). In particular McDuff’s [34, Theorem 1.2],
anditsgeneralizationbyLi–Liu[26,Proposition4.11],provesif(M4,ω)isaclosed
symplectic manifold with b+ = 1, then any deformation between cohomologous
2
symplectic forms is homotopic with fixed endpoints to an isotopy of symplectic
forms. This homotopy is done by inflation along one parameter families of em-
bedded holomorphic curves, whose existence is given by Taubes–Seiberg–Witten
theory [42, 33]. Using the relation between symplectic packings and symplectic
forms on the blow-up, McDuff [34, Corollary 1.5] used that deformation implies
isotopy to prove the space of symplectic packings is connected when b+(M)=1.
2
2.1 In the complement of a symplectic submanifold
Consider now the relative setting, where Z ⊂(M4,ω) is a closed embedded sym-
plectic surface and one considers symplectic packings Eλ¯(M\Z,ω) in the comple-
ment of Z. Biran [2, Lemma 2.1.A] worked out how to inflate a symplectic form
4
along certain symplectic surfaces C ⊂ M that intersects Z positively so that Z
stays a symplectic submanifold through the deformation, so extending the con-
nectedness of ball packing just requires one to find the appropriate holomorphic
curves to inflate along. When the Seiberg–Witten degree of Z is non-negative
d(Z)=c (Z)+Z2 ≥0, then Z will be a J-holomorphic curve for regular almost
1
complex structures on M so McDuff’s argument in [34, Corollary 1.5] generalizes
immediately.
Thesituationismoredelicatewhend(Z)<0,butrecentworkofMcDuff[36]
builds the appropriate curves in certain cases and leads to the following proposi-
tion, which is implicit in [36]. We have followed [34, Corollary 1.5] where McDuff
proves the connectedness of ball packings in the absolute case when b+(M) = 1.
2
Note that two packings φ0,φ1 ∈ Eλ¯(M,ω) are connected if and only if there is a
symplectomorphism F ∈ Symp (M,ω) in the identity component of Symp(M,ω)
0
such that F ◦φ =φ .
0 1
Proposition 2.1. Let (W4,ω) be a closed rational or ruled symplectic 4-manifold
and let Z ⊂W be a closed symplectic sphere, then the space of symplectic packing
Eλ¯(W\Z,ω) is connected.
Proof. Given two packings φ ,φ : (cid:96)k B4(λ ) → (W\Z,ω) by applying an ele-
0 1 i=1 i
ment of Sympc(W\Z,ω), the identity component of the group of compactly sup-
0
ported symplectomorphisms Sympc(W\Z,ω), we may assume that φ = φ as
0 1
maps when restricted to (cid:96)k B4(aλ ) for a>0 sufficiently small.
i=1 i
Letω¯ andω¯ bethesymplecticformsontheblow-upW =W#kCP2 associ-
0 1
ated to the ball packings φ and φ . Pick a deformation of symplectic forms ω¯ on
0 1 t
W asintheproofof[34,Corollary1.5]suchthatω¯ isconstantinaneighborhood
t
of Z and in H2(W;R)
d
(cid:88)
[ω¯ ]=[π∗ω]− (λ −ρ (t))PD(E )
t i i i
i=1
whereπ :W →W isthenaturalblow-downmap,PD(E )∈H2(W;Z)isPoincare
i
dual to the exceptional class E associated to the exceptional divisor e ⊂W, and
i i
ρ :[0,1]→[0,λ )aresmoothfunctionsequalto0inaneighborhoodoft=0,1.By
i i
[36,Proposition1.2.9]thereisacompactlysupportedisotopy{F¯} inDiff (W\Z)
t t 0
such that F¯ =id and F¯∗ω¯ =ω¯ .
0 1 1 0
From here the proof proceeds exactly as in [34, Corollary 1.5]. By blowing
down F¯ induces a symplectomorphism F ∈Sympc(W\Z,ω) so that F ◦φ =φ ,
1 0 1
so to finish the proof it suffices to show F ∈ Sympc(W\Z,ω). Note that the
0
construction of F = F(1) can be done in a family F(a) ∈ Sympc(W\Z,ω) by
startingtheconstructionwithrespecttoφ andφ beingrestrictedtothedomain
0 1
(cid:96)k B4(aλ ) for a∈(0,1]. Since we assumed φ and φ are equal on sufficiently
i=1 i 0 1
small balls, F(a) =id for a close to zero and hence F ∈Sympc(W\Z,ω).
0
5
2.2 In the complement of a spherical Lagrangian
InthissubsectionwewillproveTheorem1.1.Thesimplebutkeyobservationisto
translate to a relative setting with a symplectic sphere, which allows us to apply
Proposition 2.1. This translation will be done using the following construction.
