STABILITY OF DEGENERATE ABELIAN VARIETIES

IKU NAKAMURA

Dedicated to Professor Kunihiko Kodaira

Abstract. We compactify canonically the moduli scheme of abelian schemes over Z[ζ_{N}, 1/N] by introducing the noncommutative level structures. Any degenerate abelian scheme on the boundary of the compactification is one of our models – projectively stable quasi-abelian schemes. We prove that a degenerate abelian scheme has Kempf-stable Hilbert points if and only if it is a projectively stable quasi-abelian scheme.

0. Introduction.

The purpose of the present article is to compactify canonically the moduli scheme of abelian varieties in a way similar to the Deligne-Mumford compactification by stable curves of the moduli scheme of curves [DM69]. The problem has been considered in the complex analytic case by [Namikawa76], Ueno [unpublished] and [Nakamura75]. Recently modifying these degenerate abelian varieties slightly, Alexeev and the author [AN96] defined (torically) stable quasi-abelian varieties over any field and proved a stable reduction theorem over any complete discrete valuation ring.

In the present article we introduce the notion of polarized projectively stable quasi-abelian schemes by modifying slightly those defined in [AN96] and prove Kempf-stability of their normalized Hilbert points (Theorem 0.2) and that Kempf-stability characterizes these schemes conversely (Theorem 0.3). We also prove the existence of the projective reduced-fine-moduli scheme of projectively stable quasi-abelian schemes with certain level structures over Z[ζ_{N}, 1/N] where ζ_{N} is a primitive N-th root of unity and N ≥ 3. This is a natural geometric compactification of a moduli scheme of abelian varieties with level structures (Theorem 0.4). See [Alexeev96] for the case of principally polarized (torically) stable quasi-abelian varieties.

If we are given Faltings-Chai’s degeneration data [FC90] there are two canonical choices of flat projective degenerating families of abelian varieties (P, L) and (Q, L) where (Q, L) is the most naive choice and (P, L) is the normalization of (Q, L) after some base change. The following is the stable reduction theorem of abelian varieties proved in [AN96].

Date: 3 July 1998.

1991 Mathematics Subject Classification. Algebraic Geometry, 14J10, 14K10, 14K25.

Key words and phrases. Moduli, Abelian variety, Stable quasi-abelian variety, Projectively stable quasi-abelian scheme, Stable reduction, Stability, Hilbert point, Kempfstable, Fine moduli, Heisenberg group, Level structure, Rigid structure.

Research was supported in part by the Grant-in-aid (No. 08211201, No. 08404001) for Scientific Research, the Ministry of Education.

Theorem 0.1. Let R be a complete discrete valuation ring with the fraction field k(η) and (G_{η}, L_{η}) a polarized abelian variety over k(η). Then after a suitable finite ramified cover Spec R′ → Spec R it can be completed to a flat projective scheme (P, L) (or (Q, L)) over R′ with a relatively ample invertible sheaf L extending L_{η}.

See Section 10 for the Stable reduction theorem we need in the present article. We call the closed fibre (P_{0}, L_{0}) a torically stable quasi-abelian variety (abbr. SQAV), while we call (Q_{0}, L_{0}) a projectively stable quasi-abelian scheme (abbr. PSQAS). We note (P_{0}, L_{0}) ≃ (Q_{0}, L_{0}) if the dimension is less than five. (P_{0}, L_{0}) is always reduced, while (Q_{0}, L_{0}) can be nonreduced if the dimension is greater than four. (Q_{0}, L_{0}) determines (P_{0}, L_{0}) uniquely but the converse is unknown.

By applying [Kempf78, Corollary 5.1] we will prove

Theorem 0.2. Let (Q_{0}, L_{0}) be a projectively stable quasi-abelian scheme over the residue field k := R/I, I the maximal ideal of R. Suppose that k is algebraically closed field and the characteristic of k and N := deg L_{0}/(g!) are coprime. If Γ(Q, L)⊗k is very ample on Q_{0}, then the n-th normalized Hilbert point of (Q_{0}, L_{0}) has a closed SL(N, k)-orbit, and it is Mumford-semistable for any large n.

Theorem 0.3 shows that no similar theorem for (P_{0}, L_{0}) except semistability can be expected when P_{0} ≠ Q_{0}. See also Remark 8.4. See Section 8 for the definition of normalized Hilbert points.

Let (K, e_{K}) a finite symplectic abelian group, H a maximally isotropic subgroup of K with respect to e_{K} and N the order of H. Then ord K = N^{2}. By the uniqueness of H up to isomorphism, we define e_{min}(K) = e_{1} and e_{max}(K) = e_{g} if H ≃ ⊕_{i=1}^{g}(Z/e_{i}Z), e_{i}|e_{i+1} for some positive integers e_{i}.

Theorem 0.3. Let k be an algebraically closed field, (K, e_{K}) a finite symplectic abelian group with e_{min}(K) ≥ 3 and N = \sqrt{ord K}. Assume that the characteristic of k and N are coprime. Suppose that a k-scheme (Z, L) is smoothable into an abelian variety (A, M) with (ker λ(M), e^{M}) ≃ (K, e_{K}) where e^{M} is the Weil pairing of M. Then the following are equivalent;

1. (Z, L) is a projectively stable quasi-abelian scheme,

2. Aut(Z, L) contains a subgroup of SL_{±}(N, k) (conjugate) isomorphic to G(K),

3. the n-th normalized Hilbert point of (Z, L) has a closed SL(N, k)-orbit

for any large n where SL_{±}(N, k) = {g ∈ GL(N, k); det g = ±1} and G(K) is the finite Heisenberg group.

Compare [Gieseker82] and [Mumford77]. We say that the n-th normalized Hilbert point of (Q_{0}, L_{0}) is Kempf-stable if it has a closed SL(N, k)-orbit.

Let M = e_{max}(K). The group G(K) is noncommutative of order MN^{2} and it is a central extension of K by μ_{M} with commutator form e_{K} where μ_{M} is the group of all M-th roots of unity. The group algebra k[H^{∨}] of the dual group H^{∨} over k is an irreducible G(K)-module of weight one, called the Schrödinger representation of G(K). The k-module Γ(Q, L)⊗k is isomorphic to k[H^{∨}] as a G(K)-module. If e_{min}(K) ≥ 3, any PSQAS (Q_{0}, L_{0}) over k is embedded into the projective space P(k[H^{∨}]) (Theorem 6.3). Hence any PSQAS (Q_{0}, L_{0}) over k has a linearlized action of G(K). Thus we are led to the notion of rigid G(K)-structures and level G(K)-structures on PSQAS’s generalizing the classical level structures on abelian varieties in order to formulate the moduli problem for PSQAS’s.

