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On the Instanton Complex of Holomorphic Morse Theory

Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixed-point formulas on a possibly non-compact manifold. Finally, a number of examples and applications are given.


Introduction
Given a Morse function on a compact real manifold, the Morse inequalities bound the Betti numbers in terms of the information of critical points. However, the former can not be determined by the Morse inequalities alone unless the Morse function is perfect. If the Morse function satisfies the transversality condition [45], then there is a finite dimensional complex, called the Thom-Smale-Witten complex or the instanton complex [50], which computes the cohomology groups of the manifold. (See [10] for a historical review.) The instanton complex consists of vector spaces spanned by the critical points of the Morse function (when they are isolated), graded by their Morse indices. The coboundary operators come from counting (with orientation) the number of gradient paths between critical points whose Morse indices differ by one. The latter is related to the instanton tunneling effect in supersymmetric quantum mechanics [50].
Consider a complex manifold with a holomorphic group action and a holomorphic vector bundle over the manifold on which the group action lifts holomorphically. We want to determine the Dolbeault cohomology groups (twisted by the vector bundle) as representations of the group. When the manifold is compact, the fixed-point formula of Atiyah and Bott [2] (for isolated fixed points) and of Atiyah and Singer [3] computes 1 E-mail address: swu@maths.adelaide.edu.au the alternating sum of the characters on the cohomology groups. For holomorphic Morse theory, this (equivariant) index theorem is the counterpart of the Hopf (or Lefschetz) formula. When the manifold is compact and Kähler and the group is the circle group, Morse-type inequalities were obtained by Witten [51] using a holomorphic version of supersymmetric quantum mechanics. These (equivariant) holomorphic Morse inequalities put constraints on the sizes of Dolbeault cohomology groups but do not completely determine them. In [39], a heat kernel proof was given under the additional assumption that the fixed points are isolated. In [52], these inequalities were generalized to cases with torus and non-Abelian group actions. Furthermore, it was shown that the Kähler assumption was necessary for holomorphic Morse inequalities [52], although not so for the fixed-point theorem. In [53], these inequalities were proved analytically for compact Kähler manifolds with possibly non-isolated fixed points.
In this paper, we construct the holomorphic counterpart of the instanton complex which computes the Dolbeault cohomology groups using the combinatorial data of the group action. At the same time, we investigate more closely the condition on the complex manifold for establishing a holomorphic Morse theory. Holomorphic Morse theory differs from ordinary Morse theory in a number of ways. If the circle group acts on a compact Kähler manifold in a Hamiltonian fashion, the moment map is a perfect Morse function whose critical points have even Morse indices only, which can not differ by one. Furthermore, Smale's transversality condition fails in general and the gradient paths are never isolated because of the circular symmetry. Consequently, the techniques for holomorphic Morse theory will be quite different from those for ordinary Morse theory.
We start with a complex manifold with a holomorphic action of a (complex) torus. The action of a noncompact 1-parameter subgroup is analogous to the gradient flow of a Morse function. The group action is meromorphic if, roughly speaking, all such orbits start from and end at some points in the manifold, which must be fixed points of the torus. If so, then there is a relation on the connected components of the fixedpoint set given by the direction of the flows. The central result of this paper is that if this relation is a partial ordering, then there are two (equivariant) spectral sequences converging (equivariantly) to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences will be constructed using sheaf-theoretic techniques from a filtration of the complex manifold determined by the group action. The spectral sequences, with the natural coboundary maps, are the couterparts in holomorphic Morse theory of the instanton complex in ordinary Morse theory. The information of the E 1 -terms already implies the holomorphic Morse inequalities. But unlike ordinary Morse theory, the spectral sequences do not always degenerate at E 2 . When the manifold is compact and Kähler, the partial order condition is automatically satisfied. Thus the results of [51,39,52,53] are recovered.
The rest of the paper is organized as follows. In section 2, we establish some facts about meromorphic torus actions on a compact or a suitably non-compact complex manifold. In section 3, we construct two spectral sequences converging to Dolbeault cohomology groups and those with compact support, respectively, under the partial order condition. In particular, we obtain holomorphic Morse inequalities and fixed-point formulas for a possibly non-compact manifold. We also study the condition under which the spectral sequences degenerate to cochain complexes. In section 4, we consider various examples and applications. We first present a spectral sequence calculation using the language ofČech cohomology. The application to flag manifolds yields a geometric realization of the Bernstein-Gelfand-Gelfand resolution and its generalizations. We also study the Dolbeault cohomologies and geometric quantization on non-compact manifolds.
Throughout this paper, N, R, R ± , C and C × denote the sets of non-negative integers, real numbers, positive (negative) real numbers, complex numbers and non-zero complex numbers, respectively.

