PROJECTIVE NORMALITY OF ABELIAN VARIETIES

JAYA N.IYER

Abstract. We show that ample line bundles L on a g-dimensional simple abelian variety A, satisfying h^{0}(A, L) > 2^{g}.g!, give projective normal embeddings, for all g ≥ 1.

1. Introduction

Let A be an abelian variety of dimension g defined over the field of complex numbers and L be an ample line bundle on A. Consider the associated rational map ϕ_{L} : A ⟶ ℙ^{d−1} = ℙH^{0}(L), where d = dimH^{0}(A, L). Suppose L = M^{n} for some ample line bundle M on A. Then Koizumi has shown that L gives a projectively normal embedding if n ≥ 3 (see [2]).

When n = 2, Ohbuchi (see [7]) has shown the following.

Theorem 1.1. Suppose M is a symmetric ample line bundle on a g-dimensional abelian variety A. Then L = M^{2} gives a projectively normal embedding of A if and only if the origin 0 of A is not contained in Bs|M ⊗ P_{α}| for any α ∈ Â_{2} = {α ∈ Â : 2α = 0}, where Â is the dual abelian variety of A, P is the Poincaré bundle on A × Â, P_{α} = P_{|A×α} for α ∈ Â and Bs|M ⊗ P_{α}| is the set of all base points of M ⊗ P_{α}.

Suppose L ≠ M^{n} for any ample line bundle M on A and n > 1. When g = 2, Lazarsfeld (see [4]) has shown that if ϕ_{L} is birational onto its image then ϕ_{L} gives a projectively normal embedding, for d = 7, 9, 11 and for d ≥ 13. We showed that if the Neron Severi group NS(A) of A is ℤ, generated by L and d ≥ 7 then ϕ_{L} gives a projectively normal embedding (see [1]).

In this article, we show

Theorem 1.2. Suppose L is an ample line bundle on a g-dimensional simple abelian variety A. If d > 2^{g}.g! then L gives a projectively normal embedding, for all g ≥ 1. (Here d = dimH^{0}(A, L)).

We outline the proof of Theorem 1.2.

0Mathematics Classification Number: 14C20, 14K05, 14K25, 14N05.

2 JAYA N.IYER

For a polarized abelian variety (A, L), consider the multiplication maps

ρ_{r} : Sym^{r}H^{0}(A, L) ⟶ H^{0}(A, L^{r}).

By definition, L gives a projectively normal embedding if ρ_{r} is surjective, for all r ≥ 1. We firstly show that it suffices to show ρ_{2} is surjective. More precisely, we show that ρ_{2} is surjective implies that the maps ρ_{r} are surjective, for r ≥ 3 (see Prop.2.1).

To prove the surjectivity of the map ρ_{2} we consider a finite isogeny A ⟶ B = A/H, where H is a maximal isotropic subgroup of the fixed group K(L) of L. Then L descends down to a principal polarization M on B. Let Ĥ denote the group of characters on H. By associating to a character χ ∈ Ĥ a degree 0 line bundle L_{χ} on B one can identify Ĥ as a subgroup of the dual abelian variety Pic^{0}(B) of B. The homomorphism ψ_{M} : B ⟶ Pic^{0}(B), b ↦ t_{b}^{∗}M ⊗ M^{−1} is an isomorphism and we denote H′ = ψ_{M}^{−1}(Ĥ).

We then show that the surjectivity of the map ρ_{2} is equivalent to showing that the subgroup H′ of B generates the projective space ℙH^{0}(B, M^{2}) and its translates ℙH^{0}(t_{σ}^{∗}M^{2}), where σ ∈ B is such that ψ_{M}(2σ) = L_{χ}, L_{χ} ∈ Ĥ, i.e. the images of points of H′, under

the morphism B ⟶^{ϕ_{t^{∗}_{σ}M^{2}}} ℙH^{0}(t_{σ}^{∗}M^{2}) ≃ |t_{σ}^{∗}M^{2}|, b ↦ t_{b}^{∗}θ + t_{−b+2σ}^{∗}θ (due to Wirtinger),

have their linear span as |t_{σ}^{∗}M^{2}|. (Here we assume M is symmetric and θ is the unique symmetric divisor in |M|).

