ℂ^{∗}-ACTIONS ON GRASSMANN BUNDLES AND THE CYCLE

AT INFINITY

(MATHEMATICA SCANDINAVICA 62: 5-18, 1988)

SİNAN SERTÖZ

Abstract. We describe the Grassmann Graph construction of MacPherson in the analytic category using a ℂ^{∗}-action and the corresponding Bialynicki-Birula decomposition. It is shown that the cycle at infinity is analytic in the compact Kaehler case.

0. Introduction.

This paper describes the Grassmann Graph construction of MacPherson in the analytic category using ℂ^{∗}-actions. The details of the algebraic case can be found in [1].

In section 1 we summarize the decomposition theorem of Bialynicki-Birula in the compact Kaehler case, [2], [3]. Section 2 describes a ℂ^{∗}-action on Grassmann manifolds and gives the corresponding Bialynicki-Birula decomposition. Examples are given in the next section. In section 4 this ℂ^{∗} action is carried on to Grassmann bundles and Z_{∞}, the cycle at infinity corresponding to a bundle morphism is defined. It is shown that in the compact Kaehler case Z_{∞}is an analytic cycle. The graph construction is finally accomplished in section 5. Examples are given in section 6.

Verdier uses the existence of a closed analytic space S which contains the closure of the graph in transcribing for analytic spaces the results of MacPherson, [9, section 5, proposition], [6]. We show in theorem 1 that in the compact Kaehler case S not only contains but is equal to the closure of the graph.

1. Bialynicki-Birula decomposition

The references for this section are [2] for the algebraic case and [3] for the complex case. There is also a clear summary in [4, section Ic].

Let M be a compact Kaehler manifold with a ℂ^{∗}-action on it. Let this ℂ^{∗}-action have nontrivial fixed point set B with components B_{1}, …, B_{m}. The components of the fixed point set are complex submanifolds of M. For λ ∈ ℂ^{∗} and p ∈ M let λ · p denote the action of λ on p. The ℂ^{∗}-action extends to a meromorphic map

1991 Mathematics Subject Classification. Primary: 32M15; Secondary: 14C99 32J99 32M05 57R99.

Key words and phrases. ℂ^{∗}-action, Grassmann bundle.

Received July 24, 1985; in revised form December 12, 1986.

ℙ^{1} × {p} ⟶ M

hence lim_{λ→0} λ · p and lim_{λ→∞} λ · p exist in M. Clearly these limits are in B. There are two canonical decompositions of M into invariant complex submanifolds. Define

M_{i}^{+} = {p ∈ M| lim_{λ→0} λ · p ∈ B_{i}}

for i = 1, ..., m. Each M_{i}^{+} is a complex submanifold of M and

M = ⋃ M_{i}^{+}, 1 ≤ i ≤ m.

This is called the plus decomposition of M. Similarly the minus decomposition is defined as

M_{i}^{−} = {p ∈ M| lim_{λ→∞} λ · p ∈ B_{i}}

for i = 1, ..., m. Each M_{i}^{−} is a complex submanifold and similarly

M = ⋃ M_{i}^{−}, 1 ≤ i ≤ m.

There are two distinguished components of the fixed point set B, say B_{1} and B_{m}, which are determined by the property that M_{1}^{+} and M_{m}^{−} are open and dense in M. B_{1} is called the source and B_{m} is called the sink.

2. ℂ^{∗}-action on G(k, n).

In this section we describe a particular ℂ^{∗}-action on G(k, n), the Grassmann manifold of k-planes in n-space. Fix a coordinate system on ℂ^{n}. We will use the representation of G(k, n) by matrices. Any point p ∈ G(k, n) can be represented by a k × n-matrix A of rank k. Two such matrices A and B represent the same point in G(k, n) if there is an invertible k × k-matrix g ∈ G(k, ℂ) such that gA = B. For a k × n-matrix A of rank k set [A] = the row space of A.

Given a k × n-matrix A = (a_{ij}), 1 ≤ i ≤ k, 1 ≤ j ≤ n define two submatrices

A_{1} = (a_{ij}, 1 ≤ i, j ≤ k

and

A_{2} = (a_{ij}, 1 ≤ i ≤ k, k + 1 ≤ j ≤ n.

A_{1} is a k × k-matrix and A_{2} is a k × (n − k)-matrix and A = (A_{1}, A_{2}) is a partition of A.

