Open Gromov-Witten invariants in dimension six

0.36 MB·English

by  Jean-Yves Welschinger

Open Gromov-Witten invariants in dimension six
Jean-Yves Welschinger
January 18, 2012
2
1 Abstract:
0
LetLbeaclosedorientableLagrangiansubmanifoldofaclosedsymplecticsix-manifold
2
(X,ω). We assume that the first homology group H (L;A) with coefficients in a commu-
n 1
a tative ring A injects into the group H1(X;A) and that X contains no Maslov zero pseudo-
J
holomorphic disc with boundary on L. Then, we prove that for every generic choice of a

7 tame almost-complex structure J on X, every relative homology class d ∈ H (X,L;Z) and
1 2
adequate number of incidence conditions in L or X, the weighted number of J-holomorphic

]
G discs with boundary on L, homologous to d, and either irreducible or reducible discon-
S nected, which satisfy the conditions, does not depend on the generic choice of J, provided
. that at least one incidence condition lies in L. These numbers thus define open Gromov-
h
t Witten invariants in dimension six, taking values in the ring A.
a
m
[
Introduction

1
v
8 In the mid eighties, M. Gromov discovered that classical enumerative invariants of complex
1
geometry obtained by counting the number of curves satisfying some incidence conditions
5
3 in a smooth projective manifold actually only depend on the underlying K¨ahler form of
.
1 the manifold up to deformation and not that much on the algebraic structure. A famous
0 example of such an enumerative invariant is the number of degree d rational curves passing
2
through 3d−1 generic points in the complex projective plane, a number later computed by
1
: M.Kontsevich. ThestrategyfollowedbyGromovwastofirstintroduceanauxiliarygeneric
v
i almost-complex structure tamed by the Ka¨hler form and then to count in an appropriate
X
waythefinitenumberofJ-holomorphiccurvessatisfyingtheincidenceconditions. Hecould
r
a then prove that this number does not depend on the generic choice of the almost-complex
structure J. This approach and these results gave birth to the still developing theory of
Gromov-Witteninvariantsinsymplecticgeometry. Aquestionthenappearedtogetherwith
theseworksofGromovand(later)Witten, [6], [22]. GivenaclosedLagrangiansubmanifold
L of a closed symplectic manifold (X,ω), is it likewise possible to extract enumerative
invariants from the count of J-holomorphic discs of X with boundary on L and subject to
Keywords: holomorphic discs, Gromov-Witten invariants.
AMS Classification : 53D45.
1
given incidence conditions, in the sense that this number does not depend on the generic
choice of J? Though apparently similar, this question hides a new difficulty, namely, the
moduli spaces of J-holomorphic discs have real codimension one boundary components,
contrary to the moduli spaces of closed J-holomorphic curves. As a consequence, even the
invariance modulo two does not hold in general.
At the early 2000’s, such open Gromov-Witten invariants have been defined by C.C.
Liu and M. Katz in the presence of an action of the circle, see [7], [11], and by myself
when L is fixed by an antisymplectic involution, see [16], [17], [20] or also [3], [14]. In my
recent work [21], I define such open Gromov-Witten invariants when X is four-dimensional
and L orientable. The invariance then only holds modulo q ∈ N for discs homologous to a
class d ∈ H (X,L;Z) such that ∂d = 0 ∈ H (L;Z/qZ). I also introduce similar invariants
2 1
by counting reducible discs with a given number of irreducible components, see [21]. The
aim of this paper is to define similar open Gromov-Witten invariants in a six-dimensional
symplectic manifold.
Let thus (X,ω) be a closed symplectic six-manifold and L be a closed Lagrangian
submanifold of X. We again assume that L is orientable and moreover that the inclusion of
L into X induces an injective morphism H (L;A) → H (X;A), where A is a commutative
1 1
ring. We also assume that X contains no Maslov zero pseudo-holomorphic disc with
boundary on L, in order no to have to take into account branched cover of discs. Under
these hypotheses, given a relative homology class d ∈ H (X,L;Z) of positive Maslov index,
2
a generic almost-complex structure J tamed by the symplectic form and submanifolds of
L and X of adequate cardinality and dimensions, we prove that the weighted number of
J-holomorphic discs with boundary on L, homologous to d, either irreducible or reducible
disconnected and which meet all of the chosen submanifolds of L and X, only depends on
the homology classes of the submanifolds, of d and of ω up to deformation, while it dos not
depend on the generic choice of J. For this to be true, we nevertheless assume that at least
one submanifold has been chosen in L, which means that our discs at least contain one
markedpointontheirboundaries,seeTheorem3.1andCorollary3.2. TheseJ-holomorphic
discsarecountedwithrespecttoasign,asusualinthetheoryofGromov-Witteninvariants,
but also with respect to a weight when they have more than one irreducible component.