To see this, we show

Proposition 1.3. Let L be an ample line bundle on a simple abelian variety Z of dimension g and consider the associated rational map Z ⟶^{ϕ_{L}} ℙH^{0}(L). Then any finite subgroup G of Z of order strictly greater than h^{0}(L).g!, generates the linear system ℙH^{0}(L). More precisely, the points ϕ_{L}(h) where h runs over all elements of G not in the base locus of L span ℙH^{0}(L) (see Prop. 3.4).

We then apply above proposition to L = t_{σ}^{∗}M^{2} to obtain bounds as asserted for a polarized abelian variety (A, L) in Theorem 1.2.

Acknowledgements: We thank M.Hindry and K.Paranjape for useful conversations and A.Beauville for pointing out a gap in Prop.3.3 in an earlier version. We are grateful to A.Hirschowitz for his helpful suggestions incorporated here. This work was done at University of Paris-6 and University of Essen. Their hospitality and support from the French Ministry of Education, Research and Technology and DFG ‘Arithmetik und Geometrie’ Essen, is gratefully acknowledged.

Notations : The varieties considered in this article are defined over the complex numbers.

PROJECTIVE NORMALITY OF ABELIAN VARIETIES 3

Let L be an ample line bundle on an abelian variety Z of dimension g.

1. The fixed group of L is the group K(L) = {z ∈ Z : L ≃ t_{z}^{∗}L}, t_{z} : Z ⟶ Z, x ↦ z + x.

2. The theta group of L is the group G(L) = {(z, ϕ) : L ≃^{ϕ} t_{z}^{∗}L}.

3. The Weil form e^{L} : K (L) × K (L) ⟶ ℂ^{∗} is the commutator map (x, y) ↦ x′y′x′^{−1}y′^{−1}, for any lifts x′, y′ ∈ G(L) of x, y ∈ K(L).

4. h^{0}(L) = dimH^{0}(Z, L).

5. If G is a finite subgroup of Z then Card(G) = order(G).

2. Surjectivity of the maps ρ_{r}, r ≥ 3

Suppose L is an ample line bundle on a g-dimensional abelian variety A. Consider the multiplication maps

H^{0}(L)^{⊗r} ⟶^{ρ_{r}} H^{0}(L^{r}), for r ≥ 2.

The main result of this section is the following.

Proposition 2.1. Suppose L is an ample line bundle on an abelian variety A. If the multiplication map ρ_{2} is surjective then ρ_{r} is surjective, for all r ≥ 3.

Firstly, we recall

Proposition 2.2. Suppose L and M are ample line bundles on an abelian variety A. 1) The multiplication map

∑_{α∈U} H^{0}(L ⊗ α) ⊗ H^{0}(M ⊗ α^{−1}) ⟶ H^{0}(L ⊗ M)

is surjective, for any non-empty Zariski open subset U of Pic^{0}(A).

2) If the multiplication map H^{0}(L) ⊗ H^{0}(M) ⟶ H^{0}(L ⊗ M) is surjective then the multiplication maps

(a) H^{0}(L) ⊗ H^{0}(M ⊗ α) ⟶ H^{0}(L ⊗ M ⊗ α)

and

(b) H^{0}(L ⊗ α^{−1}) ⊗ H^{0}(M ⊗ α) ⟶ H^{0}(L ⊗ M)

are also surjective, for α in some non-empty Zariski open subset U of Pic^{0}(A).

Proof: 1) See [3], 7.3.3.

2) The proof is standard. □

4 JAYA N.IYER

Proof of Proposition 2.1 We prove by induction on r. Suppose the multiplication map ρ_{r} : H^{0}(L)^{⊗r} ⟶ H^{0}(L^{r}) is surjective, for some r ≥ 2.

Consider the composed multiplication map

H^{0}(L)^{⊗r+1} ⟶^{Id⊗ρ_{r}} H^{0}(L) ⊗ H^{0}(L^{r}) ⟶^{ρ_{1,r}} H^{0}(L^{r+1}).

