dg-ga/9602008 (February, 1996) MSRI preprint No. 1996-013 Revised (December, 1997)

To appear in J. Diff. Geom. 51 (1999)

Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian Group Actions

Siye Wu^{1}

Mathematics Section, International Centre for Theoretical Physics, Trieste I-34100, Italy and Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, USA

Abstract. We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kähler manifolds and show the necessity of the Kähler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearranged into sums over the Weyl group (having incorporated the character of the isotropy representation on the fiber) and over the orbits of the Weyl group in the fixed-point set.

1. Introduction

Index theorems express analytical indices of elliptic complexes in terms of topological invariants; information on the individual cohomology groups are usually obtained with the aid of vanishing theorems. Taking the de Rham complex for example, the Euler number is not enough to determine the Betti numbers. However, if we consider a Morse function, then the Morse inequalities bound each Betti number by the data of the critical points. In this paper, we consider a holomorphic setting in which a compact group acts holomorphically on a holomorphic vector bundle over a Kähler manifold. The index theorem is the Atiyah-Bott fixed-point formula [2] (when the fixed points are isolated), which expresses the equivariant index, the alternating sum of the characters of the Dolbeault cohomology groups, in terms of the fixed-point data. The corresponding equivariant holomorphic Morse inequalities when the group is the circle group was obtained by Witten [24] and was first proved analytically using the heat kernel method by Mathai and the present author [19] when the fixed points are isolated. In this paper, we extend the result to cases with torus and non-Abelian group actions. We also show that the Kähler condition is essential for such Morse-type inequalities although, in contrast, not necessary for the equivariant index formula of Atiyah and Bott.

1Current address: Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia. E-mail address:

swu@maths.adelaide.edu.au

Let M be a compact Kähler manifold of complex dimension n and E, a holomorphic vector bundle over M. The Dolbeault cohomology groups H^{∗}(M, O(E)) with coefficients in E are cohomologies of the twisted Dolbeault complex (Ω^{0,∗}(M, E), ¯∂_{E}) and are independent of the choice of holomorphic connections. Let G be a compact, connected Lie group whose Lie algebra is denoted by Lie(G). Let 𝔤 = \sqrt{−1} Lie(G). We assume that G acts holomorphically and effectively on M preserving the Kähler form ω. If the fixed-point set of a maximal torus in G is non-empty, then the G-action is Hamiltonian [8], i.e., there is a moment map μ: M → 𝔤^{∗} such that for any x ∈ 𝔤, the corresponding vector field V_{x} on M satisfies i_{V_{x}}ω = d⟨μ, x⟩. If the action of g ∈ G on M (still denoted by g) can be lifted holomorphically to g̃ on E, then G acts on the space of sections Γ (M, E) by g: s ↦ g̃ ∘ s ∘ g^{−1} (g ∈ G) and similarly on Ω^{∗}(M, E). In this case, G commutes with the twisted Dolbeault operator ¯∂_{E} and hence acts on the cohomology groups H^{k} = H^{k}(M, O(E)) (0 ≤ k ≤ n). The purpose of this paper is to study the decomposition of each H^{k} in terms of the irreducible representations of G.

In sections 2 and 3, the group acting on M is a torus T. Section 2 shows that there is a set of Morsetype inequalities for each choice of action chambers specifying the directions in the Lie algebra 𝔱. This is obtained by applying the result of [24, 19] to various circle subgroups of T. Section 3 applies the result of the previous section to T -invariant line bundles over toric manifolds (including projective spaces and Hirzebruch surfaces). In section 4, we demonstrate that for non-Kähler manifolds, the strong equivariant holomorphic Morse inequalities need not hold. Violation of strong inequalities is also used to show the non-existence of invariant Kähler structures. In section 5, the group G is a general compact non-Abelian group. The main result is obtained by applying that of section 2 to a maximal torus T of G, assuming that the T -fixed-point set F is discrete. The novelties are that the cohomology groups H^{k} (0 ≤ k ≤ n) are representations of the non-Abelian group G (hence the structure of H^{k} is more rigid) and that there exists an action of the Weyl group W on F (hence the set F can be organized into W -orbits).

Throughout this paper, ℂ, ℝ, ℚ, ℤ, ℕ will denote the sets of complex numbers, real numbers, rational numbers, integers, non-negative integers, respectively.

2. Equivariant holomorphic Morse inequalities with torus actions

In this section, we assume that the compact Lie group that acts on the Kähler manifold is a torus group T. Let 𝔱 = \sqrt{−1} Lie(T) be the Lie algebra of T and L, the integral lattice in 𝔱; the dual lattice L^{∗} in 𝔱^{∗} is the weight lattice. Definition 2.1 Let ℤ[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^{∗}. The support of q is the set supp(q) = {ξ ∈ L^{∗} | q_{ξ} ≠ 0}. Let Q(t) = ∑_{k=0}^{n} q_{k}t^{k} ∈ ℤ[L^{∗}][t] be a polynomial of degree n with coefficients in ℤ[L^{∗}]. We say Q(t) ≥ 0 if q_{k} ≥ 0 in ℤ[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.

For example, if V is a finite dimensional representation of T, then its character char(V) ≥ 0 in ℤ[L^{∗}]. Let the support of V, denoted by supp(V), be the set supp(char(V)) of weights whose multiplicity in V is non-zero. For any θ ∈ 𝔱, there is a homomorphism ℤ[L^{∗}] → ℂ given by e^{ξ} ↦ e^{\sqrt{−1}⟨ξ,θ⟩}. For instance, char(V) ↦ tr_{V} e^{\sqrt{−1}θ} under this homomorphism. Another important type of elements in ℤ[L^{∗}] is given by the series

\frac{e^{η}}{1 − e^{ξ}} =^{def.} ∑_{k=0}^{∞} e^{kξ+η}, ξ, η ∈ L^{∗}.  (2.1)

We emphasize here that in (2.1) the left-hand side is a notation for the formal series on the right-hand side.

Recall we assumed that T acts holomorphically and effectively on a compact Kähler manifold M with a non-empty and discrete fixed-point set F. We also assume that the T -action preserves the Kähler form ω and hence is Hamiltonian [8]; let μ: M → 𝔱^{∗} be the moment map. For any p ∈ F, let λ_{1}^{p}, ⋯, λ_{n}^{p} ∈ L^{∗}\{0} be the weights of the isotropy representation of T on T_{p}M (Figure 2.1(a)). The hyperplanes (λ_{k}^{p})^{⊥} ⊂ 𝔱 cut 𝔱 into open polyhedral cones called action chambers [12, 13, 21]. (When M is a coadjoint orbit, the action chambers are precisely the Weyl chambers.) We fix a positive action chamber C (Figure 2.1(b)). Let λ_{k}^{p,C} = ±λ_{k}^{p} be the polarized weights, with the sign chosen so that λ^{p,C} ∈ C^{∗}. (Here C^{∗} is the dual cone in 𝔱^{∗} defined by C^{∗} = {ξ ∈ 𝔱^{∗} | ⟨ξ, C⟩ > 0}.) We define the polarizing index n^{p,C} of p with respect to C as the number of weights λ_{k}^{p} ∈ −C^{∗}. We also assume that there is a holomorphic vector bundle E over M on which the T -action lifts holomorphically. The torus T acts on the fiber E_{p} over p ∈ F and on the cohomology groups H^{k} = H^{k}(M, O(E)) (0 ≤ k ≤ n). Following [24, 19], we denote E_{p}(θ) = tr_{E_{p}}e^{\sqrt{−1}θ} (p ∈ F) and H^{k}(θ) = tr_{H^{k}}e^{\sqrt{−1}θ}. Theorem 2.2 For each choice of the positive action chamber C, we have the strong equivariant holomorphic Morse inequalities

∑_{p∈F} t^{n^{p,C}} char(E_{p}) ∏_{λ_{k}^{p}∈C^{∗}} \frac{1}{1 − e^{−λ_{k}^{p}}} ∏_{λ_{k}^{p}∈−C^{∗}} \frac{e^{−λ_{k}^{p,C}}}{1 − e^{−λ_{k}^{p,C}}} = ∑_{k} t^{k} char(H^{k}) + (1 + t)Q^{C}(t)  (2.2)

for some Q^{C}(t) ≥ 0 in ℤ[L^{∗}].