Theorem 0.4. Let K be a symplectic finite abelian group with e_{min}(K) ≥ 3. Let M = e_{max}(K). The functor of g-dimensional projectively stable quasiabelian schemes with level G(K)-structures is reductively-represented by a projective scheme SQ_{g,K} over Z[ζ_{M}, 1/M].

We prove Theorem 0.4 with the help of Theorem 0.3 and Schur’s lemma for irreducible G(K)-modules of weight one.

One dimensional PSQAS’s with rigid G((Z/3Z)^{⊕2})-structures are Hesse cubic curves coupled with the Schrödinger representation of G((Z/3Z)^{⊕2}). The well known family of Hesse cubic curves over P_{Z[ζ_{3},1/3]}^{1}

μ_{0}(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) − μ_{1}x_{0}x_{1}x_{2} = 0

is the universaly family of PSQAS’s with rigid G((Z/3Z)^{⊕2})-structures. In particular, SQ_{1,(Z/3Z)^{⊕2}} ≃ P_{Z[ζ_{3},1/3]}^{1}. The Hesse cubic curves exhaust all the possible Kempf-stable cubic curves up to PGL(3)-equivalence. Theorems 0.3 and 0.4 generalize these facts.

In Section 1 we will discuss Delaunay decompositions. In Section 2 we recall the basic facts about a degenerating family of abelian varieties. In Sections 3 and 4 we recall the degeneration data of a degenerating family of abelian varieties following [FC90] and the construction of (P_{0}, L_{0}). In Section 5 we will describe the structures of (Q_{0}, L_{0}). In Section 6 we prove that Γ(Q, L) ⊗ k is very-ample if e_{min}(K) ≥ 3, which is a natural generalization of Lefschetz’s theorem for abelian varieties. In Section 7 we define group schemes G(K), G(K) and a G(K)-module V (K) and discuss central extensions of K by G_{m}. In Section 8 we prove Theorem 0.2. In Sections 9 we define rigid G(K)-structures and level G(K)-structures on a PSQAS or their family. In Section 10 we prove Stable reduction theorem for PSQAS’s with rigid G(K)-structures. In Sections 11 and 12 we prove Theorem 0.3 and Theorem 0.4. We will present two basic examples of the reduced-fine-moduli schemes of one dimensional PSQAS’s in Section 13.

Acknowledgment . We would like to thank many mathematician colleagues for their assistance and stimulating discussions during the preparation of the article. We thank V. Alexeev for collaboration in [AN96] and T. Katsura (Tokyo Univ.) for proof of Theorem 6.2 respectively. We also thank T. Kajiwara (Tohoku Univ.), S. Mori (Kyoto Univ.), T. Saito (Tokyo Univ.), T. Sekiguchi (Chuo Univ.) and T. Terasoma (Tokyo Univ.) for discussions and advices. The present article was completed during the author’s stay in 1998 at Max-Planck Institut für Mathematik in Bonn. We would like to thank all the people in the institute for their hospitality.

1. Delaunay decompositions

Definition 1.1. Let Z_{0} (resp. Z_{+}, Q_{0}, R_{0}) be the set of all nonnegative integers (resp. positive integers, nonnegative rational numbers, nonnegative real numbers). Let X be a lattice of rank g, X_{R} = X ⊗ R, and let B : X × X → Z be a positive definite symmetric integral bilinear form B : X × X → Z, which determines a distance ∥ ∥_{B} on the Euclidean space X_{R} by ∥x∥_{B} := \sqrt{B(x, x)} (x ∈ X_{R}). For an arbitrary α ∈ X_{R} we say that a lattice element a ∈ X is α-nearest if

∥a − α∥_{B} = min{∥b − α∥_{B}; b ∈ X}

We define a (closed) B-Delaunay cell σ (or simply a Delaunay cell if B is understood) to be the closed convex hull of all lattice elements which are α-nearest for some α ∈ X_{R}. Note that for a given Delaunay cell σ, α is uniquely defined only if σ has the maximal possible dimension, equal to g. In this case we call α the centre of σ. Together all the B-Delaunay cells constitute a locally finite decomposition of X_{R} into infinitely many bounded convex polyhedra which we call the B-Delaunay decomposition Del_{B}. It is clear from the definition that the Delaunay decomposition is invariant under translation by the lattice X and that the 0-dimensional cells are precisely the elements of X.

Let Del := Del_{B}, Del(c) the set of all the Delaunay cells containing c ∈ X, and Del^{(i)}(c) the the set of all i-dimensional Delaunay cells in Del(c). Let σ ∈ Del be a Delaunay cell, and nσ := {x ∈ X_{R}; x/n ∈ σ}. Let Star (0) be the union of all σ ∈ Del such that σ ∈ Del(0), and n Star (0) := {x ∈ X_{R}; x/n ∈ Star (0)}. For σ ∈ Del(0), let C(0, σ) be the closed cone over R_{+} generated by σ ∩ X.

A g-dimensional cone σ ∈ Del is called totally generating if the semi group C(0, σ)∩X is generated by Delaunay vectors σ ∩X. If a g-dimensional cone σ ∈ Del is the union of totally generating g-dimensional Delaunay cones with disjoint interiors, then we call σ simplicially generating. See [AN96].

Lemma 1.2. Let B be a symmetric positive definite integral bilinear form on X × X, Del_{B} the corresponding Delaunay decomposition of X_{R} and z (≠ 0) ∈ X.

1. Let σ ∈ Del_{B} be a Delaunay cell. Then σ ∩ (nz + σ) = ∅ for n ≥ 2.

2. Star (0) ∩ (nz + Star (0)) = ∅ if n ≥ 3.

Proof. Let α(σ) be the centre of σ. Suppose that c ∈ σ∩X and d = c+nz ∈ σ for some nonzero z ∈ X. Since σ is convex, c + z ∈ σ. Hence ∥c − α(σ)∥_{B} = ∥c + z − α(σ)∥_{B} = ∥c + nz − α(σ)∥_{B}. It follows (n^{2} − n)B(z, z) = 0, a contradiction. This proves (1).

