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Relative Flux Homomorphism in Symplectic Geometry

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We show that some quotients of this group are stable under symplectic reduction.


Introduction
In 1969 Calabi introduced the flux homomorphism for a symplectic manifold (M, ω) ( [1]). Let Symp 0 (M, ω) denote the path component of the Lie group of symplectomorphisms of (M, ω) containing the identity. Also, let Symp 0 (M, ω) denote the universal covering space of Symp 0 (M, ω). Note that an element of Symp 0 (M, ω) is represented by a homotopy class of a smooth path ψ t in Symp 0 (M, ω) connecting the identity ψ 0 = id to ψ = ψ 1 , where the homotopies fix the end points of the paths. We will denote the homotopy class of ψ t by {ψ t } ∈ Symp 0 (M, ω). Any such ψ t determines a smooth family of vector fields X t : M → T * M defined by the formula d dt ψ t = X t • ψ t . In the next section, we will define a relative flux homomorphism in the presence of a Lagrangian submanifold L ⊆ M , mainly following both the treatment and the notation of Section 10 of [4]. Most of the results and proofs of this section are analogous to those of [4] and we will omit the proofs of some of them unless the proof includes some new ingredients or makes the text more comprehensible. In the third section, we will show that some quotients of Symp 0 (M, ω) are stable under symplectic reduction.

Relative Flux Homomorphism
Throughout this section, unless stated otherwise, (M, ω) is a connected closed symplectic manifold and L a closed Lagrangian submanifold.
Let Symp(M, L, ω) denote the subgroup of Symp(M, ω) consisting of symplectomorphisms leaving the Lagrangian submanifold L invariant and Symp 0 (M, L, ω) the path component of Symp(M, L, ω) containing the identity.
Identifying the homotopy class of the constant loop at the identity element of the group Symp 0 (M, L, ω) with that of Symp 0 (M, ω), we obtain a canonical map from Symp 0 (M, L, ω) onto a subgroup of Symp 0 (M, ω), the subgroup of homotopy classes {ψ t } in Symp 0 (M, ω), where each ψ t leaves L invariant. Then we have the following result.  Let {ψ t } ∈ Symp 0 (M, L, ω) and X t the vector field defined by Since ψ t leaves L invariant, for any p ∈ L we have X t (p) ∈ T p L. Note that to prove the first assertion it suffices to show that for any smooth loop γ : S 1 → L the integral 1 0 1 0 ω(X t (γ(s),γ(s))) ds dt = 0.
However, this trivially holds since both X t (γ(s)) andγ(s) lie in T γ(s) L, and L is a Lagrangian submanifold. Now suppose that L is connected. Then the exact sequence implies that the flux homomorphism takes values in the isomorphic image of the relative cohomology group To show surjectivity let u ∈ Ω 1 (M ) be a closed 1-form which is exact on L. Then u |L = dh for some smooth function h : L → R. We can extend h first to a tubular neighborhood and then to whole M . Replacing u by u− dh we can assume that u |L = 0. The closed 1-form u gives a symplectic vector field X t on M , which is indeed constant in time. If ψ t denotes the smooth 1-parameter family of symplectomorphisms generated by X t , then clearly We need to show that ψ t leaves L invariant. Let p ∈ L and Y ∈ T p L be any vector. Then by the construction of u we have 0 = u(p)(Y ) = ı(X t )ω(p)(Y ) = ω(X t (p), Y ) for any Y ∈ T p L, which implies that X t (p) ∈ T p L, because L is a Lagrangian submanifold. This finishes the proof.
The following theorem is the relative version of Theorem 10.12 of [4], which describes the kernel of the flux homomorphism. First we define the Hamiltonian symplectomorphisms of M leaving L invariant, denoted Ham(M, L), as the subgroup of Symp(M, L) consisting of symplectomorphisms ψ such that there is a Hamiltonian isotopy ψ t : (M, L) → (M, L), t ∈ [0, 1], such that ψ 0 = id and ψ 1 = ψ; i.e., ψ t is a Hamiltonian isotopy of M such that ψ t (L) = L, for any t ∈ [0, 1].
Since L is Lagrangian a Hamiltonian H t : M → R generating the above isotopy is locally constant on L. The proof of the above theorem is almost the same as that of Theorem 10.12 of [4], where one has to observe that the symplectomorphisms and isotopies of M involved in the proof leave L invariant.
In [4] the authors define the subgroup Similarly, we define its relative version To state the next lemma we need to recall the correspondence between symplectomorphisms of (M, ω), which are C 1 −close to the identity and closed 1-forms on M , which are close to the zero form. Consider the symplec- provided that ψ ∈ Symp(M ) is sufficiently C 1 −close to the identity. Studying the proof of the existence of a such symplectomorphism Ψ, one realizes that Ψ can be chosen so that symplectomorphisms of M , C 1 −close to the identity, leaving a Lagrangian submanifold L invariant correspond to closed 1-forms which evaluates zero on T L (cf. see Theorem 3.32 and Theorem 3.14 of [4]). Indeed, for the proof of the below relative version one has to note that a symplectomorphism, C 1 −close to the identity, leaves L invariant if and only if the various 1-forms involved in the proofs of Theorem 3.32 and Theorem 3.14 of [4] vanish on (L × L) ∩ ∆ and the vector fields are parallel to L × L.
Lemma 2.4. Assume the above notation. Then there is a symplectomorphism Ψ : N (∆) → N (M 0 ) satisfying Ψ(q, q) = q for any q ∈ M , such that if ψ : (M, ω) → (M, ω) is a symplectomorphism, which is sufficiently C 1 −close to the identity map, then ψ leaves a Lagrangian submanifold L invariant if and only if σ |TqL = 0, for any q ∈ L.
Lemma 2.8. Every smooth path ψ t ∈ Ham(M, L) is generated by Hamiltonian vector fields.
The proof of the above lemma is completely analogous to that of Proposition 10.17 of [4] and thus will be omitted. The following corollary which is the relative version of Corollary 10.18 of [4] is the main result of this section. Here the third map is H → X H , the fourth map is X → [ı(X)ω], while C ∞ (M, L) denotes the algebra of smooth functions on M , which are constant on L and χ(M, L, ω) is the algebra of symplectic vector fields X on M such that X(q) ∈ T q L, for any q ∈ L.
iii) The sequence of groups The last statement is also obvious.

