The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

42(2006), 83–100

The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

By

Kohhei Yamaguchi^∗

Abstract

For a complex (quasi-) projective variety X CP^N with π2(X) = Z and an integerd≥0, let Hol^∗_d(S², X) denote the space consisting of all basepoint preserving holomorphic mapsf fromS² to X with degreed. We study the topology of certain subspaces of Hol^∗_d(S², X) defined using the concept of multiplicityof roots, and we show that the Atiyah-Jones-Segal type theorem ([1], [11]) holds for these subspaces if X is belong to a certain family of complex quasi-projective varieties.

§1. Introduction

If d ≥ 0 is an integer and X CP^m−1 is a complex (quasi-) projective variety with the condition π₂(X) = Z, we denote by Hol^∗_d(S², X) the space consisting of all base point preserving holomorphic maps (i.e. algebraic curves) f :S²→X with [f] =d∈Z=π2(X). The corresponding space of continuous maps is denoted by Map^∗_d(S², X) = Ω²_dX. The principal motivation for this paper derives from the work of G. Segal [11], in which he investigates the topology of the space Hol^∗_d(S², X) forX =CP^m−1 as follows.

Theorem 1.1 ([11]). Let m ≥ 2 and d ≥ 1 be integers. Then the inclusion id : Hol^∗_d(S²,CP^m−1)→Ω²_dCP^m−1 is a homotopy equivalence up to dimension (2m−3)d.

Communicated by K. Saito. Received May 6, 2003.

2000 Mathematics Subject Classification(s): Primary 55P10; Secondary 55P35, 55P15.

Key words: algebraic curves, holomorphic maps, homotopy equivalence

Partly supported by Grant-in-Aid for Scientific Research (C) 16540056, Japan Society of the Promotion of Science.

∗Department of Information Mathematics, University of Electro-Communications, Chofu- gaoka, Chofu, Tokyo 182-8585, Japan.

e-mail: [email protected]

Remark. We say that a mapf :X→Y is ahomotopy equivalence(resp.

homology equivalence)up to dimension N if the induced homomorphismf_∗: πi(X)→πi(Y) (resp. f_∗ :Hi(X;Z)→Hi(Y;Z)) is bijective wheni < N and is surjective wheni=N.

In this paper, we shall investigate whether a similar result hold for certain subspaces of Hol^∗_d(S², X) introduced from the concept ofmultiplicity.

For a spaceX, we denote by SP^d(X) thed-th symmetric product ofX. By definition, this is the quotient spaceX^d/Σ_d, where the symmetric group Σ_dacts on the d-fold product X^d in the natural way. An element of SP^d(X) may be identified with a formal linear combinationα=k

i=1dixi, wherex1, . . . , xkare distinct points ofXandd1, . . . , dkare positive integers such thatk

i=1 di = d.

We shall refer toα as a “configuration” of points, the point x_i having multi- plicityd_i. We shall be concerned with a subspace SP^d_n(X) of SP^d(X), defined as follows. Ifn≥2,

SP^d_n(X) = ^k

i=1

d_ix_i∈SP^d(X) :d_i< nfor alli

.

Thus, SP^d_n(X) is obtained by imposing a condition of “bounded multiplicity”, namely that all points xi (of any configuration) have multiplicity less thann.

There is a filtration

Cd(X) = SP^d₂(X)⊂ · · · ⊂SP^d_n(X)⊂ · · · ⊂SP^d_d+1(X) =· · ·= SP^d(X), where Cd(X) denotes the space of “configurations of d distinct points” inX. Note that SP^d_n(C) can be identified with the space of complex polynomials of degreedwhich aremonic, and all of whose roots havemultiplicity less thann.

(The polynomialk

i=1(z−xi)^di corresponds tok

i=1dixi.)