This weight is defined in the following way. Let D be a J-holomorphic disc with boundary
on L having n > 1 disjoint irreducible components. The boundary of D is an oriented link
in L, each component of which has trivial homology class in H (L;A). We then label every
1
vertex of the complete graph K (having n vertices) with one component of this link ∂D.
n
Every edge of K gets then decorated with the linking number of the knots associated to
n
its boundary vertices, since we have equipped L with an orientation and even actually with
a spin structure. These linking numbers take value in the ring A. Then, for every spanning
subtree T of K , we associate the product of the n−1 linking numbers associated to its
n
n−1 edges and get a number T ∈ A. The sum of all these numbers T over all spanning
∗ ∗
subtrees of K provides the weight under which we count the disc D. This weight does not
n
depend on the labeling of the vertices of K , we call it self-linking weight, see Definition
n
2.4. As a consequence, the open Gromov-Witten invariants that we define here take value
in the commutative ring A. Recall that K contains exactly nn−2 spanning subtrees, a
n
2
formula established by J. J. Sylvester and A. Cayley, see [15], [2].
The paper is organized as follows. In the first paragraph, we introduce the moduli
spaces of pseudo-holomorphic discs and discuss some of their properties, see in particular
Proposition 1.9. The secondparagraph is devoted tolinking numbers, completegraphs and
spanning subtrees. We establish there Lemma 2.3, a key property of self-linking weights.
The last paragraph is devoted to the statement and proof of our result.
Acknowledgements:
The research leading to these results has received funding from the European Com-
munity’s Seventh Framework Progamme ([FP7/2007-2013] [FP7/2007-2011]) under grant
agreement no [258204]. I am also grateful to F. Chapoton for pointing out the references
[15], [2] to me.
1 Pseudo-holomorphic discs with boundary on a La-
grangian submanifold
1.1 The automorphism group of Poincar´e’s unit disc
Let ∆ = {z ∈ C | |z| ≤ 1} be the closed complex unit disc and H = {z ∈ C | (cid:61)z > 0}
◦
be the upper half plane. Denote by ∆ the interior of ∆ and by H the closure of H in the
Riemann sphere. The homography z ∈ H (cid:55)→ z−i ∈ ∆ is an isomorphism which we fix in
z+i
order to identify ∆ and H in the sequel.
1.1.1 Orientation
The group Aut(∆) of automorphisms of ∆ is isomorphic to PSL (R) = Aut(H). It acts
2
◦
transitively and without fixed point on the product ∂∆× ∆ by (φ,z,ζ) ∈ Aut(∆) ×
◦ ◦ ◦
∂∆× ∆(cid:55)→ (φ(z),φ(ζ)) ∈ ∂∆× ∆. The product ∂∆× ∆ being canonically oriented, this
action induces an orientation on Aut(∆) which we fix once for all.
In the same way, Aut(∆) acts transitively and without fixed point on the open subset
of (∂∆)3 made of cyclically ordered triple of distinct points. The orientation induced on
Aut(∆) by this action coincides with the one we just fixed, see Lemma 1.1.
For every (cid:15) > 0, denote by r : z ∈ ∆ (cid:55)→ exp(i(cid:15))z ∈ ∆, t : z ∈ H (cid:55)→ z + (cid:15) ∈ H and
(cid:15) (cid:15)
h : z ∈ H (cid:55)→ (1 + (cid:15))z ∈ H. Denote by r. = ∂ | r , t. = ∂ | t and h. = ∂ | h the
(cid:15) 0 ∂(cid:15) (cid:15)=0 (cid:15) 0 ∂(cid:15) (cid:15)=0 (cid:15) 0 ∂(cid:15) (cid:15)=0 (cid:15)
associated holomorphic vector fields of ∆.
. . .
Lemma 1.1 The triple (r ,t ,h ) forms a direct basis of the Lie algebra aut(∆). More-
0 0 0
over, the action of Aut(∆) on the open subset of (∂∆)3 made of cyclically ordered triple of
distinct points preserves orientations.
Proof:
3
◦
Let us identify ∂∆× ∆ with the orbit of (1,0) under the action of Aut(∆). The
.
differential of this action maps r onto the positive generator of T ∂∆ whereas it maps
0 1
. .
(t ,h ) onto a direct basis of T H. The first part of the lemma follows. Now, (∞,−1,1)
0 0 i
forms a cyclically ordered triple of distinct points on ∂H. The differential of the action
of Aut(H) on this triple maps r. on the sum of positive generators of T ∂H, T ∂H and
0 ∞ −1
. .
T ∂H, whereas it maps t onto the sum of positive generators of T ∂H and T ∂H and h
1 0 −1 1 0
onto the sum of a negative generator of T ∂H and a positive generator of T ∂H, hence
−1 1
the result. (cid:3)
Remark 1.2 It follows from Lemma 1.1 that our convention of orientation of Aut(∆) is
the same as the one adopted by Fukaya, Oh, Ohta and Ono in §8 of [5].
