Theta function and Bergman metric on Abelian varieties

New York Journal of Mathematics

New York J. Math. 15(2009)19–35.

Theta function and Bergman metric on Abelian varieties

Xiaowei Wang and Hok Pun Yu

Abstract. In this note, we find explicit balanced embeddings for a principally polarized Abelian variety. As a consequence, we are able to give a very simple proof of the fact (cf. Donaldson, 2001) that balanced metrics converge to the flat metric.

Contents

1. Introduction 19

2. Abelian varieties and Theta functions 21 2.1. OA_Ω(1) and Theta functionϑ 21

2.2. Metric on OA_Ω(1) 22

3. Finite Heisenberg groups 23

4. Main theorem 27

References 35

1. Introduction

In [D], Donaldson proved the following beautiful result: Let (X,OX(1)) be a n-dimensional projective manifold, polarized by an ample line bundle OX(1). Suppose that (X,OX(1)) has no continuous automorphisms and there exists a constant scalar curvature K¨ahler (cscK) metric in the K¨ahler class c₁(OX(1)). Then the projective embedding X → PH⁰(X,OX(k))

Received September, 2007.

Mathematics Subject Classification. 11G10, 32Q20.

Key words and phrases. Theta function, Abelian variety, Bergman metric.

The work is partially supported by RGC grant CUHK403106 from the Hong Kong government.

ISSN 1076-9803/09

19

induced by OX(k) can always be balanced for k sufficiently large, that is, for each large kthere exists a basis

z⁽_i^k)

of H⁰(X,OX(k)) such that

X

z_i^(k)z⁽_j^k)

z^(k)2 − δ_ij Nk+ 1

ω_FSⁿ

n! = 0

where N_k + 1 = dimH⁰(X,OX(1)) and ω_FS is the Fubini–Study metric on P^Nk. Moreover, if we let ωk denote ω_FS/k induced from the balanced embedding, then

(1) |ωk−ω_∞|_C^r_(ω∞)=O(1/k),

where ω_∞ is the cscK metric in the class c₁(OX(1)) and |·|_C^r_(ω∞) is the C^r-norm with respect to the metric ω_∞. Since the condition of balanced embedding is equivalent to the Chow stability in the geometric invariant theory (GIT), Donaldson’s result gives an affirmative answer to one direc- tion of the conjecture raised by Yau many years ago, that is the existence of a cscK metric should be related to the stability of the polarized pair (X,OX(1)) in the GIT sense.

As Donaldson pointed out, although Mumford proved that any smooth Riemann surface is always Chow stable with respect to the canonical polar- ization [Mum2], his theorem does not give a new proof of the uniformization theorem. This is because we have to assume a priori that hyperbolic metric exists on the Riemann surface in order to apply Donaldson’s convergence result. So to get the a priori convergence of the balanced metrics for Rie- mann surfaces remains a challenge for the moment. This is the motivation of the current work. More precisely, in this note we prove the following:

Theorem 1. Let (A_Ω,OA_Ω(1)) be a principally polarized Abelian variety of dimension g with period matrix Ω = [I, Z]. For sufficiently large l ∈N, the projective embedding induced by a basis

ϑ a

b

a,b∈(Z[1/l]/Z)^g

for H⁰(A_Ω,OA_Ω(l²)) (see Section 4 for the definition) is balanced. And if we normalize the balanced metrics, each of them converges to the flat metric on A_Ω in C^r for any r >0.

It was well-known that the balanced embeddings ofP^N are always isomet- ric. There is no need to prove the convergence (1). The Abelian varieties are the first example that we do need convergence. Convergence is proven by an elementary method without using any asymptotic analysis of the Bergman kernel as in [T], [R], [Z]. By the Abel–Jacobi theorem, every Riemann sur- face can be embedded into its Jacobian which is principally polarized. This work shall be regarded as the first step toward the proof of the a priori C^∞-convergence of the balanced metrics for high genus Riemann surfaces.

Acknowledgement. The first author would like to thank professor Conan Leung for his interest.

2. Abelian varieties and Theta functions

In this section, we collect some basic facts about principally polarized Abelian varieties and properties of Theta functions, which will be used in the later sections. Good references for the materials presented here are [GH]

and [Mum1].