Define a ℂ^{∗}-action on G(k, n)

ℂ^{∗} × G(k, n) ⟶ G(k, n)

by

λ · [A] = [(A_{1}, λA_{2})].

To describe the behaviour of this action define a subset X_{ij} of G(k, n) as the set of all p in G(k, n) which can be represented by a k × n-matrix A = (A_{1}, A_{2}) such that rankA_{1} = i and rankA_{2} = j, where k − min{k, n − k} ≤ i ≤ k and 0 ≤ j ≤ min{k, n − k}. Let B = (B_{1}, B_{2}) be another k × n-matrix representing p. Then there is an invertible k × k-matrix g such that gA = B.

gA_{1} = B_{1} and gA_{2} = B_{2}.

ℂ^{∗}-ACTIONS ON GRASSMANN BUNDLES AND THE CYCLE AT INFINITY 3

Hence rankB_{1} =rank(gA_{1}) =rankA_{1} = i and similarly rankB_{2} = j, and the following definition of X_{ij} is well defines:

X_{ij} = {[A] ∈ G(k, n) | rankA_{1} = i, rankA_{2} = j }

where k − min{k, n − k} ≤ i ≤ k and 0 ≤ j ≤ min{k, n − k}. Necessarily we have i + j ≥ k; to see this, recall that A represents a point in G(k, n) hence has rank k, and if A_{1} has rank i then A_{2} must supply at least the remaining k − i ranks.

To describe the behaviour of the ℂ^{∗}-action that is defined above we prove the following lemmas.

Lemma 1. X_{i k−i} are the fixed point components of the ℂ^{∗}-action, k − min{k, n − k} ≤ i ≤ k.

Proof: Let [A] ∈ X_{i k−i}, A = (A_{1}, A_{2}). We first show that λ · [A] = [A]. If i = 0, then A_{1} = 0, and if i = k, then A_{2} = 0. In both cases λ · [A] = [A]. Assume 0 < i < k. Then there exists an invertible k × k-matrix g such that gA is of the form

(\begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix})

where B_{1} ∈ GL(iℂ) and B_{2} ∈ GL(k−i, ℂ). For λ ∈ ℂ^{∗} define h_{λ} to be the diagonal matrix [1, …, 1, 1/λ, …, 1/λ], where the number of 1/λ’s is k − i. We then have the following sequences of equalities:

λ · [A] = λ · [gA]

= λ · [(\begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix})]

= [(\begin{matrix}B_{1} & 0 \\ 0 & λB_{2} \end{matrix})]

= [h_{λ} (\begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix})]

= [(\begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix})]

= [A]

Thus we have proven that X_{ik−i} is a subset of the fixed point set. That in fact there are no other fixed points than ∪X_{ik−i}, k − min{k, n − k} ≤ i ≤ k follows from the results of the following two lemmas.

Lemma 2. If [A] ∈ X_{ij}, then lim_{λ→0} λ · [A] ∈ X_{ik−i}, where

k − min{k, n − k} ≤ i ≤ k, 0 ≤ j ≤ min{k, n − k} i + j ≥ k.

In particular X_{k0} is the source.

Proof: If i = 0 or i = k, then X_{ij} is a component of the fixed point set as in Lemma 1. Assume 0 < i < k. then there exists g ∈ GL(k, ℂ) such that

gA = (\begin{matrix}& & ⋮ & & 0 \\ B_{1} & & ⋮ \\ & & ⋮ & & B_{2} \\ . & . & . & . & . & . & . & . & . & . \\ 0 & & ⋮ & & B_{3} \end{matrix})

where B_{1} ∈ GL(i, ℂ), B_{3} ∈ GL(k − i, ℂ) and B_{2} is a (i + j − k) × (n − k)-matrix. Let h_{λ} be as in Lemma 1. then

h_{λ}λgA = (\begin{matrix}& & & ⋮ & & & 0 \\ B_{1} & & & ⋮ \\ & & & ⋮ & & & λB_{2} \\ . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & & ⋮ & & & B_{3} \end{matrix})

and since lim_{λ→0} λB_{2} = 0 we have

lim_{λ→0} λ · [A] = lim_{λ→0}[h_{λ}λgA]

= [(\begin{matrix}B_{1} & 0 \\ 0 & B_{3} \end{matrix})] .

This last matrix is clearly in X_{ik−i} as claimed.