1.1.2 Glueing of automorphisms
Denote by ∆ the nodal disc obtained as the closure of (H × {0}) ∪ ({0} × H) ⊂ C2
0
in CP2. Likewise, for every η ≥ 0, denote by ∆ the closure in CP2 of the hyperbola
η
{(x,y) ∈ H2 | xy = −η}. For every (cid:15) > 0, denote by τ : x ∈ H (cid:55)→ x ∈ H the translation
(cid:15) 1−(cid:15)x
of H fixing the origin and by κ : x ∈ H (cid:55)→ 1 x ∈ H the homothety of weight 1 , so that
(cid:15) 1+(cid:15) 1+(cid:15)
. .
the corresponding pair (τ ,κ ) of holomorphic vector fields forms a direct basis of the Lie
0 0
algebra aut(H,0), where Aut(H,0) denotes the group of automorphisms of H fixing the
boundary point 0. The group of automorphisms of ∆ is isomorphic to Aut(H,0)2 and is
0
◦
oriented in such a way that its action on the interior ∆ gets orientation preserving.
0
. . . .
Proposition 1.3 Theelements(τ ,0), (0,τ )and(− κ ,κ )oftheLiealgebraaut(∆ ) =
0 0 0 0 0
aut(H,0)2 holomorphically deform as a direct basis of aut(∆ ) for every η > 0. Moreover,
η
.
the infinitesimal action of (κ ,0) by reparametrization of maps ∆ → ∆ extends as an
0 0 0
inward normal holomorphic vector field transverse to the fibers ∆ of ∪ ∆ , lifting ∂ .
η η>0 η ∂η
The action of Aut(∆ ) by reparameterization of maps ∆ → ∆ is defined by (g,u) ∈
0 0 0
Aut(∆ )×Hol(∆ ,∆ ) (cid:55)→ u◦g−1 ∈ Hol(∆ ,∆ ).
0 0 0 0 0
Proof:
◦
Foreveryη > 0,letusparameterizetheinteriorof∆ bythemapx ∈ H (cid:55)→ (x, −η) ∈∆ .
η x η
For every (cid:15) ≥ 0, the translation τ induces through this parameterization the automor-
(cid:15)
.
phism (x,y) ∈ ∆ (cid:55)→ (τ (x),y +(cid:15)η) ∈ ∆ . The associated holomorphic vector field (τ ,η)
η (cid:15) η 0
. .
of ∆ deforms (τ ,0). By symmetry, the pair (η,τ ) defines an element of aut(∆ ) de-
η 0 0 η
.
forming (0,τ ). Likewise, the homothety κ induces through our parameterization the
0 (cid:15)
automorphism (x,y) ∈ ∆ (cid:55)→ (κ (x),h (y)) ∈ ∆ . The associated holomorphic vector field
η (cid:15) (cid:15) η
(κ. ,− κ. ) deforms the opposite of (− κ. ,κ. ). The evaluation map at i ∈ H sends the
0 0 0 0
pair (τ. ,κ. ) to a direct basis of T H. By deformation, for every η > 0 close to zero, the
0 0 i
. . . .
pairs (τ ,η) and (κ ,− κ ) evaluate as a direct basis of T ∆ . The vector field (η,τ )
0 0 0 (i,ηi) η 0
evaluates as a direct basis of T∂∆ and is close to zero at (i,ηi). As a consequence, the
η
. . . .
triple (η,τ ), (τ ,η) and (κ ,− κ ) defines a direct basis of aut(∆ ).
0 0 0 0 η
4
Finally, the biholomorphism (x,y) ∈ H2 (cid:55)→ (ηx,y) ∈ H2 maps ∆ onto ∆ for every
1 η
.
η > 0. It is obtained by integration of the vector field (h ,0). The latter is the image of
0
.
(κ ,0) under the infinitesimal action of Aut(∆ ) by reparameterization of maps ∆ → ∆ .
0 0 0 0
(cid:3)
Corollary 1.4 Let (∆ ) be the standard deformation of the nodal disc ∆ . Then , a
η η≥0 0
direct basis of the Lie algebra aut(∆ ) deforms as the concatenation of an outward normal
0
vector field lifting − ∂ and a direct basis of aut(∆ ), η > 0.
∂η η
Proof:
.
This Corollary 1.4 follows from Proposition 1.3 and the fact that the quadruple (τ ,0)
0
, (0,τ. ), (− κ. ,κ. ) and (κ. ,0) defines a direct basis of aut(∆ ). (cid:3)
0 0 0 0 0
1.2 Moduli spaces of simple discs
We denote by P(X,L) = {(u,J) ∈ C1(∆,X)×J | u(∂∆) ⊂ L and du+J| ◦du◦j = 0}
ω u st
the space of pseudo-holomorphic maps from ∆ to the pair (X,L), where j denotes the
st
standard complex structure of ∆ and J the space of almost-complex structures on X of
ω
class Cl tamed by ω, l (cid:29) 1. Note that J being of class Cl, the regularity of such pseudo-
holomorphic maps u is actually more than Cl, see [12]. More generally, for every r,s ∈ N,
◦
we denote by P (X,L) = {((u,J),z,ζ) ∈ P(X,L)×((∂∆)r \diag )×((∆)s \diag )},
r,s ∂∆ ∆
where diag = {(z ,...,z ) ∈ (∂∆)r | ∃i (cid:54)= j,z = z } and diag = {(ζ ,...,ζ ) ∈
∂∆ 1 r i j ∆ 1 s
∆s | ∃i (cid:54)= j,ζ = ζ }.
i j
Following [9], [8], [1], we define
Definition 1.5 A pseudo-holomorphic map u is said to be simple iff there is a dense
open subset ∆ ⊂ ∆ such that ∀z ∈ ∆ ,u−1(u(z)) = {z} and du| (cid:54)= 0.
inj inj z
We denote by P∗ (X,L) the subset of simple elements of P (X,L). It is a separable
r,s r,s
Banach manifold which is naturally embedded as a submanifold of class Cl−k of the space
Wk,p(∆,X)×J for every 1 (cid:28) k (cid:28) l and p > 2, see Proposition 3.2 of [12].