To see the surjectivity of the map ρ_{r+1} = ρ_{1,r} ∘ (Id ⊗ ρ_{r}) we need to show that the map ρ_{1,r} is surjective.

Using Prop.2.2 1), we can write

H^{0}(L).H^{0}(L^{r}) = ∑_{α∈U} H^{0}(L).H^{0}(L ⊗ α^{−1}).H^{0}(L^{r−1} ⊗ α)...(∗)

for any non-empty Zariski open subset U, of Pic^{0}(A).

Since ρ_{2} is surjective, by Proposition 2.2 2) (a), there exists a non-empty Zariski open subset U′ of Pic^{0}(A), such that for α^{−1} ∈ U′,

H^{0}(L).H^{0}(L ⊗ α^{−1}) = H^{0}(L^{2} ⊗ α^{−1})...(∗∗)

Now in (∗), using (∗∗) and again applying Proposition 2.2 1), we obtain

H^{0}(L).H^{0}(L^{r}) = ∑_{α^{−1}∈U′} H^{0}(L^{2} ⊗ α^{−1}).H^{0}(L^{r−1} ⊗ α)

= H^{0}(L^{r+1}).□

3. Surjectivity of the map ρ_{2}

Let Z be a g-dimensional abelian variety and D be an ample divisor on Z . We denote M = O(D) to be the ample line bundle on Z. Let G be a finite subgroup of Z. Consider the homomorphism ψ_{M} : Z ⟶ Pic^{0}(Z), z ↦ t_{z}^{∗}(M) ⊗ M^{−1}. Let G′ ⊂ Pic^{0}(Z) be the image of G under this homomorphism. Consider a finite subgroup J ⊂ Pic^{0}(Z) and containing the subgroup G′. Construct an etale cover π : X ⟶ Z corresponding to J, which is of degree equal to CardJ. Let N = O(π^{−1}D) be the ample line bundle on X.

Notice that if h ∈ G ∩ K(M) then t_{h}^{∗}M ≃ M and this implies that D + h is linearly equivalent to D on Z. If ψ_{N} : X ⟶ Pic^{0}(X) is the map x ↦ t_{x}^{∗}N ⊗ N^{−1} and π̂ : Pic^{0}(Z) ⟶ Pic^{0}(X) is the dual of the map π, then since π̂(J) = {0}, π^{−1}G ⊂ K(N) = Kerψ_{N} (since ψ_{N} = π̂ ∘ψ_{M} ∘π). This means that the divisors π^{−1}(D+h) ∈ |N|, for all h ∈ G.

Choose the subgroup J such that N is base point free. (In fact, if J contains the subgroup of 3-torsion points of Pic^{0}(Z) and G′, then by above discussion, X_{[3]} ⊂ K(N), where X_{[3]} is the subgroup of 3-torsion points of X. This implies, by [3] 2.5.6, that

PROJECTIVE NORMALITY OF ABELIAN VARIETIES 5

N = K^{3}, for some ample line bundle K on X and by a theorem of Lefschetz (see [3], 4.5.1), N is very ample).

We will use the following.

Lemma 3.1. Let V be a variety and V ⊂ Div(V) be an irreducible family of effective Cartier divisors D_{t} on V. Suppose W = ∩_{t∈V}D_{t} ⊂ V and is non-empty and r = codim(W). Then there exists divisors D_{j}, j = 1, 2, ..., r, in V, which intersect properly and dimW = dim ∩_{i=1}^{r} D_{i}.

Proof: We use induction on j. Let D_{1}, D_{2}, ..., D_{j} (j < r) be chosen in V which intersect properly in V. Now write D_{1} ∩ D_{2}... ∩ D_{j} = G_{1} ∪ G_{2}... ∪ G_{s}, where G_{1}, ..., G_{s} are irreducible components. Consider the closed subset W_{i} ⊂ V parametrizing divisors which contain G_{i} for i = 1, 2, ..., s. (Note that W_{i} ≠ V, otherwise G_{i} ⊂ W, which is not possible since dimG_{i} > dimW). Let U be the complement of ∪_{i=1}^{s}W_{i} in V, which is non-empty since V is irreducible. If D_{j+1} ∈ U, then D_{1} ∩ ...D_{j} ∩ D_{j+1} has codimension j + 1 (communicated to us by A.Hirschowitz). □.

