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. O_{A_{Ω}}(1) and Theta function ϑ 21 2.2. Metric on O_{A_{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, O_{X}(1)) be a n-dimensional projective manifold, polarized by an ample line bundle O_{X}(1). Suppose that (X, O_{X}(1)) has no continuous automorphisms and there exists a constant scalar curvature Kähler (cscK) metric in the Kähler class c_{1} (O_{X}(1)). Then the projective embedding X ↪ ℙH^{0} (X, O_{X}(k))

19^{ISSN 1076-9803/09}

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.

induced by O_{X}(k) can always be balanced for k sufficiently large, that is, for each large k there exists a basis {z_{i}^{(k)}} of H^{0} (X, O_{X}(k)) such that

∫_{X} (\frac{z_{i}^{(k)}z_{j}^{(k)}}{|z^{(k)}|^{2}} − \frac{δ_{ij}}{N_{k} + 1}) \frac{ω_{FS}^{n}}{n!} = 0

where N_{k} + 1 = dim H^{0}(X, O_{X}(1)) and ω_{FS} is the Fubini–Study metric on ℙ^{N_{k}}. Moreover, if we let ω_{k} denote ω_{FS}/k induced from the balanced embedding, then

(1)  |ω_{k} − ω_{∞}|_{C^{r}(ω_{∞})} = O(1/k),

where ω_{∞} is the cscK metric in the class c_{1}(O_{X}(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 direction 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, O_{X}(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 polarization [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 Riemann 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}, O_{A_{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 [ \begin{matrix}a \\ b \end{matrix} ]\}_{a,b∈(ℤ[1/l]/ℤ)^{g}}

for H^{0}(A_{Ω}, O_{A_{Ω}}(l^{2})) (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 of ℙ^{N} are always isometric. 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 surface 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_{A_{Ω}}(1) and Theta function ϑ. Let Λ =Span_{ℤ} {λ_{1}, ⋯, λ_{2g}} be a lattice in ℂ^{g} with its period matrix given by

Ω := [λ_{1}, ⋯, λ_{2g}] = [I, Z]

where Z satisfies Z^{t} = Z and Im Z > 0. Then A_{Ω}^{g} = ℂ^{g}/Λ is the principally polarized Abelian variety defined by Ω and the canonical factors {e_{α}, e_{g+α}} for the principal polarization O_{A_{Ω}}(1) → A_{Ω}^{g} are given by

e_{a} (z) ≡ 1 and e_{g+α} (z) = exp πi (−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)
= exp πi (−Z_{ββ} − 2z_{β} − 2Z_{βα} − Z_{αα} − 2z_{α})
= e_{g+α} (z + λ_{g+β}) e_{g+β} (z) .

Let {x_{1}, ⋯, x_{g}, y_{1}, ⋯, y_{g}} be the coordinates of the basis dual to

{λ_{1}, ⋯, λ_{2g}} ⊂ ℂ^{g}.

Then the first Chern class of O_{A_{Ω}}(1) is given by

ω := ∑_{α=1}^{g} dx_{α} ∧ dy_{α}

= ∑_{α,β} (Im Z)^{αβ} dz_{α} ∧ dz_{β} ∈ ∧^{1,1} (A_{Ω})

where

z_{α} = x_{α} + Z_{αβ}y_{β}
z_{α} = x_{α} + Z_{αβ}y_{β}

and (Im Z)^{αβ} is the inverse of Im Z. The unique global holomorphic section (up to scalar multiplication) of O_{A_{Ω}}(1) is then given by

ϑ (z, Ω) := ∑_{m∈ℤ^{g}} exp πi (m^{t}Zm + 2m^{t}z) .

In particular, ϑ satisfies

ϑ (z + m, Ω) = ϑ (z, Ω)
ϑ (z + Zm, Ω) = exp −πi (m^{t}Zm + 2m^{t}z) ϑ (z, Ω)

for any m ∈ ℤ^{g}.

2.2. Metric on O_{A_{Ω}}(1). Let h be any Hermitian metric on O_{A_{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^{t}Zm + 2m^{t}z) ϑ (z, Ω), this implies that

h (z) = h (z + Zm) |exp −πi (m^{t}Zm + 2m^{t}z)|^{2}
= h (z + Zm) exp −πi (m^{t}Zm − m^{t}Zm + 2m^{t} (z − z))
= h (z + Zm) exp 2π (m^{t}Im Zm + 2m^{t}y) .

