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Equivariant Holomorphic Morse Inequalities III: Non-Isolated Fixed Points

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the connected components of the fixed-point set and the whole manifold. We also investigate the consequences in geometric quantization, especially in the context of symplectic cutting.


Introduction
In [18] a rigorous proof of equivariant holomorphic Morse inequalities of Witten [25] was found based on the heat kernel approach. The result proved holds for circle actions on holomorphic vector bundles over compact Kähler manifolds with isolated fixed points. In [26], this result is extended to torus and non-Abelian group actions and it was shown that the Kähler condition is necessary. The purpose of this paper is to prove the general case when the fixed points can be non-isolated.
Recall that in his heat kernel proof of the Bott-Morse inequalities for the de Rham complex, Bismut [5] used a further rescaling near the critical set of a Morse function in the sense of Bott, besides the deformation of Witten [24]. In our treatment of the holomorphic analog with non-isolated fixed points, instead of using [5], we adapt the methods and techniques in [6], where a general and direct localization procedure is developed which applies to a wide range of problems on localization in index theory. For example, to prove the Bott-Morse inequalities using this method, no rescaling near the critical set is needed.
As in [26], the equivariant holomorphic Morse inequalities for circle actions imply those for torus actions; we derive the latter by considering various circle subgroups of the torus. For circle actions, we deform the Dolbeault operator as in [25]. The corresponding Laplacian is roughly of the form ∆ + |V | 2 + L V , where ∆ is the standard Laplacian, V is the vector field that generates the circle action, and L V , the Lie derivative. L V is a first order differential operator that commutes with all other operators.. The crucial step is to restrict the problem to the eigenspaces of L V , on each of which L V is a constant and therefore the techniques of [6] can be applied. This leads naturally to an equivariant Morse theory.
The equivariant holomorphic Morse inequalities relate the Dolbeault cohomologies (twisted by a holomorphic vector bundle) of the whole manifold to those of the fixed point set. We apply our result to two situations.
First, considering the exterior power of the holomorphic tangent bundle, we obtain relations of Hodge numbers of the fixed submanifolds, including some results of Carrell-Sommese [9] and Carrell-Lieberman [8]. Second, we apply the inequalities to symplectic cutting and geometric quantization. We obtain a gluing formula for the Poincaré-Hodge polynomials under the symplectic cutting precess of Lerman [17]. Applying our result to the pre-quantum line bundles, we recover and/or strengthen a few results in [10,20,22] when the symplectic manifold is Kähler. Throughout this paper, N, R, R ± and C denote the sets of non-negative integers, real numbers, positive (negative) real numbers and complex numbers, respectively.

Main Results
In this section, we state the equivariant holomorphic Morse inequalities for holomorphic torus action on holomorphic vector bundle over a compact Kähler manifold when the fixed-point set need not be isolated. We then show that the result for torus action can be deduced from that of circle action. The latter will be proved in Section 3.

