A generalization of Weierstrass’ }-function to quasi-abelian varieties

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A generalization of Weierstrass’ ℘-function to quasi-abelian varieties

理工学教育部

数理・ヒューマンシステム科学専攻学籍番号

30971003

漕江厚子

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Preface

For a given lattice Γ Weierstrass constructed a doubly periodic meromorphic function onCwith period Γ. We call it Weierstrass’℘-function. It was general- ized on abelian varieties by Zappa in 1983. LetTⁿ=Cⁿ/Γ be ann-dimentional complex torus. Γ-invariant∂-closed (0, n−1)-forms onCⁿ/Γ are considered as representatives of classes inHⁿ⁻¹(Tⁿ\ {0},O). First, Zappa constructed a Γ- invariant∂-closed (0, n−1)-form℘ⁱ, and then gave a∂-closed (n−1, n−1)-form

℘^ij onTⁿ\ {0} with the following property:

IfTⁿ is an abelian variety and Θ is a divisor onTⁿ defined by a theta function θ, then we have

∫

Θ

℘^ij(z−p) =− ∂²

∂z_i∂z_j logθ

p

+ constant.

In the case of one variable, this is just the relation between Weierstrass’ ℘- function and a theta function.

The purpose of this paper is to give a further generalization of Zappa’s result. We show that we can construct a similar (n−1, n−1)-form℘^ij even for a non-compact quasi-abelian variety.

This paper consists of three chapters. In Chapter 1, we explain in detail a part of the theory of Andreotti-Norguet which is the basis of our argument. We think that our proofs are more explicit and comprehensible than the original ones.

In Chapter 2, we state Zappa’s result. Several lemmas and propositions are stated in more general setting in order to use them later.

In Chapter 3, Weierstrass’ ℘-function is generalized on quasi-abelian vari- eties. Let X = Cⁿ/Γ be a toroidal group. We can construct a Γ-invariant

∂-closed (n−1, n−1)-form ℘^ij on X in the same manner as in the case of complex tori. But when we consider a positive divisor Θ onX, we do not know in general the convergence of the integral of ℘^ij on Θ, because Θ is not compact. Even if it converges, we are not able to apply the same argument as in the case of abelian varieties. We suppose thatX is a quasi-abelian variety of kind 0 with the standard compactificationX. We further assume that a positive divisor Θ onX is holomorphically extendable to X. Then we can prove that

℘^ij is integrable on Θ and a similar equation as the case of abelian varieties is obtained (Theorem 3.7 in Chapter 3). To prove it, we first take a family of

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relatively compact subdomains inX. We give formulas on these subdomains.

As the limit of them we obtain the expected formula.

Acknowledgement

The author would like to thank Professor Yukitaka Abe for giving her construc- tive comments and warm encouragement.

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Chapter 1

Cohomology Groups of a Punctured Polydisc

1.1 Cohomology groups

We give a proof of Lemma 2 in [5] in this section. This lemma is essentially Proposition 31.1 in [6]. The proof given here is more explicit than that in [6].

Let D = {ζ ∈ C ; |ζ| <1} be the unit disc in C. Consider a polydisc of n-dimention

Dⁿ:=D× · · · ×D.

Letz= (z₁, . . . , z_n) be coordinates ofCⁿ. Lemma 1.1([5]).

a)Hⁱ(Dⁿ\ {0}, O) = 0 fori̸= 0, n−1.

b)H⁰(Dⁿ\ {0}, O)=∼H⁰(Dⁿ,O)if n≧2.

c)Hⁿ⁻¹(Dⁿ\ {0}, O)∼= {∑

α∈Nⁿ

cαz⁻^α ; lim

|α|→∞

|α|√

|cα|= 0 }

.

Proof. If n ≧ 2, then (b) is clear, because {0} is an analytic set of Dⁿ with codimention more than 1 .

We show a). For i = 1, . . . , n we set U_i := {z ∈ Dⁿ;z_i ̸= 0}. Then U:={Ui;i= 1, . . . , n} is a Stein covering ofDⁿ\ {0}. Therefore we get

H^q(Dⁿ\ {0},O)∼=H^q(U,O)

by Leray’s theorem. For any (i₀, . . . , i_q)∈ {1, . . . , n}^q+1, we set Ui₀...i_q :=Ui₀∩ · · · ∩Ui_q.