Now if in addition we require the curvature form of h to be

ω = dz^{t} (Im Z)^{−1} dz,

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

h (z) = exp −2π (y^{t} (Im Z)^{−1} y + \frac{c^{t}y}{2} )

= exp \frac{π}{2} ((z − z)^{t} (Im Z)^{−1} (z − z) + c^{t}i (z − z))

for some c ∈ ℝ^{g}. On the other hand, the identity

h (z + Zm) = exp \frac{π}{2} ((z − z + 2iIm Zm)^{t} (Im Z)^{−1} (z − z + 2iIm Zm)
+ c^{t}i (z − z + 2iIm Zm))
= h (z) exp (−2πm^{t}Im Zm + 2πm^{t}i (z − z) − πc^{t}Im Zm)
= h (z) exp −2π (m^{t}Im Zm + 2m^{t}y)

implies

c^{t}Im Zm = 0 for all m ∈ ℤ^{g},

i.e., c = 0. Hence we obtain:

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

h (z) := exp −2πy^{t} (Im Z)^{−1} y.

Moreover, we have

h (z + Za) = h (z) exp 2π (−a^{t}Im Za − 2a^{t}y)

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^{l^{2g}−1}.

Let 𝕍 be the complex vector space of entire functions on ℂ^{g}. For a, b ∈ ℝ^{g} and f ∈ 𝕍, we introduce two operators on 𝕍:

𝔖_{b}f(z) := f(z + b),
𝔗_{α}f(z) := exp πi(a^{t}Za + 2a^{t}z)f(z + Za).

It is clear that they obey the following rules

𝔖_{b_{1}} ∘ 𝔖_{b_{2}} = 𝔖_{b_{1}+b_{2}}
𝔗_{a_{1}} ∘ 𝔗_{a_{2}} = 𝔗_{a_{1}+a_{2}}.

for a_{i}, b_{i} ∈ ℝ^{g}. On the other hand,

𝔖_{b} (𝔗_{a}f) (z) = (𝔗_{a}f) (z + b)
= exp πi (a^{t}Za + 2a^{t} (z + b)) f (z + b + Za)

and

𝔗_{a} (𝔖_{b}f) (z) = exp πi (a^{t}Za + 2a^{t}z) (𝔖_{b}f) (z + Za)
= exp πi (a^{t}Za + 2a^{t}z) f (z + b + Za) ,

implies the following commutative relation

𝔖_{b} ∘ 𝔗_{a} = exp 2πia^{t}b𝔗_{a} ∘ 𝔖_{b}.

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′) = (λλ′ exp 2πib^{t}a′, a + a′, b + b′) .

It has a natural action on 𝕍 via

((λ, a, b) · f) (z)
= λ (𝔗_{a} ∘ 𝔖_{b}f) (z)
= λ exp πi (a^{t}Za + 2a^{t}z) f (z + Za + b)

since

(λ, a, b) ((λ′, a′, b′) · f) (z)
= λλ′ (𝔗_{a} ∘ 𝔖_{b} ∘ 𝔗_{a′} ∘ 𝔖_{b′}f) (z)
= λλ′ exp 2πib^{t}a′ (𝔗_{a+a′} ∘ 𝔖_{b+b′}f) (z) .

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

∥f∥^{2} := ∫_{ℂ^{g}} |f|^{2} h (z) ω^{g}
= ∫_{ℂ^{g}} |f|^{2} exp (−2πy^{t} (Im Z)^{−1} y) ω^{g}.

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

h (z) |𝔖_{b}f|^{2} = (h |f|^{2}) (z + b) ,
h (z) |𝔗_{a}f|^{2} = h (z + Za) |f (z + Za)|^{2} .

Proof. For a, b ∈ ℝ^{g}, we have

h (z) |𝔖_{b}f|^{2} = h (z) |f (z + b)|^{2} = h (z + b) |f (z + b)|^{2}

and

h (z) |𝔗_{a}f (z)|^{2} = h (z) |exp πi (a^{t}Za + 2a^{t}z) f (z + Za)|^{2}

= h (z) exp 2π (−a^{t}Im Za − 2a^{t}y) |f (z + Za)|^{2}

= h (z + Za) |f (z + Za)|^{2} . □

Corollary 4. For any a, b ∈ ℝ^{g}, 𝔖_{b} and 𝔗_{a} are unitary operators on (𝕍, ∥·∥). In particular, the action of G on (𝕍, ∥·∥) is unitary.