Remark 2.3 The strong inequalities (2.2) imply the following weak equivariant holomorphic Morse inequalities:

char(H^{k}) ≤ ∑_{n^{p,C}=k} char(E_{p}) ∏_{λ_{j}^{p}∈C^{∗}} \frac{1}{1 − e^{−λ_{j}^{p}}} ∏_{λ_{j}^{p}∈−C^{∗}} \frac{e^{−λ_{j}^{p,C}}}{1 − e^{−λ_{j}^{p,C}}} (0 ≤ k ≤ n).  (2.3)

Setting t = −1 in (2.2), we recover the Atiyah-Bott fixed-point theorem

∑_{p∈F} \frac{char(E_{p})}{∏_{k=1}^{n}(1 − e^{−λ_{k}^{p}})} = ∑_{k=0}^{n}(−1)^{k} char(H^{k}) = Ind.  (2.4)

Remark 2.4 If T is the circle group S^{1}, then 𝔱 = \sqrt{−1} Lie(S^{1}) ≅ ℝ. There are only two action chambers C^{±} = {θ ∈ ℝ | ±θ > 0}. The polarizing index n^{p,C^{+}} of p with respect to C^{+} is the number of weights λ_{k}^{p} < 0, denoted by n_{p}. Theorem 2.2 reduces to the results in [24, 19]. In particular, for C = C^{+}, (2.2) becomes

∑_{p∈F} t^{n_{p}}E_{p}(θ) ∏_{λ_{k}^{p}>0} \frac{1}{1 − e^{−\sqrt{−1}λ_{k}^{p}θ}} ∏_{λ_{k}^{p}<0} \frac{e^{−\sqrt{−1}|λ_{k}^{p}|θ}}{1 − e^{−\sqrt{−1}|λ_{k}^{p}|θ}} = ∑_{k=0}^{n} t^{k}H^{k}(θ) + (1 + t)Q^{+}(θ, t),  (2.5)

where Q^{+}(θ, t) ≥ 0 in ℝ((e^{\sqrt{−1}θ})), the ring of formal characters of S^{1} [19].

Proof of Theorem 2.2. We notice that (2.2) is equivalent to

∑_{p∈F} t^{n^{p,C}} E_{p}(θ) ∏_{λ_{k}^{p}∈C^{∗}} \frac{1}{1 − e^{−\sqrt{−1}⟨λ_{k}^{p},θ⟩}} ∏_{λ_{k}^{p}∈−C^{∗}} \frac{e^{−\sqrt{−1}⟨λ_{k}^{p,C},θ⟩}}{1 − e^{−\sqrt{−1}⟨λ_{k}^{p,C},θ⟩}} = ∑_{k} t^{k}H^{k}(θ) + (1 + t)Q^{C}(t)(\sqrt{−1}θ) (2.6)

regarded as an equality of analytic functions in θ ∈ 𝔱^{ℂ} for Im θ ∈ −C. (2.2) implies (2.6) because for any δ ∈ C the Taylor expansions on the left-hand side of (2.6) are uniformly convergent in θ ∈ 𝔱^{ℂ} if Im θ ∈ −(δ + \overline{C}). Conversely, if (2.6) is true for complex θ ∈ 𝔱^{ℂ} with Im θ ∈ −C, we take Im θ → 0 within the cone −C. The equality (2.2) of formal series follows from the uniqueness of the Fourier expansion of tempered distributions [16]. For S^{1}-actions [24, 19], (2.5) is a true equality both in ℝ((e^{\sqrt{−1}θ})) and as analytic functions for Im θ < 0. The rest of the proof, which is similar to that of [21, Theorem 2.2], shows (2.6) using (2.5). By analyticity, it suffices to prove (2.6) for purely imaginary θ. Pick an integral point θ_{1} ∈ C ∩ L. Since ⟨λ_{k}^{p}, θ_{1}⟩ ≠ 0 for all p ∈ F and 1 ≤ k ≤ n, h = ⟨μ, θ_{1}⟩ generates a Hamiltonian S^{1}-action on M with the same fixed-point set F. Moreover the weights of S^{1} at p ∈ F are ⟨λ_{k}^{p}, θ_{1}⟩ (1 ≤ k ≤ n) and |⟨λ_{k}^{p}, θ_{1}⟩| = ⟨λ_{k}^{p,C}, θ_{1}⟩ > 0 for all p ∈ F and 1 ≤ k ≤ n. (2.5) implies (2.6) for θ = −\sqrt{−1}sθ_{1} (s > 0). The result follows from continuity since such θ’s form a dense subset of −\sqrt{−1}C. □

The inequalities (2.3) or (2.2) show that the multiplicities of weights in the cohomology groups H^{k} (0 ≤ k ≤ n) are constrained by the fixed-point data. Given an action chamber C, the support of H^{k} is contained in a suitably shifted cone −C^{∗} in L^{∗}. By choosing different chambers, it is possible to bound suppH^{∗} (0 ≤ k ≤ n) in various directions in 𝔱^{∗}. We need the following definition. (See Figure 2.1(c),(d).)

Definition 2.5 For a given choice of positive action chamber C and p ∈ F, let

Γ^{p,C} = {ξ − ∑_{k=1}^{n}r_{k}λ_{k}^{p,C} | ξ ∈ supp(E_{p}), r_{k} ≥ 0 and r_{k} > 0 if λ_{k}^{p} ∈ −C^{∗}}  (2.7)

and

Γ^{k,C} = ⋃_{p∈F,n^{p,C}=k} Γ^{p,C}.  (2.8)

We set Γ^{k} = ⋂_{C} Γ^{k,C}.

Proposition 2.6 For any 0 ≤ k ≤ n,

supp(H^{k}) ⊂ L^{∗} ∩ ⋂_{C} Γ^{k,C} = L^{∗} ∩ Γ^{k},  (2.9)
supp(H^{k}) ⊃ L^{∗} ∩ ⋃_{C} Γ^{k,C}\(Γ^{k−1,C} ∪ Γ^{k+1,C}).  (2.10)

Proof. (2.9) follows from the weak inequalities with all choices of C. (2.10) follows from the strong inequalities: If ξ ∈ L^{∗} ∩ Γ^{k,C} but ξ ∉ Γ^{k±1,C}, then the polynomial Q^{C}(t) = ∑_{k=0}^{n} ∑_{ξ∈L^{∗}} q_{ξ}^{k}e^{ξ}t^{k} in (2.2) has coefficients q_{ξ}^{k−1} = q_{ξ}^{k} = 0. This means that (1+t)Q^{C}(t) does not contain the term e^{ξ}t^{k}. Hence ξ ∈ supp(H^{k}). □

Recall that a Kähler manifold (M, ω) is quantizable if \frac{ω}{2π} represents an integral de Rham class. In this case, a pre-quantum line bundle over M is a line bundle whose curvature is \frac{ω}{\sqrt{−1}}.

Corollary 2.7 If L is a T -invariant pre-quantum line bundle of a quantizable Kähler manifold (M, ω), then suppH^{k}(M, O(L)) is contained in the moment polytope Δ for any k = 0, ⋯, n.

Proof. Let μ: M → 𝔱^{∗} be the moment map. The image Δ = μ(M) is a convex polytope [1, 14]. For any facet of Δ, there is x ∈ 𝔱 such that the hyperplane x^{⊥} contains the facet and ⟨Δ, x⟩ ≤ 0. Choose an action chamber C with an edge containing x. For any fixed point p ∈ F, the weight of the torus action on the fiber of the pre-quantum line bundle over p is μ(p) ∈ Δ, hence ⟨μ(p), x⟩ ≤ 0. Since all ⟨λ_{k}^{p,C}, x⟩ ≥ 0 we conclude that ⟨Γ^{p,C}, x⟩ ≤ 0 for all p ∈ F. By (2.9) of Proposition 2.6, ⟨ supp(H^{k}), x⟩ ≤ 0 for any facet of Δ. Since Δ is convex, we get supp(H^{k}) ⊂ Δ for any k. □

In sections 3 and 5, we will apply Theorem 2.2 and Proposition 2.6 to toric manifolds and to cases with general non-Abelian group actions.

3. Applications to toric manifolds

The toric manifolds which we consider here are smooth complex manifolds M, each equipped with an effective action of T = T^{n} (n = dim_{ℂ} M). Such a manifold can be characterized combinatorially by a fan Σ, a collection of cones in 𝔱 = \sqrt{−1} Lie(T) that satisfy certain compatibility and integrality conditions. (For reviews, see for example [5, 20, 3, 9].) M is compact if and only if the corresponding fan Σ is complete, i.e., the union of all the cones |Σ| = 𝔱. Top dimensional cones σ in the n-skeleton Σ^{(n)} are in one-to-one correspondent with the T -fixed points p_{σ} ∈ F. The weights of isotropic representations on T_{p_{σ}}M span the cone −σ^{∗} ⊂ 𝔱^{∗}. The hyperplanes containing (n − 1)-dimensional cones cut 𝔱 into action chambers. The T -equivariant holomorphic line bundles over M are characterized by (continuous) piecewise linear functions φ on Σ modulo 𝔱^{∗} (the space of globally linear functionals). For each σ ∈ Σ^{(n)}, φ|_{σ} ∈ 𝔱^{∗} is the weight of the T -action on the fiber over p_{σ}. We denote this line bundle by L_{φ}. The cohomology groups H^{k} = H^{k}(M, O(L_{φ})) (k = 0, ⋯, n) are representations of T. For each weight ξ ∈ L^{∗}, the multiplicity of ξ in H^{k} can be computed using ξ, φ and Σ (see Remark 4.3 below). The purpose of this section is to find information on such multiplicities using the results of the previous section. This method provides much geometric insight, especially on how the multiplicity varies according to the position of the weight relative to the image of the (generalized) moment map. The multiplicity in the equivariant index was studied in [18, 10] with similar considerations by using the Atiyah-Bott fixed-point formula (2.4).