Let z ∈ X. We choose all the maximal dimensional Delaunay cells σ_{i} (i ∈ I) in Del_{B} containing 0 such that z belongs to the closed cone C(0, σ_{i}) (i ∈ I). Let C be the union of C(0, σ_{i}) (i ∈ I). Then z belongs to the interior of C and z+C ⊂ C because z+C(0, σ_{i}) ⊂ C(0, σ_{i}). Hence nz+C ⊂ C for any n ≥ 1. Let σ be any maximal dimensional Delaunay cell in Star (0). Then z ∈ z + σ, hence the interior of z + σ contains an interior point of C. This implies that z + σ ⊂ C, since C is the union of (infinitely many) Delaunay cells. It follows that z + Star (0) ⊂ C, hence nz + Star (0) ⊂ (n − 1)z + C for any n ≥ 1. Meanwhile we have

Star (0) ∩ C = ∪_{i∈I} Star (0) ∩ C(0, σ_{i}) = ∪_{i∈I}σ_{i}
Star (0) ∩ (2z + C) = Star (0) ∩ C ∩ (2z + C) = ∪_{i∈I}σ_{i} ∩ (2z + C)
σ_{i} ∩ (2z + C) = σ_{i} ∩ C(0, σ_{i}) ∩ (2z + C)
= σ_{i} ∩ (2z + C(0, σ_{i})) = σ_{i} ∩ (2z + σ_{i}) = ∅

by (1). Therefore Star (0)∩(3z+Star (0)) ⊂ Star (0)∩(2z+C) = ∅. Similarly Star (0) ∩ (nz + Star (0)) ⊂ Star (0) ∩ ((n − 1)z + C) = ∅ for n ≥ 3.

Lemma 1.3. Let σ ∈ Del_{B} be a maximal Delaunay cell containing the origin 0. Let σ ∩ X = {a_{i}; 0 ≤ i ≤ r}, a_{0} = 0 and y = ∑_{i=1}^{r} n_{i}a_{i}, n_{i} ∈ Z_{0}. Then

1. B(a_{i}, a_{j}) ≥ 0. B(a_{i}, a_{j}) = 0 if and only if a_{i} + a_{j} ∈ σ ∩ X.

2. B(y, y) ≥ ∑_{i=1}^{r} n_{i}^{2}B(a_{i}, a_{i}) ≥ ∑_{i=1}^{r} n_{i}B(a_{i}, a_{i}) = 2B(α(σ), y).

3. B(y, y) = 2B(α(σ), y) holds if and only if y ∈ σ ∩ X.

4. B(y, y) > 2B(α(σ), y) if some n_{i} ≥ 2.

Proof. Since B(a_{i} + a_{j}, a_{i} + a_{j}) ≥ 2B(α(σ), a_{i} + a_{j}) = B(a_{i}, a_{i}) + B(a_{j}, a_{j}), equality holds if and only if a_{i} + a_{j} ∈ σ ∩ X. (2) follows from (1). And (3) and (4) follow from (2).

Lemma 1.4. Let σ ∈ Del(0) and b ∈ C(0, σ) ∩ X. If b ∉ σ ∩ X, then there is a ∈ σ ∩ X such that B(b − a, a) > 0.

Proof. Let b ∈ C(0, σ) ∩ X. We assume B(b − a, a) ≤ 0 for any a ∈ σ ∩ X. Then we prove b ∈ σ ∩ X. For this let b = ∑_{i=1}^{r} r_{i}a_{i} for a_{i} ∈ σ ∩ X and some r_{i} ∈ Q_{0}. We see

B(b, b) = ∑_{i=1}^{r} r_{i}B(b, a_{i}) ≤ ∑_{i=1}^{r} r_{i}B(a_{i}, a_{i}) = 2B(α(σ), b)

whence B(b, b) = 2B(α(σ), b). It follows b ∈ σ ∩ X.

1.5. Reduction theory. Let R_{+} be the set of nonnegative real numbers. We denote by Y (resp. Y_{≥0}, Y_{+}) the set of all symmetric (resp. symmetric positive semi-definite, symmetric positive definite) g×g matrices. The group GL(g, R) acts on Y and keeping Y_{≥0} and Y_{+} stable as y ↦ uyu^{t}. It is known that the action on Y_{+} is transitive.

For g ≤ 4, [Igusa67, Lemmas 4,5] gives a fundamental domain F of Y_{+} with respect to the action of GL(g, Z) explicitly. If g = 2, 3, then let

C = C_{g} := {y = (y_{ij}) ∈ Y_{+}; ∑_{j=1}^{g} y_{ij} ≥ 0, y_{ij} ≤ 0 (i ≠ j)}

Then C is a fundamental domain of Y_{+} with respect to GL(g, Z), and any Delaunay decomposition is subdivided by (a GL(g, Z)-conjugate of) Del(C). Since any maximal-dimensional cell σ ∈ Del(C) containing 0 has volume 1/(g + 1)!, it is totally generating. Hence any maximal-dimensional cell of any g (= 2, 3)-dimensional Delaunay decomposition Del_{B} is simplicially generating.

If g = 4, then with the notation of [Igusa67, Lemmas 4,5] the fundamental domain F is covered with C, F_{ab}, F_{ab,cd} or their transforms by GL(4, Z). Let Del be one of the Delaunay decompositions Del(C), Del(F_{ab}) and Del(F_{ab,cd}). Then by [Voronoi09] any maximal-dimensional Delaunay cell σ ∈ Del has volume 1/5!, whence it is totally generating. If B is a symmetric positive definite integral 4×4-matrix, then we may assume B to belong to one of the closed convex cones C, F_{ab} and F_{ab,cd} up to GL(4, Z)-conjugacy. If B is not in the interior of the cones, then one of Del(C), Del(F_{ab}) and Del(F_{ab,cd}) is a subdivision of Del_{B}, hence Del_{B} is simplicially generating.

Theorem 1.6. If the dimension is less than five, any Delaunay decomposition is simplicially generating, in particular the nilpotency [AN96, Definition 1.15] of any Delaunay decomposition is equal to one.