2.1. O_AΩ(1) and Theta function ϑ. Let Λ =Span_Z{λ₁,· · · , λ₂g} be a lattice in C^g with its period matrix given by

Ω := [λ₁,· · · , λ_2g] = [I, Z]

whereZ satisfiesZ^t=Z and ImZ >0. ThenA^g_Ω =C^g/Λ is the principally polarized Abelian variety defined by Ω and the canonical factors{eα, eg+α} for the principal polarization OA_Ω(1)→A^g_Ω are given by

e_a(z)≡1 ande_g+α(z) = expπi(−Z_αα−2z_α). Notice that the canonical factors {e_α, e_g+α} satisfy

e_g+β(z+λ_g+α)e_g+α(z)

= expπi(−Zββ−2zβ−2Zβα−Zαα−2zα)

=eg+α(z+λg+β)eg+β(z).

Let{x₁,· · · , xg, y₁,· · · , yg} be the coordinates of the basis dual to {λ₁,· · ·, λ₂g} ⊂C^g.

Then the first Chern class of OA_Ω(1) is given by ω:=

g α=1

dxα∧dyα

=

α,β

(ImZ)^αβdz_α∧dz_β ∈ ∧^1,1(A_Ω) where

zα=xα+Z_αβy_β z_α=x_α+Z_αβy_β

and (ImZ)^αβ is the inverse of ImZ. The unique global holomorphic section (up to scalar multiplication) of OA_Ω(1) is then given by

ϑ(z,Ω) :=

m∈Z^g

expπi

m^tZm+ 2m^tz .

In particular, ϑsatisfies

ϑ(z+m,Ω) =ϑ(z,Ω) ϑ(z+Zm,Ω) = exp−πi

m^tZm+ 2m^tz

ϑ(z,Ω) for any m∈Z^g.

2.2. Metric on O_AΩ(1). Lethbe any Hermitian metric onOA_Ω(1), then it must satisfy

h(z)|ϑ(z)|² =h(z+Zm)|ϑ(z+Zm)|² and h(z+m) =h(z) for all m ∈ Z^g. Since ϑ(z+Zm,Ω) = exp−πi

m^tZm+ 2m^tz

ϑ(z,Ω), this implies that

h(z) =h(z+Zm)exp−πi

m^tZm+ 2m^tz²

=h(z+Zm) exp−πi

m^tZm−m^tZm+ 2m^t(z−z)

=h(z+Zm) exp 2π

m^tImZm+ 2m^ty . Now if in addition we require the curvature form of h to be

ω=dz^t(ImZ)⁻¹dz,

that is,∂∂logh=ω, then this will force h to take the following form:

h(z) = exp−2π

y^t(ImZ)⁻¹y+c^ty 2

= expπ 2

(z−z)^t(ImZ)⁻¹(z−z) +c^ti(z−z)

for somec∈R^g. On the other hand, the identity h(z+Zm) = expπ

2

(z−z+ 2iImZm)^t(ImZ)⁻¹(z−z+ 2iImZm) +c^ti(z−z+ 2iImZm)

=h(z) exp

−2πm^tImZm+ 2πm^ti(z−z)−πc^tImZm

=h(z) exp−2π

m^tImZm+ 2m^ty implies

c^tImZm= 0 for all m∈Z^g, i.e., c= 0. Hence we obtain:

Proposition 2 (cf. [GH]). The only metric on OA_Ω(1) with curvature ω, up to a scalar multiple, is given by

h(z) := exp−2πy^t(ImZ)⁻¹y.

Moreover, we have

h(z+Za) =h(z) exp 2π

−a^tImZa−2a^ty for all a∈R^g.

3. Finite Heisenberg groups

In this section, we recall the construction of irreducible representations of finite Heisenberg groups presented in [Mum]. They will supply the key ingredient of finding a balanced embedding A_Ω⊂P^l2g−1.

LetVbe the complex vector space of entire functions onC^g. Fora, b∈R^g and f ∈V, we introduce two operators onV:

Sbf(z) :=f(z+b),

Tαf(z) := expπi(a^tZa+ 2a^tz)f(z+Za).

