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Lagrangian Non-Intersections

In this paper we use Floer theory to study topological restrictions on Lagrangian embeddings in closed symplectic manifolds. One of the phenomena arising from our results is ``homological rigidity'' of Lagrangian submanifolds. Namely, in certain symplectic manifolds, conditions on low dimensional topological invariants of a Lagrangian (such as its first homology) completely determine its entire homology. We also develop methods for studying Hamiltonian displacement of Lagrangian submanifolds and its relations to the topology of Lagrangians. Finally we present some new results on Lagrangian intersections.


Introduction and main results
The present paper is devoted to Lagrangian submanifolds of symplectic manifolds and their intersection patterns.
One of the cornerstones of symplectic topology is the rigidity of symplectic structures reflected in the behavior of Lagrangian submanifolds and their intersections. The first non-trivial restrictions on Lagrangian embeddings go back to Gromov's work of 1985 [21]. One of the many results of that paper is the following fundamental theorem: In R 2n there are no closed exact Lagrangian submanifolds, in particular there are no closed Lagrangian submanifolds L ⊂ R 2n with H 1 (L; R) = 0.
Since then, Gromov's techniques have been extended and, in combination with other methods, have led to more restrictions on the topology of Lagrangian submanifolds, most of the results being for Lagrangians submanifolds of R 2n and of cotangent bundles (see e.g. [2,10,16,24,30,29,33,32,34,35,39,40] for a partial list of older and newer results).
Only relatively recently first results on the topology of Lagrangians in closed manifolds have been obtained by Seidel [37] and later on by Biran and Cieliebak [9]. Note that when studying Lagrangians in an arbitrary manifold one encounters all Lagrangian submanifolds of R 2n . This is due to the fact that every symplectic manifold M 2n is locally modeled on R 2n , hence every Lagrangian submanifold L ⊂ R 2n can also be Lagrangianly embedded into M 2n . Thus, Lagrangian embeddings into R 2n should, in a sense, be regarded as the local case. However, our understanding of Lagrangian submanifolds of R 2n is still quite limited, in particular also that of "local" Lagrangian submanifolds in any symplectic manifold M 2n . (Thus, in this case, "local" turns out to be difficult.) In this paper we concentrate on "global" Lagrangian submanifolds. One way to "mod out" the local Lagrangians is to assume for example that the first homology of the Lagrangians is either zero or torsion. In view of the preceding theorem of Gromov such Lagrangian submanifolds must be global in the sense that they cannot lie entirely in a Darboux chart. One of the phenomena arising from our results below is that in certain symplectic manifolds the topology of Lagrangians with small first homology is extremely restricted. It turns out that in some cases (e.g. M = CP n ) certain assumptions on H 1 (L; Z) of a Lagrangian L completely determine the entire homology of L. This phenomenon is illustrated in Theorems A-C of Section 1.1 below. Note that very recently examples of this phenomenon have been discovered also for cotangent bundles of spheres by Buhovski [10] and by Seidel [35].
The second phenomenon presented in this paper belongs to the framework of Lagrangian intersections. Our results show that certain symplectic manifolds M contain some kind of "Lagrangian core" Λ ⊂ M which dominates intersections, in the sense that many global Lagrangians L ⊂ M must intersect Λ, the intersection points being irremovable by symplectic diffeomorphisms. Results in this direction are presented in Section 1.3.

Homological uniqueness of Lagrangian submanifolds.
Here and in what follows all Lagrangian submanifolds are assumed to be compact and without boundary, unless otherwise explicitly stated.
Lagrangian submanifolds of CP n . Let CP n be the complex projective space, endowed with its standard Kähler structure. It is well known that CP n has no Lagrangian submanifolds L with H 1 (L; Z) = 0 (see [37], see also [9]). Note however that there do exist Lagrangians L ⊂ CP n with H 1 (L; Z) torsion. For example, the real projective space RP n ≈ [z 0 : . . . : z n ] ∈ CP n | z i ∈ R ∀i ⊂ CP n is such a Lagrangian submanifold (for n ≥ 2). In fact Seidel proved in [37] that every Lagrangian submanifold L ⊂ CP n with H 1 (L; Z 2n+2 ) = Z 2 must satisfy H * (L; Z 2 ) ∼ = H * (RP n ; Z 2 ) as graded vector spaces. Below we shall prove a stronger statement which gives information also on the cohomology ring of L. Henceforth we say that an abelian group H is q-torsion if for every α ∈ H we have qα = 0. (This, by our conventions, includes the case when H is the trivial group.) Our first result is: Theorem A. Let L ⊂ CP n be a Lagrangian submanifold with H 1 (L; Z) 2-torsion. Then: (1) There exists an isomorphism of graded vector spaces H * (L; Z 2 ) ∼ = H * (RP n ; Z 2 ).
Other than RP n ⊂ CP n+1 we are not aware of any example of Lagrangians L with 2-torsion H 1 (L; Z). However, Chiang [11] constructed an example of a Lagrangian in L ⊂ CP 3 with H i (L; Z 2 ) = Z 2 for every i. This Lagrangian is a quotient of RP 3 by the dihedral group D 3 . It satisfies H 1 (L; Z) = Z 4 . See [11] for the details as well as other interesting examples of Lagrangian submanifolds of complex projective spaces of various dimensions.
Lagrangian submanifolds of CP n × X. It has been proved in [9] that if X is a closed symplectic manifold with π 2 (X) = 0 then for n ≥ dim C X, CP n × X has no simply connected Lagrangian submanifolds. On the other hand, if n < dim C X, then CP n × X may have such Lagrangians. Indeed, for any symplectic manifold X of dim C X = n + 1, CP n × X has a Lagrangian sphere, after a possible rescaling of the symplectic form on the X factor (see [9] for details). The following theorem shows that homologically this is the only example.
In contrast to the result of [9] mentioned above, if one drops the condition "π 2 (X) = 0", then CP n × X may have simply connected Lagrangian submanifolds even if n ≥ dim C X. For example take X to be CP n and consider CP n × CP n endowed with the equally weighted split standard symplectic structure. Then L = CP n embeds Lagrangianly as the "anti-diagonal" in CP n × CP n , namely The following theorem shows that, again, homologically this is the only example.
Theorem C. Let L ⊂ CP n × CP n be a Lagrangian submanifold with H 1 (L; Z) = 0. Then H * (L; Z 2 ) ∼ = H * (CP n ; Z 2 ) as graded algebras. Moreover, if a ∈ H 2 (CP n × CP n ; Z 2 ) is the generator of H 2 of either factor of CP n × CP n then a| L ∈ H 2 (L; Z 2 ) generates H * (L; Z 2 ) as an algebra.
Again, besides L = CP n we are not aware of any other examples of Lagrangian submanifolds in CP n × CP n with H 1 (L; Z) = 0.

Lagrangian spheres.
Here we present new restrictions on Lagrangian embeddings of spheres. Let us mention that Lagrangian spheres appear in various mathematical contexts other than symplectic geometry (e.g. singularity theory [4]) and thus deserve special attention beyond the scope of symplectic geometry.
In view of the fact that manifolds of the type CP n × X with dim C X = n + 1 have Lagrangian spheres it makes sense to ask what happens for X of other dimensions. The next theorem gives a partial answer to this question.
Theorem D. Let X be a symplectic manifold with π 2 (X) = 0 which is either closed or has a symplectically convex end. If CP n × X has a Lagrangian sphere (where n, dim X > 0) then dim C X ≡ n + 1(mod2n + 2). Theorem E. Let X be a symplectic manifold that can be covered by a symplectic manifold which is symplectomorphic to a domain in C m . Let M be a spherically monotone closed symplectic manifold. Assume dim X, dim M > 0. If M × X has a Lagrangian sphere then Examples of manifolds X satisfying the conditions of Theorem E include symplectic tori, and ball quotients, both endowed with Kähler symplectic structures. Theorem E is in fact a special case of the more general Theorem 7.A which will be proved in Section 7.
Theorem F. Let (M, ω) be a spherically monotone closed symplectic manifold with [ω] = c M 1 on π 2 (M) and denote m = dim C M. Let CP n × M be endowed with the symplectic form Ω = (n+1)σ ⊕ω, where σ is the standard symplectic Kähler form of CP n normalized so that the area of a projective line is 1. Suppose that (CP n × M, Ω) has a Lagrangian sphere, where n + m ≥ 3. Then 2 gcd (n + 1, N M ) | n + m + 1.
Remark. Embarrassingly, the only example known to us of Lagrangian spheres in manifolds of the types appearing in Theorems D-F, are all in manifolds of the type CP n × X where dim C X = n + 1. It would be interesting to figure out for example whether or not CP n × X admits a Lagrangian sphere (or even a homology sphere) when dim C X = (2k + 1)(n + 1), k ≥ 1.

Lagrangian intersections.
Here we present new results on Lagrangian intersections. The pattern that stands out in all the examples below is the existence of a "core" Λ consisting of a finite union of (possibly) immersed Lagrangian spheres with the property that every Lagrangian submanifold with prescribed topological properties (e.g. simply connected) must intersect Λ.
Intersections in the quadric. Let Q n ⊂ CP n+1 be the complex n-dimensional smooth quadric defined by the equation: endowed with the symplectic structure induced from CP n+1 . Let Λ Q ⊂ Q n be the corresponding "real" quadric, namely: It is not hard to see that Λ Q ⊂ Q n is in fact a Lagrangian sphere.