2. The flat closure of K(L_{η})

The purpose of this section is to recall the Raynaud sequence and the flat closure of a certain finite commutative group scheme K(L_{η}) for a degenerating family of abelian varieties. We quote these from Faltings-Chai [FC90, II.4.1,5.1] and Moret-Bailly [MB85, IV-VI]. We follow mainly the notation of [FC90].

2.1. Grothendieck’s stable reduction. Let R be a complete discrete valuation ring, I the maximal ideal of R and S = Spec R. Let η be the generic point of S, k(η) the fraction field of R and k = R/I the residue field.

Suppose that we are given an abelian scheme G_{η} over k(η). Then by Grothendieck’s stable reduction theorem [SGA7] (See also [SGA7, Expóse I pp. 1-24]) G_{η} can be extended to a semiabelian S-scheme G as the connected Néron model of G_{η} by taking a finite extension K′ of k(η) if necessary, ^{1} in other words, by taking the integral closure of R in K′ and by repeating the same construction. Moreover by taking a finite extension of k(η) if necessary there exists H ∈ Pic^{0}(G_{η}) such that L_{η} ⊗ H is symmetric, namely i^{∗}(L_{η} ⊗ H) = L_{η} ⊗ H for the involution i = [−1]_{G_{η}} of G_{η}.

1We mean by the connected Néron model the identity connected component of the Néron model (with connected closed fibre).

Therefore we may assume from the start that L_{η} is symmetric, ample and rigidified along the unit section. The invertible sheaf L_{η} extends to G because it is associated with some Cartier divisor, which always extends to a smooth scheme G. The extension of L_{η} is unique because G_{0} is irreducible. See [MB85, II, 1.1]. On the other hand by [Raynaud70, p.158 XI, 1.2 and p.170 XI 1.13] L_{η}^{⊗n} extends to G as an ample invertible sheaf for some n > 0 if L_{η} is symmetric and ample. Since L_{η} satisfies the condition in this case, the extension L is symmetric and ample.

Thus for a given an abelian scheme G_{η} over k(η) of relative dimension g we have a semiabelian S-scheme G with generic fibre G_{η} (with a chosen unit section), L a rigidified relatively ample invertible sheaf on G. The special fibre G_{0} is a semiabelian scheme over k, namely an extension of an abelian scheme A_{0} of relative dimension g′ by a torus T_{0} of relative dimension g′′, g′ + g′′ = g. We may assume T_{0} to be split by taking a finite base change of S.

2.2. The Raynaud sequence. Associated to G and L are the formal scheme G_{for} = lim G ⊗ R/I^{n} and an invertible sheaf L_{for} = lim L ⊗ R/I^{n}. The scheme G_{for} fits into an exact sequence

(1) 1 → T_{for} → G_{for} →^{π_{for}} A_{for} → 0

where A_{for} is a formal abelian scheme. By the theory of cubical structures [Breen83] [MB85, II, 1.1 (ii)] there exists a unique cubical structure on L (viewed as a G_{m}-torsor), which induces a cubical structure of the sheaf L_{for}. Then L_{for} is descended to a unique cubical ample invertible sheaf M_{for} on A_{for} because T_{for} is a split torus, that is, L_{for} = π_{for}^{∗}(M_{for}). Since there exists an ample sheaf on A_{for}, A_{for} is algebraizable. In other words by [EGA, III, 5.4.5] there exists an abelian S-scheme A with an ample invertible sheaf M such that the formal completion of (A, M) along a closed fibre is (A_{for}, M_{for}).

By our assumption that T_{0} is a k-split torus, T_{for} is a formal S-split torus by [SGA3, IX, 3.6], [FC90, Chap. I, 2.2]. Let X := Hom_{Z}(T_{for}, G_{m,for}) be the character group of T_{for} and T := Hom_{Z}(X, G_{m,S}). Then T algebraizes T_{for}.

The sequence 1 → T_{for} → G_{for}→A_{for} → 0 is algebraizable because we have Ext(A_{for}, T_{for}) ≃ Ext(A, T) [FC90, p.34]. The dual abelian scheme G_{η}^{t} is also extended to a semiabelian S-scheme G^{t} in a similar manner. Moreover there is a similar exact sequence for G_{for}^{t}; 1 → T_{for}^{t} → G_{for}^{t} → A_{for}^{t} → 0. Then G_{for}^{t}, T_{for}^{t} and A_{for}^{t} are algebraizable. Let Y := Hom(T^{t}, G_{m,S}). Thus there exist two group schemes G̃ and G̃^{t} with (G̃)_{for} ≃ G_{for} and (G̃^{t})_{for} ≃ G_{for}^{t} and the following exact sequences (called Raynaud sequences) which algebraize the above sequences of the formal group schemes

1 → T → G̃ →^{π} A → 0 (2)

1 → T^{t} → G̃^{t} →^{π^{t}} A^{t} → 0 (3)

plus the homomorphisms c : X → A^{t}(S), c^{t} : Y → A(S) decoding them. In other words, c ∈ Hom(X, A^{t}(S)) ≃ Ext(A, T) and c^{t} ∈ Hom(Y, A(S)) ≃ Ext(A^{t}, T^{t}) describe the extension classes of semiabelian schemes G̃ and G̃^{t} respectively. Moreover A^{t} is a dual abelian scheme of A by [FC90, p. 35]. See also [MB81, Appendice, Théoréme].

Let λ(L_{η}) : G_{η} → G_{η}^{t} be the polarization morphism. Then by the universal property of the (connected) Néron model G^{t} of G_{η}^{t} we have an extension λ : G → G^{t} of λ(L_{η}). Since G is semi-stable, λ is surjective, flat and quasi-finite by [MB85, IV, 3.4]. This gives rise to a formal surjective flat morphism λ_{for} : G_{for} → G_{for}^{t}, which induces a formal morphism λ̃_{for} : G̃_{for} → G̃_{for}^{t} as well as surjective formal morphisms λ_{T_{for}} : T_{for} → T_{for}^{t} and λ_{A_{for}} = λ(M_{for}) : A_{for} → A_{for}^{t}. Hence we have an injective homomorphism ϕ : Y → X and we can algebraize λ_{T_{for}} into a morphism λ_{T} : T → T^{t}. We also have a morphism λ_{A} := λ(M) : A → A^{t} which algebraizes λ_{A_{for}}. Since λ̃_{for} is a formal morphism, it follows that c_{for}ϕ = λ_{A_{for}}c_{for}^{t}, hence cϕ = λ_{A}c^{t}. Thus λ̃_{for} is algebraized into a morphism λ̃ : G̃ → G̃^{t} compatible with λ_{T} and λ_{A}, that is, λ_{T} = λ̃_{|T}, λ_{A}π = (π^{t})λ̃.