Since U consists of n open sets, it is obvious that Hⁿ(U,O) = 0. For any j∈ {1, . . . , n}and (i₀, . . . , i_q)∈ {1, . . . , n}^q+1, we define a map

φⁱ_j⁰^...i^q : Γ(Uji₀...i_q,O)→Γ(Ui₀...i_q,O)

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by the following way:

When j = iµ for some µ = 0, . . . , q we set φ_jⁱ⁰^...i^q = id. When j ̸= iµ for all µ = 0, . . . , q, we take R with 0 < R < 1 such that |zj| < R for any z = (z1, . . . , zn)∈ Uji0...iq. LetCj(R) be a circle with center 0 and radiusR. For anyf ∈Γ(Uji0...iq,O) we define

(φⁱ_j⁰^...i^qf)(z) := 1 2π√

−1

∫

C_j(R)

f(z₁, . . . z_j₋₁, t_j, z_j+1, . . . , z_n) tj−zj

dt_j. Then we haveφⁱ_j⁰^...i^qf ∈Γ(U_i₀_...i_q,O).

If z_j ̸= 0, then we take r such that 0 < r < |z_j| < R < 1 and denote by C_j(r) a circle with center 0 and radiusr. By Cauchy’s integral theorem we can representf ∈Γ(U_ji₀_...i_q,O) as

f(z) = 1 2π√

−1

∫

C_j(R)

f(z1, . . . zj−1, tj, zj+1, . . . , zn) t_j−z_j dt_j

− 1

2π√

−1

∫

Cj(r)

f(z1, . . . zj−1, tj, zj+1, . . . , zn) t_j−z_j dtj. Therefore we note that

(φⁱ_j⁰^...i^qf)(z) = 1 2π√

−1

∫

C_j(r)

f(z1, . . . zj−1, tj, zj+1, . . . , zn)

t_j−z_j dt_j+f(z) forz= (z₁, . . . , z_n) withz_j̸= 0.

Let rⁱ⁰^...i^q

i₀...iˆ_µ...i_q be the restriction map from functions on U_i

0...iˆµ...iq to those onU_i₀_...i_q, where ˆi_µ means thati_µ shall be omitted. Then it follows that



 rⁱ⁰^...i^q

i0...iˆµ...iqφⁱ_j⁰^...ⁱ^ˆ^µ^···^q =φⁱ_j⁰^...i^q, φⁱ_j⁰^...i^qr^ji_i ⁰^...i^q

0...i_q =id on Γ(Ui₀...i_q,O).

For anyj∈ {1, . . . , n}, we define a map

k_j :C^q(U,O)→C^q⁻¹(U,O) as follows:

For anyf = (fi₀...i_q)∈C^q(U,O) we set

(kjf)i₀...i_q−1 :=φ_jⁱ⁰^...i^q−1fji₀...i_q−1.

Letδ be the usual coboundary operater. We take anyf ∈C^q(U,O). For any multiindex (i0, . . . , iq)∈ {1, . . . , n}^q+1we calculate

(f −δk_jf−k_jδf)_i₀_...i_q =f_i₀_...i_q−(δk_jf)_i₀_...i_q−(k_jδf)_i₀_...i_q.

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Since

(δkjf)i₀...i_q =

∑q µ=0

(−1)^µ(kjf)_i

0...ib_µ...i_q

=

∑q µ=0

(−1)^µφⁱ_j⁰^...ⁱ^b^µ^...i^qf_ji

0...ib_µ...i_q

and

(k_jδf)_i₀_...i_q =φ_jⁱ⁰^...i^q(δf)_ji₀_...i_q

=φⁱ_j⁰^...i^q (

fi₀...i_q−

∑q µ=0

(−1)^µφⁱ_j⁰^...i^qf_ji

0...ibµ...iq

)

=φⁱ_j⁰^...i^qfi₀...i_q−

∑q µ=0

(−1)^µφⁱ_j⁰^...i^qf_ji

0...ibµ...iq, we have

fi₀...i_q−(δkjf)i₀...i_q−(kjδf)i₀...i_q = fi₀...i_q−φ_jⁱ⁰^...i^qfi₀...i_q+

∑q µ=0

(−1)^µφⁱ_j⁰^...ⁱ^b^µ^...i^qf_ji

0...ibµ...iq

−

∑q µ=0

(−1)^µφⁱ_j⁰^...i^qf_ji

0...ib_µ...i_q. Whenj /∈ {i0, . . . , iq}we have φ_jⁱ⁰^...i^qfi₀...i_q =fi₀...i_q. Since