Holomorphic torus actions
We first recall from [52,53] some notations of holomorphic torus actions without making the compact or Kähler assumption.
Let T be a complex torus with Lie algebra t. Let T R be the (real) maximal compact torus subgroup of T and t R = √ −1 Lie(T R ). Let ℓ be the integral lattice in t R , and ℓ * ⊂ t * R , the dual lattice. If T = C × , the multiplicative group of non-zero complex numbers, then T R = S 1 , t R = R, and ℓ = Z. In general, for any If every weight ξ ∈ ℓ * of R has a finite multiplicity r ξ , then the character char R = ξ∈ℓ * r ξ e ξ ∈ Z[ℓ * ] is well-defined. Let supp R = supp char R. As in [39,52], we write We emphasize here that the left-hand side is a notation for the formal series in Z[ℓ * ] on the right-hand side.
More generally, if R is a finite dimensional representation of T , we can write Let X be a complex manifold of dimension n. Suppose T acts holomorphically and effectively on X.
The fixed-point set X T of T in X, if non-empty, is a complex submanifold of X. Let F be the set of T acts on N α preserving the base X T α pointwise. The weights of the isotropy representation on the normal fiber remain constant within open polyhedral cones called action chambers [44]. Choose an action chamber C. Let λ C α,k = ±λ α,k , with the sign chosen so that λ C α,k ∈ C * . (Here C * is the dual cone in t * R defined by C * = {ξ ∈ t * R | ξ, C > 0}. ) We define ν C α as the number of weights λ α,k ∈ C * . Let N C α be the direct sum of the sub-bundles corresponding to the weights λ α,k ∈ C * . Then which is called the polarizing index of X T α with respect to C. In subsection 2.1, we will consider holomorphic torus actions on compact manifolds; a non-compact setting will be studied in subsection 2.2.

Meromorphic torus actions on compact manifolds
Throughout this subsection, X is a compact complex manifold with a holomorphic action of the torus T .
Then F is a finite set and each component X T α (α ∈ F ) is compact.
If T = C × , the action is meromorphic if and only if for any x ∈ X, the limits π + (x) = lim u→0 ux π − (x) = lim u→∞ exist. In this case, the holomorphic map C × × X → X can be extended to a meromorphic map P 1 × X → X.

Proposition 2.2
If the T -action on X is meromorphic, then for any x ∈ X and action chamber C, the limit π v (x) for v ∈ ℓ ∩ C depends only on C and not on the choice of v.
Proof. 1. Let X ′ be a connected component of the fixed-point set of C × v . Then X ′ ∩ X T is a closed subset of X ′ . For any x ∈ X ′ ∩ X T , let X T α be the component of X T that contains x. Since the T -action is effective and λ α,k (v) = 0 for any 0 ≤ k ≤ n − n α , we have dim X T α = dim X ′ . Therefore X ′ ∩ X T is also an open subset of X ′ . Finally, choose v 1 , . . . , v r−1 ∈ ℓ (r = dim C T ) such that {v, v 1 , . . . , v r−1 } is a basis of t R . Pick any x ′ ∈ X ′ . Since the T -action is meromorphic, the iterated limit x = π v1 π v2 · · · π vr−1 (x ′ ) exists. It is clear ; this depends only on C and not on the choice of v. ✷ We denote π v (x) by π C (x) when v ∈ ℓ ∩ C.
Remark 2.3 1. If X is a compact Kähler manifold and X T = ∅, then the T R -action is Hamiltonian [25].
Let µ: X → t * R be a moment map. For v ∈ ℓ − {0}, the 1-parameter subgroup {j v (e t ) | t ∈ R} generates the gradient flows of µ, v , along which its value strictly decreases. Therefore the limit π v (x) for any x ∈ X exists and the T -action is meromorphic.

