GRADED LAGRANGIAN SUBMANIFOLDS

PAUL SEIDEL

1. Introduction

Floer theory assigns, in favourable circumstances, an abelian group HF (L_{0}, L_{1}) to a pair (L_{0}, L_{1}) of Lagrangian submanifolds of a symplectic manifold (M, ω). This group is a qualitative invariant, which remains unchanged under suitable deformations of L_{0} or L_{1}. Following Floer [7] one can equip HF (L_{0}, L_{1}) with a canonical relative ℤ/N-grading, where 1 ≤ N ≤ ∞ is a number which depends on (M, ω), L_{0} and L_{1} (for N = ∞ we set ℤ/N = ℤ). Relative mostly means that the grading is unique up to an overall shift, although there are also cases with more complicated behaviour. In this paper we take a different approach to the grading: we consider Lagrangian submanifolds equipped with certain extra structure (these are what we call graded Lagrangian submanifolds). This extra structure removes the ambiguity and defines an absolute ℤ/N-grading on Floer cohomology. There is also a parallel notion of graded symplectic automorphism, which bears the same relation to the corresponding version of Floer theory. Both concepts were first discovered by Kontsevich (at least for N = ∞). Somewhat later, the present author came upon them independently.

One way to approach the definition of graded Lagrangian submanifold is to start with the case N = 2. It is well-known that orientations of L_{0} and L_{1} determine an absolute ℤ/2-grading HF (L_{0}, L_{1}) = HF^{0}(L_{0}, L_{1}) ⊕ HF^{1}(L_{0}, L_{1}). One can reformulate this as follows: consider the natural fibre bundles L, L^{or} ⟶ M whose fibres are the unoriented resp. oriented Lagrangian Grassmannians of the tangent spaces T M_{x}. Any Lagrangian submanifold L comes with a canonical section s_{L} : L ⟶ L|L, and an orientation of L is the same as a lift of this section to L^{or}. Hence the right objects for a Floer theory with an absolute ℤ/2-grading are pairs (L, L̃) consisting of a Lagrangian submanifold and a lift L̃ : L ⟶ L^{or} of s_{L}. In order to define the absolute ℤ/N-grading one proceeds in the same way, only that L^{or} must be replaced by a ℤ/N-covering of L of a certain kind. Such coverings, which we call Maslov coverings, need not exist in general, and they are also not unique. In fact, choosing an N-fold Maslov covering is equivalent to lifting the structure group of T M from Sp(2n) to a certain finite extension Sp^{N}(2n); and the particularly simple situation for N = 2 is due to the fact that Sp^{2}(2n) ≅ Sp(2n) × ℤ/2.

In itself this ‘graded symplectic geometry’ is not particularly deep, but it does make Floer cohomology into a more powerful invariant. To put it bluntly, the advantage of the new framework is this: in passing to graded Lagrangian submanifolds there is a choice of ℤ/N for any Lagrangian submanifold L (the choice of the lift of s_{L}). In comparison, if one uses only the relative grading, there is a ℤ/N-ambiguity for any pair of Lagrangian submanifolds, and this greater amount of choice entails a loss of information. We illustrate this through three applications, which form the main part of this paper.

Date: February 25, 1999.

(a) Lagrangian submanifolds of ℂP^{n}. We prove that any Lagrangian submanifold L ⊂ ℂP^{n} must satisfy H^{1}(L; ℤ/(2n + 2)) ≠ 0 (the actual result is slightly sharper, see Theorem 3.1).

(b) Symplectically knotted Lagrangian spheres. The paper [28] provides examples of compact symplectic four-manifolds (with boundary) M with the following property: there is a family of embedded Lagrangian two-spheres L^{(k)} ⊂ M, k ∈ ℤ, such that any two of them are isotopic as smooth submanifolds, but no two are isotopic as Lagrangian submanifolds. In such a situation we say that M contains infinitely many symplectically knotted Lagrangian two-spheres. The examples in [28] were constructed using a special class of symplectic automorphisms, called generalized Dehn twists, and the main step in the proof was a Floer cohomology computation using Pozniak’s [22] Morse-Bott type spectral sequence. Both the construction and the proof can be generalized to produce Lagrangian n-spheres with the same property for all even n.

Here, using the method of graded Lagrangian submanifolds, we will first reprove the result from [28] and its generalization in a considerably simpler way. Then, by a more complicated construction, we produce similar examples of Lagrangian n-spheres for all odd n ≥ 5. The reason why the remaining case n = 3 cannot be settled in the same way is topological, and seems to have nothing to do with Floer theory.

We can also improve on [28] in a different direction, by showing that suitable K3 and Enriques surfaces contain infinitely many symplectically knotted Lagrangian two-spheres. These are the first known examples of closed symplectic manifolds with this property. As a by-product one obtains that for these manifolds the map π_{0}(Aut(M, ω)) ⟶ π_{0}(Diff(M)) has infinite kernel, sharpening a result of [26]. Unfortunately, at the present state of development in Floer theory, it is impossible for technical reasons to carry out a similar argument in dimensions > 4.

(c) Weighted homogeneous singularities. Let p ∈ ℂ[x_{0}, …, x_{n}], n ≥ 1, be a weighted homogeneous polynomial with an isolated critical point at the origin. One can introduce the Milnor fibre of p, which is a compact symplectic manifold (M^{2n}, ω) with boundary, and the symplectic monodromy f ∈ Aut(M, ∂M, ω) of the Milnor fibration. This refines the usual notion of geometric monodromy by taking into account the symplectic geometry of the situation. We will show that [f] ∈ π_{0}(Aut(M, ∂M, ω)) has infinite order whenever the sum of the weights is ≠ 1. It is not known whether the condition on the weights is really necessary.

It should be mentioned (although this will not be used later on) that this application and the previous one are related. In fact, generalized Dehn twists are maps modelled on the monodromy of the quadratic singularity p(x) = x_{0}^{2} + ⋯ + x_{n}^{2}, and the construction of odd-dimensional knotted Lagrangian spheres is inspired by the monodromy of the singularity p(x) = x_{0}^{2} + ⋯ + x_{n−1}^{2} + x_{n}^{3} of type (A_{2}).

The importance of ‘graded symplectic geometry’ for these applications varies. For (a) and (c) its role is that of a convenient language. In fact one could replace it by monodromy considerations in the style of [27] without changing the essence of the argument. For (a) there is also a more algebraic argument, based on the fact that HF (L, L) is a module over the quantum cohomology QH^{∗}(ℂP^{n}). The situation in (b) is different, since the ‘graded’ framework allows us to state a basic geometric property of generalized Dehn twists (Lemma 5.7) which it seems hard to encode in any other way.

Notation and conventions. All manifolds are usually assumed to be connected. The automorphism group of a symplectic manifold (M, ω) will be denoted by Aut(M, ω). If M is compact, we equip this group with the C^{∞}-topology. If M is a symplectic manifold with nonempty boundary, Aut(M, ∂M, ω) ⊂ Aut(M, ω) is the subgroup of automorphisms ϕ which are equal to the identity on some neighbourhood (depending on ϕ) of the boundary. Lagrangian submanifolds are always assumed to be compact; if the symplectic manifold has a boundary, any Lagrangian submanifold is assumed to lie in the interior. Lag(M, ω) stands for the space of Lagrangian submanifolds of M, with the C^{∞}-topology. S^{1} will often be identified with ℝ/ℤ. (Co)homology groups have ℤ-coefficients unless otherwise stated.

Acknowledgements. This paper is an offshoot of my joint work with Mikhail Khovanov; several of the ideas presented here arose in conversations with him. I am indebted to Maxim Kontsevich for explaining his joint work with Fukaya to me. The idea of considering symplectic manifolds with a circle action on the boundary, which is the subject of section 4, was suggested to me by Leonid Polterovich and Yakov Eliashberg. This paper was written while I was staying at the Max Planck Institute (Bonn) and ETH Zürich, and I would like to thank both institutions for their hospitality.

2. Basic notions

2a. Linear algebra. By a ℤ/N-covering (1 ≤ N ≤ ∞) of a space X we mean a covering X^{N} with covering group ℤ/N. Such coverings are classified up to isomorphism by H^{1}(X; ℤ/N). For connected X, this correspondence associates to a homomorphism π : π_{1}(X) ⟶ ℤ/N the covering X^{N} = X̃ ×_{π} ℤ/N, where X̃ is the universal cover. If X is a connected Lie group, all ℤ/N-coverings of it (even the non-connected ones) have canonical Lie group structures.

Let (V^{2n}, β) be a symplectic vector space, Sp(V, β) the linear symplectic group, and L(V, β) the Lagrangian Grassmannian, which parametrizes linear Lagrangian subspaces of V. Both Sp(V, β) and L(V, β) are connected with infinite cyclic fundamental group. Moreover, there are preferred generators δ(V, β) ∈ H^{1}(Sp(V, β)) and C(V, β) ∈ H^{1}(L(V, β)) (the second one is called the Maslov class) so that one can canonically identify the fundamental groups with ℤ. Sp(V, β) acts transitively on L(V, β), and any orbit is a map Sp(V, β) ⟶ L(V, β) which takes C(V, β) to 2δ(V, β). For 1 ≤ N ≤ ∞, let L^{N}(V, β) be the ℤ/N-covering of L(V, β) which corresponds to the image of C(V, β) in H^{1}(L(V, β); ℤ/N). The ℤ/N-action on L^{N}(V, β) will be denoted by ρ. Define Sp^{N}(V, β) to be the group of pairs (Φ, Φ̃) consisting of Φ ∈ Sp(V, β) and a ℤ/N-equivariant diffeomorphism Φ̃ of L^{N}(V, β), which is a lift of the action of Φ on L(V, β). This is a Lie group and fits into an exact sequence

1 ⟶ ℤ/N ⟶ Sp^{N}(V, β) ⟶ Sp(V, β) ⟶ 1,

where ℤ/N is the central subgroup of pairs (Φ, Φ̃) = (id, ρ(k)). It follows that Sp^{N}(V, β) must be isomorphic to some ℤ/N-covering of Sp(V, β). The next Lemma identifies that covering.

Lemma 2.1. Sp^{N}(V, β) is isomorphic (as a Lie group) to the ℤ/N-covering of Sp(V, β) associated to the image of 2δ(V, β) in H^{1}(Sp(V, β); ℤ/N).

Proof. Let S̃p(V, β) be the universal cover of Sp(V, β). Take a loop ϕ : [0; 1] ⟶ Sp(V, β) with ϕ(0) = ϕ(1) = id and ⟨δ(V, β), [ϕ]⟩ = 1, and let ϕ̃(1) ∈ S̃p(V, β) the endpoint of the lift ϕ̃ of ϕ with ϕ̃(0) = id. The action of Sp(V, β) on L(V, β) can be lifted uniquely to an action of S̃p(V, β) on L^{N}(V, β). This action commutes with the ℤ/N-action ρ, and ϕ̃(1) acts in the same way as ρ(2). Therefore one obtains a homomorphism

S̃p(V, β) ×_{π} ℤ/N ⟶ Sp^{N}(V, β),

where π : π_{1}(Sp(V, β)) = ℤ ⟶ ℤ/N is multiplication by two. It is not difficult to see that this is an isomorphism, which proves the desired result.

As an example consider the case N = 2. One can identify L^{2}(V, β) with the oriented Lagrangian Grassmannian L^{or}(V, β). Since Sp(V, β) acts naturally on L^{or}(V, β) one has Sp^{2}(V, β) ≅ ℤ/2 × Sp(V, β). More generally, one can try to compare Sp^{N}(V, β) with the more obvious covering Sp^{N}(V, β)′ of Sp(V, β) obtained from the mod N reduction of δ(V, β). One finds that Sp^{N}(V, β)′ ≅ Sp^{N}(V, β) if N is finite and odd, and that

Sp^{2N}(V, β) ≅ Sp^{N}(V, β)′ ×_{ℤ/N} ℤ/2N. (2.1)

In particular, Sp^{N}(V, β) is connected iff N is finite and odd, and has two connected components otherwise.

Let J be a β-compatible complex structure on V, and g the corresponding inner product. Recall that the unitary group U(V, J, g) ⊂ Sp(V, β) is a deformation retract, and that δ(V, β) is represented by the determinant U(V, J, g) ⟶ S^{1}. Let U^{N}(V, J, g) be the ℤ/N-covering of U(V, J, g) determined by the mod N reduction of 2δ(V, β). These coverings are clearly deformation retracts of Sp^{N}(V, β), and they are explicitly given by

(2.2) U^{N}(V, J, g) = \{ \begin{matrix}\{(Φ, q) ∈ U(V, J, g) × S^{1} | det(Φ)^{2} = q^{N}\} & for N < ∞, \\ \{(Φ, t) ∈ U(V, J, g) × ℝ | det(Φ)^{2} = e^{2πit}\} & for N = ∞. \end{matrix}

In future we will abbreviate L(ℝ^{2n}, ω_{ℝ^{2n}}) by L(2n). Similarly we will write C(2n), L^{N}(2n), Sp(2n), Sp^{N}(2n), and U^{N}(n).

2b. Graded symplectic geometry. Let (M^{2n}, ω) be a symplectic manifold, possibly with boundary. Let P ⟶ M be the principal Sp(2n)-bundle associated to the symplectic vector bundle (T M, ω), and L ⟶ M the natural fibre bundle whose fibres are the Lagrangian Grassmannians L_{x} = L(T M_{x}, ω_{x}). One can identify L = P ×_{Sp(2n)} L(2n).

(a) An Sp^{N}(2n)-structure (1 ≤ N ≤ ∞) on M is a principal Sp^{N}(2n)-bundle P^{N} ⟶ M together with an isomorphism P^{N} ×_{Sp^{N} (2n)} Sp(2n) ≅ P.

(b) An N-fold Maslov covering is a ℤ/N-covering L^{N} ⟶ L whose restriction to L_{x} = L(T M_{x}, ω_{x}), for any x ∈ M, is isomorphic to L^{N}(T M_{x}, ω_{x}). The ℤ/N-action on L^{N} will always be denoted by ρ.

(c) A global Maslov class mod N is a class C^{N} ∈ H^{1}(L; ℤ/N) whose restriction to any fibre is the mod N reduction of the ordinary Maslov class.

There are canonical bijections between (isomorphism classes of) these three kinds of objects. If P^{N} is an Sp^{N}(2n)-structure then the associated fibre bundle with fibre L^{N}(2n) is an N-fold Maslov covering. Conversely, in the presence of a Maslov covering L^{N}, the transition maps of any system of local trivializations of (T M, ω) have canonical lifts from Sp(2n) to Sp^{N}(2n) which satisfy the cocycle condition, hence define an Sp^{N}(2n)-structure. The connection between Maslov coverings and global Maslov classes is obvious. Now assume that we have chosen an ω-compatible almost complex structure J on M, and consider the line bundle Δ = Λ^{n}(T M, J)^{⊗2}.

(d) An N-th root (1 ≤ N < ∞) of Δ is a complex line bundle Z together with an isomorphism r : Z^{⊗N} ⟶ Δ. Two pairs (Z, r), (Z′, r′) are called equivalent if there is an isomorphism j : Z ⟶ Z′ such that r′j = r. In addition, we define an ∞-th root to be a trivialization of Δ, and two of them are called equivalent iff they are homotopic.

There is a canonical bijection between Sp^{N}(2n)-structures and equivalence classes of N-th roots of Δ; it is defined as follows. Let P_{U} be the principal U(n)-bundle associated to (T M, ω, J). A U^{N}(n)-structure on M is a principal U^{N}(n)-bundle together with an isomorphism of the associated U(n)-bundle with P_{U}. Because U^{N}(n) ⊂ Sp^{N}(2n) is a deformation retract, there is a canonical bijection between U^{N}(n)-structures and Sp^{N}(2n)-structures. On the other hand, by looking at (2.2) one sees that a U^{N}(n)-structure is just a choice of N-th root of Δ. Among the equivalent notions (a)–(d) we will most frequently work with Maslov coverings, since that is convenient for dealing with Lagrangian submanifolds.

Lemma 2.2. (M^{2n}, ω) admits an N-fold Maslov covering iff 2c_{1}(M, ω) goes to zero in H^{2}(M; ℤ/N). The isomorphism classes of such coverings (provided that any exist) form an affine space over H^{1}(M; ℤ/N).