Remark 3.2. Suppose D_{1}, D_{2}, ..., D_{r} are linearly equivalent effective divisors on a variety V, W = ∩_{i=1}^{r}D_{i} and is non-empty and r = codim(W). If ℙ^{k} denotes the span of the points D_{i} in the linear system |D_{1}| then W = ∩_{t∈ℙ^{k}}D_{t}. Hence, by Lemma 3.1, there are r divisors D_{j} ∈ ℙ^{k}, which intersect properly.

With notations as above we have the following.

Proposition 3.3. Let D be an ample divisor on a g-dimensional simple abelian variety Z. Let G be a finite subgroup of Z and contained in D . Then Card(G) ≤ D^{g} (which equals h^{0}(O(D)).g!, by Riemann-Roch Theorem).

Proof: We prove this in several steps.

Step 1: We reduce to the case when the divisors D and D + h, for all h ∈ G, are linearly equivalent and O(D) is base point free. Indeed, by above discussion, choose a triple (X, N, π), as above, corresponding to a subgroup J ⊂ Pic^{0}(Z) such that N is base point free and ψ_{M}(G) ⊂ J. This shows that the divisors π^{−1}D and π^{−1}(D + h), for all h ∈ G, are linearly equivalent. Then we have a morphism ϕ_{N} : X ⟶ ℙH^{0}(N). Since π is a finite morphism of degree equal to Card(J), by projection formula, one sees that deg(π^{−1}W) = Card(J).deg(W), for a subvariety W of Z. Since (π^{−1}D)^{g} = Card(J).D^{g}, if Card(π^{−1}G) ≤ (π^{−1}D)^{g} then Card(G) ≤ D^{g}.

6 JAYA N.IYER

Step 2: We can now assume that D is an ample divisor on X and G ⊂ D is a finite subgroup such that D is linearly equivalent to D + h for all h ∈ G and N = O(D) is base point free. Let Y = ∩_{h∈G}D + h and s = dim(Y). By Lemma 3.2, Y ⊂ ∩_{j=1}^{g−s}D_{j} for some g − s divisors D_{j} ∈ |N|, which intersect properly. Now deg(Y) = [Y].[D^{s}], (here deg(Y) = deg(S), where S ⊂ Y is of pure dimension s). Since Y ⊂ ∩_{j=1}^{g−s}D_{j} we see that deg(Y) ≤ D^{g}. In particular, when s = 0, since G ⊂ Y we get Card(G) ≤ D^{g}.

Step 3: Suppose that s > 0. Let Y = Y_{1} ∪ Y_{2} ∪ ... ∪ Y_{r}, where Y_{j}, 1 ≤ j ≤ r are the irreducible components of Y such that s = dimY_{1} = dimY. Then degY_{1} ≤ degY. Since Y is G-invariant, ∪_{h∈G}Y_{1} + h ⊂ Y and ∑_{h∈_{\frac{G}{G_{Y_{1}}}}} deg(Y_{1} + h) ≤ degY, where G_{Y_{1}} = {h ∈ G : Y_{1} + h = Y_{1}} is a subgroup of G. Hence we get the inequalities Card(\frac{G}{G_{Y_{1}}} ).degY_{1} ≤ degY ≤ D^{g}, i.e. Card(G) ≤ \frac{Card(G_{Y_{1}})}{degY_{1}}.D^{g}. To complete our proof, we need to show that Card(G_{Y_{1}}) ≤ degY_{1}.