2.3. Flat closures. Let _{n}G := ker(n · id_{G} : G → G), and _{n}T := ker(n · id_{T} : T → T). Since G is smooth, n id_{G} is S-flat for any positive integer n, therefore _{n}G is a flat closed subscheme of G. Hence _{n}G is the flat closure of nG_{η} in G ([EGA, IV, 2.8.5]). There exists the maximal finite flat subgroup scheme _{n}G^{f} of _{n}G by [EGA, II, 6.2.6]. In fact, since _{n}G̃ is S-finite and nG_{for} ≃ _{n}G̃_{for}, _{n}G_{for} is a finite formal subgroup scheme _{n}G_{for} of G_{for}. There exists by [EGA, III, 5.1.8] an S-subscheme _{n}G^{f} of a quasi-projective Sscheme G such that (_{n}G^{f})_{for} ≃ _{n}G_{for} ≃ _{n}G̃_{for}. It is clear that _{n}G^{f} is maximal.

Similarly we see that there exists a finite flat subgroup scheme _{n}G^{m} of nG such that _{n}G_{for}^{m} ≃ _{n}T_{for} ⊂ _{n}G̃_{for} where _{n}T_{0} := T_{0} ∩_{n}G. We see that _{n}G^{m} is maximal among finite flat subgroup S-schemes H of G with H_{0} ⊂ _{n}T_{0}. In fact, since H_{for} is S-finite (hence S-prjective), the image H̄_{for} of H_{for} in A_{for} is also an S-finite projective group subscheme of A_{for}. Since H̄_{0} is a reduced single point of A_{0} by H_{0} ⊂ T_{0}, Γ(OH̄_{for}) is an R-module of rank at most one, hence H̄_{for} is trivial. It follows that H_{for} ⊂ T_{for}. The Cartier dual of H_{for} is therefore a finite abelian group, a quotient group of X = Hom(T_{for}, G_{m}) of finite order, where we recall that the Cartier dual of H_{for} is given by Hom_{R}(H_{for}, G_{m,for}). See [Oort66, Theorem 16.1]. Hence there exists a sublattice Z of X of finite index such that H_{for} = Hom(X/Z, G_{m,S}) = Spf R[w^{x}, x ∈ X]/(w^{z} − 1, z ∈ Z). Since nX ⊂ Z by the assumption H_{0} ⊂ _{n}G_{0}^{m}, we see H_{for} ⊂ _{n}G_{for}^{m}, hence H ⊂ _{n}G^{m}. This proves that _{n}G^{m} is maximal. See also [SGA7, IX, 6.1].

Let K(L_{η}) be the kernel of λ(L_{η}), K_{S}(L_{η}) the flat closure of K(L_{η}) in G, and K_{S}(L_{η})^{f} the maximal flat finite (hence S-proper) subscheme of K_{S}(L_{η}). The subscheme K_{S}(L_{η})^{f} is flat because K_{S}(L_{η})^{f} is open and closed in K_{S}(L_{η}) by the construction of flat closures in [EGA, IV, 2.8.5]. Hence K_{S}(L_{η})^{f} is the unique maximal finite flat subgroup S-scheme of K_{S}(L_{η}). There also exists a unique finite flat subgroup scheme K_{S}(L_{η})^{m} of K_{S}(L_{η}) such that (K_{S}(L_{η})^{m})_{0} = T_{0}∩K_{S}(L_{η}) [SGA7, IX, 6.1]. See also the argument for maximality of _{n}G^{m}. Clearly K_{S}(L_{η})^{m} ⊂ K_{S}(L_{η})^{f}.

Since there exists a positive integer n such that K_{S}(L_{η}) ⊂ _{n}G we see K_{S}(L_{η})^{m} = K_{S}(L_{η}) ∩ (_{n}G)^{m} by maximality of K_{S}(L_{η})^{m} and (_{n}G)^{m}. Hence the sequence

1 → K_{S}(L_{η})^{m} = K_{S}(L_{η}) ∩ (_{n}G)^{m} → (_{n}G)^{m} → G^{t} (4)

is exact, and therefore 1 → K_{S}(L_{η})_{for}^{m} → (_{n}G)_{for}^{m} → G_{for}^{t} is exact. Since (_{n}G)_{for}^{m} ≃ (_{n}T)_{for}, we have by [EGA, III, 5.4.5] a unique finite flat subgroup scheme K_{S}(L_{η})^{♮} of T such that K_{S}(L_{η})_{for}^{♮} ≃ K_{S}(L_{η})_{for}^{m}. We also have an exact sequence

1 → K_{S}(L_{η})^{♮} → T →^{λ_{T}} T^{t} → 1 (5)

It follows from [EGA, III, 5.4.1] that

X(K_{S}(L_{η})^{m}) ≃ X((K_{S}(L_{η})_{for}^{m}) ≃ X(K_{S}(L_{η})^{♮}) ≃ X/ϕ(Y) (6)

where X(H) := Hom_{S}(H, G_{m,S}) is the Cartier dual of H.

Since G_{0} is connected, K_{S}(L_{η})^{fn} := K_{S}(L_{η})^{f} ∩ G = K_{S}(L_{η})^{f} with the notation of [MB85, IV, 2.4]. We define K_{S}(L_{η})^{ab} := K_{S}(L_{η})^{fn}/K_{S}(L_{η})^{m}. Let e^{L_{η}} : K(L_{η}) × K(L_{η}) → G_{m,η} be the Weil pairing of L_{η} and e_{S}^{L_{η}} : K_{S}(L_{η})×K_{S}(L_{η}) → G_{m,S} the natural extension of e^{L_{η}}. By [MB85, IV, 2.4] e_{S}^{L_{η}} is an alternating bilinear form.