Lemma 3. If [A] ∈ X_{ij}, then lim_{λ→∞} λ · [A] ∈ X_{k−jj}, where

k − min{k, n − k} ≤ i ≤ k, 0 ≤ j ≤ min{k, n − k}.

In Particular _{k−mm} is the sink, where m = min{k, n − k}.

Proof: If i = 0 or i = k, then X_{ij} is a fixed point component. Assume 0 < i < k. there exists g ∈ GL(k, ℂ) such that

gA = (\begin{matrix}B_{1} & & ⋮ & & 0 \\ . & . & . & . & . & . & . & . & . & . \\ B_{2} & & ⋮ \\ & & ⋮ & & B_{3} \\ 0 & & ⋮ \end{matrix})

where B_{1} ∈ GL(k − j, ℂ), B_{3} ∈ GL(j, ℂ) and B_{2} is a (i + j − k) × k-matrix. Then

lim_{λ→∞} λ · [A] = lim_{λ→∞}[λh_{λ}gA]

= lim_{λ→∞} [(\begin{matrix}B_{1} & 0 \\ λ^{−1}B_{2} \\ & B_{3} \\ 0 \end{matrix})]

= [(\begin{matrix}B_{1} & 0 \\ 0 & B_{3} \end{matrix})] .

This last matrix is in X_{k−jj} as desired.

These last two lemmas show that X_{ik−i} for k − min{k, n − k} ≤ i ≤ k are the only fixed point components and thus complete the proof of lemma 1.

We can apply these lemmas to examine the behaviour of Schubert cells under the action of ℂ^{∗} on the Grassmann manifold. we will adopt the terminology of Griffiths and Harris on Schubert cells. For details refer to [5, pp. 195-196].

Let {e_{1}, ..., V_{n}} be the standard basis for ℂ^{n} and V_{i} = span{e_{1}, ..., e_{i}}. Then {V_{1}, ..., V_{n}} defines a flag. For any nonincreasing sequence of nonnegative integers between 0 and n − k define a cell

W_{a} = {[Λ] ∈ G(k, n) | dim(Λ ∩ V_{n−k+i−a_{i}}) = i } .

The sequence of nonincreasing integers a = (a_{1}, ..., a_{k}) with 0 ≤ a_{i} ≤ n − k is called a Schubert symbol. For [Λ] ∈ G(k, n), let A be a k × n-matrixsuch that [A] = [Λ]. If [A] ∈ W_{a} for some Schubert symbol a = (a_{1}, ..., a_{k}), then the rank of the first k × (n − k + i − a_{i}) minor is i and the rank of the last k × (k − i + a_{i}) minor is k − i. The closure of W_{a}

\overline{W_{a}} = \{[Λ] ∈ G(k, n) | dim(Λ ∩ V_{n−k+i−a_{i}}) ≥ i \}

is called a Schubert variety. If A is a matrix representing [Λ] as above, then [Λ] is in \overline{W_{a}} iff the rank of the first k × (n − k + i − a_{i}) minor of A is at most k − i. It is well known that \overline{W_{a}}is an analytic subvariety of G(k, n) and the homology class of \overline{W_{a}}, denoted by σ_{a}, is independent of the flag used in its definition, [5, p. 196]. σ_{a} is called the Schubert cycle corresponding to a = (a_{1}, ..., a_{k}). Regarding the behaviour of Schubert cycles under the ℂ^{∗}-actionwe give the following corollary to the above lemmas:

Corollary 1. All Schubert cycles of positive codimension in G(k, 2k) lie in X_{ij}’s where j < k. In particular they do not flow to the sink, i.e. if p ∈ \overline{W_{a}} then lim_{λ→∞} λ · p is not in the sink.

Proof: The codimension of \overline{W_{a}}for a = (a_{1}, ..., a_{k}) is ∑ a_{i}, [5, p. 196]. It suffices to prove the corollary for a = (1, 0, ..., 0). For [Λ] ∈ W_{a} let A = (A_{1}, A_{2}) be a matrix representation where A is a k × n-matrixof rank k, and A_{1}, A_{2} are k × kmatrices. The rank of the last k × k minor of A is of rank at most k − 1. Hence in particular the rank of A_{2} is not k, therefore [A] is not in X_{ik}. Since the only points that flow to the sink belong to the components of the form X_{ik}, [Λ] does not flow to the sink. In general if a = (a_{1}, ..., a_{k}) with a_{1} ≥ 1 then the last k × (k + a_{1} − 1) minor has rank at most k − 1. Since k + a_{1} − 1 ≥ k, the rank of A_{2} cannot be k.