2.2.1 The symplectic cutting construction
Let (M2n,ω) be a symplectic manifold and let L ⊂ M be a closed Lagrangian
admittingametricwithperiodicgeodesicflow.LetN beaWeinsteinneighborhood
of L ⊂ M, then by using the periodic geodesic flow we can perform a symplectic
cut [23] along ∂N. This cut creates a pair of new symplectic manifolds W+ and
N
W− each having Z as a codimension 2 symplectic submanifold, where the Euler
N N
classes of Z ’s normal bundles in W± satisfy e(ν+) = −e(ν−) ∈ H2(Z ). The
N N N
manifolds are
W+ =(M\IntN)/∼ W− =N/∼ Z =∂N/∼
N N N
where the equivalence relations are given by identifying point on ∂N that lie on
thesamegeodesic,andthesymplecticmanifoldM canberecoveredbytakingthe
symplectic sum [14] of W± along Z
N N
M =W+# W−. (2.1)
N ZN N
Remark 2.2. Let us record what arises when L is a Lagrangian S2 or RP2.
(1) Lagrangian S2: We have W− = (S2 ×S2,σ ⊕σ) with L ⊂ W− being the
anti-diagonal Lagrangian sphere, where Z ⊂ W− is the diagonal symplectic
sphere and Z ⊂W+ is a (−2) symplectic sphere.
(2) Lagrangian RP2: We have W− = CP2 with L ⊂ W− being the the standard
RP2, where Z ⊂ W− is the quadric Q = {z2+z2+z2 = 0} and Z ⊂ W+ is
0 1 2
a (−4) symplectic sphere.
For more details and other examples see [1].
In [15, Theorem 1.1] Hausmann–Knutson determined the effect of symplec-
tic cutting in general on the rational cohomology ring and it leads to the fol-
lowing lemma. Note that [15, Theorem 1.1] assumes the symplectic cut is by a
global Hamiltonian S1-action, which need not be true in our case. However since
H∗(M;Q)→H∗(N;Q) is surjective if N ⊂M is a Weinstein neighborhood for a
Lagrangian S2 or RP2, the proof of [15, Theorem 1.1] still applies in our case.
Lemma 2.3. Let (M4,ω) be a closed symplectic manifold with L ⊂ M a La-
grangian S2 or RP2 and let W+ be the symplectic manifold built by cutting out a
N
Weinstein neighborhood N of L, then we have the following:
(1) b+(W+)=b+(M).
2 N 2
6
(2) If L=S2, then b−(W+)=b−(M). If L=RP2, then b−(W+)=b−(M)+1.
2 N 2 2 N 2
(3) If M is rational or ruled, then W+ is rational or ruled respectively.
N
Proof. For (1) and (2): If L is a Lagrangian S2, then H∗(W+;Q) ∼= H∗(M;Q)
N
as rings by [15]. If L is a Lagrangian RP2, then by [15] the intersection forms are
related by
Q =Q ⊕Q =(−1)⊕Q (2.2)
W+ CP2 M M
N
where PD(Z )∈H2(W+;Q) is identified with 2h∈H2(CP2;Q).
N N
For(3):LetκdenotethesymplecticKodairadimension,thenby[6,Theorem
1.1] and (2.1) if follows that κ(W+)≤κ(M). For closed 4-dimensional symplectic
N
manifolds having κ = −∞ is equivalent to being rational or ruled [29]. Since [15]
gives H (W+;Q)∼=H (M;Q) it follows that W+ is rational if M is rational, and
1 N 1 N
likewise for ruled.
In the case of a Lagrangian sphere, Lemma 2.3 can also be proved by noting that
W+ is a symplectic deformation of M.
N
2.2.2 Proving Theorem 1.1
In the setting of the symplectic cutting construction, by design W+\Z is sym-
N N
plectomorphic to M\N. Therefore we immediately have the following observation
on symplectic packings of M\L.
Lemma 2.4. The space of ball packings Eλ¯(M\L) in M\L is connected if and
only if the space of ball packings Eλ¯(WN+\ZN) in WN+\ZN is connected for all
sufficiently small Weinstein neighborhoods N of L⊂M.
This in turn leads to the proof of Theorem 1.1.
Proof of Theorem 1.1. Let(W+,ω)betheresultofcuttingoutaWeinsteinneigh-
N
borhood N of L in M, with Z ⊂ W+ being the resulting symplectic sphere.
N N
By Lemma 2.3 we have that W+ is rational or ruled, so by Proposition 2.1 the
N
space of ball packings in W+\Z is connected. The theorem now follows from
N N
Lemma 2.4.