Proof. This is immediate if one uses an almost complex structure and the description (d). Alternatively one can use (a) and an argument based on the exact sequence

0 → H^{1}(M; ℤ/N) → H^{1}(M; Sp^{N}(2n)) → H^{1}(M; Sp(2n)) → H^{2}(M; ℤ/N)

of non-abelian cohomology groups, just as in the classification of spin structures in [14, Appendix A].

Let L^{N} be an N-fold Maslov covering on (M, ω). For every Lagrangian submanifold L ⊂ M there is a natural section s_{L} : L ⟶ L|L, s_{L}(x) = T L_{x} ∈ L(T M_{x}, ω_{x}). An L^{N}-grading of L is a lift L̃ : L ⟶ L^{N} of s_{L}. The pair (L, L̃) is called an L^{N}-graded Lagrangian submanifold. We write Lag^{gr}(M, ω; L^{N}) for the set of such pairs, and equip it with the topology which comes from the space of compact submanifolds of L^{N} (by considering the image of L̃). This topology defines the notion of an isotopy of graded Lagrangian submanifolds. Clearly, if L̃ is an L^{N}-grading of L then so is ρ(k) ∘ L̃ for any k ∈ ℤ/N. This defines a free ℤ/N-action on Lag^{gr}(M, ω; L^{N}).

Lemma 2.3. The forgetful map Lag^{gr}(M, ω; L^{N}) ⟶ Lag(M, ω) is a ℤ/N-covering of its image. The image itself consists of those L such that s_{L}^{∗}(C^{N}) ∈ H^{1}(L; ℤ/N) is zero, where C^{N} is the global Maslov class corresponding to L^{N}. In particular, a Lagrangian submanifold with H^{1}(L; ℤ/N) = 0 always admits an L^{N}-grading.

Proof. By definition, L admits an L^{N}-grading iff s_{L}^{∗}(L^{N}) ⟶ L is a trivial covering, which is equivalent to s_{L}^{∗}(C^{N}) = 0. The rest is obvious.

Let L^{N} be an N-fold Maslov covering on (M, ω) and ϕ a symplectic automorphism. There is a natural map ϕ^{L} : L ⟶ L, ϕ^{L}(Λ) = Dϕ(Λ), which covers ϕ. An L^{N}- grading of ϕ is a ℤ/N-equivariant diffeomorphism ϕ̃ of L^{N} which is a lift of ϕ^{L}. The pair (ϕ, ϕ̃) is called an L^{N}-graded symplectic automorphism. Such pairs form a group which we denote by Aut^{gr}(M, ω; L^{N}). If M is compact, we equip this group with the topology induced from embedding it into Diff(L^{N})^{ℤ/N}. The pairs (ϕ, ϕ̃) = (id, ρ(k)) form a central subgroup ℤ/N ⊂ Aut^{gr}(M, ω; L^{N}). L^{N}-graded symplectic automorphisms act naturally on L^{N}-graded Lagrangian submanifolds by (ϕ, ϕ̃)(L, L̃) = (ϕ(L), ϕ̃ ∘ L̃ ∘ ϕ^{−1}).

Lemma 2.4. Let L^{N} be an N-fold Maslov covering. The forgetful homomorphism Aut^{gr}(M, ω; L^{N}) ⟶ Aut(M, ω) fits into an exact sequence

1 ⟶ ℤ/N ⟶ Aut^{gr}(M, ω; L^{N}) ⟶ Aut(M, ω) ⟶^{∂} H^{1}(M; ℤ/N).

Here ∂ is not a group homomorphism, but it satisfies ∂(ϕψ) = ψ^{∗}∂(ϕ) + ∂(ψ), so that its kernel is a subgroup of Aut(M, ω). If M is compact then this is a sequence of topological groups, with ℤ/N and H^{1}(M; ℤ/N) discrete.

Proof. By definition, a symplectic automorphism ϕ admits an L^{N}-grading iff the two Maslov coverings (ϕ^{L})^{∗}(L^{N}) and L^{N} are isomorphic. By Lemma 2.2 the difference between these two coverings can be measured by a class in H^{1}(M; ℤ/N). We define ∂(ϕ) to be this class. The rest is easy.

Remark 2.5. Assume that M has nonempty boundary. Then one can consider the subgroup Aut^{gr}(M, ∂M, ω; L^{N}) ⊂ Aut^{gr}(M, ω; L^{N}) consisting of pairs with ϕ ∈ Aut(M, ∂M, ω) and where ϕ̃|L_{x}^{N} = id for any x ∈ ∂M (here L_{x}^{N} is the part of L^{N} which lies over L_{x}). The central elements (id, ρ(k)), k ≠ 0, do not lie in Aut^{gr}(M, ∂M, ω; L^{N}). In fact one has an exact sequence (with ∂ defined in a similar way as before)

1 ⟶ Aut^{gr}(M, ∂M, ω; L^{N}) ⟶ Aut(M, ∂M, ω) ⟶^{∂} H^{1}(M, ∂M; ℤ/N).

The minimal Chern number N_{M} of (M, ω) is defined to be the positive generator of the group ⟨c_{1}(M), H_{2}(M)⟩ ⊂ ℤ. Similarly, the relative minimal Chern number N_{L} of a Lagrangian submanifold L ⊂ M is the positive generator of the group ⟨2c_{1}(M, L), H_{2}(M, L)⟩, where now 2c_{1}(M, L) ∈ H^{2}(M, L) is the relative first Chern class. These numbers, or variants of them, are familiar from the definition of the relative grading on Floer cohomology. Their relationship to the concepts introduced here, at least in the case H_{1}(M) = 0, is as follows.

Lemma 2.6. A symplectic manifold (M, ω) with H_{1}(M) = 0 admits a Maslov covering L^{N} of order N iff N divides 2N_{M}. Moreover, this covering is unique up to isomorphism. A Lagrangian submanifold L ⊂ M admits an L^{N}-grading iff N divides N_{L}.

Proof. Because H_{1}(M) = 0, it follows from the universal coefficient sequence that 2c_{1}(M) goes to zero in H^{2}(M; ℤ/N) iff ⟨2c_{1}(M), x⟩ is a multiple of N for any x ∈ H_{2}(M). In view of Lemma 2.2, this proves the first part.

Take a compact oriented surface Σ and a map w : (Σ, ∂Σ) ⟶ (M, L). The number ⟨2c_{1}(M, L), [w]⟩ ∈ ℤ can be computed as follows: choose a trivialization of the pullback w^{∗}L, that is to say, a fibre bundle map

\xymatrix{L(2n) × Σ \ar[r]^{τ} \ar[d] & L \ar[d] \\ Σ \ar[r]^{w} & M.}

One has τ^{−1} ∘ s_{L} ∘ (w|∂Σ)(x) = (λ(x), x) for some map λ : ∂Σ ⟶ L(2n), and ⟨2c_{1}(M, L), [w]⟩ = ⟨C(2n), λ_{∗}[∂Σ]⟩.

Let L^{N} be the unique Maslov covering of some order N on (M, ω) and C^{N} its global Maslov class. The pullback τ^{∗}(C^{N}) ∈ H^{1}(L(2n) × Σ; ℤ/N) is of the form C(2n) + y for some y ∈ H^{1}(Σ; ℤ/N). Hence in ℤ/N one has

⟨s_{L}^{∗}(C^{N}), w_{∗}[∂Σ]⟩ =
= ⟨C^{N}, (s_{L} ∘ w|∂Σ)_{∗}[∂Σ]⟩ = ⟨τ^{∗}(C^{N}), (τ^{−1} ∘ s_{L} ∘ w|∂Σ)_{∗}[∂Σ]⟩ =
= ⟨C(2n), λ_{∗}[∂Σ]⟩ + ⟨y, [∂Σ]⟩ = ⟨2c_{1}(M, L), [w]⟩.

If N|N_{L} then ⟨2c_{1}(M, L), [w]⟩ is always a multiple of N. Since one can choose w in such a way that w_{∗}[∂Σ] is an arbitrary class of H_{1}(L), it follows that s_{L}^{∗}(C^{N}) = 0, which means that L admits an L^{N}-grading. The converse is equally simple.

In future we will use the following notation. Instead of (ϕ, ϕ̃) and (L, L̃) we will often write only ϕ̃ and L̃. The action of Aut^{gr}(M, ω; L^{N}) on Lag^{gr}(M, ω; L^{N}) will be written as ϕ̃(L̃). We will denote (id, ρ(−k)) ∈ Aut^{gr}(M, ω; L^{N}) by [k] and call it the k-fold shift operator. The graded Lagrangian submanifold ρ(−k) ∘ L̃, which is obtained from L̃ by the action of [k], will be denoted by L̃[k]. The similarity with homological algebra is intentional, and the sign in the definition of [k] has been introduced with that in mind.

2c. Examples. We will now complement the basic definitions by several examples and remarks, some of which will be used later on.

Example 2.7. Since Sp^{2}(2n) ≅ Sp(2n) × ℤ/2, an Sp^{2}(2n)-structure is just the choice of a real line bundle ξ on M. The corresponding two-fold Maslov covering, which we denote by L^{or,ξ}, is the space of pairs (Λ, o), where Λ ∈ L is a Lagrangian subspace of T M_{x} and o is an orientation of the vector space Λ ⊗_{ℝ} ξ_{x}. An L^{or,ξ}- grading of a Lagrangian submanifold L ⊂ M is the same as an orientation of T L ⊗ (ξ|L). An L^{or,ξ}-grading of a symplectic automorphism ϕ is the same as a bundle isomorphism ϕ^{∗}(ξ) ⟶ ξ. In particular, the trivial line bundle yields a two-fold Maslov covering L^{or} for which a grading of a Lagrangian submanifold is just an orientation, and such that Aut^{gr}(M, ω; L^{or}) ≅ Aut(M, ω) × ℤ/2.

Example 2.8. One can associate to any spin structure on M an Sp^{4}(2n)-structure. The reason, in the notation of (2.1), is that the restriction of the universal cover of GL^{+}(2n) to the subgroup Sp(2n) is again a nontrivial double cover, hence isomorphic to Sp^{2}(2n)′, and that Sp^{4}(2n) ≅ Sp^{2}(2n)′ ×_{ℤ/2} ℤ/4. Note that not all Sp^{4}(2n)-structures arise in this way.

Example 2.9. The following discussion relates our point of view to that of Kontsevich. Let (V, β) be a 2n-dimensional symplectic vector space, J a compatible complex structure, and g the corresponding inner product. Set Δ(V, J) = Λ^{n}(V, J)^{⊗2}, and let SΔ(V, J) ⊂ Δ(V, J) be the unit circle (with respect to the metric induced by g). One can define a fibration with simply-connected fibres

det^{2} : L(V, β) ⟶ SΔ(V, J)

by det^{2}(Λ) = (e_{1} ∧ ⋯ ∧ e_{n})^{⊗2}, where (e_{j}) is any orthonormal basis of (Λ, g|Λ). After choosing an element Θ ∈ Δ(V, J)^{∗} of unit length one can identify SΔ(V, J) with S^{1}. In this way one obtains a map det_{Θ}^{2} : L(V, β) ⟶ S^{1}. The Maslov class C(V, β) is equal to the pullback of the standard generator [S^{1}]. Hence L^{∞}(V, β) is isomorphic to the pullback of the standard covering ℝ ⟶ S^{1}.

Now let (M, ω) be a symplectic manifold and J a compatible almost complex structure. Assume that 2c_{1}(M, ω) = 0, which means that Δ = Λ^{n}(T M, J)^{⊗2} is trivial. Choose a section Θ of Δ^{∗} (in other words, a quadratic complex n-form) which has length one everywhere. As before this determines a map det_{Θ}^{2} : L ⟶ S^{1}, and one can define an ∞-fold Maslov covering by

L^{∞} = {(Λ, t) ∈ L × ℝ | det_{Θ}^{2}(Λ) = e^{2πit}}. (2.3)

An L^{∞}-grading of a Lagrangian submanifold L is just a lift of the map det_{Θ}^{2} ∘ s_{L} : L ⟶ S^{1} to ℝ. This approach is particularly useful in complex geometry: let (M, ω, J) be a Calabi-Yau manifold, take a covariantly constant holomorphic nform θ of unit length, and set Θ = θ^{⊗2}. A Lagrangian submanifold L ⊂ M is called special if (im θ)|L = 0. This condition is equivalent to det_{Θ}^{2} ∘ s_{L} ≡ 1 ∈ S^{1}. It follows that special Lagrangian submanifolds have a canonical L^{∞}-grading.

Example 2.10. Let (M, ω) be a symplectic manifold which admits a Lagrangian distribution S ⊂ T M. Then one can define an ∞-fold Maslov covering L^{∞} simply by putting together the universal covers of L(T M_{x}, ω_{x}) with base point S_{x}, for all x. The global Maslov class of this covering is represented by the fibrewise Maslov cycle I = ⋃_{x∈M}{Λ ∈ L_{x} | Λ ∩ S_{x} ≠ 0} with its canonical co-orientation. Any Lagrangian submanifold which is either tangent or transverse to S admits an L^{∞}- grading (there is even a preferred one). Typical examples are cotangent bundles M = T^{∗}X with either the vertical distribution (tangent spaces along the fibres) or the horizontal one (with respect to some Riemannian metric on X).

Example 2.11. Let M be an oriented closed surface of genus g ≥ 2, and ω a volume form on it. Using Lemma 2.2 one can see that (M, ω) admits many different Maslov coverings of any order which divides 4g − 4. However, in a sense there is no really good choice of covering:

Proposition 2.12. For every Maslov covering L^{N} of order N > 2 on M, there is an automorphism ϕ ∈ Aut(M, ω) which does not admit an L^{N}-grading.

Proof. Choose a Maslov covering L^{N} and let C^{N} be the corresponding global Maslov class. One can associate to any oriented embedded curve L ⊂ M a

Figure 1.

number R(L^{N}, L) = ⟨s_{L}^{∗}(C^{N}), [L]⟩ ∈ ℤ/N. L admits an L^{N}-grading iff it has zero rotation number, and a symplectic automorphism ϕ ∈ Aut(M, ω) admits an L^{N}-grading iff R(L^{N}, L) = R(L^{N}, ϕ(L)) for all curves L. We need two more facts, whose proofs we leave to the reader: (a) if Σ ⊂ M is a surface whose boundary is formed by the curves L_{1}, …, L_{k} then ∑_{j} R(L^{N}, L_{j}) ≡ −2χ(Σ) mod N. (b) If t_{L} is the positive Dehn twist along a curve L, then R(L^{N}, t_{L}(L′)) = R(L^{N}, L′) − ([L′] · [L])R(L^{N}, L) for any L′. Now consider the curves L_{k} shown in fig. 1. By (a) we have R(L^{N}, L_{1}) + R(L^{N}, L_{2}) + R(L^{N}, L_{3}) = 2. Hence there is a ν ∈ {1, 2, 3} such that R(L^{N}, L_{ν}) ≠ 0. Take a curve L′ with [L′] · [L_{ν}] = ±1. Property (b) shows that R(L^{N}, t_{L_{ν}} (L′)) ≠ R(L^{N}, L′), which means that t_{L_{ν}} does not admit an L^{N}-grading. This is for g = 2, but one sees immediately that the argument generalizes to all g > 2.

In contrast, for M = T^{2} there is exactly one ∞-fold Maslov covering L^{∞} (namely, the one coming from the standard trivialization of T M) with the property that any ϕ ∈ Aut(M, ω) admits an L^{∞}-grading.

2d. Two kinds of index. Let (V, β) be a 2n-dimensional symplectic vector space. The Maslov index for paths [29] [23] assigns a half-integer μ(λ_{0}, λ_{1}) ∈ \frac{1}{2}ℤ to any pair of paths λ_{0}, λ_{1} : [a; b] ⟶ L(V, β). μ(λ_{0}, λ_{1}) is an integer iff dim(λ_{0}(a) ∩ λ_{1}(a)) ≡ dim(λ_{0}(b) ∩ λ_{1}(b)) mod 2. We will now adapt this invariant to our situation. Fix some 1 ≤ N ≤ ∞. Let Λ̃_{0}, Λ̃_{1} ∈ L^{N}(V, β) be a pair of points whose images in L(V, β) intersect transversely. Choose two paths λ̃_{0}, λ̃_{1} : [0; 1] ⟶ L^{N}(V, β) with λ̃_{0}(0) = λ̃_{1}(0) and λ̃_{0}(1) = Λ̃_{0}, λ̃_{1}(1) = Λ̃_{1}. Let λ_{0}, λ_{1} be the projections of these paths to L(V, β). The absolute Maslov index of (Λ̃_{0}, Λ̃_{1}) is defined by

μ̃(Λ̃_{0}, Λ̃_{1}) = n/2 − μ(λ_{0}, λ_{1}) ∈ ℤ/N.