If we choose the point [1 : · · · : 1] as the base point of CP^m−1, the space Hol^∗_d(S²,CP^m−1) is identified with the space consisting of allm-tuples (p1(z), . . . , pm(z))∈C[z]^mof monic polynomials of degreedsuch that the poly- nomials p₁(z), . . . , p_m(z) have no common roots. It is also identified with the space ofm-tuples of positive disjoint divisors,

Hol^∗_d(S²,CP^m−1) ={(ξ1, . . . , ξm)∈SP^d(C)^m:ξ1∩ · · · ∩ξm=∅}. In this situation, let Holⁿ_d(S²,CP^m−1) denote the subspace defined by

Holⁿ_d(S²,CP^m−1) = Hol^∗_d(S²,CP^m−1)∩SP^d_n(C)^m

={(ξ₁, . . . , ξ_m)∈SP^d_n(C)^m:ξ₁∩ · · · ∩ξ_m=∅}. Now we recall the following result.

Theorem 1.2 ([13]). If n ≥ 2, there is a map Holⁿ_d(S²,CP^m−1) → Ω²₀W_m(CPⁿ⁻¹) which is a homology equivalence up to[d/2]whenn= 2and a homotopy equivalence up to dimension D1(d;m, n) =d−2m+ 2when n≥3, where [x] denotes the integer part of a number x and Wm(X) ⊂X^m denotes the m-th fat wedge.

In this paper, we would like to generalize the above result for another wider complex projective varieties. For this purpose, letm, n≥2 be integers and let X_m⊂CP^m−1denote the quasi-projective variety defined by

Xm=CP^m−1−

1≤i<j≤m

Hi,j,

whereH_i,j={[z₁:· · ·:z_m]∈CP^m−1:z_i=z_j= 0}.

It is known [5] thatX_m is simply connected and thatπ₂(X_m) =Z. If we choose the point [1 :· · ·: 1] as the base point ofX_m, Hol^∗_d(S², X_m) is identified with the space consisting of allm-tuples (p1(z), . . . , pm(z))∈C[z]^mofmutually coprime monic polynomials of degreed. We can also identify

Hol^∗_d(S², Xm) ={(ξ1, . . . , ξm)∈SP^d(C)^m:ξi∩ξj=∅ifi=j}. We denote by Holⁿ_d(S², Xm) the subspace of Hol^∗_d(S², Xm) defined by

Holⁿ_d(S², X_m) = Hol^∗_d(S², X_m)∩SP^d_n(C)^m

={(ξ1, . . . , ξm)∈SP^d_n(C)^m:ξi∩ξj=∅ifi=j}. There is a filtration

∅= Hol¹_d(S², Xm)⊂Hol²_d(S², Xm)⊂Hol³_d(S², Xm)⊂ · · ·

⊂Hol^d+1_d (S², Xm) = Hol^d+2_d (S², Xm) =· · ·= Hol^∗_d(S², Xm).

Let∨^mX denote them-times wedge ofX,∨^mX =X∨X∨ · · · ∨X (m-times).

Then our main result is stated as follows.

Theorem 1.3. If m, n ≥ 2, there is a map Holⁿ_d(S², Xm) → Ω²₀∨^m CPⁿ⁻¹which is a homotopy equivalence up to dimensionD(d;m, n)whenn≥3 and a homology equivalence up to dimension[d/2]whenn= 2. Here the number D(d;m, n)is given by

(1.3.1) D(d;m, n) =

d+n−4 if d≥n d if d < n.

Remark. (1) Since lim

d→∞D(d;m, n) = ∞, the space Holⁿ_d(S², X_m) may be regarded as a finite dimensional model for the infinite dimensional space Ω²₀(∨^mCPⁿ⁻¹).

(2) It follows from [5] that there is a fibration sequence Tⁿ → Xm →

∨^mCP^∞. Hence, using an easy diagram chasing as in [[13]; page 251] we ob- tain a homotopy equivalence lim

d,n→∞Holⁿ_d(S², X_m)Ω²₀X_m. So we obtain the stabilized version of the main theorem of [5] ifn→ ∞.

Next we shall indicate the following generalization of Theorem 1.3 to the case of certain quasi-projective varietiesX_I.