Step 4: Now G_{Y_{1}} ⊂ Stab(Y_{1}) = {a ∈ X : Y_{1} + a = Y_{1}}. Observe that Stab(Y_{1}) = ∩_{y∈Y_{1}}Y_{1} − y. Now for a point y_{0} ∈ Y_{1}, Stab(Y_{1}) = (Y_{1} − y_{0}) ∩_{y∈Y_{1}} Y_{1} − y ⊂ (Y_{1} − y_{0}) ∩_{h∈G,y∈Y_{1}} D + h − y. Let P = (Y_{1} − y_{0}) ∩_{h∈G,y∈Y_{1}} D + h − y. We proceed to show that deg(Stab(Y_{1})) ≤ deg(P). This will be true if Stab(Y_{1}) and P have same dimension. Now, we have

∩_{h∈G,y∈Y_{1}}D + h − y = ∩_{y∈Y_{1}}Y − y
= ∩_{y∈Y_{1}}((Y_{1} ∪ Y_{2} ∪ ... ∪ Y_{r}) − y)
= (∩_{y∈Y_{1}}Y_{1} − y) ∪ (∩_{y∈Y_{1}}Y_{2} − y) ∪ ... ∪ (∩_{y∈Y_{1}}Y_{r} − y).

(To see the above last equality: if x ∈ ∩_{y∈Y_{1}}(Y_{1} ∪ Y_{2} ∪ ... ∪ Y_{r}) − y, then x + y ∈ Y_{1} ∪ Y_{2} ∪ ... ∪ Y_{r}, ∀y ∈ Y_{1}. Via the translation map Y_{1} ⟶ Y_{1} ∪ Y_{2}... ∪ Y_{r}, y ↦ y + x and since Y_{1} is irreducible, x + y ∈ Y_{j}, for some j and for all y ∈ Y_{1}, i.e x ∈ ∩_{y∈Y_{1}}Y_{j} − y showing one way inclusion, the other inclusion being obvious).

We now see that if j ≠ 1 and x ∈ ∩_{y∈Y_{1}}Y_{j} −y then Y_{1} +x ⊂ Y_{j}. If dimY_{j} < dimY_{1} then this is absurd and so ∩_{y∈Y_{1}}Y_{j} − y is empty. If dimY_{j} ≥ dimY_{1}, since Y_{1} is of maximal dimension in Y, dimY_{j} = dimY_{1} and Y_{1} + x = Y_{j}. This implies that ∩_{y∈Y_{1}}Y_{j} − y = ∩_{y∈Y_{1}}Y_{1} + x − y = Stab(Y_{1}) + x. Hence ∩_{h∈G,y∈Y_{1}}D + h − y, P and Stab(Y_{1}) are of equal dimension, say equal to m and deg(Stab(Y_{1})) ≤ degP.

Step 5: We proceed to show that deg(P) ≤ deg(Y_{1}). Consider the Poincaré line bundle P on X × Pic^{0}(X). Let p_{1} and p_{2} denote the projections onto X and Pic^{0}(X) respectively from X × Pic^{0}(X). Consider the sheaf E = p_{2}_{∗}(p_{1}^{∗}N ⊗ P) on Pic^{0}(X). Since the vector spaces H^{0}(N ⊗ α) are of constant dimension for all α ∈ Pic^{0}(X), by

PROJECTIVE NORMALITY OF ABELIAN VARIETIES 7

Grauert’s theorem, E is a vector bundle on Pic^{0}(X). Let ℙ(E) denote the associated projective bundle on Pic^{0}(X). Consider the natural morphism p_{2}^{∗}(E) ⟶ p_{1}^{∗}N ⊗ P. This is surjective, since on any fibre X × α, (p_{1}^{∗}N ⊗ P)_{α} ≃ N ⊗ α which is globally generated (as N is globally generated) and E(α) ≃ H^{0}(N ⊗ α). Hence this defines a morphism δ_{N} : X × Pic^{0}(X) ⟶ ℙ(E). Let ℙ(Eˇ) denote the dual projective bundle over Pic^{0}(X). In general, the parameter space V ⊂ ℙ(Eˇ) of the family {D + h − y}_{h∈G,y∈Y_{1}} may not form an irreducible variety (unless G_{Y_{1}} = G) but we construct an irreducible subvariety F ⊂ ℙ(Eˇ) such that V ⊂ F and ∩_{h∈G,y∈Y_{1}}D + h − y = ∩_{t∈F}D_{t}, where D_{t} denotes the divisor corresponding to t, in ℙ(E) ...(∗∗)