It is clear that they obey the following rules Sb₁ ◦Sb₂ =Sb₁+b₂

Ta₁◦Ta₂ =Ta₁+a₂. forai, bi ∈R^g. On the other hand,

Sb(Taf) (z) = (Taf) (z+b)

= expπi

a^tZa+ 2a^t(z+b)

f(z+b+Za) and

Ta(Sbf) (z) = expπi

a^tZa+ 2a^tz

(Sbf) (z+Za)

= expπi

a^tZa+ 2a^tz

f(z+b+Za), implies the following commutative relation

Sb◦Ta= exp 2πia^tbTa◦Sb.

Recall that the 2g+ 1 dimensional Heisenberg group G is U(1)×R^g ×R^g with the multiplication defined by

(λ, a, b)

λ, a, b

=

λλexp 2πib^ta, a+a, b+b . It has a natural action on Vvia

((λ, a, b)·f) (z)

=λ(Ta◦Sbf) (z)

=λexpπi

a^tZa+ 2a^tz

f(z+Za+b) since

(λ, a, b)

λ, a, b

·f (z)

=λλ(Ta◦Sb◦Ta◦Sbf) (z)

=λλexp 2πib^ta(Ta+a◦Sb+bf) (z).

To make this representation unitary, we introduce a norm on V f²:=

C^g|f|²h(z)ω^g

=

C^g|f|²exp

−2πy^t(ImZ)⁻¹y

ω^g. Proposition 3. For∀a, b∈R^g, we have

h(z)|Sbf|² =

h|f|²

(z+b),

h(z)|Taf|² =h(z+Za)|f(z+Za)|². Proof. For a, b∈R^g, we have

h(z)|Sbf|²=h(z)|f(z+b)|² =h(z+b)|f(z+b)|² and

h(z)|Taf(z)|²=h(z)expπi

a^tZa+ 2a^tz

f(z+Za)²

=h(z) exp 2π

−a^tImZa−2a^ty

|f(z+Za)|²

=h(z+Za)|f(z+Za)|².

Corollary 4. For anya, b∈R^g,SbandTaare unitary operators on(V,·).

In particular, the action of G on (V,·) is unitary.

Proof. The only thing we need to check is that Taf²=

C^g|Taf(z)|²h(z)ω^g

=

C^g|f(z+Za)|²h(z+Za)ω^g

=

C^g|f(z)|²h(z)ω^g

=f².

Remark 5. TheG action onV is the classical Stone–Von Neumann repre- sentation.

Let us introduce a discrete subgroup ofG

Γ :={(1, a, b) ∈ G|a, b∈Z^g}.

Now ϑ(z,Ω), up to scalars, can be characterized as the unique Γ-invariant entire function on C^g. For a fixedl∈N, let

lΓ :={(1, a, b)∈ G |a, b∈(lZ)^g} ⊂Γ

and let Vl be the set of entire functions f(z) on C^g, invariant under the action of lΓ. That is,f ∈Vl if and only if for anya, b∈(lZ)^g, we have

f(z) =Sbf(z) :=f(z+b), f(z) =Tαf(z) := expπi

a^tZa+ 2a^tz

f(z+Za). To get a good basis of Vl, we need the following:

Proposition 6. f ∈Vl if and only if

f(z) =

n∈(Z[1/l])^g

c_nexpπi

n^tZn+ 2n^tz such that cn=cm if n−m∈(lZ)^g. In particular, dimVl=l^2g.

Proof. For anyf ∈Vl, by the invariance off under the action ofSl, it has expansion

f(z) =

n∈(Z[1/l])^g

cnexp 2πin^tz.

On the other hand, for m∈(lZ)^g,Tmf =f implies that Tmf(z) = expπi

m^tZm+ 2m^tz

f(z+Zm)

= expπi

m^tZm+ 2m^tz

n∈(Z[1/l])^g

c_nexp 2πin^t(z+Zm)

=

n∈(Z[1/l])^g

cnexpπi(m+ 2n)^tZmexp 2πi(n+m)^tz

=f(z)

=

n∈(Z[1/l])^g

cn+mexp 2πi(n+m)^tz hence we have

c_n+m=c_nexpπi(m+ 2n)^tZm

=c_nexp

πim^tZm+ 2n^tZm

=c_nexp

πi(n+m)^tZ(n+m)−n^tZn which means

cn+mexp

−πi(n+m)^tZ(n+m)

=cnexp

−πin^tZn . So if we definec_n:=cnexp

−πin^tZn then

f(z) =

n∈(Z[1/l])^g

c_nexp 2πin^tz

=

n∈(Z[1/l])^g

c_nexpπi

n^tZn+ 2n^tz

and c_n=c_m forn−m∈(lZ)^g.