3 Symplectic (−4)-spheres in rational manifolds
In this section we provide a classification of a special type of classes which can be
represented by symplectic (−4)-spheres in symplectic rational manifold (W4,ω).
ThiswillbecrucialtoourstudyofLagrangianRP2’sduetothesymplecticcutting
construction.
We first briefly establish some notation. If M = S2 ×S2, then we will let
A,B ∈H (M;Z) be the homology classes for each factor. If M =CP2#kCP2, we
2
7
will represent its second homology classes by the basis {H,E ,...,E }, where H
1 k
is the line class and E are orthogonal exceptional classes. Denote the class
i
K =−3H +E +···+E
0 1 k
and by [26, Theorem 1] we may always assume that this is the canonical class for
symplectic rational manifolds. Define the K -exceptional classes as the spherical
0
classes C satisfying K ·C =−1, and by [26] these classes must be represented by
0
an symplectic exceptional sphere.
There is a special kind of transformation on the second homology group of
rational manifolds called the Cremona transform, which are reflections
A(cid:55)→A+(A·L )L where L =H −E −E −E for i>j >k
ijk ijk ijk i j k
thatpreservetheclassK .Whenk ≤3,wealsoincludethereflectionwithrespect
0
to L = E −E . Cremona transformations can always be realized by diffeomor-
ij i j
phisms [24] (explicitly this can be done by a smooth version of Seidel’s Dehn
twist). See [38, 28] for more detailed discussions on these transformations. If two
classesareconnectedbyaseriesofCremonatransforms,wesaytheyareCremona
equivalent.
LetZ ⊂W beasymplectic(−4)-sphere,thentherationalblow-downalong Z
is the symplectic manifold built by performing a symplectic sum [14] of W along
Z with the standard quadric Q in CP2 and is denoted M = W # CP2. This
Z Q
operation is the inverse of the symplectic cutting construction from Section 2.2 in
the case of a Lagrangian RP2 ⊂M.
Remark 3.1. Unlike the symplectic cutting operation that keeps the manifold
rational or ruled in our case by Lemma 2.3, the same is not true when we blow-
down(−4)symplecticspheres.Forexampletheclass2K =6H−2(E +···+E )
0 1 10
isrepresentedbyanembedded(−4)-symplecticsphereforanappropriatesymplectic
form on CP2#10CP2, but the symplectic blow-down is not a rational manifold [7,
Section 4.2].
Proposition3.2. Let(W4,ω)beasymplecticrationalmanifoldandZ ⊂W bean
embedded symplectic (−4)-sphere. If W =S2×S2, then [Z]=A−2B or B−2A.
If W = CP2#kCP2, then k ≥ 2 and furthermore [Z] ∈ H (W,Z) is Cremona
2
equivalent to −H+2E −E provided the rational blow-down M of W along Z is
1 2
a rational manifold.
Proof. A direct computation with the adjunction formula implies all but the case
where W = CP2#kCP2 with k ≥ 4. From the relation between rational blow-
down and symplectic cutting from Section 2.2, it follows from equality (2.2) that
b−(W) = 1+b−(M). It follows that M is not minimal since it is rational and
2 2
b−(M)=k−1≥3.
2
By [7, Theorem 1.2] it follows that either
8
(i) there exists a symplectic exceptional sphere disjoint from Z, or
(ii) thereexisttwodisjointsymplecticexceptionalspheresC andC eachinter-
1 2
secting Z exactly once and positively.
Therefore we can assume after blowing down some number of exceptional spheres
disjointfromZ thattheresultingmanifoldW isminimalor(ii)occursforZ ⊂W.
IfwereachaminimalmanifoldthenW =S2×S2,sincetheotherpossibility,
CP2, does not have a symplectic (−4)-curve. So [Z] is either A−2B or B−2A in
H (W;Z) and since we needed to have blow-down at least 3 exceptional spheres
2
in W to reach W, by blowing back up to W it follows that the classes A−2B
and B −2A can be represented in the form of part (ii) in our assertion, by an
appropriate change of basis from classes A,B to the standard basis consisting of
{H,E ,...,E }.
