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Strong Lefschetz property under reduction

Let n>1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property.


Introduction
Let (M, Ω) be a compact symplectic manifold of dimension 2m. For each 0 ≤ k ≤ m, the symplectic form Ω induces a map The manifold (M, Ω) is called Lefschetz if Ω m−1 is an isomorphism. It is called strong Lefschetz if Ω k is an isomorphism for all 0 ≤ k ≤ m. By Poincaré Duality, the Lefschetz maps are surjective if and only if they are injective.
The simplest examples of symplectic manifolds are Kähler manifolds. By the Strong Lefschetz Theorem, all compact Kähler manifolds are strong Lefschetz. On the other hand, Thurston and others constructed compact symplectic manifolds that do not admit Kähler structures [1,5,6,16,20], either by showing that these manifolds have an odd Betti number which is odd, or that they do not have the strong Lefschetz property.
Another interesting fact about the Lefschetz property concerns Hamiltonian actions. A Theorem of Ono [18] asserts that if a symplectic manifold (M, Ω) is Lefschetz, then a symplectic circle action on (M, Ω) is Hamiltonian if and only if it has fixed points.
We recall that an action of a compact Lie group G on (M, Ω) is symplectic if it preserves the symplectic form Ω. Let g be the Lie algebra of G and g * its dual. For any ξ ∈ g, denote by ξ M the vector field on M induced by the group action of exp(tξ). A symplectic action is Hamiltonian if there exists a moment map Φ : M → g * , which is equivariant with respect to the group action on M and the coadjoint action on g * such that for all ξ ∈ g. The orbit space Φ −1 (0)/G is called the symplectic quotient, and often denoted by M G. If G acts freely on Φ −1 (0), the symplectic quotient M G is a smooth symplectic manifold. Its symplectic form ω is defined by where π : Φ −1 (0) → M G is the orbit map, and i : Φ −1 (0) ֒→ M is the inclusion. In general, the symplectic quotient is a symplectic stratified space [13]. Suppose that the original manifold (M, Ω) is Kähler, and that the action preserves both the Kähler metric and the symplectic form, hence, the complex structure. Then the symplectic quotient M G is Kähler with Kähler form ω [9,11]. Products of compact non-Lefschetz symplectic manifolds with a twosphere rotated about a fixed axis are trivial examples such that the existence of a non-Lefschetz symplectic quotient implies the original manifold is not Lefschetz. However, Lin [14] constructed examples of Hamiltonian S 1manifolds (M, Ω) that are strong Lefschetz but their symplectic quotients (M S 1 , ω) are not. In this paper, we generalize Lin's result.
Let ω C n = i 2 dz j ∧ dz j = dx j ∧ dy j be the standard symplectic form on C n . Let G be a compact Lie group with an n-dimensional unitary representation τ : G → U(n). Then (C n , ω C n ) is a Hamiltonian G-manifold. Theorem 1.1. Suppose C n G consists of a single point. Then there exists a compact connected Hamiltonian G-manifold (M, Ω) that is strong Lefschetz, but its symplectic quotient (M G, ω) is not.
As an application, let m, k = 2, 3, 4, . . . and G be either SU(m) or Sp(k). Then the moment map for the standard G-representation on C n , for n = m and n = 2k, respectively, is where we identify the Lie algebra g with its dual g * via the inner product A, B = trace(A * B). A direct calculation shows that ψ −1 (0) = {0}. Theorem 1.1 then implies Theorem 1.2. Let n > 1 and let G be either SU(n) or Sp(n). Then there exists a compact connected Hamiltonian G-manifold (M, Ω) that is strong Lefschetz, but its symplectic quotient (M G, ω) is not.
In Section 2, we spell out the construction for the manifold (M, Ω) in Theorem 1.1. In Section 3, we compute the Lefschetz maps. In our proof, we use a result of Gompf [6] as our symplectic quotient: The strong Lefschetz property also has an equivalent description in terms of symplectic harmonic forms. For details, see [3,15,21]. The Hamiltonian G-manifolds (M, Ω) we construct here have the property that each de Rham cohomology class contains a symplectic harmonic representative while the same is not true for (M G, ω).