Proof. The only thing we need to check is that

\begin{aligned}∥𝔗_{a}f∥^{2} & = ∫_{ℂ^{g}} |𝔗_{a}f (z)|^{2} h (z) ω^{g} \\ & = ∫_{ℂ^{g}} |f (z + Za)|^{2} h (z + Za) ω^{g} \\ & = ∫_{ℂ^{g}} |f (z)|^{2} h (z) ω^{g} \\ & = ∥f∥^{2} . \end{aligned}  □

Remark 5. The G action on 𝕍 is the classical Stone–Von Neumann representation.

Let us introduce a discrete subgroup of G

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 fixed l ∈ ℕ, let

lΓ := {(1, a, b) ∈ G | a, b ∈ (lℤ)^{g}} ⊂ Γ

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

f (z) = 𝔖_{b}f (z) := f (z + b) ,
f (z) = 𝔗_{α}f (z) := exp πi (a^{t}Za + 2a^{t}z) f (z + Za) .

To get a good basis of 𝕍_{l}, we need the following: Proposition 6. f ∈ 𝕍_{l} if and only if

f (z) = ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi (n^{t}Zn + 2n^{t}z)

such that c_{n} = c_{m} if n − m ∈ (lℤ)^{g}. In particular, dim 𝕍_{l} = l^{2g}. Proof. For any f ∈ 𝕍_{l}, by the invariance of f under the action of 𝔖_{l}, it has expansion

f (z) = ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp 2πin^{t}z.

On the other hand, for m ∈ (lℤ)^{g}, 𝔗_{m}f = f implies that

𝔗_{m}f (z) = exp πi (m^{t}Zm + 2m^{t}z) f (z + Zm)

= exp πi (m^{t}Zm + 2m^{t}z) ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp 2πin^{t} (z + Zm)

= ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi (m + 2n)^{t} Zm exp 2πi (n + m)^{t} z

= f (z)

= ∑_{n∈(ℤ[1/l])^{g}} c_{n+m} exp 2πi (n + m)^{t} z

hence we have

c_{n+m} = c_{n} exp πi (m + 2n)^{t} Zm
= c_{n} exp (πim^{t}Zm + 2n^{t}Zm)
= c_{n} exp (πi (n + m)^{t} Z (n + m) − n^{t}Zn)

which means

c_{n+m} exp (−πi (n + m)^{t} Z (n + m)) = c_{n} exp (−πin^{t}Zn) .

So if we define c′_{n} := c_{n} exp (−πin^{t}Zn) then

f (z) = ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp 2πin^{t}z

= ∑_{n∈(ℤ[1/l])^{g}} c′_{n} exp πi (n^{t}Zn + 2n^{t}z)

and c′_{n} = c′_{m} for n − m ∈ (lℤ)^{g}. □

Let μ_{m} ⊂ U(1) denote the subgroup of m^{th} roots of 1. For l ∈ ℕ, let

G_{l} := {(λ, a, b) ∈ G | λ ∈ μ_{l^{2}}; a, b ∈ (ℤ [1/l])^{g}} /lΓ

= μ_{l^{2}} × (\frac{ℤ [1/l]}{lℤ})^{g} × (\frac{ℤ [1/l]}{lℤ})^{g}

with the multiplication induced from G. Now for a, b ∈ (ℤ [1/l])^{g}, the elements 𝔖_{b}, 𝔗_{a} ∈ G commute with lΓ, (since for a′ ∈ (ℤ [1/l])^{g} and b ∈ lΓ we have exp 2πib^{t}a′ = 1) hence they act on 𝕍_{l}. This descends to an action of G_{l} on 𝕍_{l}, and the generators 𝔖_{b}, 𝔗_{a}, ∀a, b ∈ (ℤ [1/l])^{g} act on 𝕍_{l} as follows:

𝔖_{b} ( ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi (n^{t}Zn + 2n^{t}z))

= ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi (n^{t}Zn + 2n^{t} (z + b))

= ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp 2πin^{t}b exp πi (n^{t}Zn + 2n^{t}z)

and

𝔗_{a} ( ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi (n^{t}Zn + 2n^{t}z))

= exp πi (a^{t}Za + 2a^{t}z) ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi (n^{t}Zn + 2n^{t} (z + Za))

= ∑_{n∈(ℤ[1/l])^{g}} c_{n} exp πi ((n + a)^{t} Z (n + a) + 2 (n + a)^{t} z)

= ∑_{n∈(ℤ[1/l])^{g}} c_{n−a} exp πi (n^{t}Zn + 2n^{t}z) .