2.
A holomorphic action on X may not be meromorphic even if X is compact and Kähler. For example, let Z act on C − {0} by k: z → 2 k z (k ∈ Z, z ∈ C − {0}) and let X = (C − {0})/Z be the quotient. Then the standard multiplication of C × on C − {0} induces a holomorphic action on X which has no fixed points and hence is not meromorphic.
In order to capture the topology of X by the fixed-point information, it is necessary to assume that the T -action is meromorphic. If so, then X has a cell decomposition according to the connected components of X T that π C maps to.
is called the Bia lynicki-Birula decomposition with respect to C.
Consider the case T = C × . If the C × -action is meromorphic, set X ± α = (π ± ) −1 (X T α ). The decompositions X = α∈F X ± α are called the plus (minus) decompositions, respectively. For example, X = P 1 is the union of C (with the standard multiplication of C × ) and {∞}. The fixed-point set is X T = {0, ∞}. We have consist of a 0-cell and a 2-cell.
The cells X C α are T -invariant. If the transversality condition is satisfied, then the decomposition (2.3) is a stratification [8,Theorem 5]. In general, this is not true even when X is Kähler. An example is the Hirzebruch surface (the blow-up of P 2 at one point) [8, Example 1]. Definition 2.5 For α, β ∈ F , we write α → β if there is x ∈ X such that π C (x) ∈ X T α and π −C (x) ∈ X T β . We write α ≺ β if either α = β or there is a chain from α to β, i.e., a finite sequence α 0 = α, a 1 , . . . , α r−1 , α r = β in F such that α i−1 → α i for all 1 ≤ i ≤ r (r > 0). Such a chain is called a quasicycle of length r if α = β.
Obviously, the relation ≺ on F depends on the choice of C.
It is easy to see that the following statements are equivalent: 1. (F, ≺) is a partially ordered set; 2. There is no quasicycle in (F, ≺); 3. There is a strictly decreasing function on (F, ≺), i.e., a function f : Consequently, (F, ≺) is a partially ordered set if one of the following is true: 1. X is Kähler; 2. ν C α > ν C β if α ≺ β and α = β; 3. The Bia lynicki-Birula decomposition is a stratification.
In each of the above cases, the moment map (projected along some direction in C), ν C · , and dim C X C · , respectively, provides a strictly decreasing function on (F, ≺).
Example 2.7 Jurkiewicz [32] constructed a smooth compact toric 3-manifold with a meromorphic T 3 -action that has 22 isolated fixed points. Choosing an appropriate action chamber, there is a quasicycle of length 6 [32]. Therefore (F, ≺) is not a partially ordered set. In [52, § 4], it is shown that there exists a T 3 -equivariant holomorphic line bundle such that the holomorphic Morse inequalities fail. This shows that the holomorphic Morse inequalities are not valid on an arbitrary complex manifold [52], though the fixed-point theorems in [2,3] requires no further assumptions. In section 3, we construct the analog of the instanton complex in holomorphic Morse theory when (F, ≺) is a partially ordered set. The existence of such a construction implies the holomorphic Morse inequalities. Moreover, the partial order condition is weaker than the Kähler condition.
Definition 2.8 Suppose X has a C-meromorphic T -action. The Bia lynicki-Birula decomposition with respect to C is filterable if there is a descending sequence of T -invariant subvarieties Notice that we allow Z p − Z p+1 to be a union of cells labeled by elements in F unrelated by ≺. In [8, Definition 2], Z p − Z p+1 is required to be a single cell. Since X C α X C β = ∅ implies α ≺ β [19, Lemma 1], the two notions are equivalent. Notice that the function α → p(α) where α ∈ F p(α) is strictly increasing on (F, ≺).
Alternatively, (2.4) can be written as We return to the simple example X = P 1 with two fixed points 0, ∞ under the meromorphic C × -action. Let So the plus decomposition of X = P 1 is filterable. For the same reason, so is the minus decomposition. The projection π C : X C α → X T α is a T -equivariant holomorphic fibration and the fiber (π C ) −1 (x) over any The closure X C α in X is analytic and contains X C α as a Zariski open set. Consequently, X C α is locally closed in X.

Proof.
If T = C × , the necessary and sufficient condition for (2.3) to be filterable was proved in [19].
Properties 1 and 2 follow from the arguments of [17]. Property 3 follows from the arguments in [18,§ IIb], where the Kähler assumption was not made. The generalization to a higher rank torus T is straightforward.

✷
The three properties of Proposition 2.9 were shown to be valid when X is a Kähler manifold [17,18,26,35] or a complete normal algebraic variety [7,8,34], prior to the work of [19]. Without any of these assumptions, one or more of the properties in Proposition 2.9 could fail [46]. Example 2.7 was originally constructed to provide a non-filterable Bia lynicki-Birula decomposition [32].

Remark 2.10
The restriction of π −C to X C α − X T α may be discontinuous and the image π −C (X C α − X T α ) may fall into more than one connected components of X T . For example, let X = P 1 × P 1 with the diagonal The reason is that the holomorphic embedding X C α → X extend only meromorphically at infinity [18,Lemma 2], where it can be discontinuous.
Notice that despite of part 2 of Proposition 2.9, a tubular neighborhood of X T α in X C α can not be identified holomorphically with that in N C α in general [17]. There is an infinite series of obstruction to this [28,22]. However, an identification is possible locally on X T α . Consider a holomorphic vector bundle E over X on which the T -action lifts holomorphically. For future applications, we also put E into a standard local form.
for y ∈ W x and t ≥ 0. Clearly, the extension is well-defined, T -equivariant and holomorphic. Next, there is a holomorphic isomorphismψ x : W x × E x → E| Wx of vector bundles, perhaps on a smaller neighborhood W x . By [17, The extension is again well-defined and is a T -equivariant holomorphic isomorphism of vector bundles. ✷

A non-compact setting
In this subsection, we consider a class of non-compact complex manifolds with holomorphic torus actions.
We hope that this class is broad enough to include many interesting examples.
Let X be a (possibly non-compact) complex manifold with a holomorphic action of the torus T .
2. there is a compact complex orbifoldX with a meromorphic T -action and a T -equivariant holomorphic embedding of X onto a Zariski open set ofX.
The simplest example is X = C with the standard multiplication by C × . The action is plus-meromorphic and X has a compactificationX = P 1 . The plus-decomposition is X = X + 0 (a single 2-cell) and there is no minus-decomposition.