Remarks.
(1) Note that the intersection between L and Λ Q is in general not due to topological reasons but rather to symplectic ones. For example when n is odd, every Lagrangian sphere (in particular Λ Q ) can be displaced from itself via an arbitrary small (non symplectic) diffeomorphism.
(2) The quadric Q n has many different Lagrangian submanifolds with H 1 (L; Z) either zero or 2-torsion. First of all it has simply connected Lagrangians (e.g. Λ Q itself). Next, for every 0 ≤ r ≤ n consider It is not hard to see that all the L r 's are Lagrangian submanifolds of Q n and that L r is diffeomorphic to (S r × S n−r )/Z 2 , where Z 2 acts on both factors by the antipode map. It easily follows that L 0 and L n are spheres, and a simple computation shows that for every 1 < r < n − 1, we have H 1 (L r ; Z) = Z 2 but H 2 (L r ; Z 2 ) = 0. Therefore when n > 3, H * (L r ; Z 2 ) is not isomorphic to A * Q . It follows from Theorem G that for n > 3 any Lagrangian submanifold L ⊂ Q n diffeomorphic to one of the L r 's must intersect Λ Q . In particular the intersection L r ∩ Λ Q = ∅ cannot be removed via a symplectic isotopy. (4) The statement of Theorem G remains true if instead of assuming H 1 (L; Z) is 2torsion one assumes that L is monotone with minimal Maslov number N L = n.
(See the second remark at the end of the proof of the Theorem in Section 6.) It would be interesting to figure out if at least there are any monotone Lagrangians L ⊂ Q n (not necessarily with 2-torsion H 1 ) with N L = n that lie in complement of Λ Q .

Questions.
(1) Theorem G implies that (for n ≥ 3) every Lagrangian sphere L ⊂ Q n , must intersect Λ Q . Is it true that every two Lagrangian spheres L 1 , L 2 ⊂ Q n must intersect each other? Note that when n = even this easily follows by computing the intersection number [L 1 ] · [L 2 ], but for n = odd this does not seem to follow from purely topological reasons. An affirmative answer would have the following consequences in algebraic geometry: the complex quadric (of dimension ≥ 3) cannot be degenerated to a variety having two or more isolated singularities. See [6,7,4] for more details and precise statements.
(2) Note that the cohomology of the Lagrangian L 1 ⊂ Q n (taken from (2) above with r = 1) is precisely A * Q . Can L 1 be Hamiltonianly isotoped to lie in the complement of Λ Q ? It is known that L 1 can also be embedded as a monotone Lagrangian in C n with minimal Maslov number = n (see [34]). Can L 1 be embedded as a monotone Lagrangian in Q n \ Λ Q ? (3) Theorem G will not be proved by showing that the Floer homology HF (L, Λ Q ) is not zero. It would we be interesting to figure out whether this is indeed so.
Intersections in hypersurfaces of CP n+1 . Theorem G can be generalized as follows. Let Σ n d ⊂ CP n+1 be a smooth complex hypersurface of degree d, viewed as a symplectic manifold endowed with the symplectic structure induced from CP n+1 .
Theorem H. For every d > 2 there exist d n+1 (possibly) immersed Lagrangian spheres S 1 , . . . , S d n+1 ⊂ Σ n d such that their union Λ d = S 1 ∪. . . ∪S d n+1 has the following properties: and n = even, for every Lagrangian submanifold L ⊂ Σ n d with H 1 (L; Z) 2-torsion either L ∩ Λ d = ∅ or L has the following properties: and d = odd, for every Lagrangian submanifold L ⊂ Σ n d with H 1 (L; Z) 2-torsion either L ∩ Λ d = ∅ or L has the following properties: We have not been able to explicitly compute the Lagrangian spheres in Λ d , however in Section 3.6.4 below we show that S 2 , . . . , S d n+1 are all obtained from S 1 by applying suitable automorphisms of Σ n d . Examples of Lagrangian submanifolds L ⊂ Σ n d that satisfy the conditions of the Theorem H come from Picard-Lefschetz theory. Indeed Σ n d , d ≥ 2, can be included as a fibre in a degeneration with isolated singularities, hence by the Lagrangian vanishing cycle construction it must contain Lagrangian spheres (see [1,13,22,38,36], see also [4]). Here are more explicit examples of Lagrangians in Σ n d : write Σ n d as z d 0 + · · · + z d n+1 = 0 ⊂ CP n+1 . Let τ ∈ C be a root of −1 of order d. Then it is easy to see that when d = even [z 0 : . . . : z n+1 ] ∈ Σ n d z 0 ∈ τ R, z j ∈ R for every 1 ≤ j ≤ n + 1 is a Lagrangian sphere. When d = odd we also have Lagrangians homeomorphic to RP n , namely Σ n d ∩ RP n+1 . (See Appendix A of [25] for an explicit homeomorphism.) Questions.
(1) Theorem H implies in particular that (for some values of d) it is impossible to embed a Lagrangian sphere which is disjoint from the spheres S 1 , . . . , S d n+1 .
In view of this one is led to speculate that the maximal number of mutually disjoint Lagrangian spheres in Σ n d is finite, or even that this number is not bigger than d n+1 (c.f. the first question after Theorem G.) (2) Under the conditions of statement (2) of Theorem H, Lagrangians L ⊂ Σ n d \Λ d with 2-torsion H 1 (L; Z) must have the same Z 2 -coefficients cohomology as a manifold of the type where Z 2 acts on both factors of S 1 × S n+1−d by the antipode map. Note that L also satisfies the cohomological restrictions predicted by statement (2) of Theorem H. It is known that L admits a monotone Lagrangian embedding into C n with minimal Maslov number = n+2−d (see [34]). Does L admit a monotone Lagrangian embedding into Σ n d ? Into Intersections in a hypersurface of CP n × CP n . Consider CP n × CP n endowed with the standard symplectic form σ std ⊕ σ std and let Σ 2n−1 ⊂ CP n × CP n be the complex hypersurface defined by the equation where [z 0 : . . . : z n ], [w 0 : . . . : w n ] are homogeneous coordinates on CP n × CP n . We endow Σ 2n−1 with the symplectic structure induced from CP n × CP n . Put A simple computation shows that Λ Σ ⊂ Σ 2n−1 is a Lagrangian sphere.
1.4. Discussion. The phenomenon arising in the results of Section 1.3 above is the existence of certain "Lagrangian subsets" that dominate intersections in the sense that all Lagrangian submanifolds with specified topology must intersect them. A similar phenomenon is already known for cotangent bundles, where a result due to Gromov [21] implies that every Lagrangian submanifold of a cotangent bundle L ⊂ T * (X) with H 1 (L; R) = 0 must intersect the zero section. Our results show that such a phenomenon holds also in closed manifolds. It is interesting to note that, similarly to our case, also for cotangent bundles it is currently unknown whether or not the Floer homology HF (L, O X ) of L and the zero section O X is non-trivial. (Compare Question (2) after Theorem G.) Finally, let us remark that the above intersection phenomena, in general, seize to hold in the C ∞ -category. Indeed in each of the Theorems G-I one can usually remove all the intersection points by a smooth diffeomorphism.
1.5. Methods and ideas. The methods and tools used in this paper consist of two main ingredients. The first one is Floer theory for Lagrangian submanifolds. In particular we use the extension of Floer homology to monotone Lagrangian submanifolds due to Oh [31,33,32] which gives rise to an an algebraic approach for computing Floer homology in terms of a spectral sequence.
The second ingredient is a geometric technique, developed by the author in [5] and in this paper, by which it is sometimes possible to compute Floer homology in a geometric way. Our techniques enable to perform certain transformations to a given Lagrangian submanifold resulting in a new Lagrangian that can be Hamiltonianly displaced. In particular, we obtain vanishing of Floer homology. This vanishing combined with the algebraic computations mentioned above is the key point behind most of our results.
Let us now describe in some more detail the main ideas of the paper. Let Σ 2n be a closed symplectic manifold. We shall concentrate on the case when Σ can be symplectically embedded as a hyperplane section of a higher dimensional symplectic manifold, say M 2n+2 . The idea is that in view of the decomposition technique developed by the author in [5] the symplectic properties of M \ Σ can be used to study the symplectic topology of Σ itself. Consider a tubular neighbourhood U of Σ in M. Its boundary ∂U is a circle bundle π : ∂U → Σ over Σ. Now let L ⊂ Σ be a Lagrangian submanifold. As we shall see in Section 4.1 below, if we chose U carefully the restriction of this circle bundle to L, is a Lagrangian submanifold Γ L = π −1 (L) lying in M \Σ. Note that M \Σ is a Stein manifold.
The symplectic theory of Stein manifolds now comes into play. Recall that Stein manifolds are divided into subcritcal and critical ones (see Section 2 for the precise definitions). Subcritical Stein manifolds have the feature that (after a suitable completion) all compact subsets can be Hamiltonianly displaced. In particular the Floer homology of Lagrangian submanifolds must vanish.
Returning to our case, if M\Σ turns out to be subcritcal, the Floer homology HF (Γ L , Γ L ) vanishes. Thus, starting with a Lagrangian submanifold L ⊂ Σ, we have transformed it into a new Lagrangian Γ L ⊂ M \ Σ whose Floer homology vanishes due to geometric reasons. Note that in contrast to Γ L , in general L itself cannot be Hamiltonianly displaced and in fact its Floer homology might not vanish.
We now turn to algebraic computations in Floer homology. The idea is to perform the computation of HF (Γ L , Γ L ) in an alternative way. The main tool for this end is a spectral sequence based on the theory developed by Oh [33,32]. The second step of this spectral sequence is the singular cohomology of a Lagrangian, and the sequence converges to its Floer homology. Comparing this computation (performed on Γ L ) with the vanishing of HF (Γ L , Γ L ) we obtain restrictions on the cohomology of Γ L . In some cases we are even able to reproduce the entire cohomology of Γ L . Having done this, we derive information on the cohomology of L itself (recall that Γ L → L is a circle bundle). We give a rather detailed construction of this spectral sequence in Section 5 since our approach is somewhat different than Oh's original work [33,32]. Detailed computations using the spectral sequence appear throughout the proofs of the main theorems in Section 6.
Let us turn now to the case when M \ Σ is a critical Stein manifold. In this case it is no longer true that all Lagrangian submanifolds in M \ Σ are displaceable. In Section 3.5 and 4.2 we introduce a kind of geometric obstruction for displaceability in M \ Σ. This obstruction, which we call the Lagrangian trace, is a union of (possibly) immersed Lagrangian spheres Λ ⊂ Σ. It has the property that for every Lagrangian L ⊂ Σ with L∩Λ = ∅, the circle bundle Γ L ⊂ M \Σ can be Hamiltonianly displaced, in particular HF (Γ L , Γ L ) = 0. Recomputing the vanishing of HF (Γ L , Γ L ) using the spectral sequence we deduce that Lagrangian submanifolds L ⊂ Σ with certain topological properties must intersect Λ.
The choice of the coefficients. Our cohomological restrictions are with Z 2 -coefficients only. This has to do with technical reasons coming from Floer homology theory which is one of the main tools used in this paper. However, recent developments in Floer theory, due to Fukaya, Oh, Ohta and Ono [20] make it possible to define, in some cases, Floer homology with coefficients in Q or even Z. It seems very likely that several of our results above continue to hold for cohomology with Z-coefficients as well.
The title of the paper. "Lagrangian non-intersections" is derived from the following idea motivated in this paper: whenever the principle of Lagrangian intersections fails (in the sense that a Lagrangian can be Hamiltonianly displaced) we obtain restrictions on the topology of the Lagrangian via computations in Floer homology.
Organization of the paper. The rest of the paper is organized as follows. In Section 2 we collect some important facts from the symplectic theory of Stein manifolds that will be used in the sequel. We also develop in this section methods to displace Lagrangian submanifolds in both subcritical and critical Stein manifolds.
In Sections 3 and 4 we discuss symplectic manifolds Σ that appear as hyperplane sections in other manifolds M. For Lagrangian submanifolds L ⊂ Σ we introduce the Lagrangian circle bundle construction giving rise to a new Lagrangian submanifold Γ L ⊂ M \ Σ. We then study the possibilities to displace Γ L in M \ Σ and introduce the Lagrangian trace Λ ⊂ Σ which is a kind of obstruction for displacing Γ L . In Section 3.6 we present explicit calculations of Λ for various examples of Σ. Section 5 is devoted to computations in Floer homology. In Section 6 we give the proofs of the main theorems. Finally, in Section 7 we present some generalizations of the theorems of Section 1.