This motivates us to introduce a basis for 𝕍_{l}:

s_{c} (z, Ω) := ∑_{n∈c+(lℤ)^{g}} exp πi (n^{t}Zn + 2n^{t}z) for c ∈ (\frac{ℤ [1/l]}{lℤ})^{g} .

A direct calculation implies the following lemma from which we obtain the irreducibility of the G_{l} action on 𝕍_{l}.

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

𝔖_{b}s_{c} = exp (2b^{t}cπi) s_{c},
𝔗_{a}s_{c} = s_{c+a}.

Theta function and Bergman metric 27

4. Main theorem

Now we are ready to introduce the basis \{ϑ [ \begin{matrix}a \\ b \end{matrix} ]\} of H^{0} (A_{Ω}, O_{A_{Ω}} (l^{2})) whose induced projective embedding is balanced.

In [Mum], Mumford introduces the following: Definition 8. For a, b ∈ ℚ^{g},

ϑ [ \begin{matrix}a \\ b \end{matrix} ] (z, Ω) := (𝔖_{b} ∘ 𝔗_{a}) ϑ (z, Ω)

= exp πi (a^{t}Za + 2a^{t} (z + b)) ϑ (z + Za + b, Ω)

= ∑_{m∈ℤ^{g}} exp πi ((m + a)^{t} Z (m + a) + 2 (m + a)^{t} (z + b)) .

They satisfy the following properties: Proposition 9 ([Mum]). For a, a′, b, b′ ∈ (ℤ [1/l])^{g} and p, q ∈ ℤ^{g}, we have:

(1) ϑ [ \begin{matrix}0 \\ 0 \end{matrix} ] (z, Ω) = ϑ (z, Ω).

(2) 𝔖_{b′}ϑ [ \begin{matrix}a \\ b \end{matrix} ] = ϑ [ \begin{matrix}a \\ b + b′ \end{matrix} ].

(3) 𝔗_{a′}ϑ [ \begin{matrix}a \\ b \end{matrix} ] = exp (−2πib^{t}a′) ϑ [ \begin{matrix}a′ + a \\ b \end{matrix} ].

(4) ϑ [ \begin{matrix}a + p \\ b + q \end{matrix} ] = exp (2πia^{t}q) ϑ [ \begin{matrix}a \\ b \end{matrix} ].

(5)

h (z) |ϑ [ \begin{matrix}a \\ b \end{matrix} ] (z, Ω)|^{2} = h |ϑ|^{2} (z + Za + b) .

Notice that

ϑ [ \begin{matrix}0 \\ 0 \end{matrix} ] = ϑ (z)

= ∑_{m∈ℤ^{g}} exp πi (m^{t}Zm + 2m^{t}z)

= ∑_{p∈(ℤ/lℤ)^{g}} ∑_{n∈p+(lℤ)^{g}} exp πi (n^{t}Zn + 2n^{t}z)

= ∑_{p∈(ℤ/lℤ)^{g}} s_{p}

implies that

ϑ [ \begin{matrix}a \\ b \end{matrix} ] = 𝔖_{b}𝔗_{a}ϑ

= ∑_{p∈(ℤ/lℤ)^{g}} (𝔖_{b} ∘ 𝔗_{a}s_{p}) (z)

= ∑_{p∈(ℤ/lℤ)^{g}} exp (2πib^{t} (p + a)) s_{p+a}.

For a fixed k, denote ε_{k} := (0, ⋯, 1^{k^{th}}, ⋯ 0) ∈ ℤ^{g}, let

ϑ̃_{a} := \{ϑ [ \begin{matrix}a \\ q/l \end{matrix} ]\}_{q∈(ℤ/lℤ)^{g}} and s̃_{a} := \{s_{p+a}\}_{p∈(ℤ/lℤ)^{g}}

be l^{g}-dimensional vectors, then we have

ϑ̃_{a} = 𝔘_{a}s̃_{a}

with

𝔘_{a} = {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^{t}p/l = 0.