One can easily show that this is independent of all choices. From the standard properties of μ [23] one derives the following facts:

(i) Let λ_{0}, λ_{1} : [a; b] ⟶ L(V, β) be two paths such that λ_{0}(a) ∩ λ_{1}(a) = λ_{0}(b) ∩ λ_{1}(b) = 0. Lift them to paths λ̃_{0}, λ̃_{1} in L^{N}(V, β). Then

μ̃(λ̃_{0}(b), λ̃_{1}(b)) − μ̃(λ̃_{0}(a), λ̃_{1}(a)) = −μ(λ_{0}, λ_{1}).

(ii) μ̃(ρ(k)Λ̃_{0}, ρ(l)Λ̃_{1}) = μ̃(Λ̃_{0}, Λ̃_{1}) − k + l. (iii) For any (Φ, Φ̃) ∈ Sp^{N}(V, β) one has μ̃(Φ̃(Λ̃_{0}), Φ̃(Λ̃_{1})) = μ̃(Λ̃_{0}, Λ̃_{1}). (iv) μ̃(Λ̃_{1}, Λ̃_{0}) = n − μ̃(Λ̃_{0}, Λ̃_{1}).

(v) Let Λ, Λ^{⊥} ⊂ V be two complementary Lagrangian subspaces. Let B be a nondegenerate quadratic form on Λ, and A : Λ ⟶ Λ^{⊥} the unique linear map such that B(v) = β(v, Av). Take the path λ : [0; 1] ⟶ L(V, β) given by λ(t) = graph(tA), and lift it to a path λ̃ in L^{N}(V, β). Then μ̃(λ̃(1), λ̃(0)) is the Morse index of B (mod N).

(vi) For N = 2, if one identifies L^{2}(V, β) = L^{or}(V, β) then (−1)μ̃ agrees with the intersection number of oriented Lagrangian subspaces up to a constant (−1)^{n(n+1)/2}.

The application to graded symplectic geometry is as follows. Let (M, ω) be a symplectic manifold with an N-fold Maslov covering L^{N}, and (L_{0}, L̃_{0}), (L_{1}, L̃_{1}) a pair of L^{N}-graded Lagrangian submanifolds which intersect transversally. Then one can associate to any point x ∈ L_{0} ∩ L_{1} an absolute index mod N,

Ĩ(L̃_{0}, L̃_{1}; x) =^{def} μ̃(L̃_{0}(x), L̃_{1}(x)).

The Conley-Zehnder index associates to any path ϕ : [a; b] ⟶ Sp(V, β) a halfinteger ζ(ϕ) ∈ \frac{1}{2}ℤ. It can be reduced to the Maslov index for paths as follows: take (V′, β′) = (V, −β) ⊕ (V, β) and consider the two paths in L(V′, β′) given by λ_{0}(t) = graph(ϕ(t)) and λ_{1}(t) = Δ (the diagonal). Then ζ(ϕ) = μ(λ_{0}, λ_{1}) (here we are following [23, Remark 5.4]; it seems that the definition in [25] has the opposite sign). Now let (Φ, Φ̃) ∈ Sp^{N}(V, β) be a point such that det(1 − Φ) ≠ 0. Choose a path ϕ̃ : [0; 1] ⟶ Sp^{N}(V, β) from (id, ρ(k)) ∈ Sp^{N}(V, β), for some k ∈ ℤ/N, to (Φ, Φ̃), and project it to a path ϕ in Sp(V, β). The absolute Conley-Zehnder index is ζ̃(Φ, Φ̃) = n − ζ(ϕ) − k ∈ ℤ/N. This is independent of all choices and has the following properties:

(i) it is invariant under conjugation. (ii) ζ̃(Φ, Φ̃ ∘ ρ(l)) = ζ̃(Φ, Φ̃) − l. (iii) ζ̃(Φ^{−1}, Φ̃^{−1}) = 2n − ζ̃(Φ, Φ̃). (iv) Let B be a nondegenerate quadratic form on V, and A : V ⟶ V the linear

map given by ω(x, Ax) = B(x). Take the path ϕ(t) = exp(tA) in Sp(V, β) and lift it to a path ϕ̃ in Sp^{N}(V, β) with ϕ̃(0) = (id, id). Then ζ̃(ϕ(t), ϕ̃(t)) is equal to the Morse index of B (mod N) for sufficiently small t > 0.

Given a symplectic manifold (M, ω), an N-fold Maslov covering L^{N}, and an L^{N}- graded symplectic automorphism (ψ, ψ̃) which has nondegenerate fixed points, one can associate to any fixed point x an absolute index Z̃(ψ̃; x) = ζ̃(Dψ_{x}, ψ̃(x)) ∈ ℤ/N.

2e. Lagrangian surgery. We will now discuss an example which shows that the absolute Maslov index appears even in elementary questions about graded Lagrangian submanifolds.

Take an embedding γ : ℝ ⟶ ℂ with γ(t) = t for t ≤ −1/2, γ(t) = it for t ≥ 1/2, and γ(ℝ)∩−γ(ℝ) = ∅. Then H = ⋃_{t∈ℝ} γ(t)S^{n−1} ⊂ ℂ^{n} is a Lagrangian submanifold with respect to the standard symplectic form. Outside the unit ball B^{2n} ⊂ ℂ^{n} one has H ∩ (ℂ^{n} \ B^{2n}) = (ℝ^{n} ∪ iℝ^{n}) \ B^{2n}. H is called a Lagrangian handle. It is used in the following way: let (M^{2n}, ω) be a symplectic manifold and L_{1}, L_{2} ⊂ M a pair of Lagrangian submanifolds which intersect transversely in a single point {x_{0}} = L_{1} ∩ L_{2}. There is always an embedding j : B^{2n} ⟶ M with j(0) = x_{0}, j^{−1}(L_{1}) = ℝ^{n} ∩ B^{2n}, j^{−1}(L_{2}) = iℝ^{n} ∩ B^{2n}, and j^{∗}ω = ϵ ω_{ℂ^{n}} for some ϵ > 0. Then one can form the embedded connected sum

L_{1}#L_{2} = (L_{1} \ im(j)) ∪ (L_{2} \ im(j)) ∪ j(H ∩ B^{2n})

which is again a Lagrangian submanifold. This process, which is independent of all choices up to Lagrangian isotopy, is usually called Lagrangian surgery. It has been studied by Polterovich [21] and others. Our conventions are those of [28, Appendix] and differ from Polterovich’s. Note that Lagrangian surgery is not symmetric: L_{1}#L_{2} and L_{2}#L_{1} are the two possibilities of resolving the self-intersection of L_{1} ∪ L_{2} (this can be seen clearly already in the case n = 1).

Take the ∞-fold Maslov covering L^{∞} on ℂ^{n} induced by the quadratic complex nform Θ = (dz_{1}∧⋯∧dz_{n})^{⊗2} as in (2.3). An L^{∞}-grading of a Lagrangian submanifold L ⊂ ℂ^{n} is the same as a map L̃ : L ⟶ ℝ such that e^{2πiL̃(x)} = det_{Θ}^{2}(T L_{x}) for all x. For L_{1} = ℝ^{n} and L_{2} = iℝ^{n} we choose the gradings L̃_{1} ≡ 0, L̃_{2} ≡ 1 − n/2. Then the absolute index at the origin is Ĩ(L̃_{1}, L̃_{2}; 0) = 1.

Lemma 2.13. There is an L^{∞}-grading H̃ of the Lagrangian handle H which agrees with L̃_{1} on ℝ^{n} \ B^{2n} and with L̃_{2} on iℝ^{n} \ B^{2n}.

Proof. Take y_{0} ∈ S^{n−1}, and let y_{0}^{⊥} ⊂ ℝ^{n} be its orthogonal complement. The tangent space of H at a point y = γ(t)y_{0} is T H_{y} = ℝγ′(t)y_{0} ⊕ γ(t)y_{0}^{⊥}. Therefore

det_{Θ}^{2}(T H_{y}) = \frac{γ′(t)^{2}γ(t)^{2n−2}}{|γ′(t)^{2}γ(t)^{2n−2}|} ∈ S^{1}. (2.4)

As t goes from −∞ to ∞, γ(t)^{2}/|γ(t)|^{2} makes half a turn clockwise from 1 to −1, and γ′(t)^{2}/|γ′(t)|^{2} makes half a turn counterclockwise from 1 to −1. It follows that one can find a map α ∈ C^{∞}(ℝ, ℝ) with α(t) = 0 for t ≤ −1/2 and α(t) = 1 − n/2 for t ≥ 1/2, such that e^{2πiα(t)} equals the r.h.s. of (2.4). Then H̃(e^{it}y_{0}) = α(t) is a grading of H with the desired property.

From these local considerations one immediately obtains the following graded version of Lagrangian surgery.

Lemma 2.14. Let (M, ω) be a symplectic manifold with an ∞-fold Maslov covering L^{∞}. Let (L_{1}, L̃_{1}) and (L_{2}, L̃_{2}) be two L^{∞}-graded Lagrangian submanifolds which intersect transversely and in a single point x_{0} ∈ M. If Ĩ(L̃_{1}, L̃_{2}; x_{0}) = 1 then the surgery Σ = L_{1}#L_{2} has an L^{∞}-grading Σ̃ which agrees with L̃_{1} on Σ ∩ L_{1}, and with L̃_{2} on Σ ∩ L_{2}.

2f. Floer cohomology. We can now introduce the absolute grading on Floer cohomology. The exposition in this section is formal, in the sense that the conditions which are necessary to make Floer cohomology well-defined will be suppressed. Our justification is that there is no relation between these conditions and the problem of grading. Concretely, this means that if the ordinary Floer cohomology HF (L_{0}, L_{1}) of two Lagrangian submanifolds is defined, and L_{0}, L_{1} admit gradings L̃_{0}, L̃_{1}, then the graded version HF^{∗}(L̃_{0}, L̃_{1}) is also defined. The discussion of the properties of HF^{∗}(L̃_{0}, L̃_{1}) should be understood in the same way: they hold in the same generality as their ungraded analogues.

Let L_{0}, L_{1} be a pair of transversely intersecting Lagrangian submanifolds in a symplectic manifold (M^{2n}, ω), and J = (J_{t})_{0≤t≤1} a smooth family of ω-compatible almost complex structures. For any two points x_{−}, x_{+} ∈ L_{0} ∩ L_{1}, let B(x_{−}, x_{+}) be the set of smooth maps u : ℝ × [0; 1] ⟶ M with u(ℝ × {j}) ⊂ L_{j} for j = 0, 1, and lim_{s→±∞} u(s, ·) = x_{±}. Consider the subspace M(x_{−}, x_{+}; J) ⊂ B(x_{−}, x_{+}) of maps which satisfy Floer’s equation ∂_{s}u + J_{t}(u)∂_{t}u = 0. It has a natural action of ℝ by translation in the first variable. For generic choice of J, M(x_{−}, x_{+}; J) has a natural structure of a smooth manifold, with connected components of different dimensions. Let M_{k}(x_{−}, x_{+}; J) be the k-dimension part of M(x_{−}, x_{+}; J). In the simplest situation, such as in the original work of Floer and in that of Oh [19], suitable assumptions on (M, ω) and L_{0}, L_{1} ensure that the quotients M_{1}(x_{−}, x_{+}; J)/ℝ are finite sets. Then, writing n(x_{−}, x_{+}; J) ∈ ℤ/2 for the number of points mod 2 in M_{1}(x_{−}, x_{+}; J)/ℝ, one defines a chain group (CF (L_{0}, L_{1}), ∂_{J}) as follows: CF (L_{0}, L_{1}) is the ℤ/2-vector space freely generated by the points of L_{0} ∩ L_{1}, and ∂_{J}⟨x_{+}⟩ = ∑_{x_{−}} n(x_{−}, x_{+}; J)⟨x_{−}⟩. One finds that ∂_{J}^{2} = 0, and the Floer cohomology is HF (L_{0}, L_{1}; J) = ker ∂_{J}/im ∂_{J} (this is Floer cohomology because we have exchanged the usual roles of x_{−} and x_{+} in the definition of ∂_{J}).

Let L^{N} be an N-fold Maslov covering on M, and assume that L_{0}, L_{1} admit L^{N}- gradings L̃_{0}, L̃_{1}. For i ∈ ℤ/N, let CF^{i}(L̃_{0}, L̃_{1}) ⊂ CF (L_{0}, L_{1}) be the subspace generated by elements ⟨x⟩ where Ĩ(L̃_{0}, L̃_{1}; x) = i. This defines a ℤ/N-grading on CF (L_{0}, L_{1}). Floer’s index theorem [7] together with property (i) of the absolute Maslov index implies that ∂_{J} has degree one. Hence there is an induced grading on Floer cohomology. We will refer to the ℤ/N-graded group HF^{∗}(L̃_{0}, L̃_{1}; J) as graded Floer cohomology.

Let L = (L_{t})_{0≤t≤1} be an exact isotopy of Lagrangian submanifolds (exact means that it can be embedded in a Hamiltonian isotopy of M) and L a Lagrangian submanifold which intersects both L_{0} and L_{1} transversely. For any almost complex structures J^{−}, J^{+} such that HF (L, L_{0}; J^{−}) and HF (L, L_{1}; J^{+}) are defined, there is a canonical isomorphism

q(L, J^{−}, J^{+}) : HF (L, L_{0}; J^{−}) ⟶ HF (L, L_{1}; J^{+}). (2.5)

Now assume that we have a Maslov covering L^{N}, and that L_{0} and L admit L^{N}- gradings L̃_{0}, L̃. Then the isotopy (L_{t}) can be lifted to an isotopy (L̃_{t}) of L^{N}-graded Lagrangian submanifolds, and the map (2.5) has degree zero with respect to the gradings of HF^{∗}(L̃, L̃_{0}; J^{−}) and HF^{∗}(L̃, L̃_{1}; J^{+}). To complete the construction of graded Floer cohomology one follows the usual strategy: first, using the isomorphisms (2.5) for constant isotopies, one shows that graded Floer cohomology is independent of the choice of almost complex structure (we will therefore omit J from the notation from now on). Secondly, using again (2.5) but this time for C^{1}-small isotopies, one extends the definition of graded Floer cohomology to Lagrangian submanifolds which do not intersect transversally. Clearly, this extended definition is still invariant under exact isotopies of graded Lagrangian submanifolds (in both variables). Some other properties of graded Floer cohomology are: the shifting formula

HF^{j}(L̃_{0}[k], L̃_{1}[l]) ≅ HF^{j−k+l}(L̃_{0}, L̃_{1}),

invariance under graded symplectic automorphisms,

HF^{j}(ϕ̃(L̃_{0}), ϕ̃(L̃_{1})) ≅ HF^{j}(L̃_{0}, L̃_{1}),

and Poincaré duality:

HF^{j}(L̃_{1}, L̃_{0}) ≅ HF^{n−j}(L̃_{0}, L̃_{1})^{∨}.

These follow immediately from the properties of μ̃ listed in section 2d. We also need to mention the isomorphism between HF (L, L) and the ordinary cohomology H^{∗}(L; ℤ/2). This isomorphism holds in Floer’s original setup [8], but it can fail in more general situations, see [20]. Whenever it holds, so does the graded version

HF^{j}(L̃, L̃) ≅ ⨁_{i∈ℤ} H^{j+iN}(L; ℤ/2).