Remark 2.13
Consider a C-meromorphic T -action on X. We identify X with its image inX.
1. By Proposition 2.2, which applies to the non-compact setting here, the limit π v (x) (x ∈ X) does not depend on the choice of v ∈ ℓ ∩ C and is therefore denoted by π C (x). Moreover π C (x) ∈ X T . Because X is embedded into a compact spaceX, the set F of connected components of X T is finite and each component X T α (α ∈ F ) is compact. We have the Bia lynicki-Birula decomposition (2.3) with respect to C. The action chamber C of X may be divided into several action chambers ofX; letC be one of such. Then we have X C α =XC α for any α ∈ F . For x ∈ X, the limit π −C (x) exists inX but may fall intoX − X. Therefore π −C (x) is in general not defined in X.
2. As in the compact situation, there is a relation ≺ on F . If (F, ≺) is a partially ordered set, then the properties of Proposition 2.9 for X are satisfied. In particular, X C α =XC α ∩ X is a closed subvariety in X that contains X C α =XC α as a Zariski open set. Furthermore, the Bia lynicki-Birula decomposition of X with respect to C is filterable and we have filtrations of X by closed subsets (2.4) and by open subsets (2.5).
3. If E is a holomorphic vector bundle over X on which the T -action lifts holomorphically, Lemma 2.11 also holds.
Assumption 2.14 There exists an action chamber C such that the T -action on X is C-meromorphic and the set (F, ≺) is partially ordered.
In section 3, we will establish holomorphic Morse theory on a (possibly non-compact) complex manifolds satisfying Assumption 2.14. An immediate way of obtaining such non-compact manifolds comes from Definition 2.12. We start with a compact complex manifoldX with a meromorphic T -action. Suppose the Bia lynicki-Birula decomposition ofX with respect to an action chamberC is filterable and is filtered by the closed setsX Pick any m such that 0 ≤ m ≤m − 1 and let X =X −Z m+1 . T acts holomorphically on X. Let C be the action chamber that containsC. Then the T -action on X is C-meromorphic. Moreover the Bia lynicki-Birula decomposition of X with respect to C has a filtration (2.4) by closed subsets of X. The simple example X = C falls into this category, withX = P 1 .
More interestingly, the non-compact setting here is a complex analog of the symplectic setting considered in [43,44], which we now recall. Let (X, ω) be a (possibly non-compact) symplectic manifold with a Hamiltonian action of the compact torus T R , with a moment map µ: X → t * R . The fixed-point set X T of the torus T R is a symplectic submanifold of X. Let F be the set of connected components of X T .
If in addition (X, ω) is Kähler and the T R -action preserves the complex structure on X, then there is a holomorphic T -action on X.
Proposition 2.16 Let (X, ω) be a Kähler manifold with a holomorphic T -action. Suppose that the T R -action is Hamiltonian. Then Assumption 2.15 implies Assumption 2.14.
Proof. By [44,Proposition 1.6], there is an action chamber C such that for any v ∈ C, the function µ, v on X is proper and bounded from above. Therefore if v ∈ ℓ ∩ C, the limit π v (x) exists for any x ∈ X.
Pick any v ∈ ℓ ∩ C. Since F is finite, there is a ∈ R such that µ(X T ), v > a. We construct a symplectic cut X ≥a [38]. Let C × act on X × C by u: μ is a proper function on X × C and 0 is a regular value. The symplectic quotient X ≥a =μ −1 (0)/S 1 is a compact symplectic orbifold with a Hamiltonian T R -action. Since X is Kähler, X ≥a = (X × C) s /C × holomorphically and is also Kähler [29]. Here set. Using the moment map µ, v , it is easy to show that (F, ≺) is a partially ordered set. ✷

Equivariant spectral sequences in holomorphic Morse theory
We consider a holomorphic T -action on a (possibly non-compact) complex manifold X. F is the set of connected components of the fixed-point set X T . Throughout this section, we make Assumption 2.14. Then the Bia lynicki-Birula decomposition is filterable, with descending sequences of closed sets (2.4) and open sets (2.5) in X. Let E be a holomorphic vector bundle over X on which the T -action lifts holomorphically. We want to determine the Dolbeault cohomology groups H * c (X, O(E)) (with compact support) and H * (X, O(E)) as representations of T . are T -equivariant. The spectral sequence converges T -equivariantly to the representations H * if the spaces E pq ∞ are the graded components of H * as representations of T .