Applications and Stability under Symplectic Reduction
Combining Part (i) of Corollary 2.9 with its absolute version Corollary 10.18 of [4], we derive the following corollaries.  under symplectic reduction. Let S 1 act in a Hamiltonian fashion on the symplectic manifold (M, ω) and L be a Lagrangian submanifold contained in a level set µ −1 (c) of the moment map, µ : M → R, of the S 1 -action. Assume that L is S 1 invariant and the S 1 -action on the level set µ −1 (c) is free. Let M red = µ −1 (c)/S 1 be the symplectic quotient with the symplectic structure ω red and L red = L/S 1 , which is a Lagrangian submanifold of (M red , ω red ). A theorem of Li ([3]) states that the fundamental groups and hence the first cohomology groups of M , µ −1 (c) and M red are all isomorphic under the canonical homomorphisms. On the other hand, if p : L → L red denotes the quotient map then we have the following result (see [5]).
is onto for any i, and is an isomorphism for i = 1.
The above arguments together with Corollary 3.2 imply the following stability result.

3.2.
Real algebraic varieties and their complexifications. In this subsection we will mention a different kind of stability of the above groups for real algebraic varieties. Let X be a nonsingular compact real algebraic variety with a nonsingular projective complexification i : X → X C . Note that any two nonsingular projective complexifications of X are always birationally isomorphic and thus have the same fundamental group and the same first cohomology group (cf. see p.494 of [2]). Clearly X C carries a Kähler and hence a symplectic structure such that X becomes a Lagrangian submanifold. So, by Corollary 10.18 of [4] (or just let L be the empty set in Corollary 2.9(i)) the quotient group Symp 0 (X C , ω)/ Ham(X C ) is determined only by X and hence is independent of the projective complexification i : X → X C .
Define KH i (X, R) as the kernel of the induced homomorphism i * : H i (X, R) → H i (X C , R) and ImH i (X, R) as the image of the induced homomorphism In [6,7] it is shown that both KH i (X, R) and ImH i (X, R) are independent of the projective complexification i : X → X C and thus (entire rational) isomorphism invariants of X. Moreover, by Corollary 3.2, for any topological component X 0 of X the quotient group Symp 0 (X C , ω)/N (X C , X 0 ) is independent of the smooth projective complexification i : X → X C and hence is determined only by X or equally by X 0 .