Letμm ⊂U(1) denote the subgroup of m^th roots of 1. For l∈N, let Gl:={(λ, a, b)∈ G |λ∈μ_l2;a, b∈(Z[1/l])^g}/lΓ

=μ_l2×

Z[1/l]

lZ g

×

Z[1/l]

lZ g

with the multiplication induced from G. Now for a, b ∈ (Z[1/l])^g, the ele- ments Sb,Ta∈ G commute with lΓ, (since for a ∈(Z[1/l])^g and b∈lΓ we have exp 2πib^ta = 1) hence they act on Vl. This descends to an action of Gl on Vl, and the generatorsSb,Ta,∀a, b∈(Z[1/l])^g act onVl as follows:

Sb

⎛

⎝

n∈(Z[1/l])^g

c_nexpπi

n^tZn+ 2n^tz⎞

⎠

=

n∈(Z[1/l])^g

c_nexpπi

n^tZn+ 2n^t(z+b)

=

n∈(Z[1/l])^g

c_nexp 2πin^tbexpπi

n^tZn+ 2n^tz

and Ta

⎛

⎝

n∈(Z[1/l])^g

c_nexpπi

n^tZn+ 2n^tz⎞

⎠

= expπi

a^tZa+ 2a^tz

n∈(Z[1/l])^g

c_nexpπi

n^tZn+ 2n^t(z+Za)

=

n∈(Z[1/l])^g

c_nexpπi

(n+a)^tZ(n+a) + 2 (n+a)^tz

=

n∈(Z[1/l])^g

c_n−aexpπi

n^tZn+ 2n^tz .

This motivates us to introduce a basis forVl: s_c(z,Ω) :=

n∈c+(lZ)^g

expπi

n^tZn+ 2n^tz

forc∈

Z[1/l]

lZ g

.

A direct calculation implies the following lemma from which we obtain the irreducibility of theGl action on Vl.

Lemma 7. For any a, b∈Z[1/l], we have Sbsc = exp

2b^tcπi sc, Tasc =sc+a.

4. Main theorem

Now we are ready to introduce the basis

ϑ a b

ofH⁰

A_Ω,OA_Ω l² whose induced projective embedding is balanced.

In [Mum], Mumford introduces the following:

Definition 8. Fora, b∈Q^g, ϑ a

b

(z,Ω) := (Sb◦Ta)ϑ(z,Ω)

= expπi

a^tZa+ 2a^t(z+b)

ϑ(z+Za+b,Ω)

=

m∈Z^g

expπi

(m+a)^tZ(m+a) + 2 (m+a)^t(z+b) .

They satisfy the following properties:

Proposition 9 ([Mum]). Fora, a, b, b ∈(Z[1/l])^g and p, q∈Z^g, we have:

(1) ϑ 0 0

(z,Ω) =ϑ(z,Ω).

(2) Sbϑ a b

=ϑ a

b+b

. (3) Taϑ a

b

= exp

−2πib^ta

ϑ a+a b

. (4) ϑ a+p

b+q

= exp

2πia^tq ϑ a

b

. (5)

h(z) ϑ a

b

(z,Ω)

² =h|ϑ|²(z+Za+b).

Notice that ϑ 0

0

=ϑ(z)

=

m∈Z^g

expπi

m^tZm+ 2m^tz

=

p∈(Z/lZ)^g

n∈p+(lZ)^g

expπi

n^tZn+ 2n^tz

=

p∈(Z/lZ)^g

s_p

implies that

ϑ a b

=SbTaϑ

=

p∈(Z/lZ)^g

(Sb◦Tasp) (z)

=

p∈(Z/lZ)^g

exp

2πib^t(p+a) sp+a.