Step 6: Construction of F:

Consider the subspace T of H^{0}(X, N) spanned by sections s_{h}, h ∈ G such that the divisor of s_{h} is D + h. Consider the addition map m : X × X ⟶ X, (x, y) ↦ x + y. Recall the skew-Pontryagin product of the sheaves O_{X} and N, Nˆ∗O_{X} = (p_{1})_{∗}(m^{∗}N) (see [8], p.653), where p_{1}(resp.p_{2}) : X × X ⟶ X denotes the first (resp. second) projection. Then, by Grauert’s theorem, Nˆ∗O_{X} forms a vector bundle on X with fibres (Nˆ∗O_{X})_{x} ≃ H^{0}(t_{x}^{∗}N). By [8], Remark 1.2 Nˆ∗O_{X} ≃ N ∗ O_{X} where N ∗ O_{X} = m_{∗}(p_{1}^{∗}N) is the Pontryagin product and by [5], p.161 there are isomorphisms O_{X} ⊗ H^{0}(X, N) ≃^{f} Nˆ∗O_{X} ≃ ψ_{N}^{∗}E ⊗ N, (ψ_{N} : X ⟶ Pic^{0}(X) is the isogeny x ↦ t_{x}^{∗}N ⊗ N^{−1}). Consider the image F under f of the trivial subbundle O_{X} ⊗T in Nˆ∗O_{X}. Then the fibre of F at x ∈ X is the vector subspace of H^{0}(t_{x}^{∗}N) spanned by the sections t_{x}^{∗}s_{h} whose divisor is D+h−x, for h ∈ G. Now ℙ(Fˇ) is a projective subbundle of ℙ(ψ_{N}^{∗}E ⊗ Nˇ) ≃ ℙ(ψ_{N}^{∗}Eˇ) (since N is a line bundle). Since Y_{1} is irreducible, the projective bundle ℙ(Fˇ) restricted to Y_{1} is an irreducible subvariety and let F be the image of this irreducible variety in ℙ(Eˇ). Hence F is irreducible and, by construction, if R ∈ |F_{y}|, y ∈ Y_{1} then ∩_{h∈G}D + h − y ⊂ R and F satisfies property (∗∗).

Step 7: By Lemma 3.1, there exists divisors D_{1}, D_{2}, ..., D_{g−m} ∈ F such that ∩_{h∈G,y∈Y_{1}}D+ h−y ⊂ D_{1} ∩D_{2}...∩D_{g−m}. Hence P ⊂ (Y_{1} −y_{0})∩D_{1} ∩D_{2}...∩D_{g−m} ⊂ D_{1} ∩D_{2}...∩D_{g−m}. This implies that deg(P) ≤ deg(Y_{1} − y_{0})), and by Step 2 and Step 4, deg Stab(Y_{1}) ≤ degY_{1} ≤ D^{g} (since by Step 2, deg(Y_{1}) ≤ deg(Y) ≤ D^{g}). Since X is simple, Stab(Y_{1}) is zero-dimensional and G_{Y_{1}} ⊂ Stab(Y_{1}) implies that Card(G_{Y_{1}}) ≤ deg(Y_{1}), hence by Step 3, Card(G) ≤ D^{g}. This ends the proof. □

This is equivalent to the following.

Proposition 3.4. Let L be an ample line bundle on a simple abelian variety Z and consider the associated rational map Z ⟶^{ϕ_{L}} ℙH^{0}(L). Then any finite subgroup G of Z, of order strictly greater than h^{0}(L).g!, generates ℙH^{0}(L). More precisely, the points ϕ_{L}(g) where g runs over all elements of G not in the base locus of L span ℙH^{0}(L).

We recall the following result which we will need in the proof of Theorem 1.2.