LetIbe a fixed collections of subsets of{1,2, . . . , m}such that card(Λ)≥2 for any Λ∈I. We denote byXI ⊂CP^m−1 the quasi-projective variety defined by

XI =CP^m−1−

Λ∈I

HΛ,

whereH_Λ={[z₁:· · ·:z_m]∈CP^m−1:z_i= 0 for alli∈Λ}.

It is known [5] thatX_I is simply connected,π₂(X_I) =Z, and that there is a fibration sequence T^m−1 = (S¹)^m−1 →XI

qI

→ ∨^ICP^∞, where ∨^IX denotes the wedge of typeIgiven by∨^IX={(x1, . . . , xm)∈X^m: for each Λ∈I, xi=

∗ for somei∈Λ}.

Example. (1) IfI={{1,2, . . . , m}},XI =CP^m−1and∨^IX =Wm(X).

(2) IfI={{i, j}: 1≤i < j≤m},XI =Xmand∨^IX =∨^mX.

If we choose the point [1 : · · · : 1] ∈ XI as a base point ofXI, we can identify Hol^∗_d(S², XI) ={(ξ1, . . . , ξm)∈SP^d(C)^m:∩i∈Λξi=∅for any Λ∈I}. Then we denote by Holⁿ_d(S², XI) the subspace of Hol^∗_d(S², XI) defined by

Holⁿ_d(S², XI) = Hol^∗_d(S², XI)∩SP^d_n(C)^m

={(ξ₁, . . . , ξ_m)∈SP^d_n(C)^m:∩i∈Λξ_i=∅ for any Λ∈I}. There is a filtration

∅= Hol¹_d(S², X_I)⊂Hol²_d(S², X_I)⊂Hol³_d(S², X_I)⊂ · · ·

⊂Hol^d+1_d (S², XI) = Hol^d+2_d (S², XI) =· · ·= Hol^∗_d(S², XI).

Theorem 1.4. Let m, n≥2 be integers and I be a fixed collections of subsets of {1,2, . . . , m} such that card(Λ) ≥2 for each Λ ∈ I. Then there is a map Holⁿ_d(S², XI) → Ω²₀∨^I CPⁿ⁻¹ which is a homotopy equivalence up to dimension D(d;m, n)whenn≥3and a homology equivalence up to dimension [d/2]when n= 2. Here the numberD(d;m, n)is same as in (1.3.1).

Since X_I = CP^m−1 if I = {{1,2, . . . , m}} and D₁(d;m, n) < D(d;m, n) when n >2, the above theorem indicates that the main result of [13] can be improved. Moreover, although the proof given in [13] only works for the case X =CP^m−1, the proof given in this paper works for allXI (andCP^m−1, too).

So this permit us not only to sharpen the stability dimension, but also, more importantly, to obtain the stability result for wider quasi-projective varieties.

Our results may confirm this Morse theory principle for another varieties (e.g. various toric varieties cf. [8]). The explanation for these type theorems is thought to be Morse theoretic in nature, i.e. the fact that the holomorphic maps form the set of absolute minima of the energy functional on (a fixed component of) the space of smooth maps. However, the existing proofs of these results are topological in nature (cf. [3], [4]).

This paper is organized as follows. In section 2, we shall recall the stabi- lized theorem using scanning method originally invented by G. Segal [11]. In section 3, we shall give the proof of Theorem 1.3. To prove this we shall show Theorem 3.3, which is the key of this paper. In section 4, we shall indicate how the proof of Theorem 3.3 can be generalized and give the proof of Theorem 1.4.

Finally, in section 5, we shall study the stability ofs^d: SP^d_n(Ck)→SP^d+1_n (Ck) using the geometric resolution ([12]) and we shall give the proof of Theorem 5.1, which will be used in the proof of Theorem 3.3.

§2. Stabilized Spaces

In this section, we recall the scanning map and its basic properties. Be- cause this was now well explained in several papers (cf. [5], [6], [7], [8], [10], [11]), we only sketch the rough idea.