φⁱ_j⁰^...i^qf_ji

0...ib_µ...i_q=rⁱ⁰^...i^q

i0...ibµ...iq

φⁱ_j⁰^...ⁱ^b^µ^...i^qf_ji

0...ib_µ...i_q, we have in general

f_i₀_...i_q−(δk_jf)_i₀_...i_q−(k_jδf)_i₀_...i_q

=





0 if j /∈ {i0, . . . , iq} (id−rⁱ⁰

i0...ic_µj...iq

φⁱ_j⁰^...ⁱ^c^µj^...i^q)fi₀...i_q if j=iµ for someµ, whereµj isµwithj=iµ. For anyj∈ {1, . . . , n}we define a map

Φ_j :C^q(U,O)→C^q(U,O) by

Φjf :=f−δkjf−kjδf

for any f ∈ C^q(U,O). It is obvious that Φ_j maps an element in Z^q(U,O) to Z^q(U,O). Then we define a map

Φ :Z^q(U,O)→Z^q(U,O)

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by Φ := Φ1◦Φ2◦· · ·◦Φn. If 1≦q≦n−2, then for any (i0, . . . , iq)∈ {1, . . . , n}^q+1 there existsj ∈ {1, . . . , n}such thatj /∈ {i0, . . . , iq}. Hence we have Φf = 0 for anyf ∈Z^q(U,O). This means thatf ∈B^q(U,O). Therefore we obtain

H^q(U,O) = 0 for 1≦q≦n−2.

Next we show c). First we note

Zⁿ⁻¹(U,O) = Γ(U₁₂_···_n,O).

Anyf ∈Γ(U12···n,O) has the following Laurent expansion f = ∑

α∈Nⁿ

cαz⁻^α+

∑n µ=1

(−1)ⁿ⁻¹

∑∞ n_µ=0

gn_µ(z1, . . . ,czµ, . . . , zn)z_µⁿ^µ,

wheregn_µ(z1, . . . ,czµ, . . . , zn) is a holomorphic function in (z1, . . . , zµ−1, zµ+1, . . . , zn)∈ (D^∗)ⁿ⁻¹and coeﬃcientscαsatisfy

|αlim|→∞

|α√|

|cα|= 0.

Let

g₁₂_···b_µ_···_n:=

∑n µ=1

(−1)ⁿ⁻¹

∑∞ n_µ=0

g_n_µ(z₁, . . . ,cz_µ, . . . , z_n)z_µⁿ^µ.

Then we have an elementg= (g12···bµ···n) inCⁿ⁻²(U,O). Therefore we have f = ∑

α∈Nⁿ

c_αz⁻^α+δg,

which shows the isomorphism in c).

1.2 Generalized Martinelli formula

LetN0:=N∪ {0}. For anyα= (α1, . . . , αn)∈Nⁿ0 we define a (0, n−1)-form

ψα:=



 ∑

1≦j≦n

z_j^α^jzjαj





−n

∑

1≦j≦n

(−1)^j⁻¹zjαj ∧

1≦k≦n k̸=j

d(zkαk).

Lettingdz:= ∧

1≦j≦n

dzj, we consider an (n, n−1)-formKα⁽ⁿ⁾ :=dz∧ψα. For anyα= (α1, . . . , αn)∈Nⁿ0 we set

α+1:= (α₁+ 1, . . . , α_n+ 1) andα^′:= (α₂, . . . , α_n), where1= (1, . . . ,1). We also setα! :=∏

1≦j≦nα_j!.

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Let B = {z ∈ Cⁿ; ∑

1≦j≦n

zjzj < 1} be the unit ball of Cⁿ. We denote by S=∂Bthe boundary ofB. Let

∂^|^α^|

∂z^α = ∂^α¹⁺^···^+αⁿ

∂z^α₁¹· · ·∂z^αnⁿ

.

The equation in the following proposition is called the generalized Martinelli formula.

Proposition 1.2(Proposition 1 in [4]). We have

∫

S

f K_α+1⁽ⁿ⁾ = (2π√

−1)ⁿ (n−1)!

1 α!

∂^|^α^|f

∂z^α (0)

for any holomorphic functionf on a neighbourhood ofB, whereBhas the direc- tion in which

(√

−1 2

)ⁿ

dz∧dz is positive and the direction of S is compatible with the formula of Stokes.