Spectral sequence for cohomologies with compact support
In this subsection, we construct a spectral sequence converging to the Dolbeault cohomology groups Recall that if A ⊂ X is a locally closed subset, then for any sheaf F on X, there is a unique sheaf on X, For simplicity, we denote the sheaf O(E) by F from now on. If A is a T -invariant locally closed subset of X, then T acts on the sheaf F A and hence on the cohomology groups H * c (X, F A ).
Lemma 3.2 Under Assumption 2.14, there is a T -equivariant spectral sequence with Proof. From (2.5), we have a filtration of F by subsheaves and hence a filtration of the cochain complex Γ c (C * (F )) by This induces a spectral sequence that converges to H * (Γ (C * (F ))) = H * (X, F ), with Since the maps d pq are induced by the resolution, we get The support of F X C α is contained in the closed subvariety X C α . Therefore we have [ The result follows from (3.7), (3.8) and (3.9). ✷ Recall that π C : X C α → X T α is a holomorphic fibration with fiber C ν C α . The sheaf F | X C α is on the total space X C α . To calculate the right hand side of (3.6), we need another spectral sequence. We consider a general fibration π: Y → B over a compact base B with possibly non-compact fibers. For the time being, let F be an arbitrary sheaf on the total space Y . The cohomology groups with compact support 2. There is a spectral sequence with Proof.     If A = R, then (see [36] for an analytic version) (3.14) The general case is a consequence of the Künneth formula. ✷ We now return to the situation of F = O(E).
as representations of T .
Proof. Consider the holomorphic fibration π C : X C α → X T α with fiber C ν C α and the sheaf F | X C α on X C α . For any x ∈ X T α , we want to find the stalk H q c (N C , F ) x , which depends only on an open neighborhood of (π C ) −1 (x) ⊂ X C α in X. By Lemma 2.11, we can replace X C α ⊂ X by N C α | Ux ⊂ N α | Ux and E by a trivial vector bundle with fiber E x . Moreover there is a T -equivariant isomorphism (N α (3.16) So the spectral sequence of Lemma 3.4.2 degenerates at E 2 and the result follows. ✷ its sub-bundle of any given weight is of finite rank. Therefore each weight has a finite multiplicity in the cohomology groups (3.15), and their formal characters in Z[ℓ * ] exist.
Theorem 3.7 Let X be a complex manifold with a holomorphic T -action satisfying Assumption 2.14. Let E be a holomorphic vector bundle over X on which the T -action lifts holomorphically. Then where ch T and td stand for the equivariant Chern character and the Todd class, respectively.
2. Since E pq r+1 is the cohomology of (E pq r , d pq r ), we have for a character valued polynomial Q r (t) ≥ 0. Using (3.20) recursively, we get (3.18) with Q C c (t) = r≥1 Q r (t) ≥ 0.
3. By setting t = −1 in (3.18) and using nα q=0  2. The coboundary maps {d pq r } in the spectral sequence in Theorem 3.7 or Corollary 3.8 are the holomorphic counterparts of the instanton tunneling operators in [50]. Through this spectral sequence, the cohomology groups H * c (X, O(E)) are completely determined by the combinatorial data of the T -action on X. However unlike the real case, the spectral sequence of holomorphic Morse theory does not always degenerate at E 2 . A sufficient condition for degeneracy at E 2 is If so, then the spectral sequence reduces to a cochain complex {E * 0 1 , d * 0 1 }, whose cohomology is E * 0 2 = H * (X, O(E)). This would be exactly like the Thom-Smale-Witten complex [50]. For example, if X T = F is discrete, m = n in (2.5), and F p = {x ∈ F | ν −C x = p} for 0 ≤ p ≤ n, then (3.26) is satisfied.

Spectral sequence with local cohomology groups
In this subsection, we construct an alternative spectral sequence converging to the Dolbeault cohomology groups H * (X, O(E)).
Suppose for the time being that X is any topological space containing Y as a locally closed subset and that F is any sheaf on X. We want to compute the local cohomology groups H q Y (X, F ) (q ≥ 0). Let A, L * be the sheaves on B defined by the presheaves A(U ) = Γ π −1 (U) (X, F ), L * (U ) = Γ π −1 (U) (X, C * (F )), respectively, where U is any open subset of B. Then 0 → A → L * is a differential sheaf in the sence of [27,§ II.4.1]. Let H q Y (X, F ) (q ≥ 0) be the sheaves on B defined by the presheaves H q Y (X, F )(U ) = H q (L * (U )), for any open subset U ⊂ B.
2. There is a spectral sequence with Proof. 1. This is the analog of Lemma 3.4.

Lemma 3.13
H q as representations of T .
Proof. Consider the fibration π C : X C α → X T α . For any x ∈ X T α , we want to find the stalk H q which by excision [5, § II.1, Lemma 1.1] depends only on an open neighborhood of (π C ) −1 (x) ⊂ X C α in X. By Lemma 2.11, we can replace X C α ⊂ X by N C α | Ux ⊂ N α | Ux and E by a trivial vector bundle with fiber . By Lemma 3.11.1 and Lemma 3.12, td(X T α ).
(3.44) Remark 3.16 1. The same observations in Remark 3.9.1 apply to Theorem 3.14 and Corollary 3.15. In particular, all the weights of T in H * (X, O(E)) are also of finite multiplicities. When X is non-compact, the Dolbeault cohomology groups are different from those with compact support. Therefore the results of Theorem 3.7 and Theorem 3.14 are not the same. Again, it would be interesting to have an independent analytic proof of parts 2 of Theorem 3.14 and Corollary 3.15 when X is a non-compact Kähler manifold satisfying Assumption 2.15. When X is compact, Theorem 3.7 is identical to Theorem 3.14 with an opposite action chamber. It is possible that the two theorems are dual to each other in some sense; this is also reflected by the local models in Lemma 3.5 and Lemma 3.12.
2. Remark 3.9.2 applies here as well. In particular, the complex is called the global Grothendieck-Cousin complex [30,33]. If condition (3.26) is satisfied, then the complex called the local Grothendieck-Cousin complex, is a resolution of F (see for example [33,Theorem 8.7]