Equivariant holomorphic Morse inequalities for torus actions
We first recall from [26] a few notations of the Lie algebra and the formal character ring of the torus group.
Let t = √ −1 Lie(T ) be the Lie algebra of the torus group T and L, the integral lattice in t; the dual lattice L * in t * is the weight lattice. Definition 2.1 Let C[L * ] be the formal character ring of T consisting of elements q = ξ∈L * q ξ e ξ (q ξ ∈ Z).
We say q ≥ 0 if q ξ ≥ 0 for all ξ ∈ L * . Let Q(t) = n k=0 q k t k ∈ C[L * ][t] be a polynomial of degree n with coefficients in C[L * ], we say Q(t) ≥ 0 if q k ≥ 0 in C[L * ] for all k. For two such polynomials P (t) and Q(t), we say P (t) ≤ Q(t) if Q(t) − P (t) ≥ 0.
If W is a finite dimensional representation of T , We denote by mult ξ (W ) the multiplicity of the weight ξ ∈ L * in W . The character of W is char (W ) = ξ∈L * mult ξ (W ) e ξ ≥ 0 in C[L * ]. Let the support of W be supp(W ) = {ξ ∈ L * | mult ξ (W ) = 0}. For any θ ∈ t * , there is a homomorphism C[L * ] → C given by e ξ → e √ −1 ξ,θ . For instance, char (W ) → tr W e √ −1θ under this homomorphism. Now let (M, ω) be a compact Kähler manifold of complex dimension n. Suppose that T acts on M effectively preserving the complex structure and the Kähler form. If the fixed-point set F ⊂ M of the T -action is nonempty, then the action is Hamiltonian [11]. F is a finite union of connected compact Kähler submanifolds F 1 , . . . , F m ; let n 1 , . . . , n m be their complex dimensions, respectively. For each connected component F r (1 ≤ r ≤ m), the complexification of the normal bundle N r → F r in M has the decomposition N C r = N 1,0 r ⊕ N 0,1 r , and N 1,0 r is a holomorphic vector bundle over F r of rank n − n r . The torus T acts on N r preserving the base points in F r . Moreover, the weights of the isotropy representation on the normal fiber remains unchanged within any connected component of F . Let λ r,k (1 ≤ k ≤ n − n r ) be the isotropy weights on N r . N r splits, and N 1,0 r splits holomorphically, into the direct sum of various sub-bundles, each with a given weight. The hyperplanes (λ r,k ) ⊥ ⊂ t cut t into open polyhedral cones called action chambers, as in [12,21,26]. We fix a positive action chamber C. Let λ C r,k = ±λ r,k be the polarized weights, with the sign chosen so that λ C r,k ∈ C * .
(Here C * is the dual cone in t * defined by C * = {ξ ∈ t * | ξ, C > 0}.) We define the polarizing index ν C r of the component F r with respect to C as the number of weights λ r,k ∈ −C * . Let N C r be the direct sum of the sub-bundles corresponding to the weights λ r,k ∈ C * . ν C r is the rank of the holomorphic vector bundle N Finally, we define the polarized symmetric tensor product (with respect to C) of the holomorphic normal bundle N 1,0 This is bundle of eigenspaces of small eigenvalues implicitly contained in [25]. It also appeared in [7], which generalizes [12] to cases with non-isolated fixed points.
If E is a holomorphic vector bundle over M on which the T -action lifts holomorphically, then the fiber E p over each fixed point p ∈ F is a representation of T , and char (E p ) is also constant within any connected component of F . Consider an infinite dimensional bundle holomorphic K C (N r ) ⊗ E| Fr . The torus T acts on the total space while preserving the base points in F r . It is easy to see that each sub-bundle of any given weight is a holomorphic vector bundle of finite rank, i.e., is a T -invariant sub-bundle of finite rank on which the torus acts with weight ξ. The cohomology groups ) are the sum of those with coefficients in E C r,ξ , each equipped an induced T -action.
Our main result is the following Remark 2.3 1. Formula (2.2) clearly implies the corresponding weak inequalities It is easy to see that for any choice of C, (2.2) reduces to the Atiyah-Bott-Segal-Singer fixed-point theorem [2,3] after setting t = −1. In fact, where ch T , td stand for the equivariant Chern character and the Todd class, respectively.
3. If p ∈ F is an isolated fixed point, then the normal bundle N p = T p M and Therefore, (2.2) reduces to the result in [26] when F is discrete.
4. Though the Kähler assumption is not necessary in the fixed-point formula (2.4), it is essential for the strong inequalities (2.2) even when all the fixed points are isolated and when T is a circle group [26].
As in [26], (2.3) or (2.2) is a set of inequalities for each choice of the action chamber C. These inequalities provides bounds, along various directions in t * given by C, the multiplicities of weights in the Dolbeault cohomology group H k (M, O(E)) (0 ≤ k ≤ n) in terms of the fixed-point data, which includes F r and the bundles N r , E| Fr → F r with T -actions. The applications will be given in Section 4.
When the group acting on M is a non-Abelian group G, we can apply (2.2) to the maximal torus T of G. There is in addition an action of the Weyl group W on the fixed-point set F of T . Each w ∈ W induces an action on the set of connected components {F 1 , . . . , F m } of F . The sum over the connected components on the left hand side of (2.2) can be rearranged into sums over W (after incorporating the character of the isotropy representation of T on the bundle E| F ) and over its orbits in the set of connected components. Thus we can obtain the non-Abelian version of the holomorphic Morse inequalities like the case when F is isolated [26]. The details are left to the interested reader.

Reduction to the case of circle actions
When the torus T is the circle group S 1 , the Lie algebra is Lie(S 1 ) = √ −1R. There are two action chambers R ± , labeled by ± for simplicity. The weights of isotropy representation of S 1 on N r (1 ≤ r ≤ m) are λ r,k ∈ Z\{0} (1 ≤ k ≤ n r ). As before, N ± r is the direct sum of the sub-bundles corresponding to the positive and negative weights, respectively, and K ± (N r ) = S((N where Q ± (t) ≥ 0.
To show that the strong inequalities (2.2) of the torus case follows from (2.6) or (2.7), we proceed slightly differently from [26].