Proof. If n = 1, then it is just the integral formula of Cauchy. We prove the statement by induction onn. To prove it we temporarily take the direction of B in which ∧

1≦j≦n

(√

−1

2 dzj∧dzj

)

is positive. Moreover, we assume that the direction ofS is compatible with Stokes’s formula. We denote

θα:= 1 n−1

1 z₁^α¹



 ∑

1≦j≦n

z^α_j^jzjα_j





1−n

∑

2≦j≦n

(−1)^jzjα_j ∧

2≦k≦n k̸=j

d(zkα_k),

Lα:= (−1)ⁿdz∧θα. We can check

ψ_α=∂θ_α, K_α⁽ⁿ⁾=∂L_α=dL_α

at any point with z₁̸= 0 by straight calculation. For a multiindex β = (α₁+ 1, α₂, . . . , α_n) we have

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∫

S

f K_β⁽ⁿ⁾= lim

ε→0

∫

S∩{|z1|>ε}

d(f Lβ)

= lim

ε→0

∫

S∩{|z₁|=ε}

f Lβ

=(−1)ⁿ⁻¹ n−1 lim

ε→0

∫

−S∩{|z1|=ε}

f ∧

1≦j≦n

dzj

∧ 1 z₁^α+1



 ∑

1≦j≦n

z^β_j^jzjβj





1−n

∑

2≦j≦n

(−1)^jzjαj ∧

2≦k≦n k̸=j

d(zkαk)

=(−1)ⁿ⁻¹ n−1 lim

ε→0

∫

−S∩{|z₁|=ε}

f dz₁ z^α₁¹⁺¹ ∧



 ∧

2≦j≦n

dzj





∧

∑

2≦j≦n(−1)^jzjα_j∧

2≦k≦n k̸=j

d(zkα_k) (

ε^2(α¹⁺¹⁾+∑

2≦j≦nz_j^α^jzjα_j)n−1

=(−1)ⁿ⁻¹ n−1

2π√

−1 α1! lim

ε→0

∫

S∩{z₁=0}

∂^α¹f

∂z₁^α¹



 ∧

2≦j≦n

dz_j





∧

∑

2≦j≦n(−1)^jzjα_j∧

2≦k≦n k̸=j

d(zkα_k) (

ε^2(α¹⁺¹⁾+∑

2≦j≦nz_j^α^jzjα_j)n−1

= (−1)ⁿ⁻¹ n−1

2π√

−1 α1!

∫

S∩{z₁=0}

∂^α¹f

∂z^α₁¹K_α⁽ⁿ′⁻¹⁾. By the assumption of induction we have

∫

S∩{z₁=0}

∂^α¹f

∂z₁^α¹K_α⁽ⁿ_′₊₁⁻¹⁾_′ = (2π√

−1)ⁿ⁻¹ (n−2)!

1 α^′!

∂^|^α^′^|

∂z₂^α^′²· · ·∂zn^α^′ⁿ

(∂^α¹f

∂z₁^α¹ )

(0).

We note that the relation between the original direction ofBand one given here to prove the statement is as follows:

(√

−1 2

)ⁿ

ω∧ω= (−1)ⁿ⁽ⁿ²⁻¹⁾ ∧

1≦j≦n

(√

−1 2 dzj

∧dzj

) .

Using the above result, we obtain the integral

∫

S

f K_α+1⁽ⁿ⁾ in the original direction

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as follows:

∫

S

f K_α+1⁽ⁿ⁾ = (−1)ⁿ⁽ⁿ⁻¹⁾² (−1)ⁿ⁻¹ n−1

2π√

−1 α₁!

(2π√

−1)ⁿ⁻¹ (n−2)!

1 α^′!

∂^|^α^′^|

∂z^α₂^′²· · ·∂zn^α^′ⁿ

(∂^α¹f

∂z₁^α¹ )

(0)

= (2π√

−1)ⁿ (n−1)!

1 α!

∂^|^α^|f

∂z^α (0).

Then the proof is completed.

1.3 Dolbeault isomorphism

We have seen cohomology groups of a punctured polydisc in Section 1.1. Lemma 1.1 c) shows thatHⁿ⁻¹(Dⁿ\{0},O) is a Fr´echet space generated by cohomology classes

z⁻₁^α¹⁻¹· · ·z⁻_n^αⁿ⁻¹∈Γ(

∩n i=1

U_i,O), α= (α₁, . . . , α_n)∈Nⁿ0.

In this section, we study∂-closed (0, n−1)-forms corresponding to cohomology classesz₁⁻^α⁻¹· · ·z_n⁻^αⁿ⁻¹ by the Dolbeault isomorphism.