ω
For every d ∈ H (X,L;Z), we denote by Pd (X,L) = {(u,J) ∈ P∗ (X,L) | u [∆] = d}
2 r,s r,s ∗
and by Md (X,L) = Pd (X,L)/Aut(∆), where Aut(∆) acts by composition on the right,
r,s r,s
see §1.1. The latter is equipped with a projection π : [u,J,z,ζ] ∈ Md (X,L) (cid:55)→ J ∈ J
r,s ω
and an evaluation map ev : [u,J,z,ζ] ∈ Md (X,L) (cid:55)→ (u(z),u(ζ)) ∈ Lr ×Xs.
r,s
We recall the following classical result due to Gromov (see [6], [12], [5]).
Theorem 1.6 For every closed Lagrangian submanifold L of a six-dimensional closed
symplectic manifold (X,ω) and every d ∈ H (X,L;Z), r,s ∈ N, the space Md (X,L) is a
2 r,s
separable Banach manifold and the projection π : Md (X,L) → J is Fredholm of index
r,s ω
µ (d)+r+2s. (cid:3)
L
Note that from Sard-Smale’s theorem [13], the set of regular values of π is dense of the
second category. As a consequence, for a generic choice of J ∈ J , the moduli space
ω
Md (X,L;J) = π−1(J) is a manifold of dimension µ (d) as soon as it is not empty.
0,0 L
5
Forgettingthemarkedpointsdefinesamap(u,J,z,ζ) ∈ Pd (X,L) (cid:55)→ (u,J) ∈ Pd (X,L)
r,s 0,0
which quotients as a forgetful map f : Md (X,L) → Md (X,L) whose fibers are canon-
r,s r,s 0,0
ically oriented. When L is equipped with a Spin structure, the manifolds Md (X,L;J),
r,s
r,s ∈ N, inherit canonical orientations, see Theorem 8.1.1 of [5]. We adopt the follow-
ing convention for orienting these moduli spaces. When J is a generic almost-complex
structure tamed by ω, the manifold Pd (X,L;J) inherits an orientation from the Spin
0,0
structure of L, see Theorem 8.1.1 of [5]. This orientation induces an orientation on
the manifold Md (X,L;J) = Pd (X,L;J)/Aut(∆) such that at every point (u,J) ∈
0,0 0,0
Pd (X,L;J), the concatenation of a direct basis of T Md (X,L;J) in a horizontal
0,0 [u,J] 0,0
space of T Pd (X,L;J) followed by a direct basis of the orbit of Aut(∆) at (u,J) de-
(u,J) 0,0
fines a direct basis of T Pd (X,L;J). This is the convention 8.2.1.2 adopted by Fukaya,
(u,J) 0,0
Oh, Ohta and Ono in [5], so that our orientation of Md (X,L;J), which we call the quo-
0,0
tient orientation, coincides with the one introduced in [5]. The manifold Md (X,L;J)
r,s
is then oriented in such a way that at every point, the concatenation of a direct basis of
T Md (X,L;J) with a direct basis of the fiber of the forgetful map f provides a direct
[u,J] 0,0 r,s
basis of T Md (X,L;J). (The latter convention differs in general from the one adopted
[u,J] r,s
by Fukaya, Oh, Ohta and Ono in [5].)
1.3 Moduli spaces of nodal discs
Let d ,d ∈ H (X,L;Z) be such that µ (d ) > 1, µ (d ) > 1. For every z ∈ ∂∆ and
1 2 2 L 1 L 2
i ∈ {1,2}, denote by ev : (u,J) ∈ Pdi (X,L) (cid:55)→ u(z) ∈ L the evaluation map at the
z 0,0
point z. As soon as J is generic, the restrictions of these evaluation maps to Pdi (X,L;J)
0,0
are submersions. We then denote by P(d1,d2)(X,L) the fiber product Pd1(X,L) ×
0,0 0,0 ev1 ev−1
Pd2(X,L). As soon as J is generic enough, P(d1,d2)(X,L;J) is a manifold of dimension
0,0 0,0
µ (d + d ) + 3 since X is six-dimentional throughout this paper. When L is equipped
L 1 2
with a Spin structure, it is canonically oriented from [5]. Let us recall the convention of
orientation of this manifold.