Reduction of Theorem 2.2 to Theorem 2.4:
For any action chamber C, there is λ 0 ∈ L * such that where Q C ξ (t) ≥ 0 for all ξ ∈ (λ 0 − C * ) ∩ L * . For any v ∈ C ∩ L, let S 1 be the circle subgroup generated by v. Consider this circle action on M and E. The isotropy weights on representation of T such that supp(W ) ⊂ (λ 0 − C * ) ∩ L * and the multiplicities of weights are finite, then mult ξ (W ) = mult ξ,v (W ). Therefore (2.9) follows from (2.6). ✷

The case of holomorphic circle actions
The entire section is devoted to proving (2.7) in Theorem 2.4. In subsection 3.1, we study Witten's deformation of the Dolbeault operator on a compact Kähler manifold as well as on flat spaces. In subsection 3.2, we establish the Taylor expansion of the deformed operator near the fixed-point set. In subsection 3.3, we split the space of sections into a subspace corresponding to small eigenvalues and its complement, and estimate the deformed operator and its resolvent on these subspaces. It is crucial that these estimates are done on each eigenspace of L V . Finally, in subsection 3.4, we prove Theorem 2.4.

Witten's deformation of the Dolbeault operator
We consider a holomorphic S 1 -action on a Kähler manifold M which preserves the Kähler structure. Let ω, g T M and J be the Kähler form, the Kähler metric and the complex structure on M , respectively. We assume that the fixed point set F is non-empty. In this case, there is a moment map h: where V is the vector field on M that generates the S 1 -action [11]. We further assume that the S 1 -action can is an element of Γ (M, End(E)). Over the fiber of a fixed point p ∈ F , r V (p) is simply the representation of Lie(S 1 ) on E p ; this is independent of the choice of the connections on E.
Next we study Witten's deformation on flat spaces [25]. Let W be a complex vector space of (complex) dimension n with an Hermitian form. Let ρ be a unitary representation of the circle group S 1 on W such that all the weights are non-zero. Suppose W ± are the subspaces of W corresponding to positive (negative) weights, respectively, and dim C W − = ν, dim C W + = n−ν. Let z = {z k } be the complex linear coordinates on W such that the Hermitian structure on W takes the standard form and such that ρ is diagonal with weights The Lie algebra action is given by the vector field where D is the Dirac operator with the same Clifford action (3.3). We . Let E be a finite dimensional complex vector space with a Hermitian form and suppose E carries a unitary representation of S 1 .
The direct sum of these weight spaces is isomorphic to 2. When restricted to an eigenspace of L V , the operator D + √ −1 c(V ) has discrete eigenvalues.
Proof. It suffices to prove the case when E ∼ = C is a trivial representation.
1. Since W decomposes into a direct sum of 1-dimensional representations and since the solution space on the direct sum corresponds to a tensor product, it suffices to prove the result when dim C W = 1 with weight λ > 0, then W = W + , and a solution of a given weight is proportional to of unit norm and weight (k isomorphism is induced by the Hermitian form on W ∼ = C. 2. This follows from (3.6) after setting L V to a constant and from the standard properties of harmonic oscillators. ✷