We consider a toric manifold M with a T -invariant Kähler structure. This is equivalent to the existence of a T -invariant ample line bundle over M; the Kähler structure is then induced by the projective embedding. Such a line bundle corresponds to a strictly convex piecewise linear function. From the symplectic point of view, if M admits a (non-degenerate) symplectic form, then the image of the moment map is a convex polytope which does define a strictly convex piecewise linear function on its fan (with a possible perturbation of the Kähler form). Therefore a symplectic toric manifold is Kähler. An explicit construction of a Kähler structure using the Delzant construction [6] was given by [11].

Proposition 3.1 Let M be a compact smooth toric manifold of complex dimension n with a T -invariant Kähler structure and let H^{k} = H^{k}(M, O(L_{φ})) (k = 0, ⋯, n), where the line bundle is determined by a piecewise linear function φ on the fan. Then the following holds:

1. If 0 < k < n, the sets supp(H^{0}), supp(H^{k}) and supp(H^{n}) are mutually disjoint;

2. supp(H^{0}) = L^{∗} ∩ Γ^{0}, supp(H^{n}) = L^{∗} ∩ Γ^{n}, where each weight is of multiplicity 1.

Proof. Choose any action chamber C. Let σ_{C} ∈ Σ^{(n)} be the unique cone containing −C. It is clear that the polarizing indices n^{σ,C} of p_{σ} ∈ F are given by n^{σ_{C},C} = 0 and n^{σ,C} > 0 for all σ ≠ σ_{C}. Therefore in the weak inequality (2.3) for k = 0, there is only one term that bounds char(H^{0}). Since dim_{ℂ} M = dim 𝔱, the number of isotropy weights at p_{σ} is equal to n, hence the multiplicity of any weight in H^{0} is not greater than 1. Let C run through all action chambers. Since Γ^{σ_{C},C} = φ|_{σ_{C}} + σ_{C}^{∗}, we obtain Γ^{0} = ⋂_{σ∈Σ^{(n)}}(φ|_{σ} + σ^{∗}). Using again dim_{ℂ} M = dim 𝔱 = n, for any σ ∈ Σ^{(n)} and any action chamber C such that σ_{C} ≠ σ, we have (φ|_{σ} + σ^{∗}) ∩ Γ^{σ,C} = ∅. Therefore Γ^{0} ∩ Γ^{σ,C} = ∅. Taking the union of σ ∈ Σ^{(n)} with n^{σ,C} = k > 0, we get Γ^{0} ∩ Γ^{k,C} = ∅, hence Γ^{0} ∩ Γ^{k} = ∅ for k > 0. By (2.9) of Proposition 2.6, supp(H^{0}) ∩ supp(H^{k}) = ∅ for k > 0. Also, since Γ^{0} ∩ Γ^{k,C} = ∅ for any k > 0 and any C, by the Atiyah-Bott fixed-point theorem (2.4) or by (2.10) of Proposition 2.6, we have supp(H^{0}) ⊃ L^{∗} ∩ Γ^{0}. Hence supp(H^{0}) = L^{∗} ∩ Γ^{0}, each weight with multiplicity 1. The results on H^{n} can be proved similarly. □

Corollary 3.2 If φ is a strictly convex piecewise linear function, then H^{k} = 0 for k > 0 and Ind = char(H^{0}); if −φ is convex, then H^{k} = 0 for k < n and Ind = (−1)^{n} char(H^{n}).

Proof. In the first case, there is a T -invariant Kähler form such that L_{φ} is the pre-quantum line bundle and μ(p) = φ|_{p_{σ}} for all σ ∈ Σ^{(n)}. The moment polytope Δ = ⋂_{σ∈Σ^{(n)}}(μ(p_{σ}) + σ^{∗}) = Γ^{0}. From Corollary 2.7 and Proposition 3.1, we obtain Γ^{k} ⊂ Δ for all k, and Γ^{0} ∩ Γ^{k} = ∅ for k > 0. Therefore H^{k} = 0 for k > 0, and Ind = char(H^{0}), with supp(H^{0}) = L^{∗} ∩ Δ. The second case can be proved similarly. □

Example 3.3 Consider the projective space ℂP^{n} = (ℂ^{n+1}\{0})/ℂ^{∗} with the standard T^{n}-action given by (u_{1}, ⋯, u_{n}): [z_{1}, ⋯, z_{n}.z_{n+1}] ↦ [u_{1}z_{1}, · · · .u_{n}z_{n}, z_{n+1}] in homogeneous coordinates. 𝔱 = \sqrt{−1} Lie(T^{n}) ≅ ℝ^{n} is spanned by e_{1}, ⋯, e_{n}.

1. The case n = 1 has been considered in [24] using the inequalities in (2.5) and their counterparts for C = C^{−}. If L is a line bundle over ℂP^{1} with c_{1}(L) = r, then dim H^{0}(ℂP^{1}, O(L)) = r + 1, H^{1} = 0 if r ≥ 0, and H^{0} = 0, dim H^{1} = |r| − 1 if r < 0.

2. It is instructive to work out the case n = 2 directly using the results of the previous section. There are three fixed points p_{1} = [1, 0, 0], p_{2} = [0, 1, 0], p_{3} = [0, 0, 1] with isotropy weights {e_{1}^{∗}, e_{1}^{∗}−e_{2}^{∗}}, {e_{2}^{∗}, e_{2}^{∗}−e_{1}^{∗}}, {−e_{1}^{∗}, −e_{2}^{∗}}, respectively. There are six action chambers. Let C be a chamber spanned by {e_{1}, e_{1} + e_{2}} (Figure 3.1(a)). The line bundle L = (ℂ^{3}\{0} × ℂ)/ℂ^{∗}, where the quotient is (z_{1}, z_{2}, z_{3}, w) ∼ (uz_{1}, uz_{2}, uz_{3}, u^{r}w), u ∈ ℂ^{∗}, has c_{1}(L) = r ∈ ℤ. The T^{2}-action lifts to L by (u_{1}, u_{2}): [z_{1}, z_{2}, z_{3}, w] ↦ [u_{1}z_{1}, u_{2}z_{2}, z_{3}, w]. The weights on L_{p_{1}}, L_{p_{2}}, L_{p_{3}} are ξ_{1} = re_{1}^{∗}, ξ_{2} = re_{2}^{∗}, ξ_{3} = 0, respectively (Figure 3.1(b)). It is easy to see that n^{p,C} = 0, 1, 2 for p = p_{1}, p_{2}, p_{3}, respectively and that Γ^{p_{1},C} = {x_{1}e_{1}^{∗} + x_{2}e_{2}^{∗} | x_{2} ≥ 0, x_{1} + x_{2} ≤ r}, Γ^{p_{2},C} = {x_{1} < 0, x_{1} + x_{2} ≤ r}, Γ^{p_{3},C} = {x_{1}, x_{2} < 0}. On the other hand, n^{p,−C} = 2, 1, 0 for p = p_{1}, p_{2}, p_{3}, respectively and Γ^{p_{1},−C} = {x_{1} + x_{2} > r, x_{2} < 0}, Γ^{p_{2},−C} = {x_{1} ≥ 0, x_{1} + x_{2} > r}, Γ^{p_{3},−C} = {x_{1}, x_{2} ≥ 0}. If r ≥ 0, by (2.9) of Proposition 2.6, Γ^{0} ⊂ Γ^{p_{1},C} ∩ Γ^{p_{3},−C} = {x_{1}, x_{2} ≥ 0, x_{1} + x_{2} ≤ r} (Figure 3.1(c)), Γ^{1} ⊂ Γ^{p_{2},C} ∩ Γ^{p_{2},−C} = ∅ (Figure 3.1(d)), and Γ^{2} ⊂ Γ^{p_{1},C} ∩ Γ^{p_{3},−C} = ∅ (Figure 3.1(e)). By (2.10) of Proposition 2.6, supp(H^{0}) ⊃ ℤ^{2} ∩ (Γ^{p_{1},C}\Γ^{p_{2},C}). Hence supp(H^{0}) = ℤ^{2} ∩ Γ^{0}, where Γ^{0} = {x_{1}, x_{2} ≥ 0, x_{1} + x_{2} ≤ r}, and H^{1} = H^{2} = 0. Similarly, if r < 0, then H^{0} = H^{1} = 0, and H^{2} = ℤ^{2} ∩ Γ^{2}, where Γ^{2} = {x_{1}, x_{2} < 0, x_{1} + x_{2} > r}.