Symplectic geometry of Stein manifolds
Here we briefly recall some basic facts on Stein manifolds from the symplectic viewpoint. The reader is referred to [14,15] for the foundations of the symplectic theory of Stein manifolds. Apart from the contents of Subsection 2.4 most of the material below can be found in [17,14,15,8].
A Stein manifold is a triple (V, J, ϕ) where (V, J) is an open complex manifold and ϕ : V → R is a smooth exhausting plurisubharmonic function. The term "exhausting" means that ϕ is proper and bounded from below. "Plurisubharmonic" means that the 2-form ω ϕ = −dd C ϕ is a J-positive symplectic form, i.e. −dd C ϕ(v, Jv) > 0 for every 0 = v ∈ T (V ). (Here and in what follows d C stands for operator that takes a smooth function ϕ to the 1-form dϕ • J.) We denote by g ϕ (·, ·) = ω ϕ (·, J·) the associated Riemannian Kähler metric.
Given a Stein manifold (V, J, ϕ) we have the gradient vector field X ϕ = grad gϕ ϕ of ϕ with respect to the metric g ϕ . A simple computation shows that L Xϕ ω ϕ = ω ϕ , hence the flow X t ϕ of X ϕ is conformally symplectic, (X t ϕ ) * ω ϕ = e t ω ϕ . We remark that in contrast to other texts (e.g. [17]), we do not assume that the flow of X ϕ is complete, unless explicitly stated. Note however that since ϕ is exhausting the flow X t ϕ does exist for all negative times.
2.1. Canonical symplectic structures on Stein manifolds. Given a Stein manifold (V, J, ϕ) and R ∈ R, we denote by V ϕ≤R the sublevel set ϕ −1 ((−∞, R]). We write Crit(ϕ) for the set of critical points of the function ϕ.
Remark 2.1.C. In view of this lemma, we shall sometimes denote the symplectic manifold associated to the completion of a Stein manifold (V, J, ϕ) by (V, ω) (since it does not depend on the choice of the plurisubharmonic function). Note that for every open subset W ⊂ V with compact closure we have a symplectic embedding (W, ω ϕ ) ֒→ (V, ω).

2.2.
The skeleton of a Stein manifold (V, J, ϕ). By this we mean the subset ∆ ϕ ⊂ V which is formed from the union of the stable submanifolds of the flow X t ϕ , namely: Lemma 2.2.A. Let (V, J, ϕ) be a Stein manifold and assume that ϕ is a Morse-Bott function. Then for every critical submanifold C ⊂ V of ϕ we have: (1) C is isotropic with respect to ω ϕ .
Proof. For ϕ being Morse a proof can be found in [15] (see also [17], and see [5] Section 8). The case of Morse-Bott ϕ is an obvious extension of the "Morse case". [5]). Let (V, J, ϕ) be a Stein manifold, and assume that all the critical points of ϕ lie in the subset {ϕ < R} for some R ∈ R. Then arbitrarily close to ϕ ′ in the C 2 -topology there exists an exhausting plurisubharmonic function ϕ ′ : V → R such that: In particular all trajectories of X ϕ ′ go from critical points of ϕ ′ to either critical points of strictly higher index or to "infinity" (i.e. do not go to any other critical point). Moreover the skeleton ∆ ϕ ′ is an isotropic CW-complex (see below).
Let (Y, ω) be a symplectic manifold and ∆ ⊂ Y a subset. We call ∆ an isotropic CWcomplex if there exists an abstract CW-complex K and a homeomorphism i : is an isotropic embedding. We refer the reader to [5] for more details on this notion.
2.3. Subcritical Stein manifolds. Let (V, J, ϕ) be a Stein manifold. It is well known [15] that if ϕ is Morse then for every critical point p we have ind p ϕ ≤ 1 2 dim R V . We say that (V, J, ϕ) is subcritical if ϕ is Morse with finite number of critical points and for every p ∈ Critϕ, ind p ϕ < 1 2 dim R V . Note that in this case dim ∆ ϕ < 1 2 dim R V (hence the skeleton does not contain Lagrangian cells).
The following lemma shows that in subcritical Stein manifolds any compact subset can be Hamiltonianly displaced. See [8] for the proof.
2.4. The critical coskeleton. Let (V, J, ϕ) be a Stein manifold, and suppose that ϕ is a Morse-Bott function with finitely many critical submanifolds. We denote by p 1 , . . . , p N the isolated critical points of ϕ (if there are any). We say that ϕ has property (S 0 ) if one of the following two conditions is satisfied: (2) ϕ has no isolated critical points and only one positive dimensional critical manifold it may happen that r = 0.) In the second case pick a point p ′ 1 ∈ S and put r = 1. We define the critical coskeleton ∇ crit ϕ ⊂ V to be the union of the unstable submanifolds of the p ′ i 's, namely: Remark. Property (S 0 ) is purely technical and may look somewhat artificial. Note that if ϕ is Morse then it automatically has property (S 0 ) (since all its critical points are isolated). Property (S 0 ) was created to accommodate a slightly more general situation than that. It covers two different (and unrelated) possibilities. The first possibility means that among the unstable submanifolds, those that are Lagrangian (i.e. have minimal dimension) all come from isolated critical points. The second possibility, roughly speaking, means that (V, ω ϕ ) looks like a neighbourhood of the zero section in T * (S).