For a fixed k, denoteε_k:= (0,· · ·,^k

th

1,· · ·0)∈Z^g, let ϑa:=

ϑ a

q/l

q∈(Z/lZ)^g

and sa:={sp+a}_p∈(Z/lZ)^g bel^g-dimensional vectors, then we have

ϑ_a=Uas_a with

Ua=

exp 2πiq^t(p+a)/l

q,p∈(Z/lZ)^g. Lemma 10. For any r ∈(Z/lZ)^g

p∈(Z/lZ)^g

exp 2πir^tp/l= 0.

Proof. For simplicity let us first assume that r is primitive, that isr is not a multiple of some element in (Z/lZ)^g, then we have the following exact sequence of Abelian groups

0 −→ kerr −→ (Z/lZ)^g −→^r· Z/lZ −→ 0

and kerr ∼= (Z/lZ)^g−1. For any p ∈ (Z/lZ)^g, we have a decomposition of p=p+p^⊥ with p∈kerr and p^⊥ ∈kerr^⊥. Thus we obtain

p∈(Z/lZ)^g

exp 2πir^tp/l=

q∈kerr

p∈kerr^⊥

exp 2πir^t(q+p)/l

=

q∈kerr

p∈kerr^⊥

exp 2πir^tp/l

=l^g−1

p∈kerr^⊥

exp 2πir^tp/l

= 0.

where we have used the fact that

l−1

k=0

exp 2πiak/l= 0 for any 0≤a≤l−1.

for the last identity.

Ifris not primitive, then it is a multiple of a primitive vectorr₀ ∈(Z/lZ)^g. We may replace r by r₀ and use the same argument. The details are left to

the readers.

Corollary 11. Ua/l^g/2 is a l^g×l^g-unitary matrix, i.e., Ua/l^g/2 ∈U(l^g).

Proof. Since Ua= diag

exp 2πiq^ta/l

U_awithU_a:=

exp 2πiq^tp/l

, all we need to show is thatU_ais unitary, which follows from the above lemma.

In conclusion, we have:

Proposition 12.

⎡

⎢⎢

⎢⎢

⎣

ϑ₀ ϑε₁

... ϑ_(1−1/l)^Pg

i=1εi

⎤

⎥⎥

⎥⎥

⎦=

⎡

⎢⎢

⎢⎣

U₀ 0 0

0 Uε₁

. ..

0 U_(1−1/l)^Pg_i=1εg

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

s₀ sε₁

...

s_(1−1/l)^Pg_i

=1εg

⎤

⎥⎥

⎥⎦

where Ua∈U(l^g) for each a∈

Z[1/l] Z

g

. In particular, we have

a,b∈“_Z[1/l]

Z

”g

ϑ a b

(z,Ω)

²=l^g

p∈(Z/lZ)^g

a∈“_Z[1/l]

Z

”|sp+a(z)|². The next lemma gives a geometric interpretation of the spaceVl. Lemma 13. Let

l^∗ :A_Ω −→A_Ω z−→lz be the rescaling map. Then degl^∗OA_Ω(1) =l² and

Vl=H⁰(l^∗OA_Ω(1)) =H⁰ OA_Ω

l² . Proof. Let

l^∗ :C/Λ−→C/Λ z−→lz be the rescaling map and let

e_α, e_g+α

be the canonical factor associated withl^∗OA_Ω(1). Then we have

e_α(z) = 1 =e^l_α²(z) ande_g+α(z) = expπi

−Zαal²−2zαl²

=e^l_g²_+α(z). In fact,OA_Ω(1) is a symmetric line bundle, which means

l^∗OA_Ω(1) =OA_Ω l²

. On the other hand, for f ∈Vl we have

f(l(z+m)) =f(lz+lm) =f(lz) for m∈Z^g

and

f(l(z+λg+α)) = expπi

−Zααl²−2zαl² f(lz)

=e_g+α(z)f(lz)

=e^l_g²_+α(z)f(lz). Thusf ∈H⁰(l^∗OA_Ω(1)) =H⁰

OA_Ω l²

. A dimension count then implies ϕ:Vl−→H⁰

A_Ω,O l² f(z)−→f(lz)

is an isomorphism.

Now we introduce an inner product on Vl. Definition 14. For f(z)∈Vl we define

|f(z)|²_h:=

DlΛ

|f(z)|²h(z)ωⁿ

whereD_Λ is the fundamental domain associated to Λ. In particular,

|ϑ_0,0(z)|²_h :=

DlΛ

|ϑ(z)|²h(z)ω^g=l^2g

DΛ

|ϑ(z)|²h(z)ω^g. It possesses the following properties.