Proposition 3.5. (Wirtinger) Let (Z, Θ) be a principally polarized abelian variety and L = O(Θ), (here Θ is assumed to be a symmetric divisor). There is a nondegenerate inner product R : H^{0}(L^{2}) ⊗ H^{0}(L^{2}) ⟶ ℂ (which is symmetric or skew-symmetric depending on whether the multiplicity of the zero element 0 on Θ, mult_{0}Θ, is even or odd) such that if R induces the isomorphism R′,

ℙ(H^{0}(L^{2})) ≃ ℙ(H^{0}(L^{2})^{∗}) = |2Θ|,

then the composed morphism

Z ⟶^{ϕ_{L^{2}}} ℙ(H^{0}(L^{2})) ⟶^{R′} |2Θ|

is the morphism

ϕ : Z ⟶ |2Θ|, x ↦ Θ_{x} + Θ_{−x},

where Θ_{x} is the translate of Θ by x on Z.

Proof: See [6], Proposition, p.335. □

Proof of Theorem1.2 :

Consider a polarized simple abelian variety (A, L) of dimension g such that h^{0}(L) > 2^{g}.g!.

Consider the multiplication map

H^{0}(L) ⊗ H^{0}(L) ⟶^{ρ_{2}} H^{0}(L^{2}).

This map factors via

Sym^{2}H^{0}(L) ⟶^{ρ_{2}} H^{0}(L^{2})

Let H ⊂ K(L) be a maximal isotropic subgroup, for the Weil form e^{L}. Consider the isogeny A ⟶^{π} B = \frac{A}{H}. Then L descends down to a principal polarization M on B. We may assume that M is symmetric , i.e. M ≃ i^{∗}M, i(b) = −b, b ∈ B. Using the fact that π_{∗}O_{A} = ⊕_{χ∈Ĥ}L_{χ}, where L_{χ} denotes the degree 0 line bundle on B corresponding to the character χ on H, by projection formula, π_{∗}L = ⊕_{χ∈Ĥ}M ⊗L_{χ} and π_{∗}L^{2} = ⊕_{χ∈Ĥ}M^{2}⊗L_{χ}. Hence we obtain the following decompositions

H^{0}(L) = ⊕_{χ∈Ĥ}H^{0}(M ⊗ L_{χ})
H^{0}(L^{2}) = ⊕_{χ∈Ĥ}H^{0}(M^{2} ⊗ L_{χ}).

Write Sym^{2}H^{0}(L) = ∑_{χ,χ′∈Ĥ} H^{0}(M ⊗ L_{χ′}).H^{0}(M ⊗ L_{χ.χ′^{−1}}). Consider the multiplication maps

∑_{χ′∈Ĥ} H^{0}(M ⊗ L_{χ′}).H^{0}(M ⊗ L_{χ.χ′^{−1}}) ⟶^{ρ_{χ}} H^{0}(M^{2} ⊗ L_{χ}).

Since ρ_{2} = ⊕_{χ∈Ĥ}ρ_{χ} it will suffice to show the surjectivity of ρ_{χ}, for each χ ∈ Ĥ.

Since the pair (B, M) is principally polarized the homomorphism ψ_{M} : B ⟶ P ic^{0}(B) is an isomorphism. Let H′ = ψ_{M}^{−1}(Ĥ) and θ ∈ |M| be the unique symmetric divisor.

Case 1: Suppose χ is trivial.

We see that the surjectivity of the map ρ_{triv} is equivalent to showing that the reducible divisors θ_{h} + θ_{−h} generate the linear system |M^{2}|, for h ∈ H′. By Prop.3.5, using the morphism ϕ : B ⟶ |M^{2}|, this is same as saying that the image of the subgroup H′ under the morphism ϕ_{M^{2}} generates the projective space ℙH^{0}(M^{2}).

Case 2: Suppose χ is non-trivial.