Examples.
(1) If (V, J, ϕ) is a subcritical Stein manifold then clearly r = 0, hence ∇ crit ϕ = ∅. (2) Let M be a closed manifold and V = T * (M) be its cotangent bundle. Denote by q ∈ M local coordinates along X and by p ∈ T * q (M) the dual coordinates along the cotangent fibres. It is well known (see [15]) that V can be endowed with the structure of a Stein manifold with ϕ(q, p) = |p| 2 and X ϕ = p ∂ ∂p . Here | · | is a norm along the cotangent fibers (coming from a Riemannian metric on M). In this case the only critical submanifold is the zero section, and the critical coskeleton is just one fibre, ∇ crit ϕ = T * q (M).

2.4.1.
Property (S). Let (V, J, ϕ) be a Stein manifold. We say that ϕ has property (S) if it has property (S 0 ) above and in addition for every 1 ≤ i ≤ r all gradient trajectories of X ϕ emanating from the points p ′ i go to "infinity". Note that this definition does not depend on the choice of the point p ′ 1 in the case when ϕ has only one critical submanifold S with dim S = 1 2 dim R V . Indeed, in that case S is the minimum of ϕ, hence all gradient trajectories emanating from points of S go to "infinity". Lemma 2.4.A. Let (V, J, ϕ) be a complete Stein manifold and assume that ϕ has property Examples.
(1) If (V, J, ϕ) is subcritical then the the lemma reduces to Lemma 2.3.A since in this case ∇ crit ϕ = ∅, hence any compact subset can be Hamiltonianly displaced.
(2) Let V = T * (M). As we have just seen above ∇ crit ϕ = T * q (M) for some q ∈ M. Hence we recover a statement due to Lalonde and Sikorav [24] that any compact subset of T * (M) lying in the complement of a fibre T * q (M) can be Hamiltonianly displaced.
Before we prove Lemma 2.4.A we shall need some preparations. Given a Morse function ϕ : V → R, denote by Crit ≤k (ϕ) the set of critical points of ϕ of index ≤ k. Denote by ∆ k ϕ the subskeleton We shall need the following Proposition for the proof of Lemma 2.4.A.
Assume that: (1) ϕ is a Morse function with finitely many critical points There are no trajectories of X ϕ that connect any of the critical points x l+1 , . . . , x ν with one of the critical points x 1 , . . . , x l .
Fix mutually disjoint neighbourhood U 1 , . . . , U l of x 1 , . . . , x l respectively. Then arbitrarily close to ϕ in the C 2 -topology there exists an exhausting plurisubharmonic function ϕ ′ with the following properties: (3) The flow of X ϕ ′ connects any of the critical points x 1 , . . . , x l either to a point of strictly higher index or to "infinity". Moreover the subskeleton ∆ k ϕ ′ is an isotropic CW-complex.
We omit the proof as it is a straightforward adaptation of the arguments from Section 9 of [5].
Assume first that ϕ is a Morse function, hence it has no positive dimensional critical submanifolds. Put k = 1 2 dim R V − 1. Applying Proposition 2.4.B we obtain a new plurisubharmonic function ϕ ′ for which ∆ k ϕ ′ is a CW-complex. Note that if we choose the neighborhoods U 1 , . . . , U l of the points in Crit ≤k (ϕ) to be small enough we can arrange that ∇ crit Thus by replacing ϕ by ϕ ′ we may assume without loss of generality that ∆ k ϕ is a CW-complex.
is a Hamiltonian isotopy (see [8], Lemma 3.2). Assume now that ϕ has also positive dimensional submanifolds say S 1 , . . . , S q with ind S i (ϕ) + dim S i < 1 2 dim R V for every i. Pick for every i a generic Morse function f i : S i → R and a cut off function ρ i which is identically 1 near S i and identically 0 outside a small neighbourhood W i of S i . Consider now the function ϕ ǫ = ϕ + ǫ q i=1 ρ i f i . Clearly for small ǫ, ϕ ǫ is plurisubharmonic. Moreover ϕ ǫ is Morse and its critical points consists of the isolated critical points of ϕ and the critical points of the Morse functions f 1 , . . . , f q . Moreover, for every critical point Therefore the critical points of ϕ ǫ of index 1 2 dim R V are exactly the same as those of ϕ. Moreover ϕ ǫ = ϕ near these points. Next, note that if the perturbation above is in small enough neighborhoods W i of the S i 's then due to assumption (S) we have ∇ crit Replacing ω ϕ by ω ϕǫ and A by f (A) we arrive to the case from the beginning of the proof.
Finally assume that the only critical points of ϕ consist of one critical submanifolds S of dimension 1 2 Since ϕ is Morse-Bott this map is a locally trivial fibration in a neighbourhood of S (see [3]). Note that ∇ crit ϕ is just the preimage under F of a point p ∈ S. Therefore since Since S is Lagrangian a small neighbourhood of S can be identified with a neighbourhood of T * (S). Pick a Morse function G on S having all its critical points in S \ F (A). Clearly the Hamiltonian isotopy g t of G (viewed as a Hamiltonian in T * (S)) will displace F (A) away of S within arbitrary small time, say t = ǫ (compare [24]). The result now follows in the same way as in the beginning of the proof. Namely for large enough T > 0 the Hamiltonian diffeomorphism

Polarizations and decompositions of Kähler manifolds
A basic tool that we shall use throughout this work is a decomposition technique for Kähler manifolds that was developed in [5]. In Subsections 3.1-3.4 we briefly summarize the necessary facts from [5] where more details can be found. In Subsection 3.5 we introduce the concept of Lagrangian trace and in 3.6 we compute some examples.
3.1. Polarized Kähler manifolds. Throughout this paper, by a Kähler manifold we mean a triple (M, ω, J) where (M, ω) is a closed symplectic manifold and J is an (integrable) complex structure compatible with ω.
A polarized Kähler manifold P = (M 2n , ω, J; Σ) is a Kähler manifold (M, ω, J) with [ω] ∈ H 2 (M; Z) together with a smooth and reduced complex hypersurface Σ ⊂ M whose homology class The number k will be called the degree of the polarization P and denoted by k P . Note that our notion of polarized Kähler manifolds is slightly different from the one common in algebraic geometry.

3.2.
Additional structures associated with a polarization. We shall now define a distinguished plurisubharmonic function ϕ P : M \ Σ → R which is canonically associated with the polarization P. For this purpose let L = O M (Σ) be the holomorphic line bundle defined by the divisor Σ. Denote by s : M → L the (unique up to a constant factor) holomorphic section whose zero set {s = 0} is Σ. Choose a hermitian metric · on L, and a compatible connection ∇ with curvature R ∇ = 2πik P ω. Finally, define ϕ P : Moreover, it is not hard to see that ϕ P is exhausting and that it has no critical points outside some compact subset of V (see [5]). It is important to remark that the function ϕ P is canonically determined by the polarization P up to an additive constant and does not depend on any of the choices made for · , s or ∇. This is due to the requirement on the curvature R ∇ and the fact that J is integrable (see [5] for more details). Next, let g ω,J = ω(·, J·) be the Kähler Riemannian metric associated with the pair (ω, J). Finally denote by X t P the gradient flow of ϕ P with respect to g ω,J . (Note that X t P is not complete for t > 0, since (V, ω) has finite volume.) Consider the Stein manifold (V = M \ Σ, J, ϕ P ). We denote by ∆ P ⊂ V its skeleton (see 2.2 above). Note that ∆ P ⊂ M \ Σ is compact since the flow X t P is complete at −∞ and Crit(ϕ P ) is a compact subset of M \ Σ. We remark that ∆ P is completely determined by the polarization P without any further choices since the function ϕ P is determined (up to an additive constant) by P. We shall therefore call ∆ P the skeleton associated with the polarization P.