Hence \overline{W_{a}}does not flow to the sink. If a_{1} = 0, then a = (0, ..., 0) and \overline{W_{a}}does not have positive codimension.

Using the same notation as in the previous corollary we can generalize as follows:

Corollary 2. Let \overline{W_{a}}, a = (a_{1}, ..., a_{k}), be a Schubert variety in G(k, n), where a_{1} ≥ n − 2k + 1. Then \overline{W_{a}}does not flow to the sink if n ≥ 2k.

Proof: Let A = (A_{1}, A_{2}) be a k × n-matrixwith rank k representing a point [A] in \overline{W_{a}}. A_{1} is a k×(n−k)-matrix and [A] will flow to the sink if rank A_{2} is maximal. Since n ≥ 2k means n − k ≥ k, the maximal rank of A_{2} is k. the rank of the last k × (k + a_{1} − 1) minor of A is at most k − 1. By assumption k + a_{1} − 1 ≥ n − k, therefore the rank of A_{2} cannot be k. Hence \overline{W_{a}}does not flow to the sink.

3. Examples

In examples 1 and 2 we assume that the ℂ^{∗}-actionof the previous section is defined on the spaces G(2, 4) and G(4, 9).

1) G(2, 4). In G(2, 4) we have defined the following sets:

X_{20}, X_{11}, X_{02}, X_{22}, X_{12}, X_{21}.

The first three sets are the fixed point sets. As λ → 0 the elements of X_{21} and X_{22} flow to the source X_{20}, and the elements of X_{12} flow to X_{11}. As λ → ∞ the elements of X_{22} and X_{12} flow to the sink X_{02}, and the elements of X_{21} flow to X_{11}.

See Figure 1 for the direction of these flows for each X_{ij} as λ → ∞.

[figure: X_{11}; X_{12}; X_{02}; X_{21}; X_{22}; X_{20}; Fig. 1]

2) G(4, 9). For the direction of flow as λ → ∞ see Figure 2. From the decomposition of G(4, 9) into X_{ij} it can be seen that the points that lie in

X_{13} ∪ X_{33} ∩ X_{31} ∩ X_{32} ∩ X_{22} ∩ X_{23}

do not flow to the sink or the source under the action of ℂ^{∗}.

[figure: X_{40}; X_{41}; X_{31}; X_{34}; X_{04}; X_{43}; X_{13}; X_{14}; X_{33}; X_{22}; X_{42}; X_{24}; X_{32}; X_{23}; Fig. 2]

4. ℂ^{∗}-actions on Grassmann bundles

This section defines in the compact Kaehler case the Grassmann Graph construction of [1, pp. 120-121].

Let E, F be vector bundles of ranks k and n, respectively, on an analytic space M. Let G(k, E ⊕F) → M denote the Grassmann bundle whose fibre at each x ∈ M is G(k, E_{x} ⊕ F_{x}), the Grassmannian of k-planes in E_{x} ⊕ F_{x}. Define a ℂ^{∗}-action on G(k, E ⊕ F) as the fibrewise ℂ^{∗}-action. Let

π_{1} : E ⊕ F ⟶ E
π_{2} : E ⊕ F ⟶ F

and

π : G(k, E ⊕ F) ⟶ M

be the projections. Any p ∈ G(k, E ⊕ F) is represented by a k-plane H in E_{x} ⊕ F_{x} where x = π(x). π_{1}(H) and π_{2}(H) are linear subspaces of E_{x} and F_{x}, respectively. The total space G(k, E ⊕ F) can be decomposed into ℂ^{∗}-equivariant subbundles

X_{ij} = {[H] ∈ G(k, E ⊕ F) | dim π_{1}(H) = i, dim π_{2}(H) = j }

where k − min(k, n) ≤ i ≤ k, 0 ≤ j ≤ min(k, n), and i + j ≥ k. It is easy to see that

X_{ij} ≅ G(i, E) × G(j, F) if i + j = k,

which are the fixed point sets of the ℂ^{∗}-action. Let

Hom(E, F) ⟶ M

be the bundle of morphisms from E to F and let

j : Hom(E, F) ⟶ G(k, E ⊕ F)

be the natural inclusion defined fibrewise as

j_{x}(Φ) = graph(Φ | E_{x}) = {(e, Φ(e)) ∈ E_{x} ⊕ F_{x}}.