Lemma 15. For a, b∈(Z[1/l])^g, we have:

(1)

h(z)|(Sbf) (z)|² =

h|f|²

(z+b), h(z)|(Taf) (z)|² =

h|f|²

(z+Za).

(2) The actions{Sb,Ta}are unitary with respect to theL²-metric induced from h.

(3) sc

l^g/2|ϑ|h

c∈“_Z[1/l]

lZ

”g forms an orthonormal basis of (Vl,|·|_h).

Proof. Part (1) is a special case of Proposition 3.

For part (2), we have Sbf(z)²_h =

DlΛ

|f(z+b)|²h(z+b)ω^g =

DlΛ

|f(z)|²h(z)ω^g Taf(z)²_h =

DlΛ

|f(z+Za)|²h(z+Za)ω^g =

DlΛ

|f(z)|²h(z)ω^g. For part (3), since {Sc}_c∈^“Z[1/l]

lZ

”g forms a family of commuting operators, the eigenvectors {sc}_c∈^“Z[1/l]

lZ

”g are mutually orthogonal. On the other hand Lemma7 implies that for all a, c∈(Z[1/l])^g

|sc+a|²_h =|Tasc|²_h =|sc|²_h.

If we set c=−athen we have

|s_c|²_h =|s₀|²_h. This implies

|ϑ|²_h =

p∈(Z/lZ)^g

|sp|²_h =l^g|s₀|²_h.

Now let

Φ :A_Ω−→P^l2g−1 z−→[s_c(lz)]

c∈“_Z[1/l]

lZ

”g

be the projective embedding induced by {s_c}_c∈^“Z[1/l] lZ

”g. Then Φ^∗ω_FS is in- variant under the action ofSb andTa fora, b∈

Z[1/l] lZ

g

since bothSb and Ta are unitary. The Fubini–Study metric is given by

f(lz)²_FS:=

A_Ω

|f(lz)|²

"

c|sc(lz)|²Φ^∗ω_FS. This together with the identities

h(z)|(Sbf) (z)|² =

h|f|² (z+b) h(z)|(Taf) (z)|² =

h|f|²

(z+Za) and

|Sbs_c|=|s_c|

c

|Tas_c|² =

c

|s_c|²

would imply

(Sbf) (lz)²_FS =

A_Ω

|(Sbf) (lz)|²

"

c|s_c(lz)|²Φ^∗ω_FS

=

A_Ω

|(Slbf) (lz)|²h(lz)

"

c|(Sbs_c) (lz)|²h(lz)Φ^∗ω_FS

=

A_Ω

h|f|²

(lz+b)

"

ch|s_c|²

(lz+b)

(Sb◦Φ)^∗ω_FS

=

A_Ω

# |f|²

"

c|s_c|²Φ^∗ω_FS

$

(lz+b)

=f(lz)²_FS,

and

(Taf) (lz)²_FS=

A_Ω

|(Taf) (lz)|²

"

c|s_c(lz)|²Φ^∗ω_FS

=

A_Ω

|(Taf) (lz)|²h(lz)

"

c|(Tas_c) (lz)|²h(lz)(Ta◦Φ)^∗ω_FS

=

A_Ω

# |f|²

"

c|s_c|²Φ^∗ω_FS

$

(lz+Za)

=f(lz)²_FS. Hence we obtain the following:

Proposition 16. For a, b∈(Z[1/l])^g:

(1) Sb,Ta are unitary operators acting on(Vl,·_FS).

(2) The action {sc}_c∈^“Z[1/l] lZ

”g is an orthonormal basis for (Vl,·_FS).

Recall from [D] that a projective embedding of ϕ:A_Ω→P^l2g−1 is called balanced if we have

A_Ω

z_iz_j

|z|² −δ_ij l^2g

ϕ^∗ω_FS^g = 0.

Corollary 17. The embedding defined by Θ :A_Ω −→P^l2g−1

z−→ ϑ a b

(z,Ω)

a,b∈“_Z[1/l]

Z

”g

is balanced.