3.3.
The decomposition associated to a polarization. In this section we explain how to decompose a Kähler manifold into two basic building blocks. The first piece is a standard symplectic disc bundle over a complex hypersurface. The second piece is the isotropic skeleton of the Stein manifold which is the complement of this hypersurface.
3.3.1. Standard symplectic disc bundles. Let P = (M, ω, J; Σ) be a polarization of degree k P of a Kähler manifold.
Put ω Σ = ω| T (Σ) and let π : N Σ → Σ be the (complex) normal line bundle of Σ in M with first Chern class c N Σ 1 = k P [ω Σ ] ∈ H 2 (Σ). Let · be any hermitian metric on N Σ and denote by E Σ = {v ∈ N Σ v < 1} the open unit disc bundle of N Σ . Choose a connection ∇ on N Σ with curvature R ∇ = 2πik P ω Σ and denote by α ∇ the associated transgression 1-form on N Σ \ 0 defined by: With this normalization of α ∇ we have dα ∇ = −π * (k P ω Σ ). Define now the following symplectic form ω can on E Σ : where r is the radial coordinate along the fibres induced by · . It is easy to check that ω can is well defined, that it is symplectic, and has the following three properties: (1) All fibres of π : E Σ → Σ are symplectic with respect to ω can and have area 1.
(2) The restriction of ω can to the zero section Σ ⊂ E Σ equals k P ω Σ .
(3) ω can is S 1 -invariant with respect to the obvious circle action on E Σ .
Although ω can a priori depends on · and ∇, different choices of these structures in fact lead to symplectically equivalent results (see [27], see also [5,9]). We shall henceforth call (E Σ , ω can ) the standard symplectic disc bundle over (Σ, ω Σ ) modeled on N Σ . Often we shall multiply ω can by a positive number c > 0 (usually by c = 1 k P ) and refer to (E Σ , cω can ) as the standard symplectic disc bundle with fibres of area c. (Note that now the restriction of this symplectic form to Σ ⊂ E Σ equals ck P ω Σ , not k P ω Σ .) 3.3.2. The decomposition. Let P = (M, ω, J; Σ) be a polarized Kähler manifold. Denote by ρ P : M → R the function ρ P (x) = s(x) 2 , and let Z P be the gradient vector field of −ρ P . Note that since ρ P = e −4πk P ϕ P the vector fields Z P and X P are positively proportional on M \ Σ. Theorem 3.3.A (See [5]). Let P = (M, ω, J; Σ) be a polarized Kähler manifold. Then, the complement of the skeleton (M \ ∆ P , ω) is symplectomorphic to the following standard symplectic disc bundle over Σ which is modeled on the normal bundle N Σ , and with fibres of area 1 k P . In fact, there exists a canonical symplectomorphism F P , which depends only on P, such that the following diagram commutes: Moreover, F P sends the flow lines of Z P to the lines of the negative radial flow on E Σ , namely DF P (Z P ) is negatively proportional to the radial vector field r ∂ ∂r on E Σ . The proof of this theorem appears in [5]. The "Moreover" statement, is not stated explicitly in [5] as a theorem but is proved there (see proof of Proposition 7.B in [5]).

Subcritical polarizations.
A polarization P = (M, ω, J; Σ) is called subcritical if there exists a plurisubharmonic function ϕ : (V = M \ Σ, J) → R such that (V, J, ϕ) is a subcritical Stein manifold (namely ϕ is Morse and for every p ∈ Crit(ϕ), ind p (ϕ) < 1 2 dim R V ). Note that we do not assume ϕ to be the canonical function ϕ P . We refer the reader to [9] for more information on subcritical polarizations, examples and criteria for identifying them.
3.5. The Lagrangian trace. Let P = (M, ω, J; Σ) be a polarized Kähler manifold. We say that the polarization P has property (S) if ϕ P : M \ Σ → R has property (S) of Section 2.4.1. We denote by ∇ crit P ⊂ M \ Σ the critical coskeleton of ϕ P . Let ρ P : M → R be the function defined by ρ P (x) = s(x) 2 . Note that the critical points of ρ P consist of those of ϕ P and Σ which is a non-degenerate critical submanifold (of index 0). On M \ Σ the gradients of ϕ P and of −ρ P have the same (oriented) flow lines. Denote by Z t P : M → M the gradient flow of −ρ P . Assume now that P has property (S) and let ∇ crit P ⊂ M \ Σ be its critical coskeleton. Denote by Λ P ⊂ Σ the subset obtained from ∇ crit P by "projecting" it using lim t→∞ Z t P to Σ, namely In case ∇ crit P = ∅ (e.g. if ϕ P is subcritical) we put Λ P = ∅. We call Λ P ⊂ Σ the Lagrangian trace of the polarization P. This term is justified by the following proposition. Proof. Denote by G : M \ ∆ P → Σ the end point map of the flow Z t P , namely G(x) = lim t→∞ Z t P (x). Clearly We shall prove that for every i, G(W ′ i ) is an immersed Lagrangian sphere in Σ. Denote by π : E Σ → Σ the standard symplectic disc bundle, endowed with the symplectic structure 1 k P ω can . Using Theorem 3.3.A we may identify (E Σ , 1 k P ω can ) with (M \∆ P , ω). We shall view from now on W ′ i as a submanifold of E Σ and G as a map G : E Σ → Σ. It follows from Theorem 3.3.A that the map G coincides with projection map π : E Σ → Σ.
Denote P ǫ = {v ∈ E Σ | v = ǫ}. Note that W ′ i intersects P ǫ transversely because the vector field ∂ ∂r is tangent to W ′ i . Put L i = W ′ i ∩ P ǫ . We claim that L i is diffeomorphic to a sphere. Indeed, pick a small ball B i ⊂ W u p ′ i centered around p ′ i whose boundary ∂B i is transverse to Z P . Taking B i to be small enough we may assume that every flow line of Z P intersects ∂B i exactly once. Denote by ν : E Σ \ Σ → P ǫ the map ν(v) = ǫ v v . By our choice of B i we have that ν sends the sphere ∂B i diffeomorphically onto L i .
Next, we claim that π| L i : L i → Σ is an immersion. To prove this, note that for i is Lagrangian (it is coisotropic and has half the dimension of M). As ∂ ∂r is tangent to Thus v = 0. This proves that π| L i is an immersion.
It remains to prove that π(L i ) is Lagrangian. Let ξ = ker(α ∇ | T (Pǫ) ) be the contact distribution on P ǫ . Then T (P ǫ ) = ξ ⊕ iR ∂ ∂r . We first claim that T In this case ∇ crit ϕ = ∅, hence Λ P = ∅. The simplest example of a subcritical polarization consists of M = CP n endowed with its standard symplectic Kähler form σ and Σ ≈ CP n−1 being a linear hyperplane. The skeleton in this case is a point ∆ P = pt (see [5] for more details).
Another example of a subcritical polarization is M = CP n × CP n+r , r ≥ 1, endowed with the split symplectic structure σ ⊕ σ, and In this case the skeleton turns out to be an isotropic copy of CP n (see [5] for more details).
We refer the reader to [9] for more details and examples of subcritical polarizations.
Note also that ϕ P has property (S) (see 2. 3.6.4. Lagrangian trace of hypersurfaces in CP n+1 . Generalizing Example 3.6.2 above, consider P = (M, ω, J; Σ) with M = CP n+1 and Σ the degree d > 2 hypersurface: A simple computation shows that the vector field X P is invariant under the action of each of the maps R i . Therefore the unstable submanifold W u [1:ξ i 1 :...:ξ i n+1 ] coincides with R i (W u [1:... :1] ). We conclude that

Lagrangian submanifolds and polarizations
In this section we consider Lagrangian submanifolds of manifolds Σ which appear as hyperplane sections in some polarization P = (M, ω, J; Σ). Given a Lagrangian L ⊂ Σ our strategy will be to go one dimension up and construct a new Lagrangian Γ L ⊂ M \ Σ. The advantage is that, sometimes, due to the ambient geometry of M \ Σ it is easier to compute symplectic invariants of Γ L than those of L. The basic construction is presented in detail in subsection 4.1 below. Before we continue we remark again that all Lagrangian submanifolds are assumed to be compact and without boundary, unless explicitly otherwise stated.
Let us briefly recall now the notions of monotone symplectic manifold and monotone Lagrangian. Given a symplectic manifold (X, ω) we denote by c X 1 ∈ H 2 (X) the first Chern class of its tangent bundle (viewed as a complex vector bundle). A symplectic manifold (X, ω) is called spherically monotone if the following two conditions are satisfied: • c X 1 does not vanish on π 2 (X). • There exists λ > 0 such that for every A ∈ π 2 (X), ω(A) = λc X 1 (A). Denote by N X ∈ Z + the positive generator of the subgroup c X 1 (π 2 (X)) ⊂ Z. We call N X the minimal Chern number of (X, ω).
A Lagrangian submanifold K ⊂ (X, ω) is called monotone if there exists η > 0 such that the following two conditions are satisfied: • The Maslov class of K, µ K : π 2 (X, K) → Z is not zero.
Denote by N K ∈ Z + the positive generator of the subgroup µ K (π 2 (X, K)) ⊂ Z. We call N K the minimal Maslov number of K.

4.1.
The Lagrangian circle bundle construction. Let P = (M, ω, J; Σ) be a polarized Kähler manifold. Put ω Σ = ω| T (Σ) . Let L ⊂ (Σ, ω Σ ) be a Lagrangian submanifold. Consider the standard symplectic disc bundle E Σ → Σ endowed with the symplectic structure 1 k P ω can (see 3.3.1). By Theorem 3.3.A we can identify (E Σ , 1 k P ω can ) with (M \ ∆ P , ω). Pick 0 < ǫ < 1, and consider the circle bundle P = {v ∈ E Σ | v = ǫ} over Σ. Denote by π P : P → Σ the projection. Finally define Γ L = π −1 P (L) ⊂ P to be the total space of the restriction of P to L. As we shall see in a moment is a Lagrangian submanifold. We call Γ L the Lagrangian circle bundle over L. More generally, let P = (M, ω, J; Σ) be a polarized Kähler manifold and (X, ω X ) be another symplectic manifold. Let L ⊂ (Σ × X, ω Σ ⊕ ω X ) be a Lagrangian submanifold. Pick 0 < ǫ < 1 and let P = {v ∈ E Σ | v = ǫ} be as before. Consider now the circle bundle π P ×X : P × X → Σ × X. Define is a Lagrangian submanifold. This follows immediately from the definition of ω can (see 3.3.1) Note the (1 − ǫ 2 ) rescaling of the symplectic structure along the X factor. Note also that this construction coincides with the previous one when X is a point. We shall call this Γ L too the Lagrangian circle bundle over L. Note that Γ L depends on the parameter ǫ (since P does). In fact Γ L 's corresponding to different choices of ǫ are not Hamiltonianly isotopic (even when X = pt), however they are conformally symplectic in (M \ Σ) × X. This dependence on ǫ will not trouble us in the sequel as Γ L will be used only as an auxiliary object for studying the topology of L.