3. For general n, ℂP^{n} as a toric manifold can be described by a fan Σ: any k vectors in {v_{0} = − ∑_{i=1}^{n} e_{i}, v_{i} = e_{i} (1 ≤ i ≤ n)} span a k-dimensional cone in the fan Σ. For any piecewise linear function φ on Σ, either φ or −φ is convex. In fact by shifting a linear functional, φ can be brought to the standard form φ(v_{0}) = −r, φ(v_{i}) = 0 (1 ≤ i ≤ n); r is the first Chern number of the line bundle. φ is convex if and only if r ≥ 0. In this case, Corollary 3.2 implies that H^{k} = 0 for k > 0 and that supp(H^{0}) contains the integer points in the simplex {∑_{i=1}^{n} x_{i}e_{i}^{∗} | x_{i} ≥ 0 (1 ≤ i ≤ n), ∑_{i=1}^{n} x_{i} ≤ r} in 𝔱^{∗} ≅ ℝ^{n}, each with multiplicity 1. If r < 0, then H^{k} = 0 for k < n, and supp(H^{n}) = ℤ^{2} ∩ {∑_{i=1}^{n} x_{i}e_{i}^{∗} | x_{i} < 0 (1 ≤ i ≤ n), ∑_{i=1}^{n} x_{i} > r}, with multiplicity 1.

Corollary 3.4 If dim_{ℂ} M = 2, then

supp(Ind) = supp(H^{0}) ∪ supp(H^{1}) ∪ supp(H^{2})  (3.1)

is a disjoint union.

Proof. This follows from part 1 of Proposition 3.1 since the only choice of k is k = 1. □

Example 3.5 We consider the Hirzebruch surface, given by a fan Σ in ℝ^{2} whose 1-skeleton Σ^{(1)} is spanned by vectors v_{1} = e_{1}, v_{2} = e_{2}, v_{3} = −e_{1}, v_{4} = −ae_{1} − e_{2} (a ∈ ℕ). We choose an action chamber C spanned by {−e_{1}, −ae_{1} − e_{2}} (Figure 3.2(a)). Consider a piecewise linear function given by φ(v_{1}) = φ(v_{2}) = 0, φ(v_{3}) = −r and φ(v_{4}) = −s. We assume that r, s > 0. It is easy to see that if s ≥ ar, then φ is convex, hence H^{1} = H^{2} = 0, supp(H^{0}) = ℤ^{2} ∩ Γ^{0}, with multiplicity 1, where Γ^{0} = {0 ≤ x_{1} ≤ s, 0 ≤ x_{2} ≤ r − ax_{1}} is a 4-gon. If s < ar, since isotropy weights of the four fixed points p_{i} (1 ≤ i ≤ 4) are the negatives of {e_{1}^{∗}, e_{2}^{∗}}, {−e_{1}^{∗}, e_{2}^{∗}}, {−e_{2}^{∗}, −e_{1}^{∗} + ae_{2}^{∗}}, {−e_{2}^{∗}, e_{1}^{∗} − ae_{2}^{∗}}, respectively (Figure 3.2(b)), we have n^{p,C} = 0, 1, 2, 1 for p = p_{1}, p_{2}, p_{3}, p_{4}, and Γ^{p_{1},C} = {x_{1}, x_{2} ≥ 0}, Γ^{p_{2},C} = {x_{1} > r, x_{2} ≥ 0}, Γ^{p_{3},C} = {x_{1} > r, ax_{1} + x_{2} > s}, Γ^{p_{4},C} = {x_{1} ≥ 0, ax_{1} + x_{2} > s}. On the other hand, n^{p,−C} = 2, 1, 0, 1 for p = p_{1}, p_{2}, p_{3}, p_{4}, respectively, and Γ^{p_{1},−C} = {x_{1}, x_{2} < 0}, Γ^{p_{2},−C} = {x_{1} ≤ r, x_{2} < 0}, Γ^{p_{3},−C} = {x_{1} ≤ r, ax_{1} + x_{2} ≤ s}, Γ^{p_{4},−C} = {x_{1} < 0, ax_{1} + x_{2} ≤ s}. Using Proposition 2.6, we obtain supp(H^{0}) = ℤ^{2} ∩ Γ^{0}, supp(H^{1}) = ℤ^{2} ∩ Γ^{1} (the equalities follow from part 2), where Γ^{0} = Γ^{p_{1},C} ∩ Γ^{p_{3},−C} = {x_{1}, x_{2} ≥ 0, ax_{1} + x_{2} ≤ r} (Figure 3.2(c)), Γ^{1} = (Γ^{p_{2},C} ∪ Γ^{p_{4},C}) ∩ (Γ^{p_{2},−C} ∪ Γ^{p_{4},−C}) = {x_{1} ≤ r, x_{2} < 0, ax_{1} + x_{2} > s} (Figure 3.2(d)), and H^{2} = 0 since Γ^{2} ⊂ Γ^{p_{3},C} ∩ Γ^{p_{1},−C} = ∅ (Figure 3.2(e)). The results are in accord with Corollary 3.4. An alternative way of obtaining the same results is to calculate the equivariant index (2.4) and use Corollary 3.4.

Remark 3.6 We make several observations on the multiplicity of weights in H^{k} when 0 < k < n.

1. First, the multiplicity of a weight in H^{k} (0 < k < n) is not necessarily 1, even when dim_{ℂ} M = 2. For example, it was shown in [18, Example 5.6] that for a certain line bundle over the blow-up of ℂP^{2} at three points, a weight can appear in the index with multiplicity −2. Using Corollary 3.4, one concludes that the multiplicity of that weight in H^{1} is 2.

2. When dim_{ℂ} M ≥ 3, a weight can appear simultaneously in cohomology groups of different degrees. To see this, we construct a fan Σ in ℝ^{3}. Set e_{+} = e_{1} + e_{2} + e_{3}. Let the top-dimensional cones be spanned by {e_{1}, e_{2}, e_{+}}, {e_{1}, e_{2}, −e_{3}}, {−e_{1}, −e_{2}, e_{3}}, {−e_{1}, −e_{2}, −e_{+}}, and those obtained by cyclic permutations of the basis. Define a piecewise linear function φ on Σ by φ(e_{+}) = φ(−e_{i}) = 1, φ(−e_{+}) = φ(e_{i}) = −1 (i = 1, 2, 3). Then φ determines a line bundle such that 0 is a weight of both H^{1} and H^{2}.

3. It would be interesting to investigate the general conditions on the fan under which any weight is of multiplicity 1 or can appear in only one of the cohomology groups. One knows that if M_{1} and M_{2} are toric manifolds of one of these types, then so is any M_{1}-fibered toric manifolds over M_{2}. An interesting class of examples is the Bott-Samelson manifolds studied from the symplectic point of view in [10].

4. Necessity of the Kähler condition

It is well-known that the Hirzebruch-Riemann-Roch theorem or its equivariant counterpart, the Atiyah-Bott fixed-point theorem, are valid for holomorphic vector bundles over arbitrary compact complex manifolds.

Therefore it comes as a surprise that the strong equivariant holomorphic Morse inequalities holds for Kähler manifolds only. In this section, we show that the strong inequalities (2.2) are violated on a suitable non-Kähler toric manifold. We also use the violation of the strong inequalities to show the non-existence of invariant Kähler structures on certain symplectic manifolds.

Recall that a toric manifold of complex dimension n is Kähler if and only if its fan Σ admits strictly convex piecewise linear function. When the complex dimension n = 2, such a convex function always exists. Therefore every toric 2-manifold is Kähler. Notice that in this dimension, the strong inequalities (2.2) are equivalent to the weak ones (2.3). When n = 3, there are fans in 𝔱 ≅ ℝ^{3} which does not admit any convex piecewise linear functions. An example is the fan with 8 top-dimensional cones σ_{i} (0 ≤ i ≤ 6) and σ′, whose stereographic projection is given by Figure 4.1(a) (see for example [5, 9]). Suppose that there were a strictly convex piecewise linear function φ on Σ. For the choice of coordinates in Figure 4.1(a), convexity on the adjacent cones σ_{1} and σ_{4} means that 2φ(a) + φ(f) > \frac{1}{2}φ(c) + \frac{1}{2}φ(d). Summing over this and the other two inequalities obtained by the cyclic permutations of (a, d), (b, e), (c, f), we get 3(φ(a) + φ(b) + φ(c)) + φ(d) + φ(e) + φ(f) > 0. On the other hand, convexity on σ_{4} and σ_{2} implies that −2φ(a) > φ(b) + φ(f), and hence 3(φ(a) + φ(b) + φ(c)) + φ(d) + φ(e) + φ(f) < 0, a contradiction. The corresponding toric variety has an orbifold singularity because the cone σ′ is not spanned by a ℤ-basis. (In general, a toric variety has at most orbifold-type singularities if all the top-dimensional cones in the fan are simplicial. It is smooth if these cones are spanned by a ℤ-basis.) The singularity can be avoided by a further triangulation of σ′ into 15 cones σ_{i} (7 ≤ i ≤ 21) by Jurkiewicz [17], as shown in Figure 4.1(b). This results a smooth toric 3-manifold with 22 fixed points.

Theorem 4.1 There exists a T^{3}-invariant line bundle over the smooth toric 3-manifold corresponding to the fan of Figure 4.1 such that the strong equivariant holomorphic Morse inequalities are not satisfied.