Firstly, notice that if b ∈ B then ψ_{M^{2}}(b) = ψ_{M}(2b). Let σ ∈ B be such that ψ_{M^{2}}(σ) = L_{χ}, i.e. ψ_{M}(2σ) = L_{χ}. Hence the map ρ_{χ} is surjective if the reducible divisors θ_{h} + θ_{−h+2σ} span the linear system |t_{σ}^{∗}M^{2}|, for h ∈ H′ = ψ_{M}^{−1}(Ĥ). Now if b ∈ B then θ_{b} + θ_{−b+2σ} = (θ_{σ})_{b−σ} + (θ_{σ})_{−b+σ} which is the image of the divisor θ_{b−σ} + θ_{−b+σ} under the morphism |M^{2}| ⟶ |t_{σ}^{∗}M^{2}| given by the translation map B ⟶^{t_{σ}} B. Hence the morphism ϕ_{σ} : A ⟶ |t_{σ}^{∗}M^{2}| is given as b ↦ θ_{b} + θ_{−x+2σ}. This implies that ρ_{χ} is surjective if and only if the points in ϕ_{σ}(H′) generate the linear system |t_{σ}^{∗}M^{2}|.

Since the pair (A, L) is a simple polarized abelian variety with h^{0}(L) = Card(H′) > 2^{g}.g! = h^{0}(t_{σ}^{∗}M^{2}).g!, by Proposition 3.4, ρ_{χ} is surjective, for all χ ∈ Ĥ. Hence, by Proposition 2.1, our proof is now complete.

Remark 3.6. 1) Suppose g = 1. Then any line bundle of degree strictly greater than 2 on an elliptic curve, gives a projectively normal embedding. Hence the bound is sharp.

2) Suppose g = 2. If L ≃ N^{2}, where N is an ample symmetric line bundle with h^{0}(N) = 2 on an abelian surface A then it follows that h^{0}(L) = 8 (in terms of ‘type’ of an ample line bundle, N is of type (1, 2) and hence L is of type (2, 4) and h^{0}(L) = 8). By [3], 10.1.4, N has 4 base points say x_{1}, x_{2}, x_{3} and x_{4} which are 4-torsion points on A and moreover 2x_{i} ∈ K(N) = Kerψ_{N} where ψ_{N} : A ⟶ Pic^{0}(A), a ↦ t_{a}^{∗}N ⊗ N^{−1}. Let α_{i} = ψ_{N}(x_{i}), for i = 1, 2, 3, 4. Now the points x_{i} are base points for N, for i = 1, 2, 3, 4 is equivalent to saying that the origin 0 ∈ A is a base point for N ⊗ α_{i}, for i = 1, 2, 3, 4. Also 2x_{i} ∈ K(N) implies that the points α_{i} are 2-torsion points in Pic^{0}(A), hence by

10 JAYA N.IYER

Ohbuchi’s Theorem 1.1, L does not give a projectively normal embedding. So the bound is sharp.

3) Suppose g = 3. If L ≃ N^{3}, where N is a principal polarization on an abelian threefold A then h^{0}(L) = 27. But by Koizumi’s Theorem, L gives a projectively normal embedding. So the bound is not sharp in this case.

References

[1] Iyer, J.:Projective normality of abelian surfaces given by primitive line bundles, Manuscr.Math., 98, 139-153, (1999).

[2] Koizumi, S.:Theta relations and projective normality of abelian varieties, American Journal of Mathematics, 98, 865-889 (1976).

[3] Lange, H. Birkenhake, Ch. : Complex abelian varieties, Springer-Verlag, Berlin, (1992).

[4] Lazarsfeld, R.: Projectivité normale des surface abéliennes, Rédigé par O.Debarre. Prépublication No. 14, Europroj- C.I.M.P.A., Nice, (1990).

[5] Mukai, S.: Duality between D(X) and D(X̂) with its application to Picard sheaves, Nagoya Math. J., 81, 153-175 ( 1981).

[6] Mumford, D.: Prym varieties I, in: Contributions to Analysis. Academic Press, New York, 325-350 (1974).

[7] Ohbuchi, A.: A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad., 64, 119-120 (1988).

[8] Pareschi, G.: Syzygies of abelian varieties, J.Amer.Math.Soc. 13(2000), no.3, 651-664.

Max-Planck Institut fur Mathematik Vivatsgasse 7, D-53111, Bonn, Germany (Email: jniyer@mpim-bonn.mpg.de)