Remarks.
(1) Note that although Γ L ⊂ (M \ Σ) × X is monotone (when L is), Γ L is usually not monotone in M × X.
(2) It can be easily seen from the proof below that statement on monotonicity in Proposition 4.1.A remains true when dim C Σ = 1 and P is a subcritical polarization. Note however that the only subcritical polarization P = (M, ω, J; Σ) with dim C Σ = 1 is M = CP 2 , Σ = CP 1 where Σ is embedded as a projective line in CP 2 (see [9]).
Put V = M \ Σ and let ξ = ker(α ∇ | T (Pǫ) ) be the contact distribution on P . Denote by N Σ → Σ the normal bundle of Σ in M (viewed as a complex line bundle). Throughout the proof we shall also use the following notation: given a symplectic vector bundle (W, Ω) → X over a manifold X and a map v : Y → X we shall write (v * W, v * Ω) → Y for the pulled back symplectic vector bundle, namely for every y ∈ Y , (v * W y , v * Ω y ) = (W v(y) , Ω v(y) ).

With these notation the symplectic vector bundle (T (V )|
where the last isomorphism follows from the fact that ξ, ω| ξ ∼ = π * P T (Σ), (1 − ǫ 2 )π * P ω Σ and u * π * P = u * . Consider now the loop of Lagrangian subspaces In a compatible way to the symplectic isomorphism (1) we have the following isomorphism of Lagrangian subbundles over ∂D: where τ ⊂ T (P ) is the subbundle τ = iR ∂ ∂r . The Maslov index of the loop of Lagrangian subspaces {τ u(t) } t∈∂D ⊂ u * π * P N Σ , u * π * P σ is 0 because the bundle τ is globally defined, hence this loop extends to the disc D. Thus by (2) ). This proves the equality of the Maslov indices. Now suppose that dim C Σ ≥ 2. Put n = dim C M. As ∆ P ⊂ M has dimension at most n and n > 2 we have by a general position argument that But π P * : π 2 (P, Γ L ) → π 2 (Σ, L) is an isomorphism. By the equality of Maslov indices just proved it follows that Γ L is monotone if and only if L is monotone and moreover that N Γ L = N L .
Recall that given a polarization P = (M, ω, J; Σ), (M \ Σ, J, ϕ P ) is a Stein manifold. Denote by ω the symplectic structure associated to the completion of this Stein manifold (see Section 2.1 and Remark 2.1.C). Given a Lagrangian submanifold K we denote by HF (K, K) the Floer homology of K with itself (see Section 5). Remark. The Floer homology HF (Γ L , Γ L ) when computed in (M \ Σ) × X with either of the symplectic structures ω ⊕ (1 − ǫ 2 )ω X or ω ⊕ (1 − ǫ 2 )ω X is the same. This follows from convexity at infinity of M \ Σ. See [8] for more details on this type of argument.
If necessary one can Hamiltonianly cut off 1l X , in case X is not compact, in order to obtain a compactly supported h.
The statement on Floer homology follows immediately because by Proposition 4.

Computations in Floer homology
In this section we summarize necessary facts from Floer theory for monotone Lagrangian submanifolds. This extension of Floer's work was developed by Oh [31]. In section 5.2 we describe a spectral sequence which converges to the Floer homology of a Lagrangian. This spectral sequence is based on the theory developed by Oh [33,32], however our construction is somewhat different. We refer the reader to [19,18,31,33,32,37] for more details on Floer theory.
Let (M, ω) be a tame symplectic manifold (see [2], chapter 10), and let L ⊂ (M, ω) be a monotone Lagrangian submanifold with N L ≥ 2. In this situation one can define the Floer homology of L which is an invariant of the Hamiltonian isotopy class of L. Let L ′ = φ(L), φ ∈ Ham(M, ω), be a Hamiltonianly isotopic copy of L, and assume that L ⋔ L ′ . Let  [33]) shows that the differential d J increases grading by 1, d J : CF * ( mod N L ) (L, L ′ ; x 0 ) → CF * +1( mod N L ) (L, L ′ ; x 0 ). Thus the Floer homology has a Z/N L grading. Again, this grading is relative as different choices of the base point x 0 result in a shifted Z/N L grading. Note that there exists a more sophisticated approach to grading, due to Seidel [37], which overcomes the relativity problem.
The main feature of the Floer homology is its invariance under the choice of L ′ (and of J), namely, for every L ′′ = ψ(L), ψ ∈ Ham(M, ω), intersecting L transversely and any generic almost complex structures J ′ , J ′′ there is an isomorphism HF (L, L ′ ; J ′ ) ∼ = HF (L, L ′′ ; J ′′ ). Moreover, this isomorphism preserves the Z/N L grading up to a shift, namely for given choices x ′ 0 ∈ L ∩ L ′ and x ′′ 0 ∈ L ∩ L ′′ there exists s such that Finally, in case L, L ′ do not intersect transversely we define HF (L, L ′ ) = HF (L, L ′ ǫ ), where L ′ ǫ is a small Hamiltonian perturbation of L ′ with L ′ ǫ ⋔ L.
5.1. The case of HF (L, L). Let L ǫ be a perturbation of L built in a Weinstein neighbourhood U of L using a C 2 -small Hamiltonian Morse function f : L → R. Assume also that f has exactly one relative minimum x 0 . Denote by C * f the Morse complex of f . We shall use x 0 as a base intersection point for the Floer complex. From now on we shall drop J and x 0 from the notation of the Floer complex and Floer homology and simply write CF = CF (L, L ǫ ), d = d J and HF * ( mod N L ) = HF * ( mod N L ) (L, L; J, x 0 ). It is shown in [33] that f . An index computation shows that ∂ j = 0 for every j < 0 and, due to dimension reasons, ∂ j = 0 also for every j > ν, where ν = [ dim L+1 N L ]. Thus d = ∂ 0 + · · · + ∂ ν .
Roughly speaking ∂ 0 counts the Floer trajectories that lie in the small neighbourhood U of L, while ∂ 1 , . . . , ∂ ν count the "fat" trajectories which go out of U. Oh proves in [33] that (for suitable choices of J and Riemannian metric on L) the operator ∂ 0 : C * f → C * +1 f can be identified with the Morse complex differential, hence H * (C f , ∂ 0 ) ∼ = H * (L; Z 2 ). (Note however that ∂ j , j ≥ 1, are not differentials in general, namely ∂ j • ∂ j may not be zero.)

5.2.
A spectral sequence. We shall now present a spectral sequence which enables to calculate the Floer homology HF (L, L) using the operators ∂ 1 , . . . , ∂ ν . We continue to use the shortened notation omitting J and x 0 in CF , d and HF .
Let A = Z 2 [T, T −1 ] be the algebra of Laurent polynomials over Z 2 in the variable T . We define the degree of T to be N L . Thus and let d : C * → C * +1 be d = ∂ 0 ⊗ 1 + ∂ 1 ⊗ τ + · · · + ∂ ν ⊗ τ ν , where τ i : A * → A * +iN L is multiplication by T i . A simple algebraic computation shows that: Next we define a decreasing filtration · · · ⊂ F p+1 C ⊂ F p C ⊂ F p−1 C ⊂ · · · on C. For every p ∈ Z let A p = k≥p A kN L be the space of Laurent polynomials of the form k≥p α k T k . Define Note that since C j f = 0 for every j > dim L and j < 0, the filtration F p C is bounded. Denote by {E p,q r , d r } the spectral sequence defined by this filtration.
Theorem 5.2.A. The spectral sequence {E p,q r , d r } has the following properties: is induced from ∂ 1 . .
(For r = 0, 1 we have V p,q r , d r } collapses at the ν + 1 step, namely d r = 0 for every r ≥ ν + 1 (and so E p,q r = E p,q ∞ for every r ≥ ν + 1). Moreover, the sequence converges to HF , i.e. p+q=l E p,q ∞ ∼ = HF l( mod N L ) for every l ∈ Z.