Proof. Let {e_{1}, e_{2}, e_{3}} be the standard basis in 𝔱 ≅ ℝ^{3}, and {e_{1}^{∗}, e_{2}^{∗}, e_{3}^{∗}}, its dual basis in 𝔱^{∗}. We choose a piecewise linear function φ on the fan in Figure 4.1 such that φ = −(e_{1}^{∗} + e_{2}^{∗} + e_{3}^{∗}) in the cones σ_{i}, 7 ≤ i ≤ 21 (those inside σ′), φ = 0 in σ_{0}, and φ is given by linear interpolations between the cones σ′ and σ_{0}, for example, φ|_{σ_{1}} = −3e_{2}^{∗} + 3e_{3}^{∗}, φ|_{σ_{4}} = −3e_{3}^{∗}. Choose the action chamber C = σ_{7} (shown in Figure 4.1(b)). It is straightforward to check that

1) 0 ∈ Γ^{k,C} for k = 0, 7 only and n^{p_{0},C} = 0, n^{p_{7},C} = 3,

2) 0 ∉ Γ^{k,−C} for any k. If the strong inequalities (2.2) were to hold, then according to Proposition 2.6, the above claims imply the following:

1) 0 is a weight in H^{0} and H^{3} with multiplicity 1,

2) 0 is not a weight in H^{0} or in H^{3}, respectively. This is a contradiction. □

We actually showed that the strong inequalities (2.5) are violated for the action of any circle subgroup of T^{3} whose generator is in C = σ_{7}. The example is consistent with the Atiyah-Bott fixed-point formula (2.4) since possible contributions to H^{0} and H^{3} cancel in the equivariant index.

Remark 4.2 In [26], the author shows that the strong holomorphic Morse inequalities hold for any meromorphic group action on a complex manifold with a filterable Białynicki-Birula decomposition, which is a filtration of the manifold by closed subvarieties compactible with the group action. This condition is weaker than the Kähler assumption made in the current paper. The toric manifold in Theorem 4.1 was initially constructed to show that Białynicki-Birula decompositions need not be filterable [17]. Therefore the work of [26] provides a deeper understanding this counterexample of strong inequalities.

We now make some remarks on the weak inequalities.

Remark 4.3 Proposition 3.1 on toric manifolds was derived from the weak inequalities (2.3), together with the Atiyah-Bott fixed-point formula (2.4). For a general (possibly non-Kähler) toric variety M with a T -invariant holomorphic line bundle L_{φ}, Demazure’s result [7] (see also [5, 20, 9]) states that the multiplicity of ξ ∈ L^{∗} in H^{k} is equal to the dimension of the local cohomology group H_{Z(ξ)}^{k}(𝔱), where Z(ξ) = {x ∈ 𝔱 | ξ(x) ≥ φ(x)}. In fact, Demazure’s result implies Proposition 3.1. (ξ ∈ L^{∗} ∩Γ^{0} if and only if Z(ξ) = 𝔱, in which case H^{0} = ℂ and H^{k} = 0 for k > 0. That the multiplicity in H^{0} is not greater than 1 also follows from the existence of a dense open T^{ℂ}-orbit in M.) This suggests that (at least in the completely integrable cases) the weak holomorphic Morse inequalities could be valid for a larger class of complex manifolds. For example, the weak inequalities are not violated for the line bundle in Theorem 4.1. Notice also that the lowest complex dimensions for a toric manifold to be non-Kähler is 3; this is when the weak Morse inequalities become weaker than the strong ones. This leaves open the possibility that weak inequalities might hold when the manifold is complex but not Kähler.

Remark 4.4 More generally, consider a complex manifold M with a holomorphic torus T -action. Assume that the fixed-point set F is discrete. For any p ∈ F, let λ_{1}^{p}, ⋯, λ_{n}^{p} ∈ L^{∗}\{0} be the weights of T on T_{p}M. Let E be a holomorphic vector bundle over M with a lifted T -action. Let e_{1}^{p}, ⋯, e_{r}^{p} form a basis of E_{p}, with weights μ_{1}^{p}, ⋯, μ_{r}^{p} ∈ L^{∗}, respectively. Near a fixed point p ∈ F, the T -action on M has the local model ℂ^{n} equipped with the action e^{\sqrt{−1}θ}: (z_{1}, ⋯, z_{n}) ↦ (e^{\sqrt{−1}⟨λ_{1}^{p},θ⟩}z_{1}, ⋯, e^{\sqrt{−1}⟨λ_{n}^{p},θ⟩}z_{n}), θ ∈ 𝔱. Holomorphic sections of E in a neighborhood of p correspond to linear combinations of z_{1}^{m_{1}} ⋯ z_{n}^{m_{n}} ⊗e_{i}^{p} (m_{1}, ⋯, m_{n} ∈ ℕ, 1 ≤ i ≤ r), whose weight under the T -action is μ_{i}^{p} − ∑_{k=0}^{n} m_{k}λ_{k}^{p}. (The minus sign is explained in [19] in S^{1}-cases.) Not all such local sections extend to M. So supp(H^{0}) ⊂ {μ_{i}^{p} − ∑_{k=0}^{n} m_{k}λ_{k}^{p} | m_{1}, ⋯, m_{n} ∈ ℕ, 1 ≤ i ≤ r}, counting multiplicities. This is part of the weak Morse inequalities. For similar reasons, other weak inequalities in (2.3) are likely to be true when the manifold M admits a T -invariant complex structure.

Violation of strong holomorphic Morse inequalities can be used as an obstruction to the existence of T - invariant Kähler structures. Recently, Tolman [22] constructed a simply-connected six dimensional symplectic manifold (M, ω) that has a Hamiltonian T^{2}-action with isolated fixed points but does not admit any T^{2}- invariant Kähler structure. (See [25] for another perspective.) Let {e_{1}, e_{2}} be the standard basis in 𝔱 ≅ ℝ^{2} and {e_{1}^{∗}, e_{2}^{∗}}, the dual basis in 𝔱^{∗}. The six fixed points p_{i} (1 ≤ i ≤ 6) in M correspond to the six vertices μ_{i} = μ(p_{i}) of the moment polytope in Figure 4.2(a) (reproduced from [22, Figure 2]). Their isotropy weights are the negatives of {e_{1}^{∗}, e_{2}^{∗}, e_{1}^{∗}+e_{2}^{∗}}, {e_{1}^{∗}, e_{2}^{∗}, −e_{1}^{∗}−e_{2}^{∗}}, {e_{1}^{∗}, −e_{2}^{∗}, e_{1}^{∗}−e_{2}^{∗}}, {−e_{1}^{∗}, −e_{2}^{∗}, e_{1}^{∗}−e_{2}^{∗}}, {−e_{1}^{∗}, −e_{1}^{∗}+e_{2}^{∗}, −2e_{1}^{∗}+e_{2}^{∗}}, {−e_{1}^{∗}, −e_{1}^{∗} + e_{2}^{∗}, 2e_{1}^{∗} − e_{2}^{∗}}, respectively. It can be shown that the strong inequalities (2.2) do not hold in this example, therefore giving an alternative proof of the non-existence of T^{2}-invariant Kähler structures. (Most of this comes up in a discussion with S. Tolman and C. Woodward.)

Proposition 4.5 (Tolman [22]) There is no T^{2}-invariant Kähler structure on Tolman’s manifold M.

Proof. There is a T^{2}-invariant line bundle L such that the weights on L_{p_{i}} (1 ≤ i ≤ 6) are ξ_{1} = 0, ξ_{2} = 3e_{1}^{∗} + 3e_{2}^{∗}, ξ_{3} = 2e_{2}^{∗}, ξ_{4} = 3e_{1}^{∗} + 2e_{2}^{∗}, ξ_{5} = 5e_{1}^{∗}, ξ_{6} = −e_{1}^{∗} + 3e_{2}^{∗} (Figure 4.2(b), from [22, Figure 3, Case 0 < t < s]). If there is a T^{2}-invariant Kähler form ω′, which may be different from the original symplectic form ω, then Figure 4.2(b) is a deformation of the moment polytope associated to ω′. Since all the symplectic quotients of M by the T^{2}-action are ℂP^{1}, where the complex structure is unique, the curvature of L is still a (1, 1)-form on M. Therefore L can be equipped with a holomorphic structure, and it is invariant under the T^{2}-action. Let C be the cone spanned by {e_{1} − e_{2}, −e_{2}}. Then e_{1}^{∗} + 2e_{2}^{∗} ∈ Γ^{p,C} for p = p_{1}, p_{5} only and n^{p_{1},C} = 1, n^{p_{5},C} = 0 (Figure 4.2(c)). If the strong inequalities (2.2) were to hold, by (2.9) of Proposition 2.6, e_{1}^{∗} + 2e_{2}^{∗} ∉ supp(H^{2}). On the other hand, e_{1}^{∗} + 2e_{2}^{∗} ∈ Γ^{p,−C} only for p = p_{3}, p_{6} and n^{p_{3},−C} = 0, n^{p_{6},−C} = 2 (Figure 4.2(d)). By (2.10) of Proposition 2.6, e_{1}^{∗} + 2e_{2}^{∗} ∈ supp(H^{2}), a contradiction. □

In fact, this argument shows further that there is no Kähler structure invariant under any S^{1}-subgroup of T^{2} whose generator lies in the cone C.