Lemma 1.3 (Lemma 4 in [5]). a) By the Dolbeault isomorphism the cohomol- ogy class of z⁻₁^α¹⁻¹· · ·z_n⁻^αⁿ⁻¹ ((α1, . . . , αn)∈Nⁿ0)corresponds to the ∂-closed (0, n−1)-form (up to a sign)

(n−1)!ψα+1= (n−1)!

∑n

k=1(−1)^kzkα_k+1d(z1α₁+1)∧ · · · ∧d(z\kα_k+1)∧ · · · ∧d(znα_n+1) (∑n

j=1|z_j^α^j⁺¹|²)n . b) The Dolbeault representative so chosen is such that to the class (z1· · ·zn)⁻¹∑

α∈Nⁿ0 cαz⁻^αwithlim_|_α_|→∞ ^|α|√

cα= 0, corresponds to the∂-closed (0, n−1)-form expressed by absolutely convergent series

(n−1)! ∑

α∈Nⁿ0

cαψα+1 onCⁿ\ {0}.

Proof. a) Let ε^(a,b) be the sheaf of germs ofC^∞ (a, b)-forms. We particularly setε=ε^(0,0)when (a, b) = (0,0). Sinceε^(a,b)is a fine sheaf, it follows that

H^q(Dⁿ\ {0}, ε^(a,b)) = 0 for anyq≧1.

There existsφ= (φ₁_···_b_i_···_n)∈Cⁿ⁻²(U, ε) such thatz⁻^α⁻¹=δφ, i.e.

z⁻^α⁻¹=

∑n i=1

(−1)ⁱ⁻¹φ₁_···_b_i_···_n,

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for we can considerz⁻^α⁻¹∈Cⁿ⁻¹(U, ε) for all α∈Nⁿ0. Then we have

∑n i=1

(−1)ⁱ⁻¹φ₁_···_b_i_···_n= 0.

Therefore we have∂φ= (∂φ₁_···_b_i_···_n)∈Zⁿ⁻²(U, ε^(0,1)). Since Hⁿ⁻²(U, ε^(0,1))∼=Hⁿ⁻²(Dⁿ\ {0}, ε^(0,1)) = 0, there exists (φ₁_···_b_i_···_b_j_···_n)∈Cⁿ⁻³(U, ε^(0,1)) such that

∂φ=δ((φ₁_···_b_i_···_b_j_···_n)) =(∑

±φ₁_···_b_i_···_b_j_···_n )

. Repeating this procedure, we obtain (φj)∈C⁰(U, ε^(0,n⁻²⁾) such that

∂φi =∂φj for anyi, j withUi∩Uj ̸=∅. Therefore we can define aC^∞ (0, n−1)-form Ψ onDⁿ\ {0}by

Ψ :=∂φi onUi.

It is clear that∂Ψ = 0. This form Ψ is the Dolbeault representative corresponding toz⁻^α⁻¹.

We set

Pε:={|zi|≦ε;i= 1, . . . , n} for suﬃciently smallε >0. Then we have

∫

|z1|=ε,···,|zn|=ε

z⁻^α⁻¹∑

cβz^βdz1· · ·dzn= (2π√

−1)ⁿcα

for any convergent seriesf =∑

β∈Nⁿ0c_βz^β in a neighbourhood of 0 in whichP_ε is contained. Namelyz⁻^α⁻¹are generators characterized by the above property.

Using Stokes’s theorem and repeating correspondence between ˇCech cohomology classes and∂-cohomology classes, we obtain

∫

|z₁|=ε,···,|z_n|=ε

z⁻^α⁻¹f dz₁∧ · · · ∧dz_n

=

∑n i=1

∫

|z₁|=ε,···,|z_n|=ε

(−1)ⁱ⁻¹φ₁_···_b_i_···_nf dz₁∧ · · · ∧dz_n

=

∑n i=1

(−1)ⁱ⁻¹

∫

|z1|=ε,···,|zi|≦ε,···,|zn|=ε

∂φ₁_···_b_i_···_nf dz1∧ · · · ∧dzn

=· · ·

=±∑ ∫

|z_i|=ε,|z_j|≦ε(j̸=i)

Ψf dz₁∧ · · · ∧dz_n

=±

∫

∂P_ε

Ψf dz1∧ · · · ∧dzn,

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for

z⁻^α⁻¹=

∑n i=1

(−1)ⁱ⁻¹φ₁_···_b_i_···_n. Hence we have

±

∫

∂P_ε

Ψf dz1∧ · · · ∧dzn= (2π√

−1)ⁿcα.