The other version of Floer cohomology, that for symplectic automorphisms, will not be used in this paper. Nevertheless, it seems appropriate to outline briefly the parallel story about the grading. Let ϕ be an automorphism of (M, ω) which has non-degenerate fixed points. One considers a chain group CF (ϕ) which has one basis element ⟨x⟩ for any fixed point x ∈ M of ϕ, and a boundary operator ∂_{J} defined as before, but this time using the space of maps u : ℝ^{2} ⟶ M which satisfy

\{ \begin{matrix}∂_{s}u + J_{t}(u)∂_{t}u = 0, \\ u(s, t) = ϕ(u(s, t + 1)), \end{matrix}

where J = (J_{t})_{t∈ℝ} is a family of compatible almost complex structures satisfying a suitable periodicity condition. If (M, ω) has a Maslov covering L^{N} and ϕ an L^{N}-grading ϕ̃ then, using the absolute Conley-Zehnder index ζ̃, one can define a ℤ/N-grading on CF (ϕ). The index theorem of [25, section 4] implies that ∂_{J} has degree one. Therefore one obtains a ℤ/N-graded Floer cohomology group HF^{∗}(ϕ̃). These groups are invariant under Hamiltonian isotopies and satisfy HF^{∗}(ϕ̃ ∘ [k]) = HF^{∗−k}(ϕ̃), HF^{∗}(ϕ̃^{−1}) = HF^{2n−∗}(ϕ)^{∨}, HF^{∗}(id) = H^{∗}(M). For the special case of monotone symplectic manifolds, a more extensive account of this kind of Floer theory (without mention of the absolute grading) can be found in [5].

3. Lagrangian submanifolds of ℂP^{n}

In this section we use graded Floer cohomology to obtain some restrictions on the topology of Lagrangian submanifolds of ℂP^{n}. Note that, by starting with the familiar embeddings of ℝP^{n} and T^{n} and applying Lagrangian surgery, one can construct many different Lagrangian submanifolds of ℂP^{n}.

Theorem 3.1. Any Lagrangian submanifold L ⊂ ℂP^{n} satisfies

(a) H^{1}(L; ℤ/(2n + 2)) ≠ 0,

(b) H^{1}(L; ℤ/(2n + 2)) ≅ (ℤ/2)^{g} for any g ≥ 2,

(c) if H^{1}(L; ℤ/(2n + 2)) ≅ ℤ/2 then H^{i}(L; ℤ/2) ≅ ℤ/2 for all i = 0, …, n.

By Lemma 2.6, ℂP^{n} admits a unique Maslov covering L^{N} of any order N which divides 2N_{M} = 2n + 2. Consider the Hamiltonian circle action given by σ(t) = diag(e^{2πit}, 1, …, 1) ∈ U(n + 1). One can lift σ uniquely to a map σ̃ : [0; 1] ⟶ Aut^{gr}(M, ω; L^{N}) with σ̃(0) = id. By looking at any fixed point one sees that σ̃(1) = [−2]. It follows that every L^{N}-graded Lagrangian submanifold L̃ is graded Lagrangian isotopic to L̃[−2], by an isotopy which is also exact. This implies that HF^{∗}(L̃, L̃), whenever defined, is periodic with period two. From this fact we will derive Theorem 3.1.

Before explaining the proof in detail, we need to recall the Floer cohomology for monotone symplectic manifolds as developed by Oh. The basic references are [19] and [20]. Technical issues are discussed further in [18] and [13]; see also [15]. We will present the theory in a slightly simplified form. Let (M^{2n}, ω) be a closed symplectic manifold which is monotone, that is, [ω] = λ c_{1}(M, ω) for some λ > 0. For any Lagrangian submanifold L ⊂ M, let N_{L} ≥ 1 be the number defined in section 2b. Oh shows that HF (L_{0}, L_{1}) is well-defined, and invariant under Lagrangian isotopy, for all pairs (L_{0}, L_{1}) such that N_{L_{0}}, N_{L_{1}} ≥ 3 and H^{1}(L_{0}; ℝ) = H^{1}(L_{1}; ℝ) = 0. Moreover one has

Theorem 3.2 (Oh [20, Theorem 5.1]). Let L ⊂ M be a Lagrangian submanifold with H^{1}(L; ℝ) = 0 and N_{L} ≥ 3.

(a) If N_{L} ≥ n + 2 then HF (L, L) ≅ ⨁_{i} H^{i}(L; ℤ/2).

(b) If N_{L} = n + 1 then HF (L, L) is either ⨁_{i} H^{i}(L; ℤ/2) or ⨁_{i≠0,n} H^{i}(L; ℤ/2).

The proof of this goes as follow [20, p. 332]. Let H be a Morse function on L, and L′ a small Lagrangian perturbation of L constructed using H and a Darboux chart. We may assume that H has only one local minimum x_{−} and local maximum x_{+}. The intersection points of L and L′ are the critical points of H. One can write the boundary operator on CF (L, L′) as ∂_{J} = ∂_{0} + ∂_{1} + …, where ∂_{k} takes critical points of Morse index i to those of Morse index i + 1 − kN_{L}. Floer [8] proved that, for a suitable choice of J, ∂_{0} can be identified with the boundary operator in a Morse cohomology complex for H. Therefore the homology of (CF (L, L′), ∂_{0}), which is sometimes called the local Floer cohomology of L, is always isomorphic to H^{∗}(L; ℤ/2). If N_{L} ≥ n + 2 then for dimension reasons ∂_{k} = 0 for all k > 0, which proves (a). If N_{L} = n + 1 then ∂_{k} = 0 for k ≥ 2, ∂_{1}⟨x⟩ = 0 for all x ≠ x_{+}, and ∂_{1}⟨x_{+}⟩ can be either zero or ⟨x_{−}⟩. This leads to the two possibilities in (b).

Now let L^{N} be a Maslov covering of order N ≥ 3 on M. Lemma 2.6 shows that any L^{N}-graded Lagrangian submanifold (L, L̃) automatically satisfies N_{L} ≥ N ≥ 3. Hence the graded Floer cohomology groups HF^{∗}(L̃_{0}, L̃_{1}) are well-defined for L^{N}- graded Lagrangian submanifolds with zero first Betti number. Moreover, as a look at the proof shows, the obvious graded analogue of Theorem 3.2 holds.

Proof of Theorem 3.1. (a) Assume that L ⊂ ℂP^{n} (n ≥ 2) is a Lagrangian submanifold with H^{1}(L; ℤ/(2n + 2)) = 0. This implies that H^{1}(L; ℝ) = 0. By Lemma 2.3, L admits a grading L̃ with respect to the unique Maslov covering L^{2n+2} of order 2n + 2. Hence the graded Floer cohomology HF^{∗}(L̃, L̃) is well-defined. Lemma 2.6 shows that N_{L} ≥ 2n + 2, and by applying Theorem 3.2(a) one finds that

HF^{i}(L̃, L̃) = \begin{cases}H^{i}(L; ℤ/2) & 0 ≤ i ≤ n, \\ 0 & n + 1 ≤ i ≤ 2n + 1. \end{cases} (3.1)

As discussed above, the circle action σ on ℂP^{n} provides a graded Lagrangian isotopy between L̃ and L̃[−2], which implies that HF^{∗}(L̃, L̃) ≅ HF^{∗}(L̃, L̃[−2]) = HF^{∗−2}(L̃, L̃). This is a contradiction, since (3.1) is not two-periodic.

(b,c) Let L ⊂ ℂP^{n} (n ≥ 2) be a Lagrangian submanifold with H^{1}(L; ℤ/(2n + 2)) = (ℤ/2)^{g} for some g ≥ 1. This implies that H^{1}(L) = 0, that H^{1}(L; ℤ/2) ≅ (ℤ/2)^{g}, and that the homomorphism ρ : H^{1}(L; ℤ/(2n + 2)) ⟶ H^{1}(L; ℤ/(n + 1)) induced by the projection ℤ/(2n + 2) ⟶ ℤ/(n + 1) is zero.

Let C^{2n+2} ∈ H^{1}(L; ℤ/(2n + 2)) and C^{n+1} ∈ H^{1}(L; ℤ/(n + 1)) be the global Maslov classes of the Maslov coverings L^{2n+2}, L^{n+1} on ℂP^{n}. Clearly C^{n+1} is obtained from C^{2n+2} by reducing mod (n + 1). We conclude that s_{L}^{∗}(C^{n+1}) = ρ(s_{L}^{∗}(C^{2n+2})) = 0, which means that L admits an L^{n+1}-grading L̃. The same argument as before shows that the ℤ/(n + 1)-graded Floer cohomology HF^{∗}(L̃, L̃) is two-periodic.

Lemma 2.6 says that N_{L} ≥ n + 1. By Theorem 3.2(b) there are two possibilities for the Floer cohomology. One is that HF^{∗}(L̃, L̃) = H^{∗}(L; ℤ/2). In this case it follows that H^{∗}(L; ℤ/2), with the grading reduced mod n+1, is two-periodic. Since H^{0}(L; ℤ/2) = ℤ/2 and H^{n}(L; ℤ/2) = ℤ/2, the periodicity leads to H^{i}(L; ℤ/2) = ℤ/2 for all i, which proves both (b) and (c). It remains to consider the other possibility, which is

HF^{i}(L̃, L̃) = \begin{cases}H^{i}(L; ℤ/2) & 0 < i < n, \\ 0 & i = 0, n. \end{cases}

This contradicts the two-periodicity, because HF^{1}(L̃, L̃) = H^{1}(L; ℤ/2) = (ℤ/2)^{g} while HF^{−1}(L̃, L̃) = 0. Hence this possibility cannot occur.

4. A class of symplectic automorphisms

4a. The basic result. Let (M, ω, α) be a compact symplectic manifold with contact type boundary, and (ϕ_{t}^{K}) an S^{1}-action on ∂M which preserves α. This means that the symplectic form ω ∈ Ω^{2}(M) and the contact one-form α ∈ Ω^{1}(∂M) are related by dα = ω|∂M, and that the Reeb vector field R of α satisfies ω(N, R) > 0, where N is any vector field pointing outwards along ∂M; in addition, we are given a vector field K on ∂M with L_{K}α = 0 and whose flow (ϕ_{t}^{K}) is one-periodic. One can always find a collar j : (−ϵ; 0] × ∂M ↪ M, for some ϵ > 0, such that j^{∗}ω = d(e^{r}α). Choose a function H ∈ C^{∞}(M, ℝ) with H(j(r, x)) = e^{r}(i_{K}α)(x) for all r ≥ −ϵ/2. Then the Hamiltonian flow (ϕ_{t}^{H}) satisfies ϕ_{t}^{H}(j(r, x)) = j(r, ϕ_{t}^{K}(x)), and in particular ϕ_{±1}^{H}(r, x) = (r, x), for all r ≥ −ϵ/2. Set

χ_{K} = ϕ_{−1}^{H} ∈ Aut(M, ∂M, ω). (4.1)

It is easy to see that the class [χ_{K}] ∈ π_{0}(Aut(M, ∂M, ω)) is independent of the choice of H and j. Note that by taking a suitable choice, one can achieve that χ_{K} is the identity outside an arbitrarily small neighbourhood of ∂M. The question we are interested in is: when is [χ_{K}] nontrivial, and more generally, what is its order in π_{0}(Aut(M, ∂M, ω))? Answering this can be easy or difficult, depending on the specific situation. We list a few easy cases:

Examples 4.1. (a) [χ_{K}] is trivial whenever (ϕ_{t}^{K}) can be extended to a Hamiltonian circle action on M. The simplest example is when M is the unit ball in ℂ^{n} and (ϕ_{t}^{K}) is any ℂ-linear circle action on S^{2n−1}.

(b) [χ_{K}] is often nontrivial for topological reasons. The simplest relevant topological invariant is the variation homomorphism var(f) : H^{∗}(M) ⟶ H_{∗}(M, ∂M) associated to any continuous map f : M ⟶ M with f|∂M = id. If var(f) ≠ 0 then f cannot be homotoped to the identity rel ∂M. In the special case when the circle action (ϕ_{t}^{K}) is free one can compute var(χ_{K}^{m}) (mod torsion) for any m ∈ ℤ from the formula

∫_{M} var(χ_{K}^{m})(a) ∪ b = ±m ∫_{∂M/S^{1}} p(a) ∪ p(b). (4.2)

Here a, b ∈ H^{∗}(M), and p is the map H^{∗}(M) ⟶ H^{∗}(∂M) ⟶ H^{∗−1}(∂M/S^{1}). The sign in (4.2) depends on orientation conventions, and is irrelevant for our purpose. For example, let H ⊂ ℂP^{n} (n ≥ 2) be a smooth complex hypersurface of degree d. If one takes a suitable tubular neighbourhood U of H, the complement M = ℂP^{n} \ U has the structure of a symplectic manifold with contact type boundary, and ∂M is a circle bundle over H. Let χ_{K} be the automorphism of M obtained from the obvious circle action on ∂M. Using (4.2) one can see that [χ_{K}] has infinite order whenever n is odd or d ≥ 3. On the other hand, [χ_{K}] is trivial for d = 1, see (a). The remaining case (d = 2 and n even) is more difficult; we will solve it later.

(c) Take M to be a compact surface with (∂M, α) = (S^{1}, dt). Consider the standard circle action on ∂M. Then χ_{K} is a positive Dehn twist along a curve parallel to ∂M. [χ_{K}] is trivial if M is a disc, and has infinite order (for topological reasons) otherwise.

From now on assume that ∂M is connected, H^{1}(M) = 0, and that 2c_{1}(M, ω) = 0. Then there is a unique ∞-fold Maslov covering L^{∞} on M. There are two ways to choose an L^{∞}-grading for χ_{K}. One way is to use Remark 2.5 which says that there is a unique grading χ̃_{K} ∈ Aut^{gr}(M, ∂M, ω; L^{∞}). Alternatively one can lift (ϕ_{t}^{H})_{−1≤t≤0} to an isotopy of L^{∞}-graded symplectic automorphisms (ϕ̃_{t}^{H}) with ϕ̃_{0}^{H} = id, and then ϕ̃_{−1}^{H} is again a grading of χ_{K}. These two approaches do not necessarily lead to the same grading. Let σ_{K} ∈ ℤ be the difference, that is to say

χ̃_{K} = ϕ̃_{−1}^{H} ∘ [σ_{K}]. (4.3)

Lemma 4.2. Let L be a Lagrangian submanifold of M which admits an L^{∞}- grading L̃. If the class [χ_{K}^{m}] ∈ π_{0}(Aut(M, ∂M, ω)) is trivial for some m ≥ 1, then L̃ is isotopic to L̃[mσ_{K}] as an L^{∞}-graded Lagrangian submanifold.

Proof. Since the statement is independent of the choices made in the definition of χ_{K}, we can assume that the embedding j satisfies L ∩ im(j) = ∅. This means that ϕ_{t}^{H}(L) = L for all t. Since ϕ̃_{0}^{H} = id, it follows that ϕ̃_{t}^{H}(L̃) = L̃ for all t. Because of (4.3) one has χ̃_{K}^{m}(L̃) = L̃[mσ_{K}]. By assumption there is an isotopy (ψ_{t}) in Aut(M, ∂M, ω) from ψ_{0} = id to ψ_{1} = χ_{K}^{m}. One can lift this to an isotopy (ψ̃_{t}) of L^{∞}-graded symplectic automorphisms with ψ̃_{0} = id. This isotopy will remain inside Aut^{gr}(M, ∂M, ω; L^{∞}), which implies that ψ̃_{1} = χ̃_{K}^{m}. Hence ψ̃_{t}(L̃) is an isotopy of L^{∞}-graded Lagrangian submanifolds from L̃ to χ̃_{K}^{m}(L̃) = L̃[mσ_{K}].

Theorem 4.3. Let (M, ω, α) be a compact symplectic manifold with contact type boundary, and (ϕ_{t}^{K}) a circle action on ∂M which preserves α. Assume that ∂M is connected, that H^{1}(M) = 0, that 2c_{1}(M, ω) = 0, and that [ω] ∈ H^{2}(M; ℝ) is zero. Furthermore, assume that M contains a Lagrangian submanifold L with H^{1}(L) = 0. Let χ_{K} be the automorphism of M defined in (4.1), and σ_{K} the number from (4.3). If σ_{K} ≠ 0 then [χ_{K}] ∈ π_{0}(Aut(M, ∂M, ω)) is an element of infinite order.