LetI be a collection of subsets of{1,2, . . . , m}such that card(Λ)≥2 for any Λ∈I. For a connected spaceX, letE_d^I,n(X) denote the space defined by E_d^I,n(X) ={(ξ1, . . . , ξm)∈SP^d_n(X)^m:∩i∈Λξi=∅ for any Λ∈I}.

Remark that E_d^I,n(C) = Holⁿ_d(S², X_I) if X = C. If A ⊂ X is a closed subspace, we define the relative configuration spaceE_d^I,n(X, A) =E_d^I,n(X)/∼, where (ξ1, . . . , ξm) ∼(η1, . . . , ηm) if and only ifξj∩(X −A) =ηj∩(X−A) for each 1≤j≤m. Thus, for eachE_d^I,n(X, A), points inAare ignored. When A=∅, there is a natural inclusionE^I,n_d (X, A)→E_d+1^I,n(X, A) given by adding points inA. We defineE^I,n(X, A) =

d≥1

E_d^I,n(X, A).

Letsd :E_d^I,n(C)→E_d+1^I,n(C) denote the stabilization map given by adding a point from the edge in a usual way (see [5], [6], [8], [11]), and let lim

d→∞E_d^I,n(C) the colimit space induced from stabilization mapssd.

Let lim

d→∞s^d_n: lim

d→∞E_d^I,n(C)→Ω²₀E^I,n(S²,∞) be the scanning map defined similarly as [[6]; page 99]. If we identifyE_d^I,n(C) = Holⁿ_d(S², X_I), then we have the scanning map S: lim

d→∞Holⁿ_d(S², XI)→Ω²₀E^I,n(S²,∞).

Theorem 2.1. S : lim

d→∞Holⁿ_d(S², XI)→ Ω²₀E^I,n(S²,∞)is a homotopy equivalence when n≥3 and a homology equivalence whenn= 2.

Proof. This is similar to the proof in section 3 of [11] (cf. [7]).

Lemma 2.1. There is a homotopy equivalenceE^I,n(S²,∞) ∨^ICPⁿ⁻¹. Proof. If I = {{i, j} : 1 ≤i < j ≤n}, the proof is completely same as that of [[5]; Proposition 3.2]. For a general case, the similar method may be used and we omit the detail.

Corollary 2.1. S : lim

d→∞Holⁿ_d(S², X_I) → Ω²₀∨^ICPⁿ⁻¹ is a homotopy equivalence when n≥3 and a homology equivalence whenn= 2.

§3. Unstability Result

Theorem 3.1. Let k≥1 be an integer and letCk =C− {1,2, . . . , k}. If d≥n≥3, the stabilization maps^d : SP^d_n(Ck)→SP^d+1_n (Ck) is a homotopy equivalence up to dimensionN(d, n) =d+n−4 =D(d;m, n).

The proof of Theorem 3.1 is postponed to the last section and we prove the following result.

Theorem 3.2. If m, n≥2, sd : Holⁿ_d(S², Xm)→Holⁿ_d+1(S², Xm)is a homotopy equivalence up to dimensionD(d;m, n)whenn≥3 and a homology equivalence up to dimension [d/2] whenn= 2. Here the numberD(d;m, n) is same as in (1.3.1).

Before proving Theorem 3.2, we complete the proof of Theorem 1.3.

Proof of Theorem1.3. The assertion easily follows from Corollary 2.1 and Theorem 3.2.

Definition. LetV ⊂C be an open set andd1, . . . , dm≥1 be integers.

LetE_dⁿ

1,...,dm(V) be the subspace of SP^d_n¹(V)× · · · ×SP^d_n^m(V) defined by E_dⁿ_1,...,dm(V) ={(ξ1, . . . , ξm) :ξi∈SP^d_nⁱ(V) for each i, ξi∩ξj =∅ifi=j}.

Ifd₁=d₂=· · ·=d_m=d, we writeE_dⁿ(V) =Eⁿ_d

1,...,dm(V). IfV =C, we also writeE_dⁿ

1,...,dm =E_dⁿ

1,...,dm(C).