A Taylor expansion of the operator near the fixed-point set
Since the fixed point set F is the zero set of the Killing vector field V , F is a totally geodesic compact Kähler submanifold of M . We denote by n F and ν F the complex dimension and the polarizing index of F , respectively, which are locally constant. Let g T F be the induced metric and dv F , the volume element on F . Section 8e)], we now describe a coordinate system on M near F .
If y ∈ F , Z ∈ N y , let t ∈ R → x t = exp M y (tZ) ∈ M be the geodesic in M with x 0 = y, dx dt | t=0 = Z. For ǫ > 0, set B ǫ = {Z ∈ N ; |Z| < ǫ}. Since M and F are compact, there exists ǫ 0 > 0 such that for 0 ≤ ǫ < ǫ 0 , the map (y, Z) ∈ N → exp M y (Z) ∈ M is a diffeomorphism from B ǫ onto a tubular neighborhood U ǫ of F in M . From now on, we identify B ǫ with U ǫ and use the notation x = (y, Z) instead of x = exp M y (Z). Finally, we identify y ∈ F with (y, 0) ∈ N .
Let dv N denote the volume element of the fibres in N . Then dv F (y)dv Ny (Z) is a natural volume element on the total space of N . Let k(y, Z) be the smooth positive function defined on B ǫ0 by the equation dv M (y, Z) = k(y, Z)dv F (y)dv Ny (Z). The function k has a positive lower bound on U ǫ0/2 . Also, k(y) = 1 and ∂k ∂Z (y) = 0 for y ∈ F ; the latter follows from [6,Proposition 8.9] and the fact that F is totally geodesic in M .
As in [6, Section 8g)], for x = (y, Z) ∈ U ǫ0 , we will identify E x with E y and ∧(T * (0,1) Clearly, D N acts along the fibers of N . Let∂ N be the∂-operator along the fibers of N , and let (∂ N ) * be its formal adjoint under (3.14). It is easy to see that D N = √ 2(∂ N + (∂ N ) * ). Both D N and D H are self-adjoint with respect to (3.14).
For a first order differential operator if there is a constant C > 0 such that for any T ≥ 1, (y, Z) ∈ B ǫ0 √ T , we have Let J V be the representation of Lie(S 1 ) on N . Then Z → J V Z is a Killing vector field on N . We have the following analog of [6, Theorem 8.18].

Proposition 3.3 As T → ∞,
Proof. Following the proof of [6, Theorem 8.18], we get In fact the proof is much easier here because F is totally geodesic, hence the second fundamental form in [6, (8.18)] vanishes. Next, we observe that V = 0 on F and that the vector field J V Z on N is the linear approximation of V on M near F . Further, since the actions of S 1 on N and M commute with the exponential map, V (y, Z) is odd in Z, and hence the second order terms vanish. Therefore The result follows.
Let H 0 (F ) be the Hilbert space of square-integrable sections of ∧(T * (0,1) F ) ⊗ K − (N ) ⊗ E| F , and H 0 (N ), that of the bundleπ * ((∧(T * (0,1) M )⊗E)| F ), equipped with the Hermitian form (3.14). We define an embedding Here α ∈ Ω 0, * (F ), β ∈ L 2 (K − (N )⊗ E| F ) and τ is the isometry from L 2 (K − (N )⊗ E| F ) to L 2 (π * (∧(N * (0,1) )⊗ Proof. For simplicity, we prove in the case when E is a trivial line bundle and N is of rank 1 with isotropy weight λ ∈ Z\{0}. By the definitions of D H and D F , we need to provẽ for any β ∈ L 2 (K − (N )) and any vector field X on F . On a small neighborhood U ⊂ F , choose a unitary . If λ < 0, a straightforward calculation shows that for β(y, z) = f (y)ϕ k (z), we havẽ ∇ F X H τ (β)(y, z) = X H (f (y)ϕ k (z)) = (Xf − k i X A)(y)ϕ k (z); (3.26) this is the connection on (N * ) ⊗k . If λ > 0, for β(y, z) = f (y)ϕk(z), we havẽ ∇ F X H τ (β)(y, z) = (∇ N X (f (y)dz) + k i X A(y))ϕk(z) = (Xf + (k + 1) i X A)(y)ϕk(z); (3.27) this is the connection on N ⊗(k+1) . The result is proved. tively. The circle group S 1 acts on all these spaces; let H p ξ (M ), H p ξ (N ) and H p ξ (F ) be the corresponding weight spaces of weight ξ ∈ Z. Recall the constant ǫ 0 > 0 defined in the previous subsection. We now take ǫ ∈ (0, ǫ0 2 ], which is small enough for each eigenvalue of L V we will consider, but otherwise can be assumed fixed. Let ρ: R → [0, 1) be a smooth function such that * (0,1) F )). We define a linear map I T,ξ by Proposition 3.5 1. As T → ∞, where O( 1 √ T ) denotes a first order differential operator whose coefficients are dominated by C Proof. We proceed as in [6,Section 9]. In fact, parts 1 and 2 follow from Propositions 3.2, 3.3 and 3.4 the same way as in the proofs of [6,Theorems 9.8,9.9] with trivial modifications. Part 3 is the analog of [6, Theorem 9.14]. The key observation is that we are restricting the operators to a fixed weight space H ξ (M ), where L V is a constant. Thus the terms . This is the analog of the fact that [D, V ] is of order zero in [6]. The boundedness of (3.36) allows us to localize the problem to sufficiently small neighborhood U ǫ of F for each weight ξ ∈ Z.
Using Propositions 3. The details are left to the interested reader. ✷