Proof. The assumption [ω] = 0 implies that the Floer cohomology HF (L_{0}, L_{1}) of any pair of Lagrangian submanifolds L_{0}, L_{1} with H^{1}(L_{0}) = H^{1}(L_{1}) is well-defined and invariant under Lagrangian isotopy. For L_{0} = L_{1} one has HF (L_{0}, L_{1}) ≅ H^{∗}(L_{0}; ℤ/2). Of course, the same properties are true for graded Floer cohomology. Choose an L^{∞}-grading L̃ of L. If [χ_{K}] was trivial then by Lemma 4.2 L̃ would be graded Lagrangian isotopic to L̃[σ_{K}], and hence

HF^{∗}(L̃, L̃) ≅ HF^{∗}(L̃, L̃[σ_{K}]) = HF^{∗+σ_{K}} (L̃, L̃).

Because HF^{∗}(L̃, L̃) is nonzero and concentrated in finitely many degrees, this implies that σ_{K} must be zero. Conversely, if σ_{K} ≠ 0 then [χ_{K}] must be nontrivial.

Since the same argument works for the iterates χ_{K}^{m}, it also follows that [χ_{K}] has infinite order.

Let C^{∞} ∈ H^{1}(L) be the global Maslov class of L^{∞}. Take a point x ∈ ∂M and a Lagrangian subspace Λ ∈ L_{x}, and define λ : S^{1} ⟶ L by λ(t) = Dϕ_{t}^{H}(Λ). One can easily show that

σ_{K} = −⟨C^{∞}, [λ]⟩. (4.4)

This formula is useful for determining σ_{K}, which is important in applications of Theorem 4.3.

Remarks 4.4. (a) The conditions in Theorem 4.3 were chosen for their simplicity, and are not the most general ones. For instance, an inspection of the proof shows that the assumption H^{1}(M) = 0 can be omitted (however, for H^{1}(M) ≠ 0 the shift σ_{K} may depend on the choice of Maslov covering). The condition [ω] = 0 is there to ensure that there is a well-behaved Floer theory, and can also be weakened considerably. In contrast, the existence of the Lagrangian submanifold L, and the assumption that 2c_{1}(M, ω) = 0, are both essential parts of our argument.

(b) There is an alternative approach which dispenses with Lagrangian submanifolds altogether, and uses instead the Floer cohomology of the automorphism χ_{K}. This approach is more difficult to carry out than the one used here, but it is possibly more general.

4b. Manifolds with periodic geodesic flow. Let (N^{n}, g) be a Riemannian manifold such that any geodesic of length one is closed. Take M = T_{1}^{∗}N = {ξ ∈ T^{∗}N | |ξ|_{g} ≤ 1} with the standard symplectic form ω ∈ Ω^{2}(M) and contact form α ∈ Ω^{1}(∂M). The flow of the Reeb vector field R of α is just the geodesic flow on N, and our assumption is that ϕ_{1}^{R} = id. Hence one can define an automorphism χ_{R} ∈ Aut(M, ∂M, ω). Extend (ϕ_{t}^{R}) radially to a Hamiltonian circle action σ on M \ N (the moment map of σ is μ(ξ) = |ξ|). Choose a function r ∈ C^{∞}([0; 1], ℝ) with r(t) = 0 for t ≤ 1/3, r(t) = −1 for t ≥ 2/3. Then a possible choice for χ_{R} is

χ_{R}(ξ) = \begin{cases}σ_{r(|ξ|)}(ξ) & ξ ∉ N, \\ ξ & ξ ∈ N. \end{cases} (4.5)

M admits a Lagrangian distribution S ⊂ T M, formed by the tangent spaces along the fibres of the projection M ⟶ N. As indicated in Example 2.10, one can use S to define an ∞-fold Maslov covering L^{∞}. In order to satisfy the conditions of Theorem 4.3, we will assume that H^{1}(N) is zero, so that there is only one ∞- fold Maslov covering. The global Maslov class C^{∞} is represented by the fibrewise Maslov cycle I = ⋃_{ξ}{Λ ∈ L_{ξ} | Λ ∩ S_{ξ} ≠ 0}, and one can write (4.4) as

σ_{R} = −[I] · [λ].

By definition λ(t) = Dσ_{t}(Λ) for some Λ ∈ L_{ξ}, ξ ∈ ∂M. It is convenient to choose Λ in the following way: split T M_{ξ} into its horizontal and vertical parts, both of which are naturally isomorphic to T N_{x}, where x ∈ N is the base point of ξ. Then take Λ = {(η_{1}, η_{2}) ∈ T N_{x} × T N_{x} | η_{1} = g(ξ, η_{2})ξ}. This has the consequence that λ(t) ∈ I iff c(t) and c(0) are conjugate points. It is a familiar result, see [6, section 4] or [24, section 6], that the local intersection number at a point λ(t) ∈ I is equal to the multiplicity m_{c}(t) > 0 of c(t) as a conjugate point of c(0). Therefore

σ_{R} = − ∑_{t} m_{c}(t), (4.6)

where the sum is over all conjugate points t ∈ S^{1}. Since m_{c}(0) = n−1, σ_{R} is always negative (N = S^{1} is impossible because we have assumed that H^{1}(N) = 0). M always contains a Lagrangian submanifold (the zero-section) with zero first Betti number. Moreover, [ω] = 0. By applying Theorem 4.3 one obtains

Corollary 4.5. Let (N, g) be a closed Riemannian manifold with H^{1}(N) = 0 such that any geodesic of length one is closed. Then, for M = T_{1}^{∗}N and the Reeb vector field R, [χ_{R}] ∈ π_{0}(Aut(M, ∂M, ω)) has infinite order.

The main examples of manifolds with periodic geodesic flow are compact globally symmetric spaces of rank one [2]. These spaces are two-point homogeneous, which means that the isometry group Iso(N, g) acts transitively on ∂M. Hence all geodesics have the same minimal period. Symmetric spaces also have the property that any geodesic path c : [0; T] ⟶ N with c(T) = c(0) is a closed geodesic [12, p. 144]. It follows that the energy functional on the based loop space ΩN is a Morse-Bott function. Its critical point set consists of one point (the constant path) and infinitely many copies of S^{n−1}. The Morse index of the point is zero. The Morse indices of the (n − 1)-spheres can be computed by comparing (4.6) with Morse’s index theorem: they are −σ_{R} −n+1, −2σ_{R} −n+1, etc. This means that for n ≥ 3, ΩN has a CW-decomposition with one cell of dimension −σ_{R} −n+1 and other cells of dimension ≥ −σ_{R} − n + 3. It follows that ΩN is precisely (−σ_{R} − n)-connected, and that N is precisely (−σ_{R} − n + 1)-connected. This approach, complemented by explicit computations for ℝP^{2} and S^{2}, yields the following values for σ_{R}:

\begin{matrix}N & S^{m} & ℝP^{m} & ℂP^{m} & ℍP^{m} & F_{4}/Spin_{9} \\ σ_{R} & 2 − 2m & 1 − m & −2m & −4m + 2 & −22 \end{matrix}

Here F_{4}/Spin_{9} is the exceptional symmetric space diffeomorphic to the Cayley plane (which is 16-dimensional and 7-connected). For N = S^{2m+1} or ℝP^{2m+1} the result of Corollary 4.5 can be obtained more easily, without using any symplectic geometry, by computing the variation var(χ_{R}). For N = ℝP^{2m} the space M can be identified with the complement of a neighbourhood of a quadric H ⊂ ℂP^{m}, thus settling the remaining case in Example 4.1(b). The most interesting example is that of ℂP^{m}, since there we can show that the non-vanishing of [χ_{R}] is a genuinely symplectic phenomenon:

Proposition 4.6. Let M = T_{1}^{∗}ℂP^{m}, m ≥ 1. Then χ_{R} can be deformed to the identity in the group Diff(M, ∂M) of diffeomorphisms which act trivially on ∂M.

Lemma 4.7. Let M = T_{1}^{∗}ℂP^{m}. Then there is a smooth family σ^{s}, 0 ≤ s ≤ 1, of circle actions on ∂M with σ_{t}^{0} = ϕ_{t}^{R}, and such that σ^{1} extends to a circle action on the whole of M.

Proof. Since ℂP^{m} = S^{2m+1}/S^{1}, T^{∗}ℂP^{m} is a symplectic quotient T^{∗}S^{2m+1}//S^{1}. Explicitly M = {(u, v) ∈ ℂ^{m+1} × ℂ^{m+1} | |u| = 1, |v| ≤ 1, ⟨u, v⟩_{ℂ} = 0}/S^{1}, where S^{1} acts by t · (u, v) = (e^{2πit}u, e^{2πit}v). The diagonal SU(2)-action on ℂ^{m+1} × ℂ^{m+1} descends to an action of P U(2) = SU(2)/±1 on ∂M, which we will denote by ρ. Assume that we have rescaled the standard symplectic form to make the Reeb flow one-periodic, that is, ω = π/2 ∑_{j}(dv_{j} ∧ dū_{j} + dv̄_{j} ∧ du_{j}). Then the Reeb flow is ϕ_{t}^{R} = ρ(exp(tA)) with A = ( \begin{matrix}0 & −π \\ π & 0 \end{matrix} ) ∈ su_{2}. The family σ^{s} of circle actions is defined by σ_{t}^{s} = ρ(exp(tA^{s})), where (A^{s}) is any path in su(2) from A^{0} = A to A^{1} = diag(πi, −πi) such that exp(A^{s}) = −1 for all s. Because ∂M is a quotient one can write σ_{t}^{1}(u, v) = (u, e^{−2πit}v), and this shows that σ^{1} extends to a circle action on all of M.

Proof of Proposition 4.6. Clearly, using the family σ^{s} of circle actions one can deform χ_{R} inside Diff(M, ∂M) to χ′_{R}(ξ) = σ_{r(|ξ|)}^{1}(ξ). To deform this to the identity one uses the isotopy ψ_{s}(ξ) = σ_{(1−s)r(|ξ|)−s}^{1}(ξ).

Remark 4.8. It seems likely that the maps χ_{R} on T_{1}^{∗}ℂP^{m} are not only differentiably isotopic to the identity but also ‘fragile’ in the sense of [26]. We have not checked the details.

4c. Weighted homogeneous polynomials. A polynomial p ∈ ℂ[x_{0}, …, x_{n}], n ≥ 1, is called weighted homogeneous if there are integers β_{0}, …, β_{n}, β > 0 such that p(z^{β_{0}}x_{0}, …, z^{β_{n}}x_{n}) = z^{β}p(x_{0}, …, x_{n}). The numbers w_{i} = β_{i}/β are called the weights of p. Throughout this section p(x) will be a weighted homogeneous polynomial with an isolated critical point at x = 0; because of the homogeneity, this implies that p has no other critical points. By definition, the link of the singular point is L = p^{−1}(0)∩S^{2n+1}. This is in fact a contact submanifold of S^{2n+1} with respect to the standard contact form α_{S^{2n+1}} = \frac{i}{4} ∑_{j} z_{j}dz̄_{j} −z̄_{j}dz_{j}. Let B^{2n+2} ⊂ ℂ^{n+1} be the closed unit ball. Fix a cutoff function ψ with ψ(t^{2}) = 1 for t ≤ 1/3 and ψ(t^{2}) = 0 for t ≥ 2/3. For z ∈ ℂ \ {0} set F_{z} = {x ∈ B^{2n+2} | p(x) = ψ(|x|^{2})z}.

Lemma 4.9. There is an ϵ > 0 such that for all 0 < |z| < ϵ, F_{z} is a symplectic submanifold of B^{2n+2} with boundary ∂F_{z} = L.

Proof. The Zariski tangent space of F_{z} is (T F_{z})_{x} = ker L(x, z) with L(x, z) : ℂ^{n+1} ⟶ ℂ, L(x, z)ξ = dp(x)ξ − 2z ψ′(|x|^{2})⟨x, ξ⟩. Clearly L(x, z) is onto in the following three cases: (1) z = 0 and x ≠ 0, (2) 0 < |x| ≤ 1/3, (3) |x| ≥ 2/3. On the other hand, the set of those (x, z) for which L(x, z) is onto must be open. This implies that L(x, z) is onto for all (x, z) ∈ F_{z}, provided that z ≠ 0 is sufficiently small. An argument of the same kind shows that (T F_{z})_{x} is a symplectic subspace of ℂ^{n+1} for all small z ≠ 0.

We fix a z_{0} with 0 < |z_{0}| < ϵ and call M = F_{z_{0}} the Milnor fibre of the singular point 0 ∈ p^{−1}(0). Clearly (M, ω = ω_{ℂ^{2n+2}}|M, α = α_{S^{2n+1}}|L) is a compact symplectic manifold with contact type boundary. The choice of z_{0} is not really important since one can prove (by a standard argument using Moser’s technique) that any two choices give symplectically isomorphic Milnor fibres. During the following discussion, we will repeatedly make use of our right to pass to a smaller z_{0}. The next Lemma shows that M is diffeomorphic to what is traditionally called the Milnor fibre.

Lemma 4.10. There is an ϵ > 0 such that for all 0 < |z| < ϵ, F_{z} is diffeomorphic to p^{−1}(z) ∩ B^{2n+2}.

Proof. For (z, t) ∈ (ℂ \ {0}) × [0; 1] consider G_{(z,t)} = {x ∈ B^{2n+2} | p(x) = tψ(|x|^{2})z + (1 − t)z}. Using the same argument as before, one can prove that these are smooth manifolds for all sufficiently small z. If we fix such a z, the G_{(z,t)} form a differentiable fibre bundle over [0; 1]. Hence G_{(z,1)} = F_{z} and G_{(z,0)} = p^{−1}(z)∩B^{2n+2} are diffeomorphic.

Corollary 4.11. M is (n − 1)-connected. In fact, it is homotopy equivalent to a nontrivial wedge of n-spheres.

This follows from Lemma 4.10 and a classical result of Milnor [16, Theorem 6.5]. Denote by A = {x ∈ M | 2/3 ≤ |x| ≤ 1} the part of M on which it agrees with p^{−1}(0). Let θ_{ℂ^{n+1}} be the complex n-form on ℂ^{n+1} given by (θ_{ℂ^{n+1}})_{x}(v_{1}, …, v_{n}) = det_{ℂ}(\overline{dp}_{x} ∧ v_{1} ∧ ⋯ ∧ v_{n}).

Lemma 4.12. There is an ω-compatible almost complex structure J on M, and a nowhere vanishing J-complex n-form θ, which have the following properties: the restriction of J to A agrees with the standard complex structure on A ⊂ p^{−1}(0), and θ|A = θ_{ℂ^{n+1}}|A.

Proof. Consider the 2n-dimensional complex vector bundle K ⟶ M with fibres K_{x} = ker(dp_{x}). θ_{ℂ^{n+1}} defines a complex n-form on every fibre of K, and these n-forms are easily seen to be nonzero. Both K and T M are subbundles of the trivial bundle ℂ^{n+1} × M. The proof of Lemma 4.9 shows that, by choosing z_{0} sufficiently small, one can make these two subbundles arbitrarily close. This means that the orthogonal projection P_{K} : T M ⟶ K is a bundle isomorphism, and that the pullback of the complex structure on K is an almost complex structure J′ on T M which is ω-tame. The pullback θ′ = P_{K}^{∗}(θ_{ℂ^{n+1}}|K) is a J′-complex n-form on M which is nowhere zero. Since P_{K} is the identity over A, J′ and θ′ have all the properties required in the Lemma, except that J′ may not be ω-compatible. However, one can easily find an ω-compatible almost complex structure J which agrees with J′ on A. Moreover, J and J′ can be deformed into each other through almost complex structures, and the deformation can be chosen constant on A. This implies that there is a bundle automorphism Q : T M ⟶ T M such that Q^{∗}(J′) = J, and which is the identity on A. Now set θ = Q^{∗}(θ′).

Corollary 4.13. c_{1}(M, ω) = 0.

This is clear, since there is a nowhere vanishing complex n-form.

Lemma 4.14. (M, ω) always contains an embedded Lagrangian n-sphere.

Sketch of proof. The method which produces such spheres is to deform p by adding a linear term λ, such that p + λ has only nondegenerate critical points. The Lagrangian spheres appear as vanishing cycles associated to these critical points. We will now explain one version of this argument; for variations on this theme see [26, section 1.4]. Choose some small δ > 0, and let D = {(z, λ) ∈ ℂ × (ℂ^{n+1})^{∗} | |z| < δ, |λ| < δ}. For (z, λ) ∈ D define

F̃_{(z,λ)} = {x ∈ B^{2n+2} | p(x) + λ(x) = ψ(|x|^{2})z}.