By [MB85, IV, 2.4, 4.3] K_{S}(L_{η})^{fn} = (K_{S}(L_{η})^{m})^{⊥}, the orthogonal complement of K_{S}(L_{η})^{m} with respect to e_{S}^{L_{η}}, while we have K_{S}(L_{η})^{ab} ≃ K_{S}(M) := ker λ_{A}(M), a finite subgroup scheme of (A, M). Thus e_{S}^{L_{η}} induces a perfect pairing on K_{S}(M), which is just the Weil pairing e^{M} of (A, M) [MB85, IV, 2.4 (v)]. Since K_{S}(L_{η})^{m} and K_{S}(M) are finite and flat over S, their structure sheaves are finite R-free modules, whose ranks are defined to be d_{t} and d_{a}^{2} respectively. Let d := d_{a}d_{t}.

Since e^{L_{η}} is a perfect pairing of K(L_{η}), the quotient K(L_{η})/K_{S}(L_{η})^{fn} ⊗ k(η) is dual to K_{S}(L_{η})^{m} ⊗ k(η). Hence we have

(7) ord K(L_{η}) = ord K_{S}(L_{η})^{m} ord(K_{S}(L_{η})^{fn}).

On the other hand, ord K_{S}(L_{η})^{fn} = ord K_{S}(M) · ord K_{S}(L_{η})^{m} = d_{a}^{2}d_{t}. Hence ord K(L_{η}) = d^{2}. In particular, if d^{2} = ord K(L_{η}) is prime to the characteristic of k := R/I, then d_{a} and d_{t} are prime to it too.

By [MB85, V, 1.4.2] Γ(G, L) is a finite torsion-free R-module, hence R-free therefore we have rank_{R} Γ(G, L) = rank_{k(η)} Γ(G_{η}, L_{η}) = d by [Mumford74, §23, p. 234]. See also [Mumford74, §23, pp.221-235].

2.4. The Heisenberg group. Let A be an abelian scheme over S, and L an ample invertible sheaf on A. We define F (L) to be the set of all pairs (x, ϕ) where x ∈ A(S) and ϕ is an isomorphism L → T_{x}^{∗}(L). The group law on F (L) is defined to be (x, ϕ) · (y, ψ) = (x + y, T_{y}^{∗}(ϕ) ∘ ψ) where T_{y}^{∗}(ϕ) is understood as an isomorphism of T_{y}^{∗}(L) onto T_{x+y}^{∗}(L). For any S-scheme S′ we define also F (L_{S′}) similarly. Then we define a functor \underline{G}(L) from the category of noetherian S-schemes to the category of groups by

\underline{G}(L)(S′) := F (L_{S′}) (8)

We recall that for any S-scheme S′, K(L)(S′) is the set of all x ∈ A(S′) such that T_{x}^{∗}(L) ⊗ L^{−1} ∈ Pic(S′). Therefore there is a natural homomorphism π : \underline{G}(L)(S′) → K(L)(S′), whose kernel is easily seen to be Γ(S′, O_{S′}^{∗}) because any fibre of A over S is proper and integral.

Now we go back to the previous situation where G is a semi-abelian scheme over S with G_{η} an abelian scheme over k(η), and L is an ample invertible sheaf on G. Let L^{tsr} be the G_{m}-torsor on G associated with the invertible sheaf L. Let G(L_{η}) := (L_{η}^{tsr})_{|K(L_{η})} (which we call the Heisenberg group of L_{η}). Then G(L_{η}) is a group k(η)-scheme representing the functor \underline{G}(L_{η}) by [MB81, 3.2]. It is by [MB81, 3.2] a central extension of K(L_{η}) by the k(η)-split torus G_{m,η} with the commutator form e^{L_{η}};

e^{L_{η}}(ḡ, h̄) = [g, h] := ghg^{−1}h^{−1}, for ∀g, h ∈ G(L_{η}) (9)

where ḡ, h̄ are the images of g and h in K(L_{η}). Hence the sequence

1 → G_{m,η} → G(L_{η}) → K(L_{η}) → 0 (10)

is exact. By [Mumford66, p. 295] G(L_{η}) acts on Γ(G_{η}, L_{η}) by

U_{z}(s) = T_{−a}^{∗}(ϕ(s)) for s ∈ Γ(L_{η}) (11)

where z = (a, ϕ) ∈ G(L_{η}). By [MB85, V, 2.5.5] (See also [Mumford74, §23]), Γ(G_{η}, L_{η}) is an irreducible G(L_{η})-module of weight one, unique up to isomorphism by taking a finite extension of k(η) if necessary. The G(L_{η})- module structure of Γ(G_{η}, L_{η}) is known by [MB85, V, 3.4] and [Mumford74, §23] regardless of the characteristic of k(η) and ord K(L_{η}).

Since K_{S}(L_{η})^{m} is totally isotropic, there exists an étale finite cover S′ of S such that the cubical sheaf L_{S′} on G_{S′} descends to a cubical sheaf on the quotient scheme G_{S′}/K_{S′}(L_{η})^{m} [MB85, VI, 1.3] where K_{S′}(L_{η})^{m} := K_{S}(L_{η})^{m} ×_{S} S′. In fact, L_{S′} with K_{S′}(L_{η})^{m} action is a descent data. This means that the action of K_{S′}(L_{η})^{m} on G_{S′} lifts to the action on L_{S′} hence that on Γ(G_{S′}, L_{S′}). We (may) assume from the start that K_{S}(L_{η})^{m} acts on Γ(G, L). By [MB85, V, 3.4] Γ(G, L) is the direct sum of K_{S}(L_{η})^{m}- eigenspaces Γ(G, L)χ̄ (χ̄ ∈ X/ϕ(Y) = X(K_{S}(L_{η})^{m})) and rank_{R} Γ(G, L)χ̄ = d_{a} for any χ̄.

3. (P, L) - the totally degenerate case

3.1. Fourier expansions. First we consider the totally degenerate case, that is the case where A_{0} (and hence A) is trivial. Then G_{for} = T_{for} and G̃ = T. We choose and fix the coordinate w^{x} (x ∈ X) of T satisfying w^{x}w^{y} = w^{x+y} (∀x, y ∈ X). The invertible sheaf L_{for} is trivial on T_{for}, and therefore

Γ(G_{η}, L_{η}) = Γ(G, L) ⊗ k(η) ↪ Γ(G_{for}, L_{for}) ⊗ k(η) ↪ ∏_{x∈X} k(η) · w^{x}

Therefore, any element θ ∈ Γ(G_{η}, L_{η}) can be written as a formal Fourier series θ = ∑_{x∈X} σ_{x}(θ)w^{x} with σ_{x}(θ) ∈ k(η), which converges in the I-adic topology.