Calculations inČech coholomogy theory
We interpret some of the procedures in the last section in the language ofČech cohomology theory, which is especially suitable for calculations.
For any sheaf F on X and any open cover U = {U i | i ∈ I} of X, let H * (U, F ) be the cohomology groups of theČech cochain complex where i 0 , · · · , i q ∈ I (q ≥ 0) are not equal, with the standard coboundary maps. TheČech cohomology groupš Then F V is the sheaf associated to the presheafF V andȞ * (X, We now assume that X is a compact complex manifold with a meromorphic T -action and that there is an action chamber C such that the set F of connected components of X T is partially ordered with respect to the relation ≺. Then the Bia lynicki-Birula decomposition is filterable, with filtrations of X by closed subsets We choose U to be a T -invariant, i.e., for any U i ∈ U, g ∈ T , we have gU i ∈ U. Then we have a T -equivariant filtration of theČech complex where So there is a T -equivariant spectral sequence converging to H * (U, F ) with E pq 0 = i0,···,iqF Vp/Vp+1 (U i0 ∩ · · · ∩ U iq ) and E pq 1 = H p+q (U,F Vp/Vp+1 ). Taking the inductive limit of U, we conclude that there is a T -equivariant spectral sequence with E pq 1 =Ȟ(X, F Vp−Vp+1 ) that converges toȞ * (X, F ). This is Lemma 3.2 when X is compact. However the method outlined above is more convenient for calculations, which we now illustrate.

Choose an open covering
the open cover U already satisfies H * (U, F ) ∼ =Ȟ * (X, F ). ThereforeȞ * (X, F ) can be computed by the spectral sequence associated to the filtration (4.4). According to (4.5), the spaces F p C q are given by Consequently, the spaces E pq 0 = F p C p+q /F p+1 C p+q are given by The arrows here (and below) denote the coboundary operators. Let Γ pq r be a region in t * R ∼ = R 2 = {(x, y)} such that supp E pq r = Γ pq r ∩ ℓ * . (Recall that ℓ * ∼ = Z 2 is the dual lattice in t * R ∼ = R 2 .) Then regions Γ pq 0 are given by Here the multiplicity of any weight ξ ∈ Γ pq in which case the multiplicity is 2. The regions Γ pq 1 are given by Finally, the regions Γ pq 2 are given by if c ≥ 0 and by All the weights ξ ∈ Γ pq r ∩ ℓ * (r = 1, 2) are of multiplicity 1 in E pq r . So the spectral sequence degenerates at E 2 . We recover the well-known result that the only non-trivial cohomology groups are H 0 (P  More interestingly, the (holomorphic) instanton complex can be used to study the cohomology groups of vector bundles over spherical varieties, about which not all is known. (See [13] for an extension of the Borel-Weil theorem.) One notable exception is the flag manifold, which will be discussed in the next subsection.