If δ is sufficiently small, there are two possibilities for each (z, λ): either (1) F̃_{(z,λ)} is a smooth symplectic submanifold of B^{2n+2}, or (2) the complex hypersurface p(x) + λ(x) = z has a singular point x with |x| ≤ 1/3. The subset Δ ⊂ D where (2) occurs is a complex hypersurface. Hence D \ Δ is connected. By an application of Moser’s technique, it follows that all the symplectic manifolds F̃_{(z,λ)} occurring in case (1) are isomorphic to the Milnor fibre M = F̃_{(z_{0},0)}.

Take a generic small λ. Then p + λ has only nondegenerate (Morse-type) critical points. These critical points lie close to the origin, and there is always at least one of them (this is a well-known fact, which follows from considering the Milnor number μ of the singularity). Choose a critical point x of p+λ, such that |x| < 1/4, and set z = p(x) + λ(x). Since x is non-degenerate, one can write

(p + λ)(x + y) = z + Q(y) + (higher order terms in y)

where Q is a nondegenerate complex quadratic form. A careful application of Moser’s technique shows that for all 0 < ϵ_{2} ≪ ϵ_{1} ≪ |λ|, one can embed U = {y ∈ ℂ^{n+1} | |y| ≤ ϵ_{1}, Q(y) = ϵ_{2}} symplectically into F̃_{(z+ϵ_{2},λ)}. Now U is symplectically isomorphic to a neighbourhood of the zero-section in T^{∗}S^{n}, hence contains a Lagrangian n-sphere. It follows that F̃_{(z+ϵ_{2},λ)}, and hence the Milnor fibre, contain a Lagrangian n-sphere.

The Milnor fibration associated to the singular point 0 ∈ p^{−1}(0) is obtained by putting together the manifolds F_{z} for all |z| = |z_{0}|:

p : F = ⋃_{z=|z_{0}|} F_{z} × {z} ⟶ |z_{0}|S^{1}.

The proof of Lemma 4.9 shows that this is a smooth and proper fibration. Moreover, if we pull back ω_{ℂ^{n+1}} to F via the obvious projection, we obtain a closed two-form Ω whose restriction to any fibre is a symplectic form. Such a two-form defines a connection T F^{h} ⊂ T F, given by the Ω-orthogonal complements of the tangent spaces along the fibres:

T F_{(x,z)}^{h} = {X ∈ T F_{(x,z)} | Ω(X, Y) = 0 for all Y such that Dp(Y) = 0}.

The fact that Ω is closed implies that the parallel transport maps of this connection are symplectic isomorphisms between the fibres. In our case, since the fibres are manifolds with boundary, one has to check that the parallel transport maps exist. But this it clear because near the boundary ∂F = ⋃_{z} ∂F_{z} × {z} one has a natural trivialization A × |z_{0}| S^{1} ⊂ F, and the connection is compatible with this trivialization. This also implies that if we go once around the base |z_{0}|S^{1}, the parallel transport yields a map f ∈ Aut(M, ∂M, ω). We call f the symplectic monodromy of our singularity. One can show that the class [f] ∈ π_{0}(Aut(M, ∂M, ω)) is, in a suitable sense, independent of the choices of ψ and z_{0}.

Up to now, the fact that p is weighted homogeneous has not been of any importance. We will now begin to exploit this particular feature of our situation. Let σ be the complex circle action on ℂ^{n+1} with multiplicities β_{0}, …, β_{n}, and K the vector field generating it. σ|L is a circle action preserving α. Hence one can construct the associated automorphism χ_{K} ∈ Aut(M, ∂M, ω) of the Milnor fibre. We will use a particular choice which is χ_{K} = ϕ_{−1}^{H}, where H ∈ C^{∞}(M, ℝ) is given by H(x) = π ∑_{i} β_{i}|x_{i}|^{2}. From now on assume that n ≥ 2. Since H^{1}(M) is zero (Corollary 4.11), ∂M is connected (this follows from Corollary 4.11 by a Poincaré duality consideration), and 2c_{1}(M, ω) = 0 (Corollary 4.13), one can define the shift σ_{K} of χ_{K}.

Lemma 4.15. σ_{K} = 2(β − ∑_{j} β_{j}).

Proof. Let J, θ be as in Lemma 4.12. As explained in Example 2.9, Θ = θ^{⊗2}/|θ|^{2} defines an ∞-fold Maslov covering on M, whose global Maslov class is represented by the map det_{Θ}^{2} : L ⟶ S^{1}. Together with (4.4) this means that −σ_{K} is the degree of the map c : S^{1} ⟶ S^{1}, c(t) = det_{Θ}^{2}(Dϕ_{t}^{H}(Λ)) for some Λ ∈ L_{x}, x ∈ ∂M ⊂ A. The restriction of ϕ_{t}^{H} to A is simply the circle action σ, and θ|A agrees with θ_{ℂ^{n+1}}. An explicit computation shows that σ_{t}^{∗}(θ_{ℂ^{n+1}}) = e^{2πit(β_{0}+···+β_{n}−β)}θ_{ℂ^{n+1}}. Therefore c has the form c(t) = e^{4πit(β_{0}+···+β_{n}−β)}c(0).

Lemma 4.16. χ_{K} is equal to the β-th iterate f^{β} of the symplectic monodromy.

Proof. Let X̃ ∈ C^{∞}(T E^{h}) be the unique horizontal lift of the vector field X(z) = 2πiz on |z_{0}|S^{1}. Its flow (μ_{t}) maps F_{z} to F_{e^{2πit}z} symplectically. By definition, the symplectic monodromy is f = μ_{1}|F_{z_{0}}. Now let ρ be the circle action on F given by ρ_{t}(x, z) = (σ_{t}(x), e^{2πiβt}z). We denote the Killing vector field of ρ by Y. Since ρ preserves Ω, the connection T E^{h} is ρ-invariant. It follows that μ_{t} commutes with ρ for any t. Therefore η_{t} = ρ_{−t} ∘ μ_{βt} is the flow on F generated by βX̃ − Y. Note that η_{t} maps any fibre F_{z} to itself symplectically. Let H be the function which we have used to define χ_{K}. Clearly d(H|F_{z_{0}}) = −i_{Y} Ω|F_{z_{0}}. Since iX̃Ω vanishes on each fibre F_{z} one also has

d(−H|F_{z_{0}}) = (−i_{βX̃−Y} Ω)|F_{z_{0}}.

This means that (η_{t}|F_{z_{0}}) is the Hamiltonian flow of −H|F_{z_{0}}. Hence by definition χ_{K} = η_{1}|F_{z_{0}}. On the other hand, by the definition of η_{t}, f^{β} = η_{1}|F_{z_{0}}.

We can now prove our main result about symplectic monodromy.

Theorem 4.17. Let p ∈ ℂ[x_{0}, …, x_{n}] be a weighted homogeneous polynomial with an isolated critical point at 0. Assume that n ≥ 2 and that the sum of the weights w_{i} is not one. Then the symplectic monodromy f defines a class of infinite order in π_{0}(Aut(M, ∂M, ω)).

Proof. Since f^{β} = χ_{K}, it is sufficient to prove that [χ_{K}] has infinite order. Lemma 4.15 shows that σ_{K} = 2β(1 − ∑_{i} w_{i}) ≠ 0. Therefore one only needs to apply Theorem 4.3. The necessary assumptions about M have all been proved above except for [ω] = 0, which is obvious from the definition.

Examples and comments 4.18. (a) Let p ∈ ℂ[x_{0}, x_{1}, x_{2}] be one of the standard models for the du Val (or simple) singularities [1]. These models are weighted homogeneous, and the sum of the weights is > 1. For example, if p(x) = x_{0}^{2} + x_{1}^{3} + x_{1}x_{2}^{3} is the singularity of type (E_{7}) then w_{0} + w_{1} + w_{2} = 1/2 + 1/3 + 2/9 = 19/18. Hence Theorem 4.17 applies, showing that [f] ∈ π_{0}(Aut(M, ∂M, ω)) has infinite order. In contrast, it follows from Brieskorn’s simultaneous resolution [3] that the class of f in π_{0}(Diff(M, ∂M)) has finite order. Hence, at least in these cases, Theorem 4.17 expresses a genuinely symplectic phenomenon.

(b) It is possible that the assumption ∑_{i} w_{i} ≠ 1 might be removed. In fact, there are many cases when ∑_{i} w_{i} = 1 and in which the monodromy has infinite order for topological reasons, for example p(x_{0}, x_{1}, x_{2}, x_{3}) = x_{0}^{4} + x_{1}^{4} + x_{2}^{4} + x_{3}^{4}.

(c) Let p ∈ ℂ[x_{0}, x_{1}] be a weighted homogeneous polynomial with an isolated critical point. Then M is a connected surface and not a disc. The iterate f^{β} of the monodromy can be written as a composition of positive Dehn twists along the connected components of ∂M. Using this one can show easily that [f] ∈ π_{0}(Diff(M, ∂M)) is always of infinite order. This means that Theorem 4.17 holds also for n = 1.

5. Knotted Lagrangian spheres

5a. Generalized Dehn twists. This section contains the basic definitions and some facts, both topological and symplectic, which are used later on. Throughout (M, ω) will be a compact symplectic manifold of dimension 2n. By a Lagrangian sphere in M we will mean a Lagrangian embedding S^{n} ↪ M. Such embeddings will be denoted by the letters l, l_{1}, l_{2}, … and their images by L, L_{1}, L_{2}, . . .. An (A_{k})-configuration, k ≥ 2, is a collection of Lagrangian spheres (l_{1}, …, l_{k}) which are pairwise transverse, such that L_{i} ∩ L_{j} = ∅ for |i − j| ≥ 2 and |L_{i} ∩ L_{i±1}| = 1. The name comes from the relationship with singularity theory. In fact the Milnor fibre of the (A_{k})-singularity, which is the hypersurface

M = {x ∈ ℂ^{n+1} | x_{0}^{k+1} + x_{1}^{2} + ⋯ + x_{n}^{2} = ϵ, |x| ≤ 1} (5.1)

for sufficiently small ϵ ≠ 0, contains such a configuration. This was proved in [28, Proposition 8.1] for n = 2, and the general case can be treated in the same way (here we have used the classical form of the Milnor fibre, rather than the definition adopted in section 4c; this does not really matter, since the Milnor fibre as defined there also contains an (A_{k})-configuration).

Consider U = T_{1}^{∗}S^{n} = {ξ ∈ T^{∗}S^{n} | |ξ| ≤ 1} with its standard symplectic structure ω_{U}. The complement of the zero-section S^{n} ⊂ U carries a Hamiltonian circle action σ whose moment map is the length |ξ|. In the coordinates U = {(u, v) ∈ ℝ^{n+1} × ℝ^{n+1} | |v| = 1, |u| ≤ 1, ⟨u, v⟩ = 0} with ω_{U} = ∑_{j} du_{j} ∧ dv_{j} one has σ_{t}(u, v) = (cos(2πt)u − sin(2πt)v|u|, cos(2πt)v + sin(2πt)u/|u|). Note that σ_{1/2}(u, v) = (−u, −v) extends smoothly over the zero-section. Therefore one can define a diffeomorphism τ of U by setting

τ(ξ) = σ_{ψ(|ξ|)}(ξ),

where ψ is a function with ψ(t) = 1/2 for t ≤ 1/3 and ψ(t) = 0 for t ≥ 2/3. τ is equal to the identity near ∂U, and it acts on the zero-section as the antipodal map. An explicit computation shows that τ is symplectic. We call it a model generalized Dehn twist. Now let l be any Lagrangian sphere in (M, ω). One can always find an embedding j : U ⟶ M such that j|S^{n} = l and j^{∗}ω = δ ω_{U} for some δ > 0. By extending jτj^{−1} trivially over M \ im(j) one defines a symplectic automorphism τ_{l} of M, which we call a generalized Dehn twist along l. It is not difficult to see that the class [τ_{l}] in π_{0}(Aut(M, ω)) (or in π_{0}(Aut(M, ∂M, ω)), if M has a boundary) is independent of the choice of j and ψ. For this reason, we will often speak of τ_{l} as the generalized Dehn twist along l. For n = 1 these maps are just the ordinary (positive) Dehn twists along curves on a surface.

Remark 5.1. It is an open question whether [τ_{l}] depends only on the image L. If l and l′ are two embeddings with the same image, and such that l^{−1} ∘ l′ is isotopic to the identity in Diff(S^{n}), one can easily prove that [τ_{l}] = [τ_{l′}]. Moreover, the same holds if l^{−1} ∘ l′ is an element of O(n + 1) (this shows that, just as in the case n = 1, the choice of orientation of L is not important). For n ≤ 3 it is known that π_{0}(Diff^{+}(S^{n})) = 1 [17] [4] which implies that [τ_{l}] does indeed depend only on L, but in higher dimensions π_{0}(Diff^{+}(S^{n})) is often nonzero.

We will first look at generalized Dehn twists from a topological point of view. Since these maps are a symplectic form of the classical Picard-Lefschetz transformations, their action on homology is given by the familiar formula

(τ_{l})_{∗}(c) = c − (−1)^{n(n−1)/2}(c · [L])[L]. (5.2)

Using the fact that [L] · [L] = (−1)^{n(n−1)/2}χ(L) for any n-dimensional Lagrangian submanifold, one obtains the following

Lemma 5.2. (a) If n is even then (τ_{l})_{∗} has order two. If n is odd then (τ_{l})_{∗} has infinite order iff [L] is not a torsion class (otherwise (τ_{l})_{∗} = id).

(b) Assume that n is odd, and that l_{1}, l_{2} are two Lagrangian spheres with [L_{1}]·[L_{2}] = ±1. Set g = (τ_{l_{2}}τ_{l_{1}})^{3}. Then g^{2} induces the identity on homology.

For n = 2 it is known [28, Lemma 6.3] that τ_{l}^{2} is actually isotopic to the identity in Diff(M). It seems natural to ask whether this holds for other even n; there is also the analogous problem for the map g^{2} defined in (b). In both cases the answer is unknown to the author. However, there are some weaker topological results which are easier to obtain, and which are sufficient for our purpose.

Lemma 5.3. (a) Assume that n is even, and that there is an (A_{2})-configuration of Lagrangian spheres (l_{1}, l_{2}) in M. Then l_{1}^{(k)} = τ_{l_{2}}^{2k} ∘ l_{1} is isotopic to l_{1} through smooth embeddings S^{n} ↪ M for any k ∈ ℤ.

(b) Assume that n is odd and ≥ 5, and that there is an (A_{3})-configuration of Lagrangian spheres (l_{1}, l_{2}, l_{3}) in M. Then l_{1}^{(k)} = g^{2k} ∘ l_{1}, where g = (τ_{l_{2}}τ_{l_{3}})^{3}, is isotopic to l_{1} through smooth embeddings S^{n} ↪ M for any k ∈ ℤ.

Proof. (a) If n = 2 then τ_{l_{2}}^{2} is isotopic to the identity in Diff(M), which implies our result. Hence we can assume that n ≥ 4. let W ⊂ M be a regular neighbourhood of L_{1}∪L_{2}. Since W retracts onto L_{1}∪L_{2}, it is (n−1)-connected with H_{n}(W) = ℤ⟨L_{1}⟩⊕ ℤ⟨L_{2}⟩. We can assume that τ_{l_{2}} has been chosen in such a way that it preserves W. Lemma 5.2(a) shows that all the embeddings l_{1}^{(k)} represent the same homology class in W. Hence, by Hurewicz’s theorem, they are homotopic as continuous maps S^{n} ⟶ W. The proof is completed by applying a result of Haefliger [9] which shows that any two homotopic embeddings S^{n} ⟶ W are differentiably isotopic.

(b) is proved in the same way. The condition n ≥ 5 is necessary in order to use Haefliger’s result. The result is easily seen to be false for n = 1, but the author was unable to decide the remaining case n = 3. In that dimension, there are obstructions for two homotopic embeddings S^{3} ⟶ W to be differentiably isotopic. These obstructions are completely understood in principle, see [10, Corollary B], but not so easy to compute in practice.