Theorem 3.2. [Faltings–Chai90] There exists a function a : Y → k(η)^{∗} and a bilinear function b : Y × X → k(η)^{∗} with the following properties:

1. b(y, x) = a(x + y)a(x)^{−1}a(y)^{−1}, a(0) = 1 (∀x ∈ X, y ∈ Y)

2. b(y, z) = b(z, y) = a(y + z)a(y)^{−1}a(z)^{−1} (∀y, z ∈ Y)

3. b(y, y) ∈ I (∀y ≠ 0), and for every n ≥ 0, a(y) ∈ I^{n} for almost all y ∈ Y

4. Γ(G_{η}, L_{η}) is identified with the k(η) vector subspace of formal Fourier series θ that satisfy σ_{x+ϕ(y)}(θ) = a(y)b(y, x)σ_{x}(θ) and σ_{x}(θ) ∈ k(η) (∀x ∈ X).

For θ ∈ Γ(G_{η}, L_{η}), θ ∈ Γ(G, L) if and only if θ ∈ Γ(G_{for}, L_{for}). Hence Γ(G, L) is an R-submodule of Γ(G_{η}, L_{η}) consisting of formal Fourier series θ with σ_{x}(θ) ∈ R (∀x ∈ X). Moreover Γ(G, L)χ̄ is an R-free module of rank one and any element θ of Γ(G, L)χ̄ ≃ Γ(G_{for}, L_{for})χ̄ is written as θ = ∑_{x∈χ̄+ϕ(Y)} σ_{x}(θ)w^{x} for some σ_{x}(θ) ∈ R.

Definition 3.3. By taking a finite base change of S if necessary the functions b and a can be extended respectively to X × X and X so that the previous relations between b and a are still true on X × X. Then we can define the integer-valued functions A : X → Z, B : X × X → Z and b̄(y, x) ∈ R^{∗}, ā(y) ∈ R^{∗} by

B(y, x) = val_{s}(b(y, x)), dA(α)(x) = B(α, x) + r(x)/2
A(x) = val_{s}(a(x)) = B(x, x)/2 + r(x)/2
b(y, x) = b̄(y, x)s^{B(y,x)}, a(x) = ā(x)s^{(B(x,x)+r(x))/2}

for some r ∈ Hom_{Z}(X, Z). We set a_{0} = ā mod I and b_{0} = b̄ mod I. Therefore a_{0}(x), b_{0}(y, x) ∈ k^{∗} for any x, y ∈ X. B is positive definite by Theorem 3.2 (3).

Definition 3.4. For simplicity we identify ϕ : Y → X with the inclusion Y ⊂ X. We define

R̃ : = R[a(x)w^{x}ϑ; x ∈ X] ≃ R[ξ_{x}ϑ; x ∈ X],
ξ_{x} : = s^{B(x,x)/2+r(x)/2}w^{x}, ξ_{x,c} := ξ_{x+c}/ξ_{c}
ζ_{x,c} : = s^{B(α(σ),x)+r(x)/2}w^{x} (x + c ∈ C(c, σ))

where R̃ is the graded algebra with deg(a(x)w^{x}ϑ) = 1 and deg a = 0 for a ∈ R, while σ ∈ Del_{B}(c) is a maximal-dimensional Delaunay cell with x + c ∈ C(c, σ).

Let Q̃ := Proj(R̃), and P̃ the normalization of Q̃. We define an action S_{y} on Q̃ by

S_{y}^{∗}(a(x)w^{x}ϑ) = a(x + y)w^{x+y}ϑ for y ∈ Y (12)

S_{y} induces a natural action on P̃, which we denote by the same S_{y}. We denote by L̃ O_{Proj}(1) on Q̃ as well as its pull back to P̃.

Our construction of (P, L) is the same as in [Mumford72].

Theorem 3.5. The quotient (P̃_{for}, L̃_{for})/Y is a flat projective formal Sscheme. There exists a flat projective S-scheme (P, L) such that the formal completion (P_{for}, L_{for}) of it along the closed fibre is isomorphic to (P̃_{for}, L̃_{for})/Y.

Proof. This follows from [EGA, III, 5.4.5] and [Mumford72].

Definition 3.6. Let Del(P_{0}) be the Delaunay decomposition corresponding to P_{0}. By taking a finite base change of S if necessary, we may assume that dA(α(σ)) ∈ Hom(X, Z) for any Delaunay cell σ ∈ Del(P_{0}). By [AN96] this implies that P_{0} is reduced. This also follows from Theorem 3.7. Then we call the closed fibre (P_{0}, L_{0}) of the flat projective family (P, L) a polarized stable quasi-abelian variety over k := R/I.

Theorems 3.7 and 3.8 describe the structures of the closed fibres P̃_{0} and P_{0} in terms of Del(P_{0}). If P_{0} is reduced, G is realized as an open subscheme of P. See 4.12. In what follows we always assume that dA(α(σ)) ∈ Hom(X, Z) for any σ ∈ Del := Del(P_{0}). Hence P_{0} is reduced.

Theorem 3.7. Let (P̃_{0}, L̃_{0}) be the closed fibre of (P̃, L̃) and ζ̄_{x,c} := ζ_{x,c} ⊗ k the restriction to P_{0}. Then

1. P̃_{0} is covered with finitely many open affine subschemes of finite type

U_{0}(c) := Spec k[ζ̄_{x,c}; x ∈ X] (c ∈ X).

2. R_{0}(c) := k[ζ̄_{x,c}; x ∈ X] is a k-algebra of finite type. Let x_{i} ∈ X. If x_{i}’s are cellmates (resp. otherwise), then

ζ̄_{x_{1},c} ⋯ ζ̄_{x_{m},c} = ζ̄_{x_{1}+···+x_{m},c} (resp. 0)

Theorem 3.8. Let (P_{0}, L_{0}) be the closed fibre of (P, L). Let σ and τ be Delaunay cells in Del(P_{0}).

1. For each σ there exists a T -invariant subscheme O(σ) of the closed fibre P_{0} which is a torus of dimension dim σ over k,

2. σ ⊂ τ iff O(σ) is contained in O(τ),

3. For each σ there exists a unique closed subscheme V (σ) which is the closure of O(σ) and has dimension dim σ over k,

4. σ ⊂ τ iff V (σ) is contained in V (τ),

5. P_{0} = ⋃_{σ∈Del(P) mod Y} V (σ)

See [Namikawa76], [AN96].