Let (u ,u ,J) ∈ P(d1,d2)(X,L;J) and (v ,v ,v ) (resp. (w ,w ,w )) be elements of
1 2 0,0 1 2 3 1 2 3
T Pd1(X,L;J)(resp. T Pd2(X,L;J))suchthat(v (1),v (1),v (1))(resp. (w (−1),w (−1),
u1 0,0 u2 0,0 1 2 3 1 2
w (−1))) forms a direct basis of T L (resp. w (−1) = v (1), w (−1) = v (1), w (−1) =
3 u1(1) 1 1 2 2 3
v (1)). Let B (resp. B ) be an ordered family of µ (d ) (resp. µ (d )) elements of
3 1 2 L 1 L 2
T Pd1(X,L;J) (resp. T Pd2(X,L;J)) such that (B ,v ,v ,v ) (resp. (w ,w ,w ,B ))
u1 0,0 u2 0,0 1 1 2 3 1 2 3 2
defines a direct basis of T Pd1(X,L;J) (resp. T Pd2(X,L;J)). Then, the manifold
u1 0,0 u2 0,0
P(d1,d2)(X,L;J) is oriented such that the basis (B ,v +w ,v +w ,v +w ,B ) of
0,0 1 1 1 2 2 3 3 2
T P(d1,d2)(X,L;J) becomes direct. We then denote by M(d1,d2)(X,L) the quotient
(u1,u2,J) 0,0 0,0
P(d1,d2)(X,L)/Aut(∆ ), where ∆ denotes the nodal disc, see §1.1.2. When J is generic
0,0 0 0
enough, M(d1,d2)(X,L) is a manifold of dimension µ (d + d ) − 1 which we equip with
0,0 L 1 2
the quotient orientation as in §1.2. This convention of orientation coincides thus with the
one adopted in [5]. Note the the tautological involution M(d1,d2)(X,L) → M(d2,d1)(X,L)
0,0 0,0
which exchanges the discs preserves the orientation since the Maslov indices µ (d ), µ (d )
L 1 L 2
6
are even. Finally, for every r,s ∈ N, we denote by P(d1,d2)(X,L) = P(d1,d2)(X,L)×((∂∆ \
r,s 0,0 0
◦
{node})r \ diag ) × ((∆ )s \ diag )}, where diag = {(z ,...,z ) ∈ (∂∆ )r | ∃i (cid:54)=
∂∆0 0 ∆0 ∂∆0 1 r 0
j,z = z } and diag = {(ζ ,...,ζ ) ∈ ∆s | ∃i (cid:54)= j,ζ = ζ }. We then denote by
i j ∆0 1 s 0 i j
M(d1,d2)(X,L) = P(d1,d2)(X,L)/Aut(∆ ) and by f : M(d1,d2)(X,L) → M(d1,d2)(X,L) the
r,s r,s 0 r,s r,s 0,0
associatedforgetfulmap, whosefibersarecanonicallyoriented. WhenJ isagenericalmost-
complex structure tamed by ω, the manifolds M(d1,d2)(X,L;J) are oriented in such a way
r,s
that at every point [u ,u ,J,z,ζ] ∈ M(d1,d2)(X,L;J), the concatenation of a direct basis
1 2 r,s
of T M(d1,d2)(X,L;J) in a horizontal space with a direct basis of the fiber of the
[u1,u2,J,z,ζ] 0,0
forgetful map f at [u ,u ,J,z,ζ] provides a direct basis of T M(d1,d2)(X,L;J).
r,s 1 2 [u1,u2,J,z,ζ] r,s
(The latter convention differs in general from the one adopted by Fukaya, Oh, Ohta and
Ono in [5].)
From Gromov’s compactness and glueing theorems (see for example [4], [5], [1]), the
spaceM(d1,d2)(X,L;J)canonicallyidentifiesasacomponentoftheboundaryofthemoduli
r,s
space Md1+d2(X,L;J). The following Proposition 1.7, analogous to Proposition 8.3.3 of
r,s
[5], comparesthe orientationof Mr(d,s1,d2)(X,L;J) withthe one inducedbyMdr,1s+d2(X,L;J).
Proposition 1.7 Let L be a closed Lagrangian Spin submanifold of a closed symplectic
six-manifold (X,ω). Let d ,d ∈ H (X,L;Z) be such that µ (d ) ≥ 2, µ (d ) ≥ 2 and let
1 2 2 L 1 L 2
r,s ∈ N. Then, for every generic almost-complex structure J tamed by ω, the incidence
index (cid:104)∂Mdr,1s+d2(X,L;J),M(rd,s1,d2)(X,L;J)(cid:105) equals −1.