Construction
Let G be a compact Lie group acting on (C n , ω C n ) by a unitary representation with the property that the symplectic quotient C n G is a point. Extending the action to C n × C trivially on the last factor, it induces a Hamiltonian G-action on the complex projective space P(C n × C) = CP n . Let F = CP n and ω F the Fubini-Study symplectic form on F scaled by ǫ > 0. The symplectic quotient F G under the induced G-action is again a point.
Remark 2.1. This projective space (F, ω F ) can be constructed by symplectic reduction. Let S 1 denote the circle group and we identify both the Lie algebra of S 1 and its dual with R. The multiplication of S 1 on C n × C is a Hamiltonian action with moment map φ : C n × C → R given by If we choose the constant to be ǫ, the symplectic quotient is (F, ω F ). This is the same as taking a closed ball in C n of radius ǫ 1/2 centered at zero and reduce the boundary since Let (B, ω) be a symplectic manifold and let π : P → B be a principal S 1 -bundle over B with a given Chern class c and θ a connection form on P . With the Hamiltonian S 1 -action on (F, ω F ) induced from the multiplication on (C n , ω C n ), we form the associated bundle P × S 1 F .

Lemma 2.2.
For ǫ sufficiently small, the associated bundle P × S 1 F is a symplectic manifold with fiber (F, ω F ).
Proof. This is done by the technique of minimal coupling [19,23].
First identify both the Lie algebra of S 1 and its dual with R and consider the space P × R. The minimal coupling form π * ω + d pr 2 , θ is nondegenerate on P × {0}, and therefore is symplectic on a small δneighborhood P × (−δ, δ). Let I = (−δ, δ). The S 1 -action on P × I given by a · (p, η) = (pa −1 , Ad * (a)η) is Hamiltonian with minus the projection onto the second factor −pr 2 : P × I → R being its moment map. The symplectic quotient (P × I) S 1 is (B, ω). Second consider the product space P × I × F . Let Φ F : F → R be the S 1moment map on F . The diagonal S 1 -action on (P × I) × F is Hamiltonian with moment map given by If ǫ < δ, the image of F under Φ F is contained in I, and Φ −1 (0) ∼ = P × F . So the symplectic quotient is is the curvature form of θ. Since θ is a connection 1-form on the principal S 1 -bundle P with Chern class c, it follows that Finally, since the G-action on C n we start out with commutes with the S 1multiplication, the induced Hamiltonian actions on F also commute. The G-moment map on F is S 1 -invariant. Hence the manifold (P × S 1 F, Ω) inherits a fiberwise G-action with moment map Ψ : P × S 1 F → g * induced by the G-moment map on the fiber. Reduction can be carried out fiber by fiber. Since F G is a point, by construction, Ψ −1 (0)/G = (B, ω). For our purpose, we use the symplectic manifold (B, ω) and the Chern class c prescribed in Theorem 1.3.

Strong Lefschetz property
This section proves that the Hamiltonian G-manifold (M, Ω) constructed in the previous section is strong Lefschetz. Namely, we show that the map is an isomorphism for every integer 0 ≤ k ≤ n + 2. By Poincaré Duality, it suffices to show that the map Ω k is injective. This holds for k = n + 2 by the nondegeneracy of the symplectic form and also for k = 0. It remains to show injectivity for 1 ≤ k ≤ n + 1.
A typical fibre of (M, Ω) is (F, ω F ), which is isomorphic to CP n with a scaled Fubini-Study symplectic form. To simplify notations, let When there is no confusion, we use the same symbol to denote a closed form and its cohomology class. For example, x and u are both closed and we use the same symbol for both the form and its cohomology class in the subsequent computation.
The form z n restricted to each fibre of the bundle projection M → B is the volume form on (F, ω F ). Integrating along the fibre, we obtain where V (ω F ) = π n ǫ n Γ(n+1) is the volume and Γ is the Euler Γ-function. On the other hand, M (n + 1)yhz n = M (n + 1) Φ F , Θ hz n = M −(n + 1) Φ F , π * c hz n .
In this case, along each fibre, the volume form ω n F is scaled by the moment map Φ F . The volume of S 2n−1 with radius r is 2π n r 2n−1 Γ(n) . By Remark 2.1, it follows that M (n + 1)yhz n = Hence β 2 = 1 2 nǫc. Note that β 2 depends on ǫ which can be chosen anywhere between 0 and δ.
If k = n, we have α = b 2 + b 0 u. Hence, If α is in the kernel, that is, Ω n α = 0, then This implies that However the left hand side is lifted from an element in H 4 (B) and can be made non-zero with a generic ǫ. This is a contradiction. Hence for a generic ǫ, we must have b 0 = 0 and b 2 = −b 0 (β 2 + nx) = 0. We conclude that ker(Ω n ) = {0} for a generic ǫ. If 1 ≤ k < n and n − k is even, we have α = b 4 u (n−k−2)/2 + b 2 u (n−k)/2 + b 0 u (n−k+2)/2 .