1 k
Assume now (ii) occurs for Z ⊂ W, where W has no exceptional spheres
disjoint from Z. From the classification of exceptional classes in rational mani-
folds up to Cremona equivalence [38, Proposition 1.2.12], [28, Corollary 4.5], any
exceptional class is Cremona equivalent to E for some i. Also since Cremona
i
transformations can be realized by diffeomorphisms, we may now assume without
loss of generality [C ]=E for i=1,2.
i i
TakingourdivisorZ intoaccount,theeffectofblowingdownbothC canbe
i
described as performing symplectic fiber sums [14, Theorem 1.4] of (W,Z) with
(CP2,(cid:96)) along C and (cid:96) , where (cid:96) is a line intersecting the line (cid:96) transversally,
i i i
for i = 1 and 2. The result is (W(cid:48),Z(cid:48)) where Z(cid:48) is a symplectic (−2)-sphere in a
rational symplectic manifold W(cid:48) and under ι : H (W(cid:48)) → H (W) the canonical
2 2
inclusion we have ι[Z(cid:48)]=[Z]−E −E .
1 2
It follows from [28, Proposition 4.10] that [Z(cid:48)]∈H (W(cid:48)) is Cremona equiva-
2
lenttotheclassE −E .SinceCremonatransformsaregivenbyreflectionsabout
3 4
homology classes, the inclusion ι translates the Cremona equivalent of [Z(cid:48)] and
E −E in H (W(cid:48)) to a series of Cremona transforms of H (W) that fix E and
3 4 2 2 1
E while sending ι[Z(cid:48)] to the class E −E −E −E . This class is equivalent to
2 3 4 1 2
−H+2E −E byreflectionwithrespecttoH−E −E −E ,and−H+2E −E
3 4 1 2 3 3 4
is equivalent to the form in the assertion.
Corollary 3.3. If (W,ω) is a symplectic rational manifold and Z is an embedded
symplectic (−4)-sphere, then in the complement of Z there are n=b−(W)−1 dis-
2
jointexceptionalsymplecticspheres{D }n .Moreover,givenanysetoforthogonal
i i=1
exceptional classes pairing trivially with [Z], there are disjoint exceptional spheres
in W\Z representing these classes.
Proof. ThereisnothingtoproveifW =S2×S2orCP2#CP2,sobyProposition3.2
we can assume W = CP2#(n+1)CP2 with n ≥ 1 and [Z] = −H +2E −E . In
1 2
this case,
E(cid:48) =H −E −E and E(cid:48) =E for i=2,...,n.
1 1 2 i i+1
9
is a set of n orthogonal exceptional classes that pair trivially with [Z]. Now it
suffices to prove the second part, which follows from McDuff’s result [36, Propo-
sition 1.2.5] that asserts we can find a compatible almost complex structure J on
(W,ω) for which Z is a complex submanifold and the classes E(cid:48) are represented
i
by a J-holomorphic embedded spheres D . By positivity of intersections is follows
i
that the D are disjoint from each other and Z.
i
Definition 3.4. If (W4,ω) is a symplectic rational manifold and Z ⊂ W is a
symplectic (−4)-sphere, then a set of n=b−(W)−1 disjoint exceptional spheres
2
{D }n in W\Z is called a push off set of Z and the set {[D ]}n of exceptional
i i=1 i i=1
classes is called a push off system of Z.
4 Lagrangian RP2’s in a rational manifolds
Inthissection(M4,ω)willalwaysbeasymplecticCP2#kCP2 wherethecanonical
classisK =−3H+E +···+E andL⊂M willbeaLagrangianRP2.Notethat
0 1 k
S2 ×S2 does not contain Lagrangian RP2 since it does not have a Z -homology
2
class with non-trivial self-pairing, so we need not put it into consideration. We
will let (W,Z) = (W+,Z ) be the result of performing the symplectic cutting
N N
procedure from Section 2.2 on (M,L). Note by construction that Z ⊂ W is a
symplectic (−4) sphere and W is a symplectic rational manifold CP2#(k+1)CP2
by Lemma 2.3.
4.1 Push off and orthogonal systems
Translating our knowledge about symplectic (−4) spheres leads to the following
result.
Proposition 4.1. If (M4,ω) is a symplectic rational manifold with L ⊂ M a
Lagrangian RP2, then there is a set of k = b−(M) disjoint exceptional spheres
2
{D }k in M\L.
i i=1
Proof. By construction W\Z is symplectomorphic to a subset of M\L and by
Corollary 3.3 there are k = b−(W) − 1 = b−(M) disjoint exceptional spheres
2 2
{D }k in W\Z.
i i=1
In light of Proposition 4.1, we introduce the following analogue of Definition 3.4.
Definition 4.2. If (M4,ω) is a symplectic rational manifold and L ⊂ M is a
Lagrangian RP2, then a set k = b−(M) disjoint exceptional spheres {D }k in
2 i i=1
M \L is called a push off set of L and the set {[D ]}k of exceptional classes is
i i=1
called a push off system of L.
The following proposition shows the importance of the notion of a push off
system in proving Theorem 1.3.
10