Let z1, . . . , zm ∈ {w ∈ C : 1 < |w| < 2} be any fixed points such that zi=zj ifi=j. For each 1≤k≤m, define the map

j_k :E_dⁿ

1,...,d_k,...,d_m({|w|<1})→E_dⁿ

1,...,d_k−1,d_k+1,d_k+1...,d_m({|w|<2}) by (ξ1, . . . , ξm) → (ξ1, . . . , ξk−1, ξk +zk, ξk+1, . . . , ξm). Up to homotopy, jk

defines the mapj_k :E_dⁿ

1,...,d_k,...,d_m →E_dⁿ

1,...,d_k−1,d_k+1,d_k+1,...,d_m.

Theorem 3.3. Let m, n ≥ 2, 1 ≤ k ≤ m, and d1, . . . , dm ≥ 2 are integers. Then jk : E_dⁿ

1,...,d_k,...,d_m → E_dⁿ

1,...,d_k−1,d_k+1,d_k+1,...,d_m is a homotopy equivalence up to dimension D(dk;m, n)when n ≥3 and a homology equiva- lence up to dimension[d_k/2]whenn= 2.

Before giving the proof of Theorem 3.3, we complete the proof of Theo- rem 3.2.

Proof of Theorem 3.2. If we may identify Holⁿ_d(S², Xm) = E_dⁿ(C), we havesd=j1◦j2◦ · · · ◦jm(up to homotopy). Hence the assertion easily follows from Theorem 3.3.

Definition. We say that a map f :X →Y isacyclic up to dimension N if for any local coefficient system L on Y, f_∗ : H_i(X, f^∗L) → H_i(Y, L) is bijective wheni < N and is surjective wheni=N, wheref^∗Lis the induced local system onX.

Lemma 3.1 ([9]). Let f : X → Y be a continuous map between con- nected CW complexes such that π₁(X) and π₁(Y) are abelian groups. If the mapf is acyclic up to dimensionN,it is a homotopy equivalence up to dimen- sionN.

Lemma 3.2. If d₁, . . . , d_m≥2 andn≥3 are integers,π₁(Eⁿ_d

1,...,dm) is an abelian group.

Proof. Geometrically π₁(E_dⁿ

1,...,dm) may described as the group of m- tuples (d1-strings, d2-strings, · · ·, dm-strings) of braids such that i-th braids are allowed to pass until (n−1)-crossings. Hence, the assertion may be proved in a similar way as in [[5]; appendix].

Now we can give the proof of Theorem 3.3.

Proof of Theorem 3.3. It suffices to give the proof when k = 1. Let π : E_dⁿ

1,...,dm → E_dⁿ

2,...,dm be the projection map given by π(ξ₁, . . . , ξ_m) = (ξ2, . . . , ξm). Ifn≥3 and d1< n, Theorem 3.3 follows from [[5]; Theorem 1.8]

and Lemmas 3.1, 3.2. Ifn= 2, we have a homotopy commutative diagram Cd₁(Ck) ^s

−−−−→d Cd₁+1(Ck)

 

E²_d

1,...,dm

j₁

−−−−→ E_d²

1+1,d2,...,dm

π

 _π

E²_d

2,...,dm

−−−−→= E_d²

2,...,dm

where we take k=m

i=2di andCk =C− {1,2, . . . , k}, vertical sequences are fibrations ands^d is a stabilization map. Sinces^d is a homology equivalence up to dimension [d₁/2] ([[11]; appendix]), j₁ is also a homology equivalence up to dimension [d₁/2]. Hence the assertion holds whenn= 2.

So without loss of generalities, we may assume that d1 ≥ n ≥ 3. Then it follows from Lemmas 3.1, 3.2 that it suffices to show that j1 is acyclic up to dimension N(d1, n) = d1+n−4. Let L be a local coefficient system on E_dⁿ

1+1,d₂,...,d_m. We shall use the letterLto denote its restriction to any (closed or open) subspace. In this case, we also consider the commutative diagram

E_dⁿ

1,d₂,...,d_m j1

−−−−→ E_dⁿ

1+1,d₂,...,d_m π

 _π

Eⁿ_d

2,...,d_m

−−−−→= E_dⁿ

2,...,d_m

If vertical maps were fibrations, we could prove the assertion in a similar way as above. However, they are fibrations only over certain subspaces as we shall now explain.