Now consider an (A_{2})-configuration (l_{1}, l_{2}) of Lagrangian spheres in M. From (5.2) it follows that, in any dimension, [τ_{l_{2}}(L_{1})] = ±[τ_{l_{1}}^{−1}(L_{2})] ∈ H_{n}(M). In fact the following stronger result is true:

Lemma 5.4. τ_{l_{1}}(L_{2}) and τ_{l_{2}}^{−1}(L_{1}) are isotopic as (unoriented) Lagrangian submanifolds of (M, ω). In fact, both of them are Lagrangian isotopic to the surgery L_{1}#L_{2}.

This was proved in [28, Appendix] for n = 2, and the argument given there carries over to arbitrary n. For future use we need to recall one aspect of the proof: both τ_{l_{1}}(L_{2}) and L_{1}#L_{2} agree with L_{2} away from a neighbourhood of L_{1} which, by an appropriate choice of τ_{l_{1}} and of the surgery, can be made arbitrarily small. The Lagrangian isotopy constructed in [28] between them remains constant outside this neighbourhood. Similarly, the isotopy between L_{1}#L_{2} and τ_{l_{2}}^{−1}(L_{1}) is constant outside a neighbourhood of L_{2}.

Lemma 5.5. Assume that (l_{1}, l_{2}) form an (A_{2})-configuration of Lagrangian spheres in M, and set g = (τ_{l_{1}}τ_{l_{2}})^{3}. Then g(L_{1}) is Lagrangian isotopic to L_{1}, and g(L_{2}) is Lagrangian isotopic to L_{2}.

Proof. Using Lemma 5.4 and the obvious fact that τ_{l}(L) = L for any l, one sees that (τ_{l_{1}}τ_{l_{2}})^{3}(L_{2}) ≃ (τ_{l_{1}}τ_{l_{2}})^{2}τ_{l_{2}}^{−1}(L_{1}) ≃ τ_{l_{1}}τ_{l_{2}}τ_{l_{1}}(L_{1}) ≃ τ_{l_{1}}τ_{l_{1}}^{−1}(L_{2}) = L_{2}, where ≃ stands for Lagrangian isotopy. The proof for L_{1} is similar.

5b. The graded point of view. From now on we assume that (M, ω) satisfies 2c_{1}(M, ω) = 0 and that its dimension 2n is at least four (the case of classical Dehn twists, n = 1, is more complicated because T^{∗}S^{1} admits infinitely many different Maslov coverings). Choose an ∞-fold Maslov covering L^{∞} on M. Lemma 2.3 implies that any Lagrangian submanifold L ⊂ M with H^{1}(L) = 0 admits an L^{∞}- grading. Let l be a Lagrangian sphere in M and τ_{l} the generalized Dehn twist along it defined using some embedding j : U ↪ M. By definition τ_{l} is the identity outside im(j).

Lemma 5.6. There is a unique L^{∞}-grading τ̃_{l} of τ_{l} which acts trivially on the part of L^{∞} which lies over M \ im(j).

Proof. The uniqueness is obvious. To prove the existence, consider the local model U = T_{1}^{∗}S^{n}. Since c_{1}(U, ω_{U}) = 0 and H^{1}(U) = 0, U admits a unique ∞-fold Maslov covering L_{U}^{∞}. Remark 2.5 says that the model generalized Dehn twist τ has a unique L_{U}^{∞}-grading τ̃ which acts trivially on the part of L_{U}^{∞} which lies over ∂U. Now, for an arbitrary Lagrangian sphere l in M and Maslov covering L^{∞}, one can identify L^{∞}|im(j) with L_{U}^{∞} and then extend τ̃ by the identity to an L^{∞}-grading τ̃_{l} of τ_{l}.

We emphasize that this preferred grading τ̃_{l} does not depend on the choice of a grading for L.

Lemma 5.7. The preferred L^{∞}-grading τ̃_{l} satisfies τ̃_{l}(L̃) = L̃[1 − n] for any L^{∞}- grading L̃ of L.

Proof. Clearly it is enough to prove the corresponding fact for the local model τ̃. We find it convenient to use complex coordinates U = {ξ ∈ ℂ^{n+1} | |re ξ| ≤ 1, |im ξ| = 1, ⟨re ξ, im ξ⟩_{ℝ} = 0}. The tangent spaces are T U_{ξ} = {η ∈ ℂ^{n+1} | ⟨im ξ, im η⟩_{ℝ} = 0, ω_{ℂ^{n+1}}(ξ̄, η) = 0}. Hence if Λ is a Lagrangian subspace of T U_{ξ}, Λ ⊕ ℝξ̄ is a Lagrangian subspace of ℂ^{n+1}. This stabilization defines a map r : L ⟶ L(2n + 2) whose restriction to any fibre L_{ξ} induces an isomorphism of the fundamental groups. It follows that as our Maslov covering L_{U}^{∞}, we can take the pullback r^{∗}(L^{∞}(2n+2)) of the universal cover of L(2n + 2).

τ^{2} is the time-one map of the Hamiltonian flow ϕ_{t}(ξ) = σ_{t(2ψ(|ξ|)−1)}(ξ). Lift (ϕ_{t}) to an isotopy (ϕ̃_{t}) of L_{U}^{∞}-graded symplectic automorphisms, starting with ϕ̃_{0} = id. By definition, ϕ_{t} is equal to the identity in a neighbourhood of the zero-section S^{n} ⊂ U for any t. This implies that ϕ̃_{t}(S̃^{n}) = S̃^{n} for any t and any L_{U}^{∞}-grading S̃^{n} of S^{n}. On the other hand, ϕ̃_{1} acts as some shift [k] on the part of L^{∞} which lies over ∂U. Using the fact that L_{U}^{∞} is defined as a pullback, it is not difficult to see that k can be computed as follows: choose a point ξ ∈ ∂U and a Lagrangian subspace Λ ⊂ T U_{ξ}. Then λ(t) = r(Dϕ_{t}(Λ)) is a loop in L(2n + 2), and one has

k = −⟨C(2n + 2), [λ]⟩.

To determine k it is convenient to take Λ tangent to ∂U, because ϕ_{t}|∂U agrees with the inverse of the standard diagonal circle action on ℂ^{n+1}. Then λ(t) = e^{−2πit}Λ ⊕ e^{2πit}(ℝξ̄), which means that k = 2n − 2. We now know that ϕ̃_{1} ∘ [2 − 2n] is an L_{U}^{∞}- grading of τ^{2} which acts trivially on the boundary of U. Therefore it must be the square τ̃^{2} of the preferred grading τ̃. Hence τ̃^{2}(S̃^{n}) = ϕ̃_{1}(S̃^{n}[2 − 2n]) = S̃^{n}[2 − 2n] which proves that τ̃(S̃^{n}) = S̃^{n}[1 − n].

The next two results are graded analogues of Lemma 5.4 and Lemma 5.5.

Lemma 5.8. Assume that (l_{1}, l_{2}) form an (A_{2})-configuration in M. Choose L^{∞}- gradings L̃_{1}, L̃_{2} such that the index at the only intersection point {x_{0}} = L_{1} ∩ L_{2} is Ĩ(L̃_{1}, L̃_{2}; x_{0}) = 1. Then τ̃_{l_{1}}(L̃_{2}) and τ̃_{l_{2}}^{−1}(L̃_{1}) are isotopic as L^{∞}-graded Lagrangian submanifolds.

Proof. Let Σ = L_{1}#L_{2} be the Lagrangian surgery, and Σ̃ the L^{∞}-grading from Lemma 2.14. Recall that τ_{l_{1}}(L_{2}) and Σ agree outside a neighbourhood of L_{1}. The gradings τ̃_{l_{1}}(L̃_{2}) and Σ̃ also agree there: this follows from the fact that τ̃_{l_{1}} is trivial away from L_{1}, and that the grading Σ̃ agrees with L̃_{2} on Σ ∩ L_{2}. In the proof of Lemma 5.4 we have used an isotopy from τ_{l_{1}}(L_{2}) to Σ which is concentrated near L_{1}. By considering a point which remains fixed, one sees that when one lifts this to an isotopy of graded Lagrangian submanifolds, it connects τ̃_{l_{1}}(L̃_{2}) with Σ̃. The same kind of argument shows that Σ̃ is isotopic to τ̃_{l_{2}}^{−1}(L̃_{1}).

Lemma 5.9. Assume that (l_{1}, l_{2}) is an (A_{2})-configuration in M, and define g̃ = (τ̃_{l_{1}} τ̃_{l_{2}})^{3}. Then for any L^{∞}-gradings L̃_{1}, L̃_{2} one has

g̃(L̃_{1}) ≃_{gr} L̃_{1}[4 − 3n], g̃(L̃_{2}) ≃_{gr} L̃_{2}[4 − 3n],

where ≃_{gr} stands for isotopy of L^{∞}-graded Lagrangian submanifolds.

Proof. Clearly, if the result holds for some grading of L_{1}, L_{2} then it holds for all gradings. Hence we can assume that the absolute index Ĩ(L̃_{1}, L̃_{2}; x_{0}) at {x_{0}} = L_{1} ∩ L_{2} is 1. Lemma 5.8 says that τ̃_{l_{1}}(L̃_{2}) ≃_{gr} τ̃_{l_{2}}^{−1}(L̃_{1}). Applying the same result to L̃_{2} and L̃_{1}[n − 2], which satisfy Ĩ(L̃_{2}, L̃_{1}[n − 2]; x_{0}) = n − Ĩ(L̃_{1}[n − 2], L̃_{2}; x_{0}) = n − Ĩ(L̃_{1}, L̃_{2}; x_{0}) − n + 2 = 1, yields τ̃_{l_{2}}(L̃_{1}) ≃_{gr} τ̃_{l_{1}}^{−1}(L̃_{2})[2 − n]. Using these two equations and Lemma 5.6 one computes

(τ̃_{l_{1}} τ̃_{l_{2}})^{3}(L̃_{2}) = (τ̃_{l_{1}} τ̃_{l_{2}})^{2}τ̃_{l_{1}}(L̃_{2})[1 − n] ≃_{gr} (τ̃_{l_{1}} τ̃_{l_{2}})^{2}τ̃_{l_{2}}^{−1}(L̃_{1})[1 − n] =
= τ̃_{l_{1}} τ̃_{l_{2}}(L̃_{1})[2 − 2n] ≃_{gr} τ̃_{l_{1}}τ̃_{l_{1}}^{−1}(L̃_{2})[4 − 3n] = L̃_{2}[4 − 3n].

The result for L̃_{1} is proved in the same way.

5c. Symplectically knotted Lagrangian spheres. After these preparations, we can now apply the ‘graded’ techniques to the construction of symplectically knotted Lagrangian spheres. The two parallel results obtained in this way (for even-dimensional and odd-dimensional spheres, respectively) are

Theorem 5.10. Let (M^{2n}, ω) be a compact symplectic manifold with contact type boundary, with n even, which satisfies [ω] = 0 and 2c_{1}(M, ω) = 0. Assume that M contains an (A_{3})-configuration (l_{0}, l_{1}, l_{2}) of Lagrangian spheres. Then M contains infinitely many symplectically knotted Lagrangian spheres. More precisely, if one defines L_{1}^{(k)} = τ_{l_{2}}^{2k}(L_{1}) for k ∈ ℤ, then all the L_{1}^{(k)} are isotopic as smooth submanifolds of M, but no two of them are isotopic as Lagrangian submanifolds.

Theorem 5.11. Let (M^{2n}, ω) be a compact symplectic manifold with contact type boundary, with n ≥ 5 odd, which satisfies [ω] = 0 and 2c_{1}(M, ω) = 0. Assume that M contains an (A_{4})-configuration (l_{0}, l_{1}, l_{2}, l_{3}) of Lagrangian spheres. Then M contains infinitely many symplectically knotted Lagrangian spheres. More precisely, if one defines L_{1}^{(k)} = g^{2k}(L_{1}) for k ∈ ℤ, where g = (τ_{l_{2}}τ_{l_{3}})^{3}, then all the L_{1}^{(k)} are isotopic as smooth submanifolds of M, but no two of them are isotopic as Lagrangian submanifolds.

Examples of manifolds satisfying these conditions are the Milnor fibres (5.1). Together these two Theorems prove the existence of infinitely many knotted Lagrangian n-spheres in any dimension n ≠ 1, 3. As mentioned in the Introduction, the case n = 2 has been proved before in [28], and the proof given there would work in all even dimensions. Nevertheless, our new proof is substantially simpler.

Proof of Theorem 5.10. Let {x_{0}} = L_{0} ∩ L_{1} and {x_{1}} = L_{1} ∩ L_{2}. Choose an ∞-fold Maslov covering L^{∞} on M, and L^{∞}-gradings L̃_{0}, L̃_{1}, L̃_{2} in such a way that Ĩ(L̃_{0}, L̃_{1}; x_{0}) = Ĩ(L̃_{1}, L̃_{2}; x_{1}) = 0. Set L̃_{1}^{(k)} = τ̃_{l_{2}}^{2k}(L̃_{1}). The assumptions on M imply that Floer cohomology (and its graded version) are well-defined for all (graded) Lagrangian submanifolds of M with vanishing first Betti number. Because L_{0} ∩L_{1} and L_{1} ∩L_{2} intersect in a single point, which has index zero, it follows from the definition of graded Floer cohomology that

HF^{∗}(L̃_{0}, L̃_{1}) = ℤ/2^{[0]}, HF^{∗}(L̃_{1}, L̃_{2}) = ℤ/2^{[0]}.

Here ℤ/2^{[k]} stands for the ℤ-graded group which is ℤ/2 in degree k and zero in other degrees. Since L_{0} ∩ L_{2} = ∅, one can choose τ_{l_{2}} in such a way that it acts trivially in a neighbourhood of L_{0}. This implies that L_{0} ∩ L_{1}^{(k)} = L_{0} ∩ L_{1} = {x_{0}}. Because the preferred grading τ̃_{l_{2}} also acts trivially near L_{0}, one has Ĩ(L̃_{0}, L̃_{1}^{(k)}; x_{0}) = Ĩ(L̃_{0}, L̃_{1}; x_{0}) = 0 and hence (for any k)

HF^{∗}(L̃_{0}, L̃_{1}^{(k)}) = ℤ/2^{[0]}. (5.3)

On the other hand, using the invariance of graded Floer cohomology under graded symplectic automorphisms together with Lemma 5.6, one finds that

\begin{aligned}HF^{∗}(L̃_{1}^{(k)}, L̃_{2}) & ≅ HF^{∗}(L̃_{1}, τ̃_{l_{2}}^{−2k}(L̃_{2})) = \\ & = HF^{∗}(L̃_{1}, L̃_{2}[2k(n − 1)]) = ℤ/2^{[2k(1−n)]}. \end{aligned} (5.4)

Now assume that for some L_{1}^{(k)}, for some k, is Lagrangian isotopic to L_{1}. This implies that L̃_{1}^{(k)} is graded Lagrangian isotopic to L̃_{1}[r] for some r ∈ ℤ. Using the isotopy invariance property of Floer cohomology one obtains

HF^{∗}(L̃_{0}, L̃_{1}^{(k)}) ≅ HF^{∗}(L̃_{0}, L̃_{1}[r]) = ℤ/2^{[−r]} and HF^{∗}(L̃_{1}^{(k)}, L̃_{2}) ≅ ℤ/2^{[r]}.

Comparing the first part of this equation with (5.3) yields r = 0. Comparing the second part with (5.4) yields r = 2k(1 − n) ≠ 0 and hence that k = 0. This means that L_{1}^{(k)} is not Lagrangian isotopic to L_{1} unless k = 0. Because of the way in which the L_{1}^{(k)} are defined, it follows that no two of them are Lagrangian isotopic. The topological part of the theorem follows from Lemma 5.3(a).

Proof of Theorem 5.11. Since this is very similar to the proof of Theorem 5.10, we will be more brief. Take an ∞-fold Maslov covering L^{∞} and gradings L̃_{0}, L̃_{1}, L̃_{2} such that HF^{∗}(L̃_{0}, L̃_{1}) = HF^{∗}(L̃_{1}, L̃_{2}) = ℤ/2^{[0]}. Define g̃ = (τ̃_{l_{2}} τ̃_{l_{3}})^{3} and L̃_{1}^{(k)} = g̃^{2k}(L̃_{1}). One can choose τ_{l_{2}}, τ_{l_{3}} in such a way that g and g̃ act trivially near L_{0}. This implies that HF^{∗}(L̃_{0}, L̃_{1}^{(k)}) = ℤ/2^{[0]} for any k. Using Lemma 5.9 one computes HF^{∗}(L̃_{1}^{(k)}, L̃_{2}) ≅ HF^{∗}(L̃_{1}, g̃^{−2k}(L̃_{2})) ≅ HF^{∗}(L̃_{1}, L̃_{2}[2k(3n − 4)]) = ℤ/2^{[2k(4−3n)]}. The assumption that L̃_{1}^{(k)}, for some k ≠ 0, is graded Lagrangian isotopic to L̃_{1}[r], for some r ∈ ℤ, leads to the contradiction 0 = r = 2k(4 − 3n). The topological part of the theorem is Lemma 5.3(b).