Recall that ℂ can be imbedded into ℙ^{1} as

ℂ ⟶ ℙ^{1}
λ ⟶ [1 : λ],

[1, p. 120]. Define a ℂ^{∗}-action on G(k, E ⊕ F) × ℙ^{1}

ℂ^{∗} × G(k, E ⊕ F) × ℙ^{1} ⟶ G(k, E ⊕ F) × ℙ^{1}

as

(λ, p, [λ_{0} : λ_{1}]) ⟶ (λ · p, [λ_{0} : λλ_{1}])

where λ · p is the ℂ^{∗}-action which is defined above. Also define the ℂ^{∗}-action on M × ℂ,

ℂ^{∗} × M × ℂ ⟶ M × ℂ

as

(λ, x, t) ⟶ (x, λt).

Every Φ ∈ Hom(E, F) defines an equivariant imbedding \overline{s}(Φ) of M ×ℂ into G(k, E⊕ F) × ℙ^{1},

\overline{s}(Φ) : M × ℂ ⟶ G(k, E ⊕ F) × ℙ^{1}

where

\overline{s}(Φ)(x, λ) = ([j_{x}(λΦ_{x})], [1 : λ]).

Let s(Φ) = pr(\overline{s}(Φ)) where pr is the projection

pr : G(k, E ⊕ F) × ℙ^{1} ⟶ G(k, E ⊕ F).

s(Φ)(M, λ) is the graph of λΦ. Now define

Z_{∞} = lim_{λ→∞} s(Φ)(M, λ).

Theorem 1. If M is a compact Kaehler manifold, then for any Φ ∈ Hom(E, F) the corresponding Z_{∞} is an analytic cycle.

Proof: Let ρ : ℂ^{∗} × G(k, E ⊕ F) → G(k, E ⊕ F) be the ℂ^{∗}-action defined above. Consider M as a subspace of G(k, E ⊕F) by the imbedding s(Φ)(M, λ); i.e. identify M and the graph of Φ. define a holomorphic map

A : M × ℂ^{∗} ⟶ G(k, E ⊕ F)

as

A(m, t) = s(Φ)(m, t),

where m ∈ M and t ∈ ℂ^{∗}. This map is equivariant with respect to ρ and the trivial action of ℂ^{∗} on M ×ℂ^{∗}, multiplication in the second component; for if λ ∈ ℂ^{∗} then

A(m, λ · t) = s(Φ)(m, λt)
= s(λΦ)(m, t)
= λ · s(Φ)(m, t)
= ρ(λ, s(Φ)(m, t))
= ρ(λ, A(m, t))

hence equivariance. But Sommese has shown that if ψ : Y × ℂ^{∗} → X is a holomorphic map equivariant with respect to the trivial action of ℂ^{∗} on Y × ℂ^{∗} and the action of ℂ^{∗} on X with fixed points then ψ extends meromorphically to Y × ℙ^{1}, [8,p. 111 (Lemma II-B)]. Thus A extends meromorphically to

A′ : M × ℙ^{1} ⟶ G(k, E ⊕ F).

Let T be the closure of the graph of A in M × ℙ^{1} × G(k, E ⊕ F).

By the definition of a meromorphic map, T is an analytic space. Since

M × {∞} × Z_{∞} = T ∩ (M × {∞} × G(k, E ⊕ F)),

being the intersection of two analytic spaces it is analytic. If pr: M × {∞} × Z_{∞} → M is the projection, then for any m ∈ M, pr^{∗}(M) = {m}×Z_{∞} is an analytic cycle, from which it follows that Z_{∞} is analytic as desired.

Z_{∞} is called the cycle at infinity corresponding to the map Φ. Notice that there is an alternate definition of Z_{∞} see [1, p. 121];

Let W be the closure of \overline{s}(Φ)(M × ℂ) in G(k, E ⊕ F) × ℙ^{1}. Then Z_{∞} × \{∞\} is the intersection of W and G(k, E ⊕ F) × {∞}.

In the algebraic category W is an algebraic variety but in the analytic category the observation that W can be obtained through a ℂ^{∗}-action with fixed points on a compact Kaehler manifold is crucial in concluding that it is analytic.

Clearly {Z_{λ} = s(Φ)(M, λ)} defines a family of cycles which are algebraically and hence homologically equivalent.