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

0 ⟶ ker r ⟶ (ℤ/lℤ)^{g} ⟶^{r·} ℤ/lℤ ⟶ 0

and ker r ≅ (ℤ/lℤ)^{g−1}. For any p ∈ (ℤ/lℤ)^{g}, we have a decomposition of p = p^{⊤} + p^{⊥} with p^{⊤} ∈ ker r and p^{⊥} ∈ ker r^{⊥}. Thus we obtain

∑_{p∈(ℤ/lℤ)^{g}} exp 2πir^{t}p/l = ∑_{q∈ker r} ∑_{p∈ker r^{⊥}} exp 2πir^{t} (q + p) /l

= ∑_{q∈ker r} ∑_{p∈ker r^{⊥}} exp 2πir^{t}p/l

= l^{g−1} ∑_{p∈ker r^{⊥}} exp 2πir^{t}p/l

= 0.

where we have used the fact that

∑_{k=0}^{l−1} exp 2πiak/l = 0 for any 0 ≤ a ≤ l − 1.

for the last identity.

Theta function and Bergman metric 29

If r is not primitive, then it is a multiple of a primitive vector r_{0} ∈ (ℤ/lℤ)^{g}. We may replace r by r_{0} and use the same argument. The details are left to the readers. □ Corollary 11. 𝔘_{a}/l^{g/2} is a l^{g} × l^{g}-unitary matrix, i.e., 𝔘_{a}/l^{g/2} ∈ U (l^{g}). Proof. Since 𝔘_{a} = diag [exp 2πiq^{t}a/l] 𝔘′_{a} with 𝔘′_{a} := {exp 2πiq^{t}p/l}, all we need to show is that 𝔘′_{a} is unitary, which follows from the above lemma. □

In conclusion, we have: Proposition 12.

[ ϑ̃_{0} ; ϑ̃_{ε_{1}} ; ⋮ ; ϑ̃_{(1−1/l) ∑_{i=1}^{g} ε_{i}} ]=[ \begin{matrix}𝔘_{0} & 0 & & & & 0 \\ 0 & 𝔘_{ε_{1}} \\ & & . & . & . \\ 0 & & & & & 𝔘_{(1−1/l) ∑_{i=1}^{g} ε_{g}} \end{matrix} ] [ s̃_{0} ; s̃_{ε_{1}} ; ⋮ ; s_{(1−1/l) ∑_{i=1}^{g} ε_{g}} ]

where 𝔘_{a} ∈ U (l^{g}) for each a ∈ (\frac{ℤ[1/l]}{ℤ})^{g}. In particular, we have

∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} |ϑ [ \begin{matrix}a \\ b \end{matrix} ] (z, Ω)|^{2} = l^{g} ∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})} |s_{p+a} (z)|^{2} .

The next lemma gives a geometric interpretation of the space 𝕍_{l}. Lemma 13. Let

l^{∗} : A_{Ω} ⟶ A_{Ω}
z ⟼ lz

be the rescaling map. Then deg l^{∗}O_{A_{Ω}} (1) = l^{2} and

𝕍_{l} = H^{0} (l^{∗}O_{A_{Ω}}(1)) = H^{0} (O_{A_{Ω}} (l^{2})) .

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 with l^{∗}O_{A_{Ω}}(1). Then we have

e′_{α} (z) = 1 = e_{α}^{l^{2}} (z) and e′_{g+α} (z) = exp πi (−Z_{αa}l^{2} − 2z_{α}l^{2}) = e_{g+α}^{l^{2}} (z) . In fact, O_{A_{Ω}}(1) is a symmetric line bundle, which means

l^{∗}O_{A_{Ω}}(1) = O_{A_{Ω}} (l^{2}) .

On the other hand, for f ∈ 𝕍_{l} we have f (l (z + m)) = f (lz + lm) = f (lz) for m ∈ ℤ^{g} 30 Xiaowei Wang and Hok Pun Yu

and

f (l (z + λ_{g+α})) = exp πi (−Z_{αα}l^{2} − 2z_{α}l^{2}) f (lz)
= e′_{g+α} (z) f (lz)
= e_{g+α}^{l^{2}} (z) f (lz) .