Proof:
The glueing map of J-holomorphic discs preserves the orientations of the fibers of
the forgetful map f , so that from our conventions of orientations of moduli spaces, it
r,s
suffices to prove the result for r = s = 0. From Lemma 8.3.5 of [5], the glueing map
of J-holomorphic maps P(d1,d2)(X,L;J) → Pd1+d2(X,L;J) preserves orientations. Let
0,0 0,0
(u ,u ,J) ∈ P(d1,d2)(X,L;J), B be a direct basis of T M(d1,d2)(X,L;J) and B a
1 2 0,0 1 [u1,u2,J] 0,0 0
direct basis of the linearized orbit of Aut(∆ ) at (u ,u ,J). Then, by definition, the con-
0 1 2
catenation(B ,B )definesadirectbasisofT P(d1,d2)(X,L;J). Now,let(u # u )
1 0 (u1,u2,J) 0,0 1 R 2 R(cid:29)1
be a path of Pd1+d2(X,L;J) transversal to P(d1,d2)(X,L;J) at u # u = (u ,u ) given
0,0 0,0 1 ∞ 2 1 2
by the glueing map. From Corollary 1.4, B deforms for R < +∞ as a pair (n,B(cid:48)) of
0 0
T Pd1+d2(X,L;J), where B(cid:48) is a direct basis of the linearized orbit of Aut(∆) at
(u1#Ru2,J) 0,0 0
u # u and n points towards the boundary of Pd1+d2(X,L;J). As a consequence, (B ,n)
1 R 2 0,0 1
deforms as a direct basis of T Md1+d2(X,L;J). The result now follows from the
(u1#Ru2,J) 0,0
fact that the cardinality of B is odd. (cid:3)
1
1.4 Moduli spaces of reducible discs
Let n ≥ 1 and d ,...,d ∈ H (X,L;Z) be such that µ (d ) > 0, i ∈ {1,...,n}. For
1 n 2 L i
every almost-complex structure J tamed by ω, we denote by Md1,...,dn(X,L;J) the di-
0,0
rect product Md1 (X,L;J) × ··· × Mdn(X,L;J) and by Pd1,...,dn(X,L) the correspond-
0,0 0,0 0,0
7
ing fiber product Pd1(X,L) × ··· × Pdn(X,L). When J is generic enough and
0,0 Jω Jω 0,0
L Spin, Md1,...,dn(X,L;J) is a manifold of dimension µ (d + ··· + d ) equipped with
0,0 L 1 n
its product orientation. Since the manifolds Mdi (X,L;J) are even dimensional, i ∈
0,0
{1,...,n}, this product orientation coincides with the quotient orientation induced by
Pd1,...,dn(X,L;J)/Aut(∆)n, where Pd1,...,dn(X,L;J) = (cid:81)n Pdi (X,L;J) is itself equipped
0,0 0,0 i=1 0,0
with the product orientation. For every r,s ∈ N, we likewise define Md1,...,dn(X,L) as
r,s
◦ ◦
the quotient of Pd1,...,dn(X,L) × ((∂∆ ∪ ··· ∪ ∂∆)r \ diag ) × ((∆ ∪···∪ ∆)s \ diag )}
0,0 ∂∆ ∆
by Aut(∆)n, where diag = {(z ,...,z ) ∈ (∂∆ ∪ ··· ∪ ∂∆)r | ∃i (cid:54)= j,z = z } and
∂∆ 1 r i j
diag = {(ζ ,...,ζ ) ∈ (∆ ∪ ··· ∪ ∆)s | ∃i (cid:54)= j,ζ = ζ }. It is equipped with a forgetful
∆ 1 s i j
map f : Md1,...,dn(X,L) → Md1,...,dn(X,L) whose fibers are canonically oriented by the
r,s r,s 0,0
complex structure. For generic J, the manifold Md1,...,dn(X,L;J) is as before oriented in
r,s
such a way that at every point [u ,...,u ,J,z,ζ], the concatenation of a direct basis of
1 n
T Md1,...,dn(X,L;J)inahorizontalspacewithadirectbasisofthefiberofthefor-
[u1,...,un,J,z,ζ] 0,0
getful map f at [u ,...,u ,J,z,ζ] provides a direct basis of T Md1,...,dn(X,L;J).
r,s 1 n [u1,...,un,J] r,s
This orientation is not the quotient orientation of Pd1,...,dn(X,L;J)/Aut(∆)n. The tauto-
r,s
logical action of the group of permutation of {d ,...,d } on these spaces Md1,...,dn(X,L;J)
1 n r,s
preserves the orientations, since the manifolds Mdi (X,L;J) are even dimensional.
0,0
In the same way, for every n ≥ 2, we denote by M(d1,d2),d3,...,dn(X,L;J) the direct
0,0
product M(d1,d2)(X,L;J) × Md3 (X,L;J) × ··· × Mdn(X,L;J) and equip it with the
0,0 0,0 0,0
product orientation, see §1.3. We likewise define, for every r,s ∈ N, the correspond-
ing space M(d1,d2),d3,...,dn(X,L;J) and still denote by f : M(d1,d2),d3,...,dn(X,L;J) →
r,s r,s r,s
M(d1,d2),d3,...,dn(X,L;J) the forgetful map, whose fibers are canonically oriented. We orient
0,0
M(d1,d2),d3,...,dn(X,L;J) as the concatenation of the orientation of M(d1,d2),d3,...,dn(X,L;J)
r,s 0,0
with the orientation of the fibers of f .
r,s
Note that every element of M(d1,d2)(X,L;J) is parameterized by the nodal disc ∆ and
0,0 0
thus possess a special point, the node • of ∆ . We denote by f : M(d1,d2)(X,L;J) →
0 • 0,0
Md1,d2(X,L;J) the tautological map forgetting this special point • and also by f :
0,0 •
Mr(d,s1,d2),d3,...,dn(X,L;J) → Mrd,1s,...,dn(X,L;J) the induced forgetful map.