Flag manifolds and generalized Bernstein-Gelfand-Gelfand resolutions
We show that the spectral sequence for the cohomology of a flag manifold leads to geometric realizations of the Bernstein-Gelfand-Gelfand [6] and related resolutions.
Let G be a complex semi-simple Lie group and T , a maximal torus of G. Let g, t be the Lie algebras of G, T , respectively. Let g = t ⊕ α∈∆ g α be the root space decomposition, where ∆ ⊂ t * − {0} is the root system of the pair (g, t) and g α = Ce α (α ∈ ∆). Let ∆ + be a set of positive roots and let ∆ − = −∆ + . Let n ± = α∈∆± g α . Let B be the Borel subgroup corresponding to the Borel subalgebra b = t ⊕ n + . Let W be the Weyl group of the pair (g, t). Denote by w 0 the element in W of maximal length l(w 0 ) = |∆ + |.
Recall that the Verma module of highest weight λ is the U (g)-module M λ = U (g) ⊗ U(b) Cv λ , where Cv λ is the 1-dimensional U (b)-module defined by λ ∈ t * R . M λ is free over U (n − ). As a U (t)-module, M λ is determined by char M λ = e λ α∈∆ + (1−e −α ) . When λ is a dominant weight, let R λ be the (finite dimensional) irreducible module of highest weight λ. We have a resolution of R λ by Verma modules [6] where ρ = 1 2 α∈∆+ α. This is called the Bernstein-Gelfand-Gelfand resolution of R λ .
For any w ∈ W , put n w ± = wn ± w −1 . The twisted Verma module M w λ is a U (g)-module of highest weight λ that is free over U (n w + ∩ n − ) and co-free over U (n w + ∩ n + ) [23].
For example, take g = sl(2, C) = span C {h, e, f } with commutation relations The Weyl group is W = {±1}. The twisted Verma modules of highest weight λ ∈ R are M 1 λ = M * λ and The dual is When λ is not a dominant weight, i.e., when λ ∈ N, M λ ∼ = M * λ as U (g)-modules and the isomorphism is given by When λ ∈ N, M λ and M * λ are not isomorphic U (g)-modules. The irreducible module R λ is a quotient of M λ and a submodule of M * λ . We consider the non-degenerate flag manifold X = G/B − , where B − is the Borel subgroup opposite to B. The maximal torus T acts meromorphically on X. The fixed-point set is X T = {wB − | w ∈ W }. The isotropy weights at wB − are wα (α ∈ ∆ + ). The action chambers in t are the Weyl chambers. Choose the positive Weyl chamber, denoted by "+". Then the polarizing index of wB − is ν − w = |∆ − ∩ w∆ + | = l(w) for any w ∈ W . The Bia lynicki-Birula decomposition is precisely the Bruhat decomposition X = w∈W X + w , where X + w = BwB − /B − (w ∈ W ) are the Bruhat cells [1]. These cells are also the B-orbits in X. Moreover, the relation ≺ on F ∼ = W is the Chevalley-Bruhat order [20], which is a partial ordering. Consequently, the Bia lynicki-Birula decomposition is filterable, and we have the filtration (2.4), where m = |∆ + | = dim C X.
The closed sets Z p = l(w)≥p X + w (0 ≤ p ≤ |∆ + |) are the Schubert varieties. Since Z p − Z p+1 = l(w)=p X + w (0 ≤ p ≤ |∆ + |) and ν − w = l(w), the cohomology groups H * (X, F ) with coefficients in any sheaf F can be computed by the (global) Grothendieck-Cousin complex (3.45), which becomes . (4.12) So as representations of T , H l(w) X + w (X, F λ ) is the same as the Verma module M w(λ+ρ)−ρ . In fact the above local cohomology groups are U (g)-modules. This is because the canonical resolution C * (F λ ) of F λ , on which the Lie algebra g acts, is U (g)-equivariant. (Notice however that a representation of g on an infinite dimensional space may not exponentiate to that of G.) Therefore the Grothendieck-Cousin complex (4.11) is U (g)-equivariant.
If λ + ρ is regular, the cohomology groups H * (X, F λ ), hence those of the complex (4.13), are where w λ is the unique element in W such that w λ (λ + ρ) − ρ is a dominant weight [9]. The complex (4.13) is called the generalized Bernstein-Gelfand-Gelfand resolution of R w λ (λ+ρ)−ρ [23, §2.3]. If λ + ρ is singular, then all H * (X, F λ ) = 0 [9]. When λ is a dominant weight, (4.13) is the dual of the Bernstein-Gelfand-Gelfand resolution (4.6) for R λ [33,14,15]. When w 0 λ − 2ρ is dominant, (4.13) is the Bernstein-Gelfand-Gelfand resolution for R w0λ−2ρ . Let F λ = O(L λ ). The cohomology groups H * (P 1 , F λ ) as U (g)-modules can be computed from the U (g)equivariant cochain complex be two open sets in P 1 . Then the above cochain complex becomes We define two sections s i ∈ Γ (U i , F λ ) (i = 0, 1) by s i ([z 0 , z 1 ]) = [z 0 , z 1 , z λ i ]. Then Γ (U 0 , F λ ) = span C {z k s 0 | k ∈ N} and Γ (U 0 ∩ U 1 , F λ )/Γ (U 1 , F λ ) = span C {z k+1 s 1 | k ∈ N}, where z = z1 z0 on U 0 . The actions of g on the two spaces are given by where the isomorphisms are given by z k → k! ξ k λ and z k+1 Remark 4.4 Lepowsky [37] found a Bernstein-Gelfand-Gelfand-type resolution of any irreducible U (g)module by the generalized Verma modules, which are induced from representations of a parabolic subgroup P ⊂ G. In [42], a geometric realization of this resolution was constructed using the local cohomology of the P -orbits in G/B − (rather than the B-orbits in G/P − ). Let H be the Levi subgroup of P , and h, its Lie algebra. Let ∆ H be the root system of the pair (h, t), and W H , the corresponding Weyl group. Then X = G/B − decomposes into its P -orbits according to T ′ ⊂ T , whose Lie algebra t ′ ⊂ t. Consider the (meromorphic) T ′ -action on X. The fixed-point set X T ′ = w ′ ∈W/WH Hw ′ B − /B − . Choose the action chamber C ′ ⊂ t ′ such that α, C ′ > 0 for all α ∈ ∆ + − ∆ H ∩ ∆ + .
The the Bia lynicki-Birula decomposition of X with respect to C ′ is precisely (4.20). Therefore (3.45) gives the geometric realizations of Lepowsky's resolution and similar generalizations.