Remark 4.6 It is not known whether there is a T^{2}-invariant complex structure on Tolman’s manifold. If it exists, then the strong inequalities (2.2) need not be true even when there is a moment map. On the other hand since the toric 3-manifold in Theorem 4.1 is not symplectic, though generalized moment maps in the sense of [18] do exist, there is a possibility that the strong inequalities (2.2) could be true for Hamiltonian torus actions on symplectic manifolds with invariant complex structures not necessarily calibrated [3, §II.1.5] by the symplectic forms. If so, then Tolman’s example does not admit any T^{2}-invariant complex structure.

5. Non-Abelian equivariant holomorphic Morse inequalities

In this section we consider a compact Kähler manifold M of complex dimension n with a holomorphic and Hamiltonian action of a compact connected real Lie group G, assuming that the fixed-point set of a maximal torus T ⊂ G is discrete. Let E be a holomorphic vector bundle over M with a lifted holomorphic G-action. Let G^{ℂ} be the complexification of G that contains G as a maximal compact subgroup. Then G^{ℂ} acts holomorphically on M and E [15]. Choose an (isolated) T -fixed-point p ∈ F and denote the isotropy group of p in G and G^{ℂ} by H = G_{p} and U = (G^{ℂ})_{p}, respectively. (In this way, we have for simplicity dropped the subscript p unless ambiguity occurs.) Clearly H ⊃ T and U is a complex subgroup of G^{ℂ} containing H^{ℂ} such that U ∩ G = H. We need a few group-theoretic results on G^{ℂ}, U, H^{ℂ} satisfying the above constraints. These results will determine the behavior of the fixed-point set, the isotropy weights and the fiber of the vector bundle over the fixed points. For this purpose, it suffices that M is a complex manifold with a holomorphic G-action; the Kähler condition will be used only to establish the equivariant holomorphic Morse inequalities.

Let H_{0}, U_{0} be the identity components of H and U, and let W = N_{G}(T)/T = N_{G^{ℂ}}(T^{ℂ})/T^{ℂ}, W_{H}, W_{H_{0}}, W_{U}, W_{U_{0}} be the Weyl groups of G, H, H_{0}, U, U_{0}, respectively.

Lemma 5.1 W acts transitively on the T^{ℂ}-fixed-point set in G^{ℂ}/U with isotropy group W_{U}.

Proof. The fixed-point set is {gU | g ∈ G^{ℂ}, g^{−1}T^{ℂ}g ⊂ U}. It has a well-defined action of W because h ∈ N_{G^{ℂ}}(T^{ℂ}) acts on the fixed points by gU ↦ hgU and for any t ∈ T^{ℂ}, htgU = hg(g^{−1}tg)U = hgU. To show that the action is transitive, choose any fixed point gU. Since g^{−1}T^{ℂ}g and T^{ℂ} are two maximal tori in U_{0}, there is an element u ∈ U_{0} such that u^{−1}g^{−1}T^{ℂ}gu = T^{ℂ}. So gu ∈ N_{G^{ℂ}}(T^{ℂ}) and gU = (gu)U can be obtained from U by an element of W. Finally, h ∈ N_{G^{ℂ}}(T^{ℂ}) fixes U (i.e., hU = U) if and only if h ∈ U ∩ N_{G^{ℂ}}(T^{ℂ}) = N_{U}(T^{ℂ}). So the isotropy group in W is N_{U}(T^{ℂ})/T^{ℂ} = W_{U}. □

Proposition 5.2 There is an action of W on the T -fixed-point set F in M. Each W -orbit in F is the intersection of F with a G- or G^{ℂ}-orbit in M, which do not depend on the choice of T.

Proof. Lemma 5.1 implies that for p ∈ F, the T -fixed points in G^{ℂ} · p ≅ G^{ℂ}/U form a single W -orbit. For a different maximal torus gT g^{−1} (g ∈ G), the new fixed-point set gF is contained in the same G-orbits. □

Example 5.3 We consider the diagonal action of G = SO(3) on M = S^{2} × S^{2}, where G acts on S^{2} by standard rotations. The maximal torus in G is T = S^{1}. Let n, s be the poles in S^{2} which are fixed by T. Then the T -fixed-point set in M is F = {(n, n), (s, s), (n, s), (s, n)}. Though the isotropy groups G_{p} = T for all p ∈ F, the situation is different if we consider the action of the complexification G^{ℂ} = SL(2, ℂ)/ℤ_{2}. It is easy to see that the stabilizer (G^{ℂ})_{(n,n)} = B, a Borel subgroup in G^{ℂ}, whereas the stabilizer (G^{ℂ})_{(n,s)} = T^{ℂ}. Consequently, the orbit G^{ℂ} · (n, n) = {(x, x) | x ∈ S^{2}} ≅ G^{ℂ}/B is compact and contains another fixed point (s, s) related to (n, n) by W = ℤ_{2}, whereas G^{ℂ} · (n, s) = {(x, y) | x ≠ y ∈ S^{2}} ≅ G^{ℂ}/T^{ℂ} is non-compact and contains (s, n), related to (n, s) also by ℤ_{2}. In this case, M is the union of two G^{ℂ}-orbits.

Let Δ, Δ_{H_{0}}, Δ_{U_{0}} be the set of roots of the pairs (G^{ℂ}, T^{ℂ}), (H_{0}^{ℂ}, T^{ℂ}), (U_{0}, T^{ℂ}), respectively. Choose a set of positive roots Δ^{+} ⊂ Δ and let Δ^{−} = −Δ^{+}, Δ_{H_{0}}^{±} = Δ^{±} ∩ Δ_{H_{0}}. The length of w ∈ W is l(w) = |wΔ^{+} ∩ Δ^{−}|. Let l_{H_{0}}(w) = |w(Δ^{+}\Δ_{H_{0}}^{+}) ∩ Δ^{−}| be the length of w relative to the subgroup H_{0}. Notice that l_{T} (w) = l(w) (w ∈ W).

Lemma 5.4 There is a T^{ℂ}-fixed-point in G^{ℂ}/U at which the set of weights of the isotropy representation contains Δ^{+}\Δ_{H_{0}}^{+}.

Proof. Since the sets of isotropy weights at two fixed-points p and wp (w ∈ W) are related by the action of w and since any two choices of Δ^{+} are conjugate under a w ∈ W, it suffices to show that the lemma holds for any given p under a particular choice of Δ^{+} depending on p. Because U is a complex subgroup of G, Δ_{U_{0}} is a closed subset of Δ, i.e., if α, β ∈ Δ_{U_{0}} and α + β ∈ Δ, then α + β ∈ Δ_{U_{0}}. Furthermore, since U ∩ G = H, Δ_{U_{0}} ∩ (−Δ_{U_{0}}) = Δ_{H_{0}}. It is easy to see that Δ′ = Δ_{U_{0}}\Δ_{H_{0}} is also closed (if α, β ∈ Δ′ but α + β ∈ Δ_{H_{0}}, then −β = (−α − β) + α ∈ Δ_{U_{0}}, a contradiction) and satisfies Δ′ ∩ (−Δ′) = ∅. So Δ′ is contained in some choice of Δ^{−} [4]. Consequently, there is a parabolic subgroup P ⊃ U_{0} and P ∩ G = H_{0}. Therefore the set of weights of isotropy representation at p is Δ\Δ_{U_{0}} ⊃ Δ\Δ_{P} = Δ^{+}\Δ_{H_{0}}^{+}. □

For each W -orbit S ∈ F/W, choose p ∈ S and let H = G_{p} be the isotropy group in G. Denote Δ_{S} = Δ_{H_{0}}, l_{S}(w) = l_{H_{0}}(w) and det_{S} w = (−1)^{l_{S}(w)} det w for w ∈ W; they do not depend on the choice of p in S.

Proposition 5.5 In each orbit S ∈ F/W we can choose a representative, denoted by p_{S}, such that the weights of the isotropy representation at p_{S} contains Δ^{+}\Δ_{S}^{+}.

Proof. This is a direct consequence of Lemma 5.4. □

Remark 5.6 In [13] (see also [21]), under a regularity assumption that implies H = T, it was shown that the set of isotropy weights contains Δ^{+} up to signs that depend on the roots. Proposition 5.5 is a refinement of this result when M is complex and the G-action preserves the complex structure. Let λ_{k}^{S} (k = 1, ⋯, n−|Δ^{+}\Δ_{S}^{+}|) be the other weights at p_{S}. Since T_{p_{S}}M is also a representation of U, the set of weights (Δ^{+}\Δ_{S}^{+})∪{λ_{k}^{S}} is W_{U}- invariant. At another fixed point wp_{S} ∈ S, the corresponding set of isotropy weights is w((Δ^{+}\Δ_{S}^{+}) ∪ {λ_{k}^{S}}), which depends only on the coset w̄ ∈ W/W_{U}. By W -invariance, any action chamber can be transformed into one that intersects the positive Weyl chamber 𝔱^{+} = {x ∈ 𝔱 | ⟨Δ^{+}, x⟩ > 0} in an open cone; let C be one of such. At p_{S}, the polarizing index n_{p_{S}}^{C} is equal to n^{C}({λ_{k}^{S}}), that of the set {λ_{k}^{S}}. At wp_{S}, n_{wp_{S}}^{C} = l_{S}(w)+n^{C}({wλ_{k}^{S}}).