Lemma 1.8 Let L be a closed oriented Lagrangian submanifold of a closed symplectic
six-manifold (X,ω). Let r,s ∈ N, n ≥ 2 and d ,...,d ∈ H (X,L;Z) be such that µ (d ) ≥
1 n 2 L j
2, j ∈ {1,...,n}. Then, for every generic almost-complex structure J tamed by ω, the
image f (cid:0)M(d1,d2),d3,...,dn(X,L;J)(cid:1) is, outside of a codimension two subspace, a canonically
• r,s
cooriented codimension one submanifold of Md1,...,dn(X,L;J).
r,s
Note that the Lagrangian L is only supposed to be oriented in Lemma 1.8, so that the
canonical coorientation given by this lemma does not originate from orientations of the
moduli spaces M(rd,s1,d2),d3,...,dn(X,L;J) and Mdr,1s,...,dn(X,L;J).
Proof:
8
Let [u ,...,u ,J,z,ζ] ∈ M(d1,d2),d3,...,dn(X,L;J) where J is generic. Outside of a sub-
1 n r,s
space of codimension more than two of M(d1,d2),d3,...,dn(X,L;J), the differentials of u
r,s 1
and u at the node • do not vanish. Moreover, outside of a codimension two subspace
2
of M(d1,d2),d3,...,dn(X,L;J), the lines Im(d| u | ) and Im(d| u | ) are in direct sum in
r,s • 1 ∂∆ • 2 ∂∆
T L. These lines are moreover canonically oriented, so that the normal line N to the
u1(•) •
oriented plane Im(d| u | )⊕Im(d| u | ) in the oriented three-space T L inherits an
• 1 ∂∆ • 2 ∂∆ u1(•)
orientation. Now, since the Maslov index µ (d ) is greater than one and J is generic,
L 1
the map u deforms as a family (uλ) in Md1 (X,L;J) such that uλ(•) is positively
1 1 λ∈]−(cid:15),(cid:15)[ 0,0 1
transverse to Im(d| u | )⊕Im(d| u | ) at λ = 0. The family [uλ,u ,...,u ,J,z,ζ]
• 1 ∂∆ • 2 ∂∆ 1 2 n λ∈]−(cid:15),(cid:15)[
of Mdr,1s,...,dn(X,L;J) is then transversal to f• : M(rd,s1,d2),d3,...,dn(X,L;J) at λ = 0 and the
canonical coorientation of this image is defined such that this family becomes positively
transverse. (cid:3)
Note that the tautological involution which exchange the discs of class d and d
1 2
preserves the coorientations given by Lemma 1.8. When L is Spin, the orientations of
the spaces M(rd,s1,d2),d3,...,dn(X,L;J) and Mdr,1s,...,dn(X,L;J) also induce a coorientation of
f• : M(rd,s1,d2),d3,...,dn(X,L;J) in Mdr,1s,...,dn(X,L;J). We denote the incidence index between
these coorientations by (cid:104)Mdr,1s,...,dn(X,L;J),M(rd,s1,d2),d3,...,dn(X,L;J)(cid:105), so that it equals +1
if they coincide and −1 otherwise.
Proposition 1.9 Let L be a closed Spin Lagrangian submanifold of a closed symplec-
tic six-manifold (X,ω). Let r,s ∈ N, n ≥ 2 and d ,...,d ∈ H (X,L;Z) be such that
1 n 2
µ (d ) ≥ 2, j ∈ {1,...,n}. Then, for every generic almost-complex structure J tamed by
L j
ω, (cid:104)Mrd,1s,...,dn(X,L;J),M(rd,s1,d2),d3,...,dn(X,L;J)(cid:105) = +1, whereas
(cid:104)∂Md1+d2,d3,...,dn(X,L;J),M(d1,d2),d3,...,dn(X,L;J)(cid:105) = −1.
r,s r,s
Proof:
The glueing map of J-holomorphic discs preserves the orientations of the fibers of the
forgetful map f , so that from our conventions of orientations of moduli spaces, it suffices
r,s
to prove the result for r = s = 0. Moreover, the spaces Mdj (X,L;J) being even dimen-
0,0
sional, it suffices to prove the result for n = 2. The second part of Proposition 1.9 thus
follows from Proposition 1.7. Let then [u ,u ,J] ∈ M(d1,d2)(X,L;J). We may assume that
1 2 0,0
the lines Im(d| u | ) and Im(d| u | ) are in direct sum in T L and that the differen-
• 1 ∂∆ • 2 ∂∆ u1(•)
tialsoftheevaluationmapsMdj (X,L;J) → L,j ∈ {1,2},atthenode•aresurjective. Let
0,0
r. (resp. r. ) be an element of Aut(∆) whose action on Md1 (X,L;J) (resp. Md2 (X,L;J))
1 2 0,0 0,0
evaluated at the node • positively generates Im(d| u | ) (resp. Im(d| u | )). Let
• 1 ∂∆ • 2 ∂∆
r∗ (resp. r∗) be an element of T Md2 (X,L;J) (resp. T Md1 (X,L;J)) whose eval-
1 2 u2 0,0 u1 0,0
. . . .