Proof. It follows from the fact that

# l^g/2s_c ϑ_FS

$

c∈“_Z[1/l]

lZ

”g

forms an orthonormal basis for (Vl,·_FS), and the map Φ and Θ differ by

an unitary transformation.

Remark 18. Geometrically, the finite Heisenberg group Gl acts on the em- bedding leaving Im Φ invariant. And {Sb,Ta} acts on Φ (A_Ω) via transla- tions under the group law. Although we can not get a homomorphism from A_Ω to U(l²), we do have Gl⊂U(l²).

Finally, we are ready to state our main result.

Theorem 19.

a,b∈“_Z[1/l]

Z

”g

ϑ a b

(z,Ω)

²=l^g

p∈(Z/lZ)^g

a∈“_Z[1/l]

Z

”|sp+a(z)|².

llim→∞l^−g

p∈(Z/lZ)^g

a∈“_Z[1/l]

Z

”g

|s_p+a(z)|²h(z) =

A_Ω|ϑ(z)|²h(z)dx∧dy.

To prove this we need the following elementary lemma.

Lemma 20. Let f be a smooth real-valued doubly periodic function on C^g with period matrix Ω = (I, Z)∈M_g×2g(C). Let D_Ω⊂C^g be a fundamental domain. Then

det ImZ l^2g

a,b∈“_Z[1/l]

Z

”g

f(lz+Za+b)

=

DΩ

f(z) ω^g_Cg

det ImZ +O l⁻³

for l→ ∞ where ω_C^g :="

idzi∧dzi is the standard K¨ahler form on C^g.

Proof. By performing affine transformations, we may assume that Z =iI.

First, we notice thatf being smooth implies that

llim→∞

1 l^2g

a,b∈“_Z[1/l]

Z

”g

f(z+ai+b) =

DΩ

f ω^g_Cg

for any fixedz∈DΩ. Moreover we have that

1 l^2g

a,b∈“_Z[1/l]

Z

”g

f(z+ai+b)−

DΩ

f ω_C^gg

≤Cl⁻³

with C being a constant depending only on the sup-norm of the second derivative of f but not on z. In particular, this implies

1 l^2g

a,b∈“_Z[1/l]

Z

”g

f(lz+ai+b)−

DΩ

f ω^g_Cg

≤Cl⁻³. Proof of Theorem 19. By Proposition 3, we have

a,b

ϑ a b

(lz)

²h(lz)

=

a,b

|ϑ(lz+aτ +b)|²h(lz+aτ+b)

hence

l^g

p∈(Z/lZ)^g

a∈“_Z[1/l]

Z

”g

|sp+a(lz)|²h(lz)

=

a,b∈“_Z[1/l]

Z

”g

ϑ a b

(lz,Ω) ²h(lz)

=

a,b

|ϑ(lz+Za+b)|²h(lz+Za+b).

By Lemma20, we have

llim→∞

Vol (A_Ω) l^2g

a,b

|ϑ(lz+aτ +b)|²h(lz+aτ+b)

=

A_Ω|ϑ(z)|²h(z)ω^g. Hence

llim→∞l^−g

p∈(Z/lZ)^g

a∈“_Z[1/l]

Z

”|s_p+a(lz)|²h(lz)

= 1

Vol (A_Ω)

A_Ω|ϑ(z)|²h(z)dx∧dy.

Corollary 21. The balanced metricΘ^∗_lω_FS/l² converges toω₀, the flat met- ric on A_Ω, as a C^∞ function as l→ ∞.

Proof. Let h_FS denote the Fubini–Study metric on OA_Ω l²

induced via the embedding Θ. Then

h_FS= h^l2

"

p∈(Z/lZ)^g

"

a∈“_Z[1/l]

Z

”|sp+a(lz)|²h(lz). On the other hand

Θ^∗_lω_FS

l² = ∂∂logh_FS l²

=∂∂logh−

∂∂log "

p∈(Z/lZ)^g"

a∈“_Z[1/l]

Z

”|s_p+a(lz)|²h(lz)

l²

=ω₀− ∂∂log

l^gϑ_h+O l^g−3

l² →ω₀.

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The Chinese University of Hong Kong, Sha Tin, N.T., Hong Kong [email protected]

[email protected]

This paper is available via http://nyjm.albany.edu/j/2009/15-2.html.