Therefore, it holds that for any holomorphic functionf on a neighbourhood of

B ∫

∂B

Ψf dz₁∧ · · · ∧dz_n=±(2π√

−1)ⁿ(α!)⁻¹∂^|^α^|f

∂z^α (0),

whereBis an open ball with center 0 and suﬃciently small radius. ForKα+1= ω∧ψα+1it holds that

∫

∂B

f Kα+1= (2π√

−1)ⁿ (n−1)!

1 α!

∂^|^α^|f

∂z^α (0)

by Proposition 1.2 in the previous section. Therefore, the representative in the Dolbeault classes corresponding toz⁻^α⁻¹is (n−1)!ψα+1(up to sign).

b) From (a) we see that the series (n−1)! ∑

α∈Nⁿ0

cαψα+1

formally corresponds to

(z₁· · ·z_n)⁻¹ ∑

α∈Nⁿ0

c_αz⁻^α with lim

|α|→∞

|α|√

|c_α|= 0.

Then it suﬃces to show that the above series converges absolutely onCⁿ\ {0}. Takeδ with 0< δ <1. Let∑n

i=1|z_i|² > δ. Then we have |z_i|> ^δ_n for somei.

Therefore we have

∑n j=1

|zj|^2α^j⁺²≧|zi|^2αⁱ⁺²>

(δ n

)2αi+2

>

(δ n

)2|α|+2

.

Consequently the absolute value of the coeﬃcient ofdz₁∧ · · · ∧dzc_j∧ · · · ∧dz_n inψ_α+1is estimated by

∏

k̸=j

αk|zj||z1|^α¹· |zn|^αⁿ ((δ

n

)2|α|+2)₋n

=∏

k̸=j

αk|zj| 1 (_δ

n

)2n

( |z₁| (_δ

n

)2n

)α₁

· · · ( |z_n|

(_δ

n

)2n

)α_n

.

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Thus we see that the absolute value of the coeﬃcient ofdz1∧· · ·∧dzcj∧· · ·∧dzn

in∑

cαψα+1 is estimated by

|z_j| 1 (_δ

n

)2n

∑|c_α|w^α, w_i:= |z_i| (_δ

n

)2n.

Hence the series converges uniformly on compact subsets inCⁿ\ {0}.

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Chapter 2

Zappa’s Results

2.1 Cohomology groups of a punctured torus

LetTⁿ =Cⁿ/Γ be a complex torus ofn-dimension, where Γ is a lattice ofCⁿ. Letπ:Cⁿ→Tⁿ be the canonical projection. We can take a neighbourhoodV of 0 inTⁿ such that

π⁻¹(V) = ⊔

γ∈Γ

Uγ (disjoint union), π|U_γ:Uγ →V

is a biholomorphic mapping, where Uγ is a polydisc with center γ. Applying Mayer-Vietoris’ theorem toTⁿ =V ∪(Tⁿ\ {0}), we obtain a cohomology exact sequence

0→H⁰(Tⁿ,O)→H⁰(V,O)⊕H⁰(Tⁿ\ {0},O)→ · · · →H^k(Tⁿ,O)→ H^k(Tⁿ,O)⊕H^k(Tⁿ\ {0},O)→ H^k(V \ {0},O)→ · · · →Hⁿ⁻²(V \ {0},O)

→Hⁿ⁻¹(Tⁿ,O)→Hⁿ⁻¹(V,O)⊕Hⁿ⁻¹(Tⁿ\ {0},O)→Hⁿ⁻¹(V \ {0},O)

→ Hⁿ(Tⁿ,O) → Hⁿ(V,O)⊕Hⁿ(Tⁿ \ {0},O) → Hⁿ(V \ {0},O) → 0.

By Lemma1.1 a) in Chapter 1 and

Hⁱ(V,O) = 0 if i≧1, we have the following exact sequence

0→Hⁿ⁻¹(Tⁿ,O)→Hⁿ⁻¹(Tⁿ\ {0},O)→

Hⁿ⁻¹(V \ {0},O)→−^δ Hⁿ(Tⁿ,O)→0.

Therefore we obtain

Hⁿ⁻¹(Tⁿ\ {0},O)∼=Hⁿ⁻¹(Tⁿ,O)⊕Kerδ.

A generalization of Weierstrass’ }-function to quasi-abelian varieties