Cohomology and geometric quantization of non-compact manifolds
In section 3, we obtained equivariant holomorphic Morse inequalities and equivariant index theorems for non-compact complex manifolds under Assumption 2.14. In this subsection, we apply them to establish some results on the cohomology groups and on geometric quantization.
Let X be a (possibly non-compact) complex manifold of dimension n with a holomorphic T -action satisfying Assumption 2.14. Let H pq (X) = H q (M, O(∧ p T X)), H pq c (X) = H q c (M, O(∧ p T X)) (p, q = 0, 1, . . . , n) be the Dolbeault cohomology groups of X and those with compact support, respectively. Let P (X; s, t) = n p,q=0 s p t q char H pq (X), P c (X; s, t) = n p,q=0 s p t q char H pq c (X), the character-valued Poincaré-Hodge polynomials. If the cohomology groups are finite dimensional, then h pq (X) = dim C H pq (X), h pq c (X) = dim C H pq c (X) are the Hodge numbers of X and p(X; s, t) = n p,q=0 s p t q h pq (X), p c (X; s, t) = n p,q=0 s p t q h pq c (X), the (usual) Poincaré-Hodge polynomials. Notice that if X is a non-compact Kähler manifold, the Hodge numbers or the Poincaré-Hodge polynomials do not necessarily satisfy the usual symmetry relations. For example let X = C.
Without loss of generality, we assume that the moment map µ is bounded from above. Then the C × -action is plus-meromorphic. Suppose 0 is a regular value of µ. For simplicity, we assume that the S 1 -action on µ −1 (0) is free. Then the symplectic quotient X 0 = µ −1 (0)/S 1 = X s /C × is a smooth Kähler manifold. We construct the symplectic cuts (X ± , ω ± ) as the symplectic quotients of the S 1 -action on X ×C, where the weights on C are ±1, respectively [38]. The two cuts are Kähler manifolds with holomorphic C × -actions. X + is compact and X − satisfies Assumption 2.15. The sets of connected components of X C × ± are F ± = {0} ∪ {α ∈ F | µ(X C α ) ∈ R ± }, respectively, and X C × ±,0 ∼ = X 0 , X C × ±,α ∼ = X C × α as complex manifolds [53,Lemma 4.6], which we now identify. Let N 0 → X 0 be the holomorphic line bundle associate to the circle bundle µ −1 (0) → X 0 . Then C × acts on the fibers of N 0 with weight 1. The holomorphic normal bundles of X 0 in X ± are isomorphic to N ±1 0 , respectively. Since the action of C × lifts to L, the pre-quantum line bundles L 0 → X 0 and L ± → X ± exist. We have the isomorphisms L ± | X0 ∼ = L 0 and L ± | X±−X0 ∼ = L| µ −1 (R ± ) (see for example [53,Lemma 4.9]). Proof. For α ∈ F , let td(X C × α ).  The result follows. ✷ Remark 4.9 1. We can define H c (X) = n q=0 (−1) q H q c (X, O(L)) as the counterpart of (4.23) with compact support. Using However dim C H c (X) C × = dim C H(X 0 ) in general. For example, take X = C and choose the moment map µ(z) = −1 − 1 2 |z| 2 . Then dim C H c (X) C × = 1 but X 0 = ∅. 2. When (X, ω) is symplectic, the individual cohomology groups in (4.23) do not make sense, but H(X) can be defined as the index of a spin C -Dirac operator. In [21], (4.25) and (4.26) were proved for compact symplectic manifolds. (4.26) is the S 1 -case of a conjecture by Guillemin and Sternberg [29]; the cases with higher rank torus and non-Abelian group actions were proved by Meinrenken [40,41], Jeffrey and Kirwan [31], Vergne [48] and others under various generalities using localization techniques, and by Tian and Zhang [47] using an analytic approach.
3. Proposition 4.8 shows that the results of [21] holds for non-compact Kähler manifolds under Assumption 2.15. For non-compact symplectic manifolds satisfying Assumption 2.15, the validity of (4.25) and (4.26) remains open. Also it would be interesting to investigate, analytically or otherwise, whether the Tian-Zhang inequalities [47] n−1 k=0 t k dim C H k (X 0 , O(L 0 )) = n k=0 t k dim C H k (X, O(L)) C × + (1 + t)Q 0 (t) (4.35) for some Q 0 (t) ≥ 0 hold when X is non-compact and Kähler. The conjecture in [53,Remark 4.11] can also be posed in this non-compact setting.

Remark 4.10
In ordinary Morse theory, the underlying real manifold is (the bosonic part of) the configuration space of a supersymmetric system [50]. In holomorphic Morse theory, the complex manifold X, if it is Kähler, can be interpreted as the phase space of a bosonic system; this interpretation is adopted in Definition 4.7. The spectral sequence in Theorem 3.14.1 or Corollary 3.15.1 that converges to the quantum Hilbert space (4.23) is a finite dimensional model of the BRST approach in conformal field theory [24]. In [23,11], the case of flag manifolds (see subsection 4.2) was considered. Here we show that the analogy works for any quantizable Kähler manifold with a Hamiltonian S 1 -action satisfying Assumption 2.15. It would be interesting to extend the present work to infinite dimensional settings.