The following lemma and example are communicated to the author by D. Vogan.

Lemma 5.7 N(Δ_{U_{0}}) = {w ∈ W | wΔ_{U_{0}} = Δ_{U_{0}}} contains W_{H_{0}} as a normal subgroup. Moreover, U/U_{0} ≅ W_{U}/W_{H_{0}} is a subgroup of N(Δ_{U_{0}})/W_{H_{0}}.

Proof. Since H_{0}^{ℂ} is a subgroup of U, W_{H_{0}} and W_{U} preserve the set Δ_{U_{0}}, hence W_{H_{0}} < W_{U} < N(Δ_{U_{0}}). For any α ∈ Δ_{H_{0}}, w ∈ N(Δ_{U_{0}}), since wr_{α}w^{−1} = r_{wα}, and wα ∈ wΔ_{U_{0}} ∩ (−wΔ_{U_{0}}) = Δ_{H_{0}}, W_{H_{0}} is a normal subgroup of N(Δ_{U_{0}}). Finally, we prove U/U_{0} ≅ W_{U}/W_{H_{0}}. Define a homomorphism N_{U}(T^{ℂ}) → U/U_{0} by u ↦ uU_{0}. Its kernel is N_{U_{0}}(T^{ℂ}); we show that it is onto. For any u ∈ U, u^{−1}T^{ℂ}u is a maximal torus in U_{0}. So there is u_{0} ∈ U_{0} such that u_{0}^{−1}u^{−1}T^{ℂ}uu_{0} = T^{ℂ}. This implies that uu_{0} ∈ N_{U}(T^{ℂ}) and uu_{0} ↦ uU_{0}. By homomorphism theorems, U/U_{0} ≅ N_{U}(T^{ℂ})/N_{U_{0}}(T^{ℂ}) ≅ W_{U}/W_{U_{0}}. Finally, let P be a parabolic subgroup that contains U_{0} and such that P ∩ G = H_{0}. Then W_{H_{0}} < W_{U_{0}} < W_{P}. Since W_{P} = W_{H_{0}}, we conclude that W_{U_{0}} = W_{H_{0}}. □

Example 5.8 The isotropy group U can be disconnected. Let the action of G = SO(3) on M = ℂP^{2} be the projectivization of the adjoint representation on 𝔰𝔬 (3)^{ℂ} ≅ ℂ^{3}. If the root space decomposition is 𝔰𝔬 (3)^{ℂ} = ℂe_{0} ⊕ ℂe_{+} ⊕ ℂe_{−}, where ℂe_{0} = Lie(T^{ℂ}) and [e_{0}, e_{±}] = ±e_{±}, then the T -fixed-point set F = {[e_{0}], [e_{+}], [e_{−}]}. The isotropy groups of [e_{±}] in G^{ℂ} are Borel subgroups while that of [e_{0}] is the full normalizer N_{G^{ℂ}}(T^{ℂ}) of T^{ℂ}, which is disconnected.

Since G is compact, 𝔱 is equipped with an invariant inner product; this induces one on 𝔱^{∗}, which will be denoted by (·, ·). Let D = {λ ∈ L^{∗} | (λ, Δ^{+}) ≥ 0} and D_{H_{0}} = {λ ∈ L^{∗} | (λ, Δ_{H_{0}}^{+}) ≥ 0} be the sets of dominant weights with respect to G and H_{0}, respectively. We denote the irreducible representation of G (H_{0}, respectively) of the highest weight λ ∈ D (λ ∈ D_{H_{0}}, respectively) by R_{λ}^{G} (R_{λ}^{H_{0}}, respectively). Since W_{U} preserves Δ_{U_{0}}, hence Δ_{U_{0}} ∩ (−Δ_{U_{0}}) = Δ_{H_{0}}, the action of W_{U} permutes the Weyl chambers of the pair (H_{0}^{ℂ}, T^{ℂ}). Let D_{U} ⊂ D_{H_{0}} be a fundamental region of the group W_{U} in L^{∗}.

Lemma 5.9 Let V be a finite dimensional representation of U. Then

char(V) = ∑_{Λ∈D_{U}} m_{Λ} ∑_{w∈W_{U}} w \frac{e^{Λ}}{∏_{α∈Δ_{H_{0}}^{+}} (1 − e^{−α})}.  (5.1)

Here m_{Λ} ∈ ℚ; if U is connected, then m_{Λ} ∈ ℕ is the multiplicity of R_{Λ}^{H} in V.

Proof. As a representation of H_{0}, V = ⨁_{λ} R_{λ}^{H_{0}}; the direct sum is over some λ ∈ D_{H_{0}} with possible multiplicities. Since V is a representation of U, W_{U}/W_{H_{0}} acts on the set {R_{λ}^{H_{0}}}. For a given W_{U}/W_{H_{0}}-orbit through R_{Λ}^{H_{0}}, let W_{Λ} be the isotropy group. The contribution of this orbit to the character is

\frac{1}{|W_{Λ}|} ∑_{w̄∈W_{U} /W_{H_{0}}} w̄ char(R_{Λ}^{H_{0}}) = \frac{1}{|W_{Λ}|} ∑_{w∈W_{U}} w \frac{e^{Λ}}{∏_{α∈Δ_{H_{0}}^{+}} (1 − e^{−α})}.  (5.2)

Equation (5.1) follows readily. If U is connected, then W_{U}/W_{H_{0}} = {1}, hence all m_{Λ} ∈ ℕ. □

At p_{S} (S ∈ F/W) whose isotropy group in G^{ℂ} is U, set D_{S} = D_{U}. According to the above lemma,

char(E_{p_{S}}) = ∑_{Λ∈D_{S}} m_{Λ}^{S} ∑_{w∈W_{U}} w \frac{e^{Λ}}{∏_{α∈Δ_{S}^{+}}(1 − e^{−α})},  (5.3)

where m_{Λ}^{S} ∈ ℚ; if the isotropy group U is connected, then m_{Λ}^{S} ∈ ℕ is the multiplicity of R_{Λ}^{H} in E_{p_{S}}.

Theorem 5.10 Let G be a compact connected Lie group acting Hamiltonianly on a compact Kähler manifold M preserving the complex structure. Assume that the fixed-point set F of the maximal torus T is discrete. If E is a holomorphic vector bundle over M where the G-action lifts holomorphically, then G acts on the cohomology groups H^{k}(M, O(E)) (0 ≤ k ≤ n). Let H^{k}(M, O(E)) = ⨁_{Λ∈D} m_{Λ}^{k}R_{Λ}^{G} (m_{Λ}^{k} ∈ ℕ) be the decompositions into irreducible representations R_{Λ}^{G} of G. Choose an action chamber C that intersects 𝔱^{+} in an open cone. Then under the above notation, we have the following non-Abelian equivariant holomorphic Morse inequalities

∑_{S∈F/W} ∑_{w∈W} det_{S}w t^{l_{S}(w)+n^{C}(\{wλ_{k}^{S}\})} ∑_{Λ∈D_{S}} m_{Λ}^{S} e^{w(Λ+ρ)−ρ} ∏_{wλ_{k}^{S}∈C^{∗}} \frac{1}{1 − e^{−wλ_{k}^{S}}} ∏_{wλ_{k}^{S}∈−C^{∗}} \frac{e^{wλ_{k}^{S}}}{1 − e^{wλ_{k}^{S}}}

= ∑_{k=0}^{n} t^{k} ∑_{w∈W} det w ∑_{Λ∈D} m_{Λ}^{k} e^{w(Λ+ρ)−ρ} + (1 + t)Q^{C}(t) ∏_{α∈Δ^{+}}(1 − e^{−α})  (5.4)

for some Q^{C}(t) ≥ 0.