5d. K3 and Enriques surfaces. Let (M, ω) be a closed symplectic four-manifold such that c_{1}(M) is a torsion class. We want to consider Lagrangian spheres in M. This is a borderline case for Floer cohomology, where the conventional methods do not yield a completely satisfactory theory. The problems appear when one tries to prove that the Floer group is independent of the choice of almost complex structure. We will now explain what parts of Floer’s construction can be salvaged from this breakdown. Throughout the whole of this section, all Lagrangian submanifolds are assumed to be two-spheres. For such an L ⊂ M, let J^{reg}(L) be the set of ω-compatible almost complex structures J such that there are no non-constant J-holomorphic maps ℂP^{1} → M or (D^{2}, ∂D^{2}) → (M, L).

Lemma 5.12. J^{reg}(L) is a dense subset of the space of all ω-compatible almost complex structures.

Proof. The virtual dimension of the space of J-holomorphic spheres in a homology class A is 4 + 2⟨c_{1}(M), A⟩ − 6 = −2. Similarly, the virtual dimension of the space of J-holomorphic discs representing B ∈ H_{2}(M, L) is 2 + ⟨2c_{1}(M, L), B⟩ − 3 = −1. Here 2c_{1}(M, L) ∈ H^{2}(M, L) is the relative first Chern class, and we have used the assumption that H^{1}(L) = 0 to conclude that c_{1}(M, L) is a torsion class. Standard transversality results say that for generic J there are no simple J-holomorphic spheres and no somewhere injective J-holomorphic discs. The first result implies that there are no non-constant J-spheres at all, because any such sphere covers a simple one. Similarly, the second result implies that there are no J-holomorphic discs. However, this time the argument is more complicated: it relies on the structure theorem of Kwon-Oh [13] or on the simpler form given by Lazzarini [15].

Let L_{0}, L_{1} ⊂ M be two Lagrangian submanifolds which intersect transversally. Fix J_{0} ∈ J^{reg}(L_{0}), J_{1} ∈ J^{reg}(L_{1}). We define J^{reg}(L_{0}, J_{0}, L_{1}, J_{1}) to be the space of smooth families J = (J_{t})_{0≤t≤1} of compatible almost complex structures, connecting the given J_{0}, J_{1}, which satisfy the following conditions: (a) there are no non-constant J_{t}-holomorphic maps ℂP^{1} → M for any t; (b) any solution of Floer’s equation, u ∈ M(x_{−}, x_{+}; J) for x_{−}, x_{+} ∈ L_{0} ∩ L_{1}, is regular.

Lemma 5.13. J^{reg}(L_{0}, J_{0}, L_{1}, J_{1}) is a dense subset of of the space of all families J which connect J_{0} with J_{1}.

The proof is again a combination of standard transversality arguments. For J ∈ J^{reg}(L_{0}, J_{0}, L_{1}, J_{1}) one can define the Floer cohomology HF (L_{0}, J_{0}, L_{1}, J_{1}; J) in the familiar way, using a suitable Novikov ring Λ as coefficient ring. The next step in the construction is

Lemma 5.14. HF (L_{0}, J_{0}, L_{1}, J_{1}; J) is independent of J ∈ J^{reg}(L_{0}, J_{0}, L_{1}, J_{1}) up to canonical isomorphism.

The proof uses the continuation equation ∂_{s}u + Ĵ_{s,t}(u)∂_{t}u = 0 for a two-parameter family Ĵ = (Ĵ_{s,t}) of almost complex structures such that Ĵ_{s,0} = J_{0} and Ĵ_{s,1} = J_{1} for all s ∈ ℝ. There will be Ĵ_{s,t}-holomorphic spheres (for isolated values of s, t) even if Ĵ is generic, but these can be dealt with as in [11]. Following the usual custom, we will from now on omit J from the notation of Floer homology.

The problematic issue mentioned above is whether HF (J_{0}, L_{0}, J_{1}, L_{1}) is independent of J_{0}, J_{1}. We will use only a special case, in which the result is obvious:

Lemma 5.15. Assume that L_{0}, L_{1} ⊂ M intersect transversally in a single point. Then HF (L_{0}, J_{0}, L_{1}, J_{1}) ≅ Λ for all J_{0} ∈ J^{reg}(L_{0}), J_{1} ∈ J^{reg}(L_{1}).

By definition, Floer cohomology is invariant under symplectic automorphisms, in the sense that HF (ϕ(L_{0}), ϕ_{∗}(J_{0}), ϕ(L_{1}), ϕ_{∗}(J_{1})) ≅ HF (L_{0}, J_{0}, L_{1}, J_{1}). The next result is a weak form of isotopy invariance.

Lemma 5.16. Assume that L_{0}, L_{1} are transverse and choose J_{0} ∈ J^{reg}(L_{0}), J_{1} ∈ J^{reg}(L_{1}). Let ϕ ∈ Aut(M, ω) be a map which is Hamiltonian isotopic to the identity, and such that ϕ(L_{1}) intersects L_{0} transversally. Then ϕ_{∗}(J_{1}) ∈ J^{reg}(ϕ(L_{1})) and

HF (L_{0}, J_{0}, ϕ(L_{1}), ϕ_{∗}(J_{1})) ≅ HF (L_{0}, J_{0}, L_{1}, J_{1}).

Outline of the proof. take a function H ∈ C^{∞}([0; 1] × M, ℝ) with H_{t} = 0 for t ≤ 1/3 or t ≥ 2/3, such that the Hamiltonian isotopy (ϕ_{t}^{H}) generated by it satisfies ϕ_{1}^{H} = ϕ^{−1}. Let X_{t} be the Hamiltonian vector field of H_{t}, and ψ ∈ C^{∞}(ℝ, ℝ) a cutoff function with ψ(s) = 0 for s ≤ 0 and ψ(s) = 1 for s ≥ 1. One considers finite energy solutions u : ℝ × [0; 1] ⟶ M of the equation

\{ \begin{matrix}∂_{s}u + Ĵ_{s,t}(u)(∂_{t}u − ψ(s)X_{t}(u)) = 0, \\ u(s, 0) ∈ L_{0}, u(s, 1) ∈ L_{1}. \end{matrix} (5.5)

Here Ĵ = (Ĵ_{s,t}) is a two-parameter family of ω-compatible almost complex structures such that Ĵ_{s,0} = J_{0}, Ĵ_{s,1} = J_{1} for all s. If one writes u(s, t) = ϕ_{t}^{H}(w(s, t)) then (5.5) in the region s ≥ 1 reads

\{ \begin{matrix}∂_{s}w + Ĵ′_{s,t}(w)∂_{t}w = 0, \\ w(s, 0) ∈ L_{0}, w(s, 1) ∈ ϕ(L_{1}). \end{matrix}

where Ĵ′_{s,t} = (ϕ_{t}^{H})_{∗}^{−1}Ĵ_{s,t}, in particular Ĵ′_{s,1} = ϕ_{∗}J_{1}. Following the usual strategy, one can use solutions of (5.5) to define a map

HF (L_{0}, J_{0}, L_{1}, J_{1}) ⟶ HF (L_{0}, J_{0}, ϕ(L_{1}), ϕ_{∗}(J_{1})). (5.6)

The main technical point is that there can be no bubbling off of holomorphic discs, since the almost complex structures Ĵ_{s,t} for t = 0, 1 do not admit such discs. Standard arguments of a similar kind show that (5.6) is an isomorphism.

If 2c_{1}(M) = 0, one can consider graded Lagrangian submanifolds, and obtains graded Floer groups HF^{∗}(L̃_{0}, J_{0}, L̃_{1}, J_{1}) with properties analogous to those above.

Theorem 5.17. Let (M, ω) be a symplectic four-manifold with 2c_{1}(M, ω) = 0 and which contains an (A_{3})-configuration (l_{0}, l_{1}, l_{2}) of Lagrangian two-spheres. Define L_{1}^{(k)} = τ_{l_{2}}^{2k}(L_{1}). Then all the L_{1}^{(k)} are isotopic as smooth submanifolds, but no two of them are isotopic as Lagrangian submanifolds.

Proof. This is the same argument as in Theorem 5.10, except that one has to be more careful about the properties of Floer cohomology. Choose an ∞-fold Maslov covering L^{∞} and gradings L̃_{k} such that Ĩ(L̃_{0}, L̃_{1}; x_{0}) = Ĩ(L̃_{1}, L̃_{2}; x_{1}) = 0, where x_{0}, x_{1} are the unique intersection points. Fix some k ≠ 0 and write L′_{1} = τ_{l_{2}}^{2k}(L_{1}), L̃′_{1} = τ̃_{l_{2}}^{2k}(L̃_{1}). Assume that L_{1} ≃ L′_{1}, so that L̃_{1}[r] ≃_{gr} L̃′_{1} for some r ∈ ℤ. One can embed the Lagrangian isotopy into a Hamiltonian isotopy (ϕ_{t}) of M. The graded analogue of Lemma 5.16 shows that

HF^{∗}(L̃_{0}, J_{0}, L̃_{1}, J_{1}) ≅ HF^{∗−r}(L̃_{0}, J_{0}, L̃′_{1}, (ϕ_{1})_{∗}(J_{1})),
HF^{∗}(L̃_{1}, J_{1}, L̃_{2}, J_{2}) ≅ HF^{∗+r}(L̃′_{1}, (ϕ_{1})_{∗}(J_{1}), L̃_{2}, J_{2})

for all J_{m} ∈ J^{reg}(L_{m}), m = 0, 1, 2. On the other hand, by using the graded version of Lemma 5.15 and arguing as in the proof of Theorem 5.10 one finds that

HF^{∗}(L̃_{0}, J_{0}, L̃_{1}, J_{1}) = HF^{∗}(L̃_{1}, J_{1}, L̃_{2}, J_{2}) = Λ^{[0]},
HF^{∗}(L̃_{0}, J_{0}, L̃′_{1}, (ϕ_{1})_{∗}(J_{1})) = Λ^{[0]}, HF^{∗}(L̃′_{1}, (ϕ_{1})_{∗}(J_{1}), L̃_{2}, J_{2}) = Λ^{[−2k]},

which leads to a contradiction.

Corollary 5.18. For (M, ω) as in Theorem 5.17, the map π_{0}(Aut(M, ω)) ⟶ π_{0}(Diff(M)) has infinite kernel.

Proof. We have already quoted the fact that τ_{l_{2}}^{2} is trivial in π_{0}(Diff(M)) [28, Lemma 6.3]. Theorem 5.17 obviously implies that [τ_{l_{2}}^{2}] ∈ π_{0}(Aut(M, ω)) has infinite order.

Example 5.19. Recall that an Enriques surface is an algebraic surface with fundamental group ℤ/2, whose universal cover is a K3 surface. Enriques surfaces satisfy 2c_{1} = 0. We will now construct an Enriques surface which contains an (A_{3})-configuration of Lagrangian spheres. Consider the quartic surface X ⊂ ℂP^{3} defined by

(x_{0}^{2} + x_{1}^{2})^{2} + x_{0}^{2}x_{2}^{2} + x_{2}^{4} + x_{3}^{4} = 0. (5.7)

X has two singular points [1 : ±i : 0 : 0] of type (A_{3}). Let Y be the minimal resolution of singularities of X. It is a K3 surface and hence has a holomorphic symplectic form Ω. Complex conjugation defines an anti-holomorphic involution on X, which is free because (5.7) has no nonzero real solutions. Because of the uniqueness of minimal resolutions, this involution lifts to an anti-holomorphic involution on Y, which we denote by ι. Obviously ι is again free. Since the holomorphic symplectic form is unique up to a constant, it satisfies ι^{∗}Ω = zΩ̄ for some z ∈ S^{1}. By rescaling Ω we can assume that z = 1. Then ω = re Ω descends to a real symplectic form on the quotient M = Y/ι. The singular points of X give rise to two disjoint (A_{3})-configuration of rational curves in Y, which are exchanged by ι. These curves are Lagrangian with respect to ω, hence descend to a single (A_{3})-configuration of Lagrangian spheres on M. To see that M is Kähler one argues as follows: let β ∈ Ω^{1,1} be a positive form on Y such that ι^{∗}β = −β. Such a form can be constructed e.g. by averaging. Let g be the unique Ricci-flat Kähler metric which has the same Kähler class as β. The uniqueness theorem for such metrics implies that g is ι-invariant. The metric g is hyperkähler, which means that there are complex structures J and K such that Ω(v, w) = g(v, Jw) + ig(v, Kw). It follows that J is ι-invariant and hence descends to a complex structure on M which is compatible with ω. This means that M is Kähler and in fact an Enriques surface.

Example 5.20. K3 surfaces have c_{1} = 0. As an example of a K3 surface containing an (A_{3})-configuration of Lagrangian spheres one can take the manifold (Y, re Ω) constructed in the previous example.

References

[1] V. I. Arnol’d, Normal forms for functions near degenerate critical points, the Weyl groups of A_{k}, D_{k}, E_{k} and Lagrangian singularities, Funct. Anal. Appl. 6 (1972), 254–272.

[2] A. Besse, Manifolds all of whose geodesics are closed, Erg. der Mathematik und ihrer Grenzgebiete, vol. 93, Springer, 1978.

[3] E. Brieskorn, Die Auflösung rationaler Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255–270.

[4] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ_{4} = 0), Lecture Notes in Math., vol. 53, Springer, 1968.

[5] S. Dostoglou and D. Salamon, Self dual instantons and holomorphic curves, Annals of Math. 139 (1994), 581–640.

[6] J. Duistermaat, On the Morse index in variational calculus, Advances in Math. 21 (1976), 173–195.

[7] A. Floer, A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988), 393–407.

[8] , Witten’s complex and infinite dimensional Morse theory, J. Differential Geom. 30 (1989), 207–221.

[9] A. Haefliger, Plongements différentiables des variétés dans variétés, Comm. Math. Helv. 36 (1962), 47–82.

[10] , Knotted spheres and related geometric problems, Proceedings of the ICM, Moscow, 1966, pp. 437–445.

[11] H. Hofer and D. Salamon, Floer homology and Novikov rings, The Floer memorial volume (H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, eds.), Progress in Mathematics, vol. 133, Birkhäuser, 1995, pp. 483–524.

[12] W. Klingenberg, Riemannian geometry, De Gruyter, 1982.

[13] D. Kwon and Y.-G. Oh, Structure of the image of (pseudo)-holomorphic discs with totally real boundary conditions, Preprint (final version).

[14] H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton Univ. Press, 1989.

[15] L. Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Preprint, December 1998.

[16] J. Milnor, Singular points of complex hypersurfaces, Princeton Univ. Press, 1968.

[17] J. Munkres, Differentiable isotopies of the 2-sphere, Notices of the Amer. Math. Soc. 5 (1958), [18] Y.-G. Oh, On the structure of pseudo-holomorphic discs with totally real boundary conditions,

Preprint (revised version, May 1996).

[19] , Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs I, Comm. Pure Appl. Math. 46 (1993), 949–994.

[20] , Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Int. Math. Res. Notices (1996), 305–346.

[21] L. Polterovich, Surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), 198–210.

[22] M. Poźniak, Floer homology, Novikov rings and clean intersections, Ph.D. thesis, University of Warwick, 1994.

[23] J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844.

[24] , The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), 1–33.

[25] D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.

[26] P. Seidel, Floer homology and the symplectic isotopy problem, Ph.D. thesis, Oxford University, 1997.

[27] , π_{1} of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), 1046–1095.

[28] , Lagrangian two-spheres can be symplectically knotted, Preprint, 1998.

[29] C. Viterbo, Intersection des sous-variétés Lagrangiennes, fonctionelles d’action et indice des systèmes Hamiltoniens, Bull. Soc. Math. France 115 (1987), 361–390.