Thus f ∈ H^{0} (l^{∗}O_{A_{Ω}}(1)) = H^{0} (O_{A_{Ω}} (l^{2})). A dimension count then implies

φ : 𝕍_{l} ⟶ H^{0} (A_{Ω}, O (l^{2}))
f (z) ⟼ f (lz)

is an isomorphism. □

Now we introduce an inner product on 𝕍_{l}. Definition 14. For f (z) ∈ 𝕍_{l} we define

|f (z)|_{h}^{2} := ∫_{𝔻_{lΛ}} |f (z)|^{2} h (z) ω^{n}

where 𝔻_{Λ} is the fundamental domain associated to Λ. In particular, |ϑ_{0,0} (z)|_{h}^{2} := ∫_{𝔻_{lΛ}} |ϑ (z)|^{2} h (z) ω^{g} = l^{2g} ∫_{𝔻_{Λ}} |ϑ (z)|^{2} h (z) ω^{g}.

It possesses the following properties. Lemma 15. For a, b ∈ (ℤ [1/l])^{g}, we have:

(1)

h (z) |(𝔖_{b}f) (z)|^{2} = (h |f|^{2}) (z + b) ,
h (z) |(𝔗_{a}f) (z)|^{2} = (h |f|^{2}) (z + Za) .

(2) The actions {𝔖_{b}, 𝔗_{a}} are unitary with respect to the L^{2}-metric induced from h.

(3) \{\frac{s_{c}}{l^{g/2}|ϑ|_{h}} \}_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}} forms an orthonormal basis of (𝕍_{l}, |·|_{h}).

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

For part (2), we have

∥𝔖_{b}f (z)∥_{h}^{2} = ∫_{𝔻_{lΛ}} |f (z + b)|^{2} h (z + b) ω^{g} = ∫_{𝔻_{lΛ}} |f (z)|^{2} h (z) ω^{g}
∥𝔗_{a}f (z)∥_{h}^{2} = ∫_{𝔻_{lΛ}} |f (z + Za)|^{2} h (z + Za) ω^{g} = ∫_{𝔻_{lΛ}} |f (z)|^{2} h (z) ω^{g}.

For part (3), since \{𝔖_{c}\}_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}} forms a family of commuting operators, the eigenvectors \{s_{c}\}_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}} are mutually orthogonal. On the other hand Lemma 7 implies that for all a, c ∈ (ℤ [1/l])^{g}

|s_{c+a}|_{h}^{2} = |𝔗_{a}s_{c}|_{h}^{2} = |s_{c}|_{h}^{2} .

Theta function and Bergman metric 31

If we set c = −a then we have

|s_{c}|_{h}^{2} = |s_{0}|_{h}^{2} .

This implies

|ϑ|_{h}^{2} = ∑_{p∈(ℤ/lℤ)^{g}} |s_{p}|_{h}^{2} = l^{g} |s_{0}|_{h}^{2} .  □

Now let

Φ : A_{Ω} ⟶ ℙ^{l^{2g}−1}
z ⟼ [s_{c} (lz)]_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}}

be the projective embedding induced by \{s_{c}\}_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}}. Then Φ^{∗}ω_{FS} is invariant under the action of 𝔖_{b} and 𝔗_{a} for a, b ∈ (\frac{ℤ[1/l]}{lℤ})^{g} since both 𝔖_{b} and 𝔗_{a} are unitary. The Fubini–Study metric is given by

∥f (lz)∥_{FS}^{2} := ∫_{A_{Ω}} \begin{matrix}|f (lz)| & 2 \\ ∑ & c & |s_{c} (lz)| & 2 \end{matrix} Φ^{∗}ω_{FS}.

This together with the identities

h (z) |(𝔖_{b}f) (z)|^{2} = (h |f|^{2}) (z + b)
h (z) |(𝔗_{a}f) (z)|^{2} = (h |f|^{2}) (z + Za)

and

|𝔖_{b}s_{c}| = |s_{c}|
∑_{c} |𝔗_{a}s_{c}|^{2} = ∑_{c} |s_{c}|^{2}

would imply

∥(𝔖_{b}f) (lz)∥_{FS}^{2} = ∫_{A_{Ω}} \begin{matrix}|(𝔖_{b}f) (lz)| & 2 \\ ∑ & c & |s_{c} (lz)| & 2 \end{matrix} Φ^{∗}ω_{FS}

= ∫_{A_{Ω}} \frac{|(𝔖_{lb}f) (lz)|^{2} h (lz)}{∑_{c} |(𝔖_{b}s_{c}) (lz)|^{2} h (lz)}Φ^{∗}ω_{FS}