uation at the node • coincides with r (resp. r ), so that r +r∗ and r∗+ r define
1 2 1 1 2 2
elements of T M(d1,d2)(X,L;J). Let ν = ν + ν ∈ M(d1,d2)(X,L;J) be an ele-
[u1,u2,J] 0,0 1 2 0,0
ment such that ν(•) = ν (•) = ν (•) positively generates the normal N to the ori-
1 2 •
ented plane Im(d| u | ) ⊕ Im(d| u | ) in the oriented three-space T L. By con-
• 1 ∂∆ • 2 ∂∆ u1(•)
9
. .
struction, the basis (r (•),r∗(•),ν (•)) = (r∗(•),r (•),ν (•)) of T L is direct. Let
. . . . 1 2 1 1 2 2 u1(•)
(t ,h ) (resp. (t ,h )) be a pair of elements of aut(∆) which vanish at the node •,
1 1 2 2
. . . . . .
so that (r ,t ,h ) (resp. (r ,t ,h )) is a direct basis of the Lie algebra aut(∆). Let
1 1 1 2 2 2
B (resp. B ) be a family of elements of T Pd1(X,L;J) (resp. T Pd2(X,L;J))
su1ch that th2e basis (B ,t. ,h. ,r. ,r∗,ν ) (resp(u.1,(Jr)∗,0r.,0,ν ,B ,t. ,h. )) of T(u2,J) P0,d01(X,L;J)
(resp. T Pd2(X,L;1J)1) is1dir1ect2. B1y definition1, th2e b2asis2(B2 ,t.2 ,h. ,r. (u+1,rJ∗),r0∗,0+ r. ,ν +
(u2,J) 0,0 1 1 1 1 1 2 2 1
ν ,B ,t. ,h. )ofT P(d1,d2)(X,L;J)isdirect,sothat(B ,r. +r∗,r∗+ r. ,ν +ν ,B )isa
2 2 2 2 (u1,u2,J) 0,0 1 1 1 2 2 1 2 2
direct basis of T M(d1,d2)(X,L;J). Likewise, (B ,t. ,h. ,r. ,r∗,ν ,r∗,r. ,ν ,B ,t. ,h. )
[u1,u2,J] 0,0 1 1 1 1 2 1 1 2 2 2 2 2
is a direct basis of T Pd1,d2(X,L;J), so that (B ,r∗,ν ,r∗,ν ,B ) is an indirect basis
(u1,u2,J) 0,0 1 2 1 1 2 2
of T Md1,d2(X,L;J). The differential of the forgetful map f sends our basis (B ,r.
[u1,u2,J] 0,0 • 1 1
+r∗,r∗+ r. ,ν +ν ,B ) onto the family (B ,r∗,r∗,ν +ν ,B ) of T Md1,d2(X,L;J).
1 2 2 1 2 2 1 1 2 1 2 2 [u1,u2,J] 0,0
The canonical coorientation given by Lemma 1.8 is defined such that ν is inward nor-
1
mal. The incidence index (cid:104)Md1,d2(X,L;J),M(d1,d2)(X,L;J)(cid:105) thus equals +1 if and only if
0,0 0,0
the basis (−ν ,B ,r∗,r∗,ν +ν ,B ) of T Md1,d2(X,L;J) is direct. The result thus
1 1 1 2 1 2 2 [u1,u2,J] 0,0
follows from the parity of the cardinality of B . (cid:3)
1
2 Linking numbers, complete graphs and spanning
subtrees
2.1 Complete graphs
2.1.1 Definitions
For every positive integer n, denote by K the complete graph having n vertices. Denote
n
by S (resp E ) its set of vertices (resp (cid:0)n(cid:1) edges). For every e ∈ E , let c : K → K
n n 2 n+1 e n+1 n
be the contraction map of the edge e. It is defined as the quotient map of K by the
n+1
following equivalence relation R (see Figure 1):
• ∀s ,s ∈ S , s Rs if and only if s and s bound e.
1 2 n+1 1 2 1 2
• ∀e ,e ∈ E , e Re if and only if e and e have one common boundary vertex and
1 2 n+1 1 2 1 2
the other one bounding e.
The contraction map c is surjective. The graph K contains a unique vertex s whose
e n e
inverse image under c does not reduce to a singleton, but to the two bounding vertices of
e
e. The inverse image to an edge of K is made of one or two edges depending on whether
n
this edge contains s in its boundary or not.
e
Finally, for every commutative ring A and every positive integer n, we denote by A =
n
(cid:80)
A.e the free A-module generated by the edges of K . For every edge e of K , we
e∈En n 0 n+1
then denote by 1 = 1.e the associated generator of the factor A.e . The contraction map
e0 0 (cid:80) 0 (cid:80)
c induces a morphism of A-modules (c ) : a .e ∈ A (cid:55)→ a .c (e) ∈
e0 e0 ∗ e∈En+1 e n+1 e∈En+1 e e0
A which contains A.e in its kernel.
n 0
10