SinceE_dⁿ

1,d2,...,dm ⊂SP^d1(C)×· · ·×SP^dm(C)∼=C^d1×· · ·×C^dm=C^dis an open set, E_dⁿ

1,d2,...,dm is an open complex manifold of dimension d=m i=1d_i and the Poincar´e duality isomorphismHk(E_dⁿ

1,d₂,...,d_m;L)∼=H_c^2d−k(E_dⁿ

1,d₂,...,d_m; H(L)) holds, whereH(L) denotes the orientation bundle and H_cⁱ denotes the cohomology with compact supports ([2]).

From now on, we may identify E_dⁿ

1,d₂,...,d_m with the space consisting of all m-tuples (p₁(z), . . . , p_m(z)) ∈ C[z]^m of monic polynomials satisfying the following two conditions:

(a) p_i(z) is a monic polynomial of degree d_i and has no roots of multiplicity

≥nfor each 1≤i≤m.

(b) p_i(z) andp_k(z) are coprime wheneveri=k.

Let E_d^k1,k2,...,km

1,d₂,...,d_m be the subspace of E_dⁿ

1,d₂,...,d_m consisting of allm-tuples of polynomials (p1(z), . . . , pm(z)) ∈ E_dⁿ

1,d₂,...,d_m which satisfies the following condition:

(c) p_j(z) has at leastk_j distinct roots for each 1≤j≤m.

LetX_d^1,k2,...,km

1,d₂,...,d_m =E_d^1,k2,...,km

1,d₂,...,d_m− m i=2

E_d^{1,k2,...,ki+1,...,km}

1,d₂,...,d_i,...,d_m ,i.e. the subset consisting of allm-tuples (p1(z), . . . , pm(z))∈E_dⁿ

1,...,d_m such that eachpi(z) (2≤i≤m) has exactlyki distinct roots.

LetY_d^k2,...,km

2,...,dm =π(X_d^1,k2,...,km

1,d2,...,dm) =E_d^k2,...,km

2,...,dm− m i=2

E^k_d^{2,...,ki+1,...,km}

2,...,di,...,dm .The map πrestricts to a map π:X_d^1,k2,...,km

1,d2,...,dm →Y_d^k2,...,km

2,...,dm and we obtain the homotopy commutative diagram

SP^d_n¹(Ck) ^s

d1

−−−−→ SP^d_n¹⁺¹(Ck)

 

X_d^1,k2,...,km

1,d2,...,dm

j₁

−−−−→ X_d^1,k2,...,km

1+1,d2,...,dm

π

 _π

Y_d^k2,...,km

2,...,dm

−−−−→= Y_d^k2,...,km

2,...,dm

where we takek=m

i=2k_i,Ck=C− {1,2, . . . , k}, and vertical sequences are fibrations.

Sinces^d1is a homotopy equivalence up to dimensionN(d1, n) by Theorem 3.1, j1 : X_d^1,k2,...,km

1,d₂,...,d_m → X_d^1,k2,...,km

1+1,d₂,...,d_m is also a homotopy equivalence up to dimensionN(d1, n). Now we remark the following:

(†k) If 1 ≤ ki ≤di (i = 2, . . . , m), then j1 : E_d^1,k2,...,km

1,d2,... ,dm → E_d^1,k2,...,km

1+1,d2,... ,dm is acyclic up to dimensionN(d₁, n).

We postpone the poof of (†k) and complete the proof of 3.3. If (†k) is true, then j₁ : E_d^1,1,...,1

1,d2,... ,dm → E_d^1,1,...,1

1+1,d2,... ,dm is acyclic up to dimension N(d₁, n).