= ∫_{A_{Ω}} \begin{matrix}(h |f|^{2}) & (lz + b) \\ (∑_{c} h |s_{c}|^{2}) & (lz + b) \end{matrix} (𝔖_{b} ∘ Φ)^{∗} ω_{FS}

= ∫_{A_{Ω}} \begin{cases}|f| & 2 \\ ∑ & c & |s_{c}| & 2 \end{cases} Φ^{∗}ω_{FS}\} (lz + b)

= ∥f (lz)∥_{FS}^{2} , and

∥(𝔗_{a}f) (lz)∥_{FS}^{2} = ∫_{A_{Ω}} \begin{matrix}|(𝔗_{a}f) (lz)| & 2 \\ ∑ & c & |s_{c} (lz)| & 2 \end{matrix} Φ^{∗}ω_{FS}

= ∫_{A_{Ω}} \frac{|(𝔗_{a}f) (lz)|^{2} h (lz)}{∑_{c} |(𝔗_{a}s_{c}) (lz)|^{2} h (lz)} (𝔗_{a} ∘ Φ)^{∗} ω_{FS}

= ∫_{A_{Ω}} \begin{cases}|f| & 2 \\ ∑ & c & |s_{c}| & 2 \end{cases} Φ^{∗}ω_{FS}\} (lz + Za)

= ∥f (lz)∥_{FS}^{2} .

Hence we obtain the following:

Proposition 16. For a, b ∈ (ℤ [1/l])^{g}:

(1) 𝔖_{b}, 𝔗_{a} are unitary operators acting on (𝕍_{l}, ∥·∥_{FS}).

(2) The action \{s_{c}\}_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}} is an orthonormal basis for (𝕍_{l}, ∥·∥_{FS}).

Recall from [D] that a projective embedding of φ : A_{Ω} → ℙ^{l^{2g}−1} is called balanced if we have

∫_{A_{Ω}} (\frac{z_{i}z_{j}}{|z|^{2}} − \frac{δ_{ij}}{l^{2g}} ) φ^{∗}ω_{FS}^{g} = 0.

Corollary 17. The embedding defined by

Θ : A_{Ω} ⟶ ℙ^{l^{2g}−1}
z ⟼ [ϑ [ \begin{matrix}a \\ b \end{matrix} ] (z, Ω)]_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}}

is balanced.

Proof. It follows from the fact that

\{\frac{l^{g/2}s_{c}}{∥ϑ∥_{FS}} \}_{c∈( \frac{ℤ[1/l]}{lℤ})^{g}}

forms an orthonormal basis for (𝕍_{l}, ∥·∥_{FS}), and the map Φ and Θ differ by an unitary transformation. □

Remark 18. Geometrically, the finite Heisenberg group G_{l} acts on the embedding leaving Im Φ invariant. And {𝔖_{b}, 𝔗_{a}} acts on Φ (A_{Ω}) via translations under the group law. Although we can not get a homomorphism from A_{Ω} to U(l^{2}), we do have G_{l} ⊂ U(l^{2}).

Finally, we are ready to state our main result.

Theorem 19.

∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} |ϑ [ \begin{matrix}a \\ b \end{matrix} ] (z, Ω)|^{2} = l^{g} ∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})} |s_{p+a} (z)|^{2} .

lim_{l→∞} l^{−g} ∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})^{g}} |s_{p+a} (z)|^{2} h (z) = ∫_{A_{Ω}} |ϑ (z)|^{2} h (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

\frac{det Im Z}{l^{2g}} ∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} f (lz + Za + b)

= ∫_{𝔻_{Ω}} f (z) \frac{ω_{ℂ^{g}}^{g}}{det Im Z} + O (l^{−3}) for l → ∞

where ω_{ℂ^{g}} := ∑_{i} dz_{i} ∧ dz_{i} is the standard Kähler form on ℂ^{g}.