5. Graphs of complexes

In this section we define the Grassmann Graph construction and the cycle at infinity associated to a complex of vector bundles. This construction was first introduced by MacPherson and used by Baum, Fulton and MacPherson to prove Riemann-Roch theorem for singular algebraic varieties, [1] and [6].

Consider a complex of vector bundles on M,

(E.) : 0 ⟶ E_{m} ⟶ E_{m−1} ⟶ ⋯ ⟶ E_{0} ⟶ 0.

Denote the maps by γ_{i}, i.e.

γ_{i} : E_{i} ⟶ E_{i−1}

where i = 0, ..., m, E_{−1} = 0.

Assume that there is a subvariety S of M such that (E.) is exact on M − S. Let

G_{I} = G(rankE_{i}, E_{i} ⊕ E_{i−1} ), i = 1, ..., m.

and let

τ_{i} ⟶ G_{i} the tautological bundle, i = 1, ..., m.

Define

G = G_{0} ×_{M} ⋯ ×_{M} G_{m}

where ×_{M} denotes the bundle product on M. On G let τ_{i} denote the pull back of τ_{i} → G_{i} by the projection pr_{i} : G → G_{i} of the i-th component, i = 0, ..., m.

Let

τ = τ_{0} − τ_{1} + ⋯ + (−1)^{m}τ_{m}

be the virtual tautological bundle on G. Recalling the definition of s from the previous section, for any λ ∈ ℂ define an imbedding

s_{λ}^{i} : M ⟶ G_{i}

as

s_{λ}^{i}(x) = s(γ_{i})(x, λ)

where i = 0, ..., m. Then define for any λ ∈ ℂ an imbedding

s_{λ} : M ⟶ G

as

s_{λ}(x) = (s_{λ}^{0}(x), ..., s_{λ}^{m}(x)).

Using s_{λ}(M) we define

Z_{∞} = lim_{λ→∞} s_{λ}(M)

to be the cycle at infinity corresponding to the complex (E.).

Let π : G → M be the natural projection. Recalling that S is the set off which (E.) is exact we have the following result: (For proofs see [1, p. 121].)

Theorem (Baum, Fulton, MacPherson). The cycle Z_{∞} has a unique decomposition Z_{∞} = Z_{∗} + M_{∗}, where

1) π maps M meromorphically onto M.

2) π : M_{∗} − π^{−1}(S) ⟶ M − S is a biholomorphism.

3) π maps Z into S.

4) τ restricts on M_{∗} to the zero bundle.

Remark. By Theorem 1 of the previous section, Z_{∞} is a product of analytic cycles in the product bundle G, hence this theorem can be stated in the analytic category as above. Any cycle can be written as a sum of irreducible cycles. the decomposition of Z_{∞} is such a sum. For a proof of (4) see [1, p. 122].

Finally we define two residues on S. Let E be the virtual bundle E_{0} − E_{1} + ⋯ + (−1)^{m}E_{m} on M. Then τ | Z_{0} is isomorphic to E since Z_{0} ≅ M. Since Z_{)} and Z_{∞} are rationally equivalent

c(E) ∩ [M] = c(τ) ∩ Z_{0} = c(τ) ∩ Z_{∞}

where c(·) denotes the Chern class and ∩ denotes the cap product. Since Z_{∞} decomposes

c_{i}(τ) ∩ Z_{∞} = c_{i}(τ) ∩ (Z_{∗} + M_{∗})
= c_{i}(τ) ∩ Z_{∗} + c_{i}(τ) ∩ M_{∗}
= c_{i}(τ) ∩ Z_{∗}

where i > 0 and the last equality follows since τ | M_{∗} = 0 by (4) of the above theorem.

Define

c_{S}^{i}(E.) = π_{∗}(c_{i}(τ) ∩ Z_{∗}) ∈ H_{∗}(S : ℂ).

Similarly let ch(·) denote the Chern character, then

ch(E) ∩ [M] = ch(τ) ∩ Z_{0}
= ch(τ) ∩ Z_{∞}
= ch(τ) ∩ Z_{∗} + ch(τ) ∩ M_{∗}
= ch(τ) ∩ Z_{∗}.

Similarly define

ch_{S}(E.) = π_{∗}(ch(τ) ∩ Z_{∗}) ∈ H_{∗}(S; ℂ).

For basic properties of ch(E.) in the algebraic category see [1, pp. 121-126]. We will use c_{S}^{i}(E) for calculating the Baum-Bott residue of singular holomorphic foliations in [7].