For two bounded operators
We fix a constant c 0 ∈ (0, 1] such that where Spec denotes the spectrum of an operator. Then we have the following analog of [6, Theorem 9.23]. Proposition 3.6 For any ξ ∈ Z, there exists T 0 ≥ 1, such that for any T ≥ T 0 , λ ∈ C with |λ| = c 0 , λ − D T,ξ is invertible on H 0 (M ). Moreover, for any integer p ≥ 2n + 2, there exists c p > 0 such that for any T ≥ T 0 , Proof. In view of Proposition 3.5, the result can be proved using the same formal arguments in [6, Sections 9c)-9e)]. ✷

Proof of Theorem 2.4
Let γ be the circle in C of center 0 and radius c 0 , oriented counterclockwise when necessary. By Proposition 3.6, γ ∩ Spec(D T,ξ ) = ∅ for T large enough. Let K c0 T,ξ be the direct sum of eigenspaces of D T,ξ associated to the eigenvalues λ such that |λ| ≤ c 0 . For T large enough, is the orthogonal projection from H 0 ξ (M ) onto K c0 T,ξ . Integrating by parts in (3.40), we get for any p ∈ N, Using Proposition 3.6, we obtain, for some C > 0, where P F ξ is the orthogonal projection from H 0 ξ (F ) onto K ξ = ker D F ξ . Therefore for T large enough, In fact, (3.43) holds for each degree in the Dolbeault complexes. By Hodge theorem and Proposition 3.2, we have for any 0 ≤ k ≤ n F , Let m k ξ be the sum of the multiplicities of all eigenvalues λ of D T,ξ on (0, k)-forms such that |λ| < c 0 . Then On the other hand, an obvious application of (the equivariant version of) the Hodge theorem yields that m k Suppose that there is a holomorphic action of the torus group T on M preserving the Kähler form. As before, if the fixed-point set F is non-empty, it is a disjoint union of connected compact Kähler submanifolds F 1 , . . . , F m , of (complex) dimensions n 1 , . . . , n m , respectively. Recall that the weights λ r,k (1 ≤ r ≤ m, 1 ≤ k ≤ n) of isotropy representations of T on the normal bundles of F r cut the Lie algebra t into action chambers.  In particular, if all the fixed points are isolated, then for any choice of chamber C, we have and h k,l (M ) = 0 for k = l. Proof.
The T -action can be lifted holomorphically to the bundles ∧ k (T * (1,0) M ) (0 ≤ k ≤ n). Applying where Q C 0 (s, t) ≥ 0 in Z[s, t] is defined by the relation n k=0 s k Q k,C 0 (t) = (1 + s)Q C 0 (s, t). Now let s = t in (4.7). Since the moment map (after projecting to any direction in ±C) is a perfect Morse function in the sense of Bott (see for example [11]), we get Q C 0 (t, t) = 0. Therefore Q C 0 (s, t) = 0 and (4.1) follows. ✷  Proof. The first part is well-known [1,13] and is included here for completeness; in fact, F rC is the inverse image of a vertex of the moment polytope. Taking s = 0 in (4.1), we get P (M ; 0, t) = P (F rC ; 0, t). The rest follows easily. ✷ Example 4.4 Hamiltonian S 1 -actions on symplectic 4-manifolds have been classified up to S 1 -equivariant diffeomorphisms [4], and subsequently, up to S 1 -equivariant symplectomorphisms [16]. Moreover it was shown that all such manifolds are Kähler [16]. Let M be such a manifold with a Hamiltonian S 1 -action and let If V is a holomorphic Killing vector field on (M, ω), let Z 1 , . . . , Z m be the connected components of the zero-set of V , of (complex) dimensions n 1 , . . . , n m , respectively. We have the following special case of the theorem of Carrell-Lieberman [8,9], which holds for a more general (not necessarily Killing) holomorphic vector field. Proof. The 1-parameter group generated by V is a subgroup of the isometry group of M . Therefore its closure is a torus T , whose fixed-point set is precisely Z. We choose any action chamber C (for example, the one which contains the generator V ). Let s = t −1 in (4.1). We get The result follows easily. ✷ When the fixed-point set F is discrete, this result was deduced from holomorphic Morse inequalities in [25].