Remarks.
(1) Our filtration F p C is different than the filtration used by Oh [33,32]. The filtration used by Oh comes from the following filtration on the singular cohomology of L: F p H * (L; Z 2 ) = 0≤j≤n−p H j (L; Z 2 ). Thus the spectral sequence in [33,32] is different than ours.
(2) Semi-rigorous arguments suggest that the spectral sequence is multiplicative in the following sense. For every r ≥ 1 we have a product E p,q r ⊗ E p ′ ,q ′ r → E p+p ′ ,q+q ′ r and the differential d r satisfies Leibniz rule with respect to this product. Moreover the product on E 1 comes from the cup product on the cohomology H * (L; Z 2 ). Taking the product structure in consideration in combination with the other techniques of this paper should lead to new restrictions on the topology of Lagrangian submanifolds.
Proof of Theorem 5.2.A. Most of the proof is purely algebraic and follows from the basic construction of a spectral sequence associated to a filtered complex. For the readers who are not familiar with spectral sequences we shall now give a brief summary of this construction. More details can be found in any basic text on spectral sequences. Below we follow the conventions of [26].
Let ( C, d) be a complex with a decreasing filtration F p C, p ∈ Z. (We write the over C and d to be consistent with the notation of our theorem.) Assume further that the filtration is bounded, namely for every l ∈ Z, there exist s = s(l), t = t(l) such that F s C l = 0 and F t C l = C l .
Define for every p, q ∈ Z and r ≥ −1: . A simple computation shows that d maps Z p,q r into Z p+r,q−r+1 r and moreover it descends to a homomorphism d r : E p,q r → E p+r,q−r+1 r such that the following diagram commutes: Here the vertical arrows are the canonical projections. As d • d = 0 we have d r • d r = 0. Denote by F p H( C, d) the induced filtration on the homology of ( C, d), namely Note that F p H( C, d) is also a bounded filtration. The fundamental features of the above construction are the following (see [26]): (1) The homology of (E * , * r , d r ) is isomorphic to E * , * r+1 , namely H(E * , * r , d r ) ∼ = E * , * r+1 .
(4) For every p, q there exists r 0 = r 0 (p, q) such that E p,q ∞ = E p,q r for every r ≥ r 0 .
If our complex C consists of vector spaces then summing up F p H l ( C, d)/F p+1 H l ( C, d) over all p's gives us an isomorphic copy of H l ( C, d), hence by (3), p+q=l E p,q ∞ ∼ = H l ( C, d).
We now turn to the proof of our theorem, applying the above construction to our complex. A simple computation shows that To see 2, write elements x ∈ F p C p+q as finite sums . A simple computation shows that: ). Moreover we have: where ∂ 0 (x p+q−pN L ) = 0 and x ′ ∈ F p+2 C p+q . Then This completes the proof of statement 2. Statement 3 follows immediately from statement 1 by induction on r. Indeed, note that A jN L is 1-dimensional for every j. Thus the homomorphism d r : V p,q r ⊗ A pN L → V p+r,q−r+1 r ⊗ A (p+r)N L must be of the form d r = δ r ⊗ τ r where δ r is a homomorphism V p,q r → V p+r,q−r+1 r . As d r • d r = 0 we also have δ r • δ r = 0. Moreover, the homologies of d r and of δ r are related by H(V p,q To prove statement 4, note that statement 1 implies that for every r, E p,q r = 0 only if 0 ≤ p + q − pN L ≤ dim L. Let r ≥ ν + 1 and 0 ≤ p + q − pN L ≤ dim L. As d r : E p,q r → E p+r,q−r+1 r it is enough to show that (p+r)+(q−r+1)−(p+r)N L < 0, i.e. that p + q − pN L + 1 − rN L < 0. But this is immediate because rN L ≥ (ν + 1)N L > dim L + 1. The second part of statement 4 follows from the general theory of spectral sequences outlined at the beginning of the proof and the fact that H l ( C, d) ∼ = HF l( mod N L ) .
Finally, statement 5 follows from the following symmetries of the spectral sequence which are easy to check: ∞ ) for every p, q.

Proof of the main results
Proof of Theorem A. Since H 1 (L; Z) is 2-torsion it is easy to see that L ⊂ CP n must be monotone with N L = k(n + 1) for some k ∈ N.
Consider the polarization P = (M = CP n+1 , σ, J; Σ = CP n ) where σ is the standard Kähler form on CP n+1 , J is the standard complex structure and Σ ⊂ CP n+1 is a linear hyperplane. By Example 3.6.1, P is a subcritical polarization. Put V = M \ Σ and consider the Lagrangian circle bundle Γ L ⊂ V as constructed in Section 4.1. By Proposition 4.1.A Γ L is monotone and N Γ L = N L = k(n + 1). By Theorem 4.2.A we have HF (Γ L , Γ L ) = 0.
In order to recover the cohomology of L itself we use the Gysin sequence of the circle bundle Γ L → L. Note that the second Stiefel-Whitney class of the vector bundle corresponding to Γ L → L is just α = a| L ∈ H 2 (L; Z 2 ), where a ∈ H 2 (CP n ; Z 2 ) is the generator.
Next note that H 1 (L; Z) cannot be 0 since if it were than N L = 2(n + 1). Thus H 1 (L; Z) is a non-trivial 2-torsion group hence H 1 (L; Z 2 ) = 0.
Proof of Theorem B. Consider CP n ×X ⊂ CP n+1 ×X, and let Γ L ⊂ (CP n+1 \CP n )× X be the Lagrangian circle bundle corresponding to L ⊂ CP n × X. By our assumptions on L and on X, L is monotone and N L = 2(n + 1). By Proposition 4.1.A, Γ L is monotone too and N Γ L = 2(n + 1). (See the second remark after Proposition 4.1.A for the case n = 1.) The rest of the proof is very similar to that of Theorem A. From Theorem 4.2.A we obtain HF (Γ L , Γ L ) = 0. Then a similar computation via the spectral sequence gives: H i (Γ L ; Z 2 ) = 0 for every 1 < i < 2n + 1.
The proof now continues in the same way, using the Gysin sequence, only that now H 1 (L; Z 2 ) = 0. This implies that H i (L; Z 2 ) = 0 for every 0 < i < 2n + 1, which is exactly the Z 2 -cohomology of S 2n+1 .
Finally, it is easy to see that α = a| L where a ∈ H 2 (CP n × CP n ; Z 2 ) is the generator of H 2 of any of the factors of CP n × CP n . (As L is Lagrangian, it does not matter which factor we take.) Before we go on to the proofs of the rest of the theorems we shall need the following proposition.
Proposition 6.A. Let (V 2k , ω) be a tame symplectic manifold of dimension k ≥ 2 and K k ⊂ (V 2k , ω) a monotone Lagrangian submanifold with N K ≥ 2. Suppose that HF (K, K) = 0. Then: Remark. The cohomological condition in statement 2 is satisfied whenever K is a circle bundle over a sphere of dimension ≥ 3.
Proof of Proposition 6.A. We start by proving the second statement. Consider the spectral sequence {E * , * r , d r } of Section 5.2. Note that since dim K = k, E −r,k+r−1 r = 0 for every r ≥ 1. Therefore at the r'th step the differential d r behave as follows: (1) Either k + 1 − r 0 N K = 1, hence N K | k; (2) Or k + 1 − r 0 N K = 0, hence N K | k + 1.
Since H 1 (K; Z 2 ) = 0, E 0,k−1 1 = H k−1 (K; Z 2 ) ⊗ A 0 is also not 0 by Poincaré duality. Applying the same arguments as above this time to 0 dr − → E 0,k−1 r dr − → E r,k−r r and denoting by r 1 the minimal r ≥ 1 for which d r : E 0,k−1 r → E r,k−r r is not 0 we obtain: Comparing cases (1),(2) with (1'),(2') above we conclude that the only possibility is The proof of statement 1 is similar (and actually simpler).
Proof of Theorem D. Put m = dim C X. We shall first assume that n + m ≥ 3, i.e. that dim L ≥ 3. Embed CP n × X ⊂ CP n+1 × X and let Γ L ⊂ (CP n+1 \ CP n ) × X be the Lagrangian circle bundle over L. As in the proof of Theorem B, L and Γ L are both monotone with N L = N Γ L = 2n + 2. (Note that L being a sphere of dimension n + m ≥ 3 is simply connected.) By Theorem 4.2.A, HF (Γ L , Γ L ) = 0.
Proof of Theorem E. Theorem E is a special case of Theorem 7.A which will be proved in Section 7 below.
Proof of Theorem F. The proof is very similar to that of Theorem D, only that now N L = 2 gcd(n + 1, N M ).
Proof of Theorem G. Consider the polarization P = (M = CP n+1 , σ, J; Σ = Q n ), where Σ = Q n is the quadric. By Example 3.6.2, the Lagrangian trace is exactly the Lagrangian sphere Λ Q ⊂ Σ.
Let L ⊂ Σ be a Lagrangian submanifold with H 1 (L; Z) either 0 or a non-trivial 2torsion group, and assume that L ∩ Λ Q = ∅. The minimal Chern number of Σ is n, hence N L = kn for some k ∈ N. Let Γ L ⊂ M \ Σ be the Lagrangian circle bundle over L. By Proposition 4.1.A, N Γ L = N L = kn. We first claim that k = 1. Indeed, by Theorem 4.2.B, HF (Γ L , Γ L ) = 0. Hence, if k ≥ 2 then since n ≥ 3 all the differentials d r : E * , * r → E * +r, * −r+1 r of the spectral sequence must vanish for every r ≥ 1 which is impossible since E * , * ∞ = 0 and E * , * 1 is not 0. This proves N Γ L = N L = n. Note that this implies that H 1 (L; Z) = 0, for otherwise N L would be 2n.
Consider now the Gysin sequence of the circle bundle Γ L → L. The first Chern class of the normal bundle N Σ/M is c is the positive generator. Therefore the second Stiefel-Whitney class of the vector bundle associated to Γ L → L is 0. From the Gysin sequence we get H i (L; Z 2 ) = 0 for every 2 < i < n − 2, as well as the following exact sequence: Since H 1 (L; Z) = 0 is 2-torsion we conclude that H 1 (L; Z 2 ) = 0. (Actually this follows also from the fact that N L = n and N Q = n.) It now easily follows that H 1 (L; Z 2 ) = Z 2 and that H 2 (L; Z 2 ) = 0. This proves that H * (L; Z 2 ) ∼ = A * Q . Assume now that n = 3. From (4) we obtain H 2 (Γ L ; Z 2 ) = Z 2 ⊕ Z 2 . Using the Gysin sequence and similar arguments to the preceding ones we obtain H i (L; Z 2 ) = Z 2 for every 0 ≤ i ≤ 3, hence H * (L; Z 2 ) ∼ = A * Q .