6. Examples

1) Let E, F be vector bundles on M and ψ ∈ Hom(E, F).Then the graph Γ(ψ) of ψ gives rise to a cycle at infinity Z_{∞}. Let rankE = k, rankF = n, and m = min{k, n}. For i = 0, 1, ..., m, let B_{i} = X_{k−ii}, where X_{ij} is as defined in Section 4. B_{0}, ..., B_{m} are the components of the fixed point set B under the ℂ^{∗}- action on the Grassmann bundle G(k, E ⊕ F). To understand the structure of Z_{∞} we describe its intersection with B. For this purpose define the following sets

Σ_{i} = {p ∈ M | rank ψ_{p} ≤ i }, i = 0, ..., r

where r is the generic rank of ψ. the behaviour of Z_{∞} can now be described as follows:

(Z_{∞} ∩ B_{i})_{p} ≠ ∅ iff p ∈ Σ_{t} and t ≥ i ≥ r.

2) We want to show that the Hironaka Blow-up at a point can be recovered as a Grassmann Graph construction. The problem is local so let M be an open set in ℂ^{n}. Define two trivial bundles L and F as

L = M × ℂ and F = M × ℂ^{n}.

Define a morphism θ ∈ Hom(L, F) as:

θ(p, t) = (p, tp) for p ∈ ℂ^{n}, t ∈ ℂ.

The cycle at infinity Z_{∞} corresponding to θ intersects the sink of G(1, L ⊕ F) in M_{∗}, that is Z_{∞} = M_{∗} + Z_{∗}. M_{∗}is the Hironaka Blow-up of M at the origin. We can see this as follows. Let p = (p_{1}, ..., p_{n}) ∈ M = ℂ^{n}. We also identify ℙ(L ⊕ F) with ℙ^{n}. There is a ℂ^{∗}-action

ℂ^{∗} × M × ℙ^{n} ⟶ M × ℙ^{n}

given as

(λ, p, [y_{0} : y_{1} : ⋯: y_{n}]) → (p, [y_{0} : λy_{1} : ⋯: λy_{n}]).

The graph of θ has the form

Γ(θ) = {(p, [1 : p_{1} : ⋯: p_{n}]) ∈ M × ℙ^{n}}.

The ℂ^{∗}-action moves Γ(θ) as

λ · Γ(θ) = {(p, [1 : λp_{1} : ⋯: λp_{n}]) ∈ M × ℙ^{n}}.

Consider the usual imbedding of ℂ^{∗} in ℙ^{1} as λ = [1 : λ] = [λ_{0} : λ_{1}], where λ = λ_{1}/λ_{0}. Since λ → ∞ iff λ_{0} → 0 with λ_{1} ≠ 0, we have the following limit

Z_{∞} = lim_{λ→∞} λ · Γ(θ)

= lim_{λ_{0}→0}{(p, [λ_{0} : λ_{1}p_{1} : ⋯: λ_{1}p_{n}]) ∈ M × ℙ^{n}}

= {(p, [0 : λ_{1}p_{1} : ⋯: λ_{1}p_{n}]) ∈ M × ℙ^{n}}.

Clearly (p, [0 : λ_{1}p_{1} : ⋯: λ_{1}p_{n}]) ∈ M × ℙ^{n} can be considered as a point (p, [x_{1} : ⋯: x_{n}]) in M × ℙ^{n−1} such that

p_{j}x_{i} = p_{i}x_{j}, i ≠ j, 1 ≤ i, j ≤ n.

From here it is easy to see that the intersection of Z_{∞} with the sink of the ℂ^{∗}-action is the Hironaka blow-up of M at the origin.

Acknowledgment. This work constitutes part of the author’s dissertation. The author thanks his supervisor professor J. B. Carrell for his encouraging and stimulating conversations. Financial support for this work was provided by TÜBİTAK (Scientific and Technological Research Council of Turkey) and the Faculty of Graduate Studies at the University of British Columbia.

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ℂ^{∗}-ACTIONS ON GRASSMANN BUNDLES AND THE CYCLE AT INFINITY 13

9. J. L. Verdier, Chern classes for analytic spaces, Preprint, University of Paris.

Department of Mathematics, Bilkent University, 06533 Ankara, Turkey University of British Columbia, Vancouver, B.C., Canada

E-mail address: sertoz@fen.bilkent.edu.tr