Symplectic quotients, symplectic cuts and quantization
Let (M, ω) be a symplectic manifold of (real) dimension 2n. If the circle group S 1 acts Hamiltonianly on (M, ω), let V be the vector field on M that generates the action and h: M → R, the moment map such that There is a canonical symplectic form ω 0 on M 0 such that π * ω 0 = i * ω. To avoid orbifold singularities, we further assume that S 1 acts freely on h −1 (0). In this case, M 0 is a smooth manifold of dimension 2n − 2.
We now recall the notion of symplectic cutting [17]. Let the complex plane C be equipped with the standard Consider two actions of the circle group S 1 on C with weights ±1. Both actions are Hamiltonian with the moment maps ∓ 1 2 |z| 2 , respectively. The diagonal actions of S 1 on M × C are again Hamiltonian and the moment maps are h ∓ 1 2 |z| 2 , of which 0 is still a regular value. Let M ± be the symplectic quotients of M × C at level 0 by the two S 1 -actions defined above. (M ± , ω ± ) are smooth symplectic manifolds with Hamiltonian S 1 -actions and M 0 is embedded as one of the components (still denoted by M 0 ) fixed by S 1 ; the compliments M ± \M 0 are S 1 -equivariantly symplectomorphic to h −1 (R ± ) ⊂ M , respectively. Therefore the fixed-point set of M ± is the union of M 0 and all the components Moreover the circle bundle of the normal bundle N 0 of M 0 in M + is isomorphic to the bundle h −1 (0) → M 0 , with weight −1, while the normal bundle of M 0 in M − is isomorphic to N * 0 , with weight 1. When (M, ω) is a Kähler manifold with a Hamiltonian S 1 -action, then the symplectic quotient (M 0 , ω 0 ) and two symplectic cuts (M ± , ω ± ) are also Kähler. We now establish a gluing formula for Poincaré-Hodge polynomials.    M is a general symplectic (hence almost complex) manifold with a (possibly) non-Abelian group action [20,22]. The first part is the refinement of this result when M is Kähler.
Recall that a symplectic manifold (M, ω) is quantizable if the de Rham class [ ω 2π ] ∈ H 2 (M, Z). In this case, there is a complex line bundle, called the pre-quantum line bundle, with a connection ∇ whose curvature is ω √ −1 . We have the following Lemma 4.8 Suppose that a symplectic manifold (M, ω) is quantizable and is equipped with a Hamiltonian S 1 -action with moment map h. If one of h(F r ) ∈ Z, then all h(F r ) ∈ Z and the S 1 -action can be lifted to the pre-quantum line bundle L. In this case, M ± , M 0 are quantizable; let L ± , L 0 be their pre-quantum line bundles.
Proof. The generator of Lie(S 1 ) acts on the space of sections of L by −∇ V + √ −1h; this gives an R-action on L preserving ∇. If h(F r ) ∈ Z, then the R-action is an S 1 -action on L| Fr . Since ∇ is R-invariant, the parallel transport commutes with the R-action. Therefore the R-action factorizes through S 1 on the total space of L.
In particular, h(F r ) ∈ Z on any fixed component F r . The line bundle L 0 = i * L/S 1 → M 0 has a connection ∇ 0 such that π * ∇ 0 = i * ∇ so that its curvature is ω0 √ −1 [14]. Following the construction of symplectic cuts, there are S 1 -invariant pre-quantum line bundles L ± on M ± whose curvature is ω± √ −1 . The isomorphisms in the last part were proved in [10,19]. When M is not a complex manifold, each individual cohomology group in (4.14) does not make sense, however the alternating sum can be defined as the index of a spin C -Dirac operator using only an almost complex structure. In [10], it was proved that under this more general setting, we have the following relation on The last equality in (4.15) was the S 1 -case of a conjecture by Guillemin and Sternberg [14]; the case with compact non-Abelian group actions was proved by Meinrenken [19], and Vergne [23], Jeffrey and Kirwan [15] and others under various generalities using localization techniques, and by Tian and Zhang [22] using a direct analytic approach. Moreover, when M is a compact Kähler manifold with a Hamiltonian action of (possibly) non-Abelian group G, there are Morse-type inequalities which bound the invariant cohomologies of M in terms of those of the symplectic quotient M 0 (of complex dimension n 0 ) [22]. That is, for some Q 0 (t) ≥ 0. We prove similar Morse-type inequalities relating quantizations on M 0 and M ± , which will strengthen the first equality of (4.15) when M is Kähler and the group is S 1 .