Proof. By performing affine transformations, we may assume that Z = iI. First, we notice that f being smooth implies that

lim_{l→∞} \frac{1}{l^{2g}} ∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} f (z + ai + b) = ∫_{𝔻_{Ω}} fω_{ℂ^{g}}^{g}

for any fixed z ∈ 𝔻_{Ω}. Moreover we have that

|\frac{1}{l^{2g}} ∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} f (z + ai + b) − ∫_{𝔻_{Ω}} fω_{ℂ^{g}}^{g} | ≤ 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

|\frac{1}{l^{2g}} ∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} f (lz + ai + b) − ∫_{𝔻_{Ω}} fω_{ℂ^{g}}^{g} | ≤ Cl^{−3}.  □

Proof of Theorem 19. By Proposition 3, we have

∑_{a,b} |ϑ [ \begin{matrix}a \\ b \end{matrix} ] (lz)|^{2} h (lz)

= ∑_{a,b} |ϑ (lz + aτ + b)|^{2} h (lz + aτ + b)

hence

l^{g} ∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})^{g}} |s_{p+a} (lz)|^{2} h (lz)

= ∑_{a,b∈( \frac{ℤ[1/l]}{ℤ})^{g}} |ϑ [ \begin{matrix}a \\ b \end{matrix} ] (lz, Ω)|^{2} h (lz)

= ∑_{a,b} |ϑ (lz + Za + b)|^{2} h (lz + Za + b) .

By Lemma 20, we have

lim_{l→∞} \frac{Vol (A_{Ω})}{l^{2g}} ∑_{a,b} |ϑ (lz + aτ + b)|^{2} h (lz + aτ + b)

= ∫_{A_{Ω}} |ϑ (z)|^{2} h (z) ω^{g}.

Hence

lim_{l→∞} l^{−g} ∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})} |s_{p+a} (lz)|^{2} h (lz)

= \frac{1}{Vol (A_{Ω})} ∫_{A_{Ω}} |ϑ (z)|^{2} h (z) dx ∧ dy. □

Corollary 21. The balanced metric Θ_{l}^{∗}ω_{FS}/l^{2} converges to ω_{0}, the flat metric on A_{Ω}, as a C^{∞} function as l → ∞.

Proof. Let h_{FS} denote the Fubini–Study metric on O_{A_{Ω}} (l^{2}) induced via the embedding Θ. Then

h_{FS} = \frac{h^{l^{2}}}{∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})} |s_{p+a} (lz)|^{2} h (lz)}.

On the other hand

\frac{Θ_{l}^{∗}ω_{FS}}{l^{2}} = \frac{∂∂ log h_{FS}}{l^{2}}

= ∂∂ log h − \frac{∂∂ log (∑_{p∈(ℤ/lℤ)^{g}} ∑_{a∈( \frac{ℤ[1/l]}{ℤ})} |s_{p+a} (lz)|^{2} h (lz))}{l^{2}}

= ω_{0} − \frac{∂∂ log (l^{g} ∥ϑ∥_{h} + O (l^{g−3}))}{l^{2}} → ω_{0}.

□

Theta function and Bergman metric 35

References

[D] Donaldson, S. K. Scalar curvature and projective embeddings. I. J. Diffential Geom. 59 (2001), no. 3, 479–522. MR1916953 (2003j:32030), Zbl 1052.32017.

[GH] Griffiths, Phillip; Harris, Joseph. Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience, New York, 1978. xii+813 pp. ISBN: 0-471-32792-1. MR0507725 (80b:14001), Zbl 0408.14001.

[Mum] Mumford, David. Tata lectures on Theta. I. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. Progress in Mathematics, 28. Birkhäuser Boston, Inc., Boston, MA, 1983. xiii+235 pp. ISBN: 3-7643-3109-7. MR0688651 (85h:14026), Zbl 0509.14049.

[Mum1] Mumford, David. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, 5. Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970 viii+242 pp. MR0282985 (44 #219), Zbl 0223.14022.

[Mum2] Mumford, David. Stability of projective varieties. Enseignement Math. (2)23 (1977) 39–110. MR0450272 (56 #8568), Zbl 0363.14003.

[R] Ruan, Wei-Dong. Canonical coordinates and Bergmann [Bergman] metrics. Comm. Anal. Geom. 6 (1998), no. 3, 589–631. MR1638878 (2000a:32050), Zbl 0917.53026.

[T] Tian, Gang. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32 (1990) 99–130. MR1064867 (91j:32031), Zbl 0706.53036.

[Z] Zelditch, Steve. Szegö kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331. MR1616718 (99g:32055), Zbl 0922.58082.

The Chinese University of Hong Kong, Sha Tin, N.T., Hong Kong xiaowei@math.cuhk.edu.hk hpyu@math.cuhk.edu.hk

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