Remarks.
(1) Using the multiplicative structure of the spectral sequence (see the second remark after Theorem 5.2.A), it seems that the following should hold for n = 3: if a ∈ H 1 (L; Z 2 ) is the generator then a ∪ a = 0.
(2) The same proof as above with small changes actually shows that if L ⊂ Q n , n ≥ 3, is a monotone Lagrangian submanifold with N L = n then either H * (L; Proof of Theorem H. Throughout the proof we set Σ = Σ n d . Consider the polarization P = (M = CP n+1 , σ, J; Σ). By Example 3.6.4 the Lagrangian trace Λ P ⊂ Σ consists of at most d n+1 immersed Lagrangian spheres.
To prove statement 1 assume first that 2d ≤ n + 1 and let L ⊂ Σ be a Lagrangian submanifold with H 1 (L; Z) = 0. A simple computation shows that N L = 2(n + 2 − d) hence N L ≥ n + 3. As dim Γ L = n + 1 it follows that the Floer differential is d = ∂ 0 , hence HF (Γ L , Γ L ) ∼ = H * (Γ L ; Z 2 ) = 0 which is a contradiction. This proves that L ∩ Λ P = ∅. The second part of statement 1 (i.e. d ≥ 3 2 (n + 1)) will be treated towards the end of the proof.
To prove statements 2 and 3, assume that 2d ≤ n + 1 and that H 1 (L; Z) is 2-torsion. A simple computation shows that N L = k(n + 2 − d) for some k ∈ N. We first claim that k = 1. Indeed, if k ≥ 2 then N L ≥ n + 3 and we would obtain HF (Γ L , Γ L ) ∼ = H * (Γ L ; Z 2 ) contradicting the vanishing of Floer homology. Thus N L = n + 2 − d. As 2d ≤ n + 1 we have d = ∂ 0 + ∂ 1 and so E * , * 2 = E * , * ∞ = 0. Following the differentials of the spectral sequence we get: Consider now the Gysin sequence of Γ L → L. Note that the second Stiefel-Whitney class w of the vector bundle corresponding to Γ L → L is da| L ∈ H 2 (L; Z 2 ) where a ∈ H 2 (CP n+1 ; Z 2 ) is the generator. When d = even we have w = 0, hence the Gysin sequence gives the exact sequence: Combining with (5) we obtain H j (L; Z 2 ) = 0 for every d ≤ j ≤ n − d. Moreover, we have β i (L) + β i−1 (L) = β i (Γ L ). By (5) and Poincaré duality we have β d−i (Γ L ) = β i (Γ L ) for every 0 ≤ i ≤ d, hence The statement for d = odd follows at once from (5) and the fact that the Stiefel-Whitney class w = da| L equals a| L since d is odd. Statement 4 follows easily from statement 2 of Proposition 6.A applied to Γ L . To prove the second part of statement 1 and statement 5 we shall use an extension of Floer homology for so called strongly negative Lagrangian submanifolds, due to Lazzarini [25]. Let K ⊂ (V, ω) be a Lagrangian in a tame symplectic manifold. K is called strongly negative if there exists λ < 0 such that µ K = λ[ω] on π 2 (V, K) and in addition the following conditions are satisfied: Under these assumptions the Floer homology HF (K, K ′ ) is well defined for every Lagrangian K ′ which is Hamiltonianly isotopic to K and moreover The reason for this, roughly speaking, is that under the above negativity assumptions there are no J-holomorphic spheres or discs with boundary on K for generic almost complex structure J. Hence, the Floer differential for CF (K, K) is the Morse-homology differential d = ∂ 0 . The non-existence of spheres and discs is due to negative dimension of the moduli spaces of J-holomorphic spheres and discs. For the dimension formulae to hold one has to work with regular almost complex structures J. Regularity may be achieved by generic perturbations of J as long as the J-holomorphic discs/spheres in question are somewhere injective (see [28]). Thus an essential ingredient in applying the dimension argument is a procedure which enables to extract a somewhere injective disc from a given pseudo-holomorphic disc. Such procedures have been developed by Lazzarini [25] and by Kwon and Oh [23].
In a similar manner to the proofs of Theorems A and B we recover the cohomology of Γ L via the spectral sequence, obtaining: H i (Γ L ; Z 2 ) = 0 for every 1 < i < 2n − 1.
Next note that the restriction of the complex line bundle N Σ/CP n ×CP n to L is trivial, since its first Chern class is just [ω]| L and H 2 (L; Z) is torsion-free because H 1 (L; Z) = 0. Thus the circle bundle Γ L → L is trivial, hence H * (Γ L ) = H * (L) ⊗ H * (S 1 ), from which it follows that H * (L; Z 2 ) ∼ = H * (S 2n−2 ; Z 2 ).

Generalizations
Below we present miscellaneous generalizations of some of the theorems from Section 1. Since the proofs are rather analogous to those of Section 6 we shall only outline the proofs omitting repeated details.
The following theorem generalizes Theorem E.
Theorem 7.A. Let X be a symplectic manifold that has a covering which is symplectomorphic to a domain in a subcritical Stein manifold. Let M be a closed spherically monotone symplectic manifold. Assume dim M, dim X > 0. If M × X has a Lagrangian sphere then 2N M | dim C M + dim C X + 1.
Examples of subcritical Stein manifolds, other than C n , can be found in [5,9]. (Note however, that after completion all subcritical Stein manifolds are split [12].) Proof of Theorem 7.A. Let Y → X be a covering of X by a symplectic manifold Y which is (symplectomorphic to) a domain in a subcritical Stein manifold V . Let L ⊂ M × X be a Lagrangian sphere. By assumption dim L ≥ 2, hence L is simply connected. Consider the lift L ⊂ M × Y ⊂ M × V of L. Clearly L is also an embedded Lagrangian sphere, and N L = 2N M . As V is subcritical we have HF ( L, L) = 0. By Proposition 6.A, 2N M | dim C M + dim C X + 1.
Theorem 7.B. Let Σ be a closed monotone symplectic manifold (resp. spherically monotone) that participates in some polarization P (see Section 4.2). Denote n = dim C Σ.
We remark that a similar statement to 1(a) above was proved in [9] using a different approach.
Proof of Theorem 7.B. Note that statement (1) is a special case of statement (2), since for a subcritical polarization P we have Λ P = ∅. We therefore prove statement (2).
Let L ⊂ Σ be a Lagrangian submanifold with H 1 (L; Z) = 0 and L ∩ Λ P = ∅. Note that L is monotone and that N L ≥ 2N H Σ (here we continue to denote by N L the minimal Maslov number, i.e. the positive generator of the subgroup µ L (π 2 (Σ, L)) ⊂ Z.) Consider now the Lagrangian circle bundle Γ L → L in M \ Σ (where, P = (M, ω, J; Σ) is the polarization in which Σ participates). By Theorem 4.2.B, HF (Γ L , Γ L ) = 0.
As in the end of the proof of Theorem I, since H 1 (L; Z) = 0 and L ⊂ Σ is Lagrangian, the bundle Γ L → L must be trivial. Thus H 1 (Γ L ; Z 2 ) = Z 2 . In view of this, the only way the spectral sequence can converge to 0 is if N L ≤ n + 1. Thus if 2N H Σ > n + 1 we arrive at a contradiction. This proves 2(a).
We omit the proof of the statements assuming spherical monotonicity since it is completely analogous to the proof above.
The following theorem generalizes statement 4 of Theorem H.
Theorem 7.C. Let Σ be a closed spherically monotone symplectic manifold that participates in a polarization P which has property (S) (see Sections 4.2 and 3.5). Assume that dim C Σ ≥ 3 and N Σ ≥ 2. If 2N Σ ∤ dim C Σ + 1 then every Lagrangian sphere L ⊂ Σ must satisfy L ∩ Λ P = ∅.
We omit the proof as it is rather similar to that of Theorem H. The next theorem provides "Euler characteristic" type restrictions on monotone Lagrangians. Let (V, ω) be a symplectic manifold and L ⊂ (V, ω) a monotone Lagrangian submanifold with minimal Maslov number N L ≥ 2. For every j ∈ Z, denote by γ j the sum of all Z 2 -Betti numbers of indices that are congruent to j modulo N L , namely γ j = k∈Z dim Z 2 H kN L +j (L; Z 2 ).