Proof. We apply (2.2) to the maximal torus T of G. The contribution of one orbit S ∈ W/F is a sum over cosets w̄ ∈ W/W_{U}. This sum can be combined with that over W_{U} in (5.3). Since the set w((Δ^{+}\Δ_{S}^{+}) ∪ {λ_{k}^{S}}) depends only on the coset w̄ ∈ W/W_{U} and since char(E_{w̄p_{S}}) = w̄ char(E_{p_{S}}), the result is

∑_{w∈W} t^{n_{wp_{S}}^{C}} ∏_{wα∈w(Δ^{+}\Δ_{S}^{+})∩Δ^{+}} \frac{1}{1 − e^{−wα}} ∏_{wα∈w(Δ^{+}\Δ_{S}^{+})∩Δ^{−}} \frac{e^{wα}}{1 − e^{wα}} ∑_{Λ∈D_{S}} m_{Λ}^{S} w (\frac{e^{Λ}}{∏_{α∈Δ_{S}^{+}}(1 − e^{−α})}) L_{w,S}^{C}
= ∑_{w∈W}(−1)^{l(w)−l_{S}(w)}t^{l_{S}(w)+n^{C}(\{wλ_{k}^{S}\})} w ( ∏_{α∈Δ^{+}\Δ_{S}^{+}} \frac{1}{1 − e^{−α}} ) ∑_{Λ∈D_{S}} m_{Λ}^{S}w (\frac{e^{Λ}}{∏_{α∈Δ_{S}^{+}}(1 − e^{−α})}) L_{w,S}^{C}
= ∑_{w∈W} det_{S}w t^{l_{S}(w)+n^{C}(\{wλ_{k}^{S}\})} \frac{det w ∑_{Λ∈D_{p}} m_{Λ}^{S} e^{w(Λ+ρ)−ρ}}{∏_{α∈Δ^{+}}(1 − e^{−α})} L_{w,S}^{C},  (5.5)

where

L_{w,S}^{C} = ∏_{wλ_{k}^{S}∈C^{∗}} \frac{1}{1 − e^{−wλ_{k}^{S}}} ∏_{wλ_{k}^{S}∈−C^{∗}} \frac{e^{wλ_{k}^{S}}}{1 − e^{wλ_{k}^{S}}}  (5.6)

is a factor from weights not in Δ^{+}\Δ_{S}^{+}. Using Weyl’s character formula, the contribution to (2.2) from the cohomology group is

∑_{k=0}^{n} t^{k} char(H^{k}) = ∑_{k=0}^{n} t^{k} ∑_{Λ∈D} m_{Λ}^{k} \frac{∑_{w∈W} det w e^{w(Λ+ρ)−ρ}}{∏_{α∈Δ^{+}}(1 − e^{−α})} .  (5.7)

The result follows after multiplying both sides of (2.2) by ∏_{α∈Δ^{+}}(1 − e^{−α}). □

Remark 5.11 Setting t = −1 in (5.4), we obtain a fixed-point formula

∑_{S∈F/W} ∑_{w∈W} det w ∑_{Λ∈D_{S}} m_{Λ}^{S} e^{w(Λ+ρ)−ρ} ∏_{k=1}^{n−|Δ^{+}\Δ_{S}^{+}|} \frac{1}{1 − e^{−wλ_{k}^{S}}} = ∑_{k=0}^{n}(−1)^{k} ∑_{w∈W} det w ∑_{Λ∈D} m_{Λ}^{k} e^{w(Λ+ρ)−ρ} (5.8)

for non-Abelian group actions.

Acknowledgement. The author would like to thank M. Božičević, R. Da̧browski, M. Grossberg, V. Mathai, E. Prato, S. Tolman, D. Vogan, C. Woodward for illuminating discussions. He also thanks M. Vergne for kindly pointing out an error in an earlier version of this paper and the referees for helpful suggestions. This work is partially supported by NSF grants DMS-93-05578 at Columbia University, DMS-90-22140 at MSRI and by ICTP.

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[figure: e_{1}^{∗}; e_{2}^{∗}; e_{2}^{∗} − e_{1}^{∗}; ^{𝔱^{∗}}; (a) An example of isotropy weights at a fixed point p ∈ F when dim T = 2, dim_{ℂ} M = 3.; C; −C; 𝔱; (b) The hyperplanes cut 𝔱 into 6 chambers. (Weights at other fixed points may cut them further.) A chamber C is chosen.; ξ; Γ^{p,C}; (c) The cone Γ^{p,C} when suppE_{p} = {ξ}. Dotted line stands for bound- aries not included. The polarizing in- dex n^{p,C} = 1.; ξ; Γ^{p,−C}; (d) The cone Γ^{p,−C} when suppE_{p} = {ξ}. Dotted line stands for bound- aries not included. The polarizing in- dex n^{p,−C} = 2.]

Figure 2.1: Illustration of isotropy weights, action chambers and the regions Γ^{p,C}.

[figure: σ_{3}; σ_{1}; σ_{2}; C; (a) The fan of ℂP^{2} with three top dimensional cones σ_{i} (1 ≤ i ≤ 3). An action chamber C = σ_{3} ∩ (−σ_{1}) is chosen.; ξ_{3}(0, 0); ξ_{1}(r, 0); ξ_{2}(0, r); (b) Position of weights ξ_{i} in the fiber over p_{i} ∈ F. The two arrows at ξ_{i} are the negative of isotropy weights at p_{i}.; Γ^{p_{1},C}; Γ^{p_{3},−C}; (c) The two cones Γ^{p_{1},C} and Γ^{p_{3},−C}, with n^{p_{1},C} = n^{p_{3},−C} = 0, intersect at the dotted region.; Γ^{p_{2},C}; Γ^{p_{2},−C}; (d) The two cones Γ^{p_{2},C} and Γ^{p_{2},−C}, with n^{p_{2},C} = n^{p_{2},−C} = 1, have empty in- tersection.; Γ^{p_{3},C}; Γ^{p_{1},−C}; (e) The two cones Γ^{p_{3},C} and Γ^{p_{1},−C}, with n^{p_{2},C} = n^{p_{2},−C} = 2, have empty in- tersection.]

Figure 3.1: The projective space ℂP^{2} with an invariant line bundle of first Chern number r ≥ 0

[figure: v_{1}; v_{3}; v_{2}; v_{4}; σ_{1}; σ_{2}; σ_{3} = C; σ_{4}; (a) The fan of the Hirzebruch surface with four top dimen- sional cones σ_{i} (1 ≤ i ≤ 4). A chamber C = σ_{3} is chosen.; ξ_{1}(0, 0); ξ_{2}(r, 0); ξ_{4}(0, s); ξ_{3}(r, s − ar); (b) Position of weights ξ_{i} in the fiber over p_{i} ∈ F. The two arrows at ξ_{i} are the negative of isotropy weights at p_{i}.; Γ^{p_{3},−C}; Γ^{p_{1},C}; (c) The two cones Γ^{p_{1},C} and Γ^{p_{3},−C}, with n^{p_{1},C} = n^{p_{3},−C} = 0, intersect at the dotted region.; Γ^{p_{2},C}; Γ^{p_{2},−C}; Γ^{p_{4},C}; Γ^{p_{4},−C}; (d) The four cones with po- larizing indices 1. Γ^{p_{4},C} and Γ^{p_{2},−C} intersect at the dotted region.; Γ^{p_{1},−C}; Γ^{p_{3},C}; (e) The two cones Γ^{p_{3},C} and Γ^{p_{1},−C}, with n^{p_{3},C} = n^{p_{1},−C} = 2, have empty in- tersection.]

Figure 3.2: Hirzebruch surface with an invariant line bundle, case s < ar

[figure: a; b; c; d; e; f; σ′; 0; 0; 0; 1; 2; 3; 4; 5; 6; (a) A fan that does not admit a strictly convex piecewise linear function. The edges are spanned by a(1, 0, 0), b(0, 1, 0), c(0, 0, 1), d(0, −2, −1), e(−1, 0, −2), f(−2, −1, 0).; d; e; f; g; h; i; j; k; l; m; 7; (b) The triangulation of σ′ by Jurkiewicz. New edges are generated by g(−1, −2, −1), h(−1, −1, −2), i(−2, −1, −1), j(−1, −2, −2), k(−2, −1, −2), l(−2, −2, −1), m(−1, −1, −1).]

Figure 4.1: Stereographic projection of the fan Σ corresponding to a non-Kähler toric 3-manifold. There are 22 top-dimensional cones σ_{i} (0 ≤ i ≤ 21) in Σ. The cones σ_{i} (0 ≤ i ≤ 6) are shown in (a). The cones σ_{i} (7 ≤ i ≤ 21) are contained in σ′. Only σ_{7} is labeled in (b).

[figure: μ_{1}; μ_{2}; μ_{3}; μ_{4}; μ_{5}; μ_{6}; (a) The moment polytope of Tol- man’s symplectic manifold. The ver- tices μ_{i} are the values of the moment map μ at p_{i} ∈ F.; ξ_{1}(0, 0); ξ_{2}(3, 3); ξ_{3}(0, 2); ξ_{4}(3, 3); ξ_{5}(5, 0); ξ_{6}(−1, 3); (b) Position of the weights ξ_{i} in the fiber of a line bundle over p_{i} ∈ F. The negative of isotropy weights are also shown.; (1, 2); Γ^{p_{1},C}; Γ^{p_{5},C}; (c) Two cones Γ^{p_{1},C}, Γ^{p_{5},C} contain- ing e_{1}^{∗} + 2e_{2}^{∗}. The polarizing indices n^{p_{1},C} = 1, n^{p_{5},C} = 0.; (1, 2); Γ^{p_{3},−C}; Γ^{p_{6},−C}; (d) Two cones Γ^{p_{3},−C}, Γ^{p_{6},−C} con- taining e_{1}^{∗} + 2e_{2}^{∗}. The polarizing in- dices n^{p_{3},−C} = 0, n^{p_{6},−C} = 2.]

Figure 4.2: Moment polytope and a T^{2}-invariant line bundle over Tolman’s manifold