Since E_d^1,1,...,1

1,d2,... ,dm = E_dⁿ

1,d2,... ,dm, j₁ : E_dⁿ

1,d2,... ,dm → E_dⁿ

1+1,d2,... ,dm is acyclic up to dimension N(d1, n). This completes the proof of Theorem 3.3.

Proof of(†k): The proof of (†k) is by downwards induction onk=m i=2k_i. The induction begins withk=m

i=2d_i. Since E_d^1,d2,...,dm

1,d₂,... ,d_m =X_d^1,d2,...,dm

1,d₂,... ,d_m, the assertion is clearly true. As a next step, we assume that (†l) holds for anyl > k and we shall prove that (†k) is true. We remark that for any 2≤j ≤mthe following statement holds:

(‡j) j₁: j i=1

E_d^{1,l2(i),...,lm(i)}

1,d₂,... ,d_m → j i=1

E_d^{1,l2(i),...,lm(i)}

1+1,d₂,... ,d_m is a homology equivalence up to dimension N(d1, n) wheneverm

t=2lt(i)> kfor any 1≤j≤m.

Proof of (‡j): We can prove (‡j) easily by induction on j. In fact, if j = 1, (‡j) holds, because (†l) is true when l > k. Now assume (‡j−1) is true and we shall show that (‡j) is true. Since E_d^1,k2,...,km

1,d2,... ,dm ∩E_d^1,k2,...,km

1,d2,... ,dm = E^1,max(k2,k2),...,max(km,k_m)

d₁,d₂,... ,d_m , using the Mayer-Vietoris exact sequence and 5-lemma, we can prove the assertion (‡j) easily.

It follows from (‡m) thatj₁: m i=2

E_d^{1,k2,...,ki+1,... ,km}

1,d2,...,di,... ,dm → m i=2

E_d^{1,k2,...,ki+1,...,km}

1+1,d2,...,di,... ,dm

is a homology equivalence up to dimensionN(d1, n).

SinceE_d^{1,k2,...,ki+1,...,km}

1,d₂,...,d_i,... ,d_m ⊂Eⁿ_d

1,d₂,...,d_i,... ,d_m is an open subspace, the space m

i=2

E_d^{1,k2,...,ki+1,... ,km}

1,d₂,...,d_i,... ,d_m is an open complex manifold of dimension d. Hence it follows from Poincar´e duality that there is a commutative diagram

Hj

_m

i=2

E_d^{1,k2,...,ki+1,... ,km}

1,d₂,...,d_i,... ,dm ;L

j_1∗

−−−−→ Hj

_m

i=2

E_d^{1,k2,...,ki+1,...,km}

1+1,d₂,...,d_i,... ,dm;L

∼=



^∼=

H_c^2d−j _m

i=2

E_d^{1,k2,...,ki+1,... ,km}

1,d₂,...,d_i,... ,d_m

j₁^∗

−−−−→ H_c^2d+2−j _m

i=2

E_d^{1,k2,...,ki+1,...,km}

1+1,d₂,...,d_i,... ,d_m

whereH_c^k(Y) =H_c^k(Y;H(L)). Hence, the induced homomorphism (i) j₁^∗:H_c^j

_m

i=2

E_d^{1,k2,...,ki+1,...,km}

1,d₂,...,d_i,...,d_m

→H_c^j+2 _m

i=2

E_d^{1,k2,...,ki+1,...,km}

1+1,d₂,...,d_i,... ,d_m

is bijective when j >2d−N(d₁, n) and surjective whenj = 2d−N(d₁, n).

Similarly, because j1 : X_d^1,k2,...,km

1,d₂,...,d_m → X_d^1,k2,...,km

1+1,d₂,...,d_m is also a homotopy equivalence up to dimensionN(d1, n), the induced homomorphism

(ii) j1∗:H_c^j(X_d^1,k2,...,km

1,d2,...,dm)→H_c^j+2(X_d^1,k2,...,km

1+1,d2,...,dm)

is bijective when j >2d−N(d1, n) and surjective whenj = 2d−N(d1, n).