We will describe Γ(P_{0}, L_{0}^{n}) explicitly. Compare [AN96].

Theorem 3.9. If n > 0, h^{0}(P_{0}, L_{0}^{n}) = [X : Y]n^{g}, h^{i}(P_{0}, L_{0}^{n}) = 0 (i > 0), and

Γ(P_{0}, L_{0}^{n}) = \{∑_{x∈X} c(x)η_{n}(x); \begin{matrix}c(x + ny) = b_{0}(y, x)a_{0}(y)^{n}c(x) \\ c(x) ∈ k, ∀x ∈ X, ∀y ∈ Y \end{matrix}\}

Proof. Let Del(P_{0}) = Del_{B} for some integral matrix B. By Theorem 3.7 P̃_{0} is covered with open affine sets U_{0}(c) (c ∈ X). For any Delaunay cell σ ∈ Del, let V (σ) be the closure of a torus orbit O(σ). It is a closed subscheme of P̃_{0} and a normal complete torus embedding by Theorem 3.7 or Theorem 3.8. Let V (σ, c) := V (σ) ∩ U_{0}(c) for c ∈ σ and C(c, σ) the cone spanned by edges of σ at c. In order to avoid complication we write ζ_{x,c} instead of ζ̄_{x,c}. Then we easily see V (σ, c) = Spec k [ζ_{x,c}; x + c ∈ C(c, σ) ∩ X]. We define

η_{n}(x) := ζ_{x−nc,c}ξ_{c}^{n} = s^{B(α(σ),x−nc)+nB(c,c)/2+r(x)/2}w^{x}

for \frac{x}{n} ∈ σ ∩ \frac{X}{n}. Note that (x − nc) + c ∈ C(c, σ). The definition of η_{n}(x) depends on x and n, but does not depend on the choice of a Delaunay cell σ containing \frac{x}{n} and a vertex c ∈ X of σ. If we need to specify dependence on n or L^{n}, we write η_{L^{n}}(x) or simply η_{n}(x). We note η_{1}(x) = η_{L}(x) = ξ_{x} := s^{(B(x,x)+r(x))/2}w^{x} (∀x ∈ Star (0) ∩ X) and that ζ_{x,c} = s^{B(α(σ),x)+r(x)/2}w^{x} if x + c ∈ C(c, σ).

Now we fix n and compute H^{i}(V (σ), L_{0}^{n}). Let i = 0. The line bundle L restricted to V (σ) is defined by a one-cocycle {ξ_{d}/ξ_{c}}_{c,d∈σ∩X} with respect to an affine covering {V (σ) ∩ U_{0}(c); c ∈ σ ∩ X}. Let ϕ = {ϕ_{c}} ∈ H^{0}(V (σ), L_{0}^{n}) with ϕ_{c} = (ξ_{d}/ξ_{c})^{n}ϕ_{d}. Let ϕ = ϕ_{c}ξ_{c}^{n} = ϕ_{d}ξ_{d}^{n}. Then we easily see that

ϕ = ∑_{\frac{x}{n} ∈C(c,σ)∩ \frac{X}{n}} c(x)ζ_{x−nc,c}ξ_{c}^{n} = ∑_{\frac{x}{n} ∈ ∩_{d∈σ∩X}C(d,σ)∩ \frac{X}{n}} c(x)ζ_{x−nd,d}ξ_{d}^{n}

for some constants c(x) ∈ k independent of d ∈ σ. It follows that

H^{0}(V (σ), L_{0}^{n}) = \{ ∑_{\frac{x}{n} ∈σ∩ \frac{X}{n}} c(x)η_{n}(x); c(x) ∈ k\}

because σ = ∩_{d∈σ∩X}C(d, σ). It follows that L_{0}^{n} (n ≥ 1) is generated by global sections on V (σ). We see that the restriction L_{0} on V (σ) is ample, because L_{0} on the inverse image of V (σ) in P̃_{0}/Y′ is very ample by taking a smaller sublattice Y′ of Y if necessary. By [Oda88], H^{i}(V (σ), L_{0}^{n}) = 0 for i > 0.

Let Del^{(i)} be the set of all i-dimensional cells of Del. We see that there is an exact sequence on P̃_{0}

0 → OP̃_{0} → ⊕O_{V (σ_{g})} →^{∂_{g}} ⋯ →^{∂_{2}} ⊕O_{V (σ_{1})} →^{∂_{1}} ⊕O_{V (σ_{0})} → 0

where σ_{i} ∈ Del^{(i)}. The homomorphism ∂_{i} : ⊕O_{V (σ_{i})} → ⊕O_{V (σ_{i−1})} is defined by ∂_{i}(⨁_{σ} ϕ_{σ}) = ⨁_{τ} ∑_{τ⊂σ}[σ : τ]ϕ_{σ} where the summation runs over the set of all σ containing a fixed τ as a face of codimension one, while the direct sum ⨁_{τ} is over the set Del^{(i−1)}. Any Delaunay cell σ is here assumed to be oriented and [σ : τ] (= ±1) is the incidence number of σ relative to its face τ of codimension one.

From now we assume by choosing a sublattice Y′ of Y of finite index if necessary that S_{y}(V (σ))∩V (σ) = ∅ for any Delaunay cell σ and y ∈ Y \{0}. We prove H^{i}(P_{0}, L_{0}^{n}) = 0 for i > 0 under the assumption. We note that the general case follows from it easily.

From the above sequence tensored by L_{0}^{n}, we obtain a complex with a Y -action

0 → ⊕H^{0}(V (σ_{g}), L_{0}^{n}) →^{∂_{g}} ⋯ →^{∂_{1}} ⊕H^{0}(V (σ_{0}), L_{0}^{n}) → 0

Let W_{i} := ⨁_{σ∈Del^{(g−i)}} H^{0}(V (σ), L_{0}^{n}). By the assumption on Y, we have

H^{i}(⊕_{σ_{j} mod Y} L_{0}^{n} ⊗ O_{V (σ_{j})}) ≃ ⊕_{σ_{j} mod Y} H^{i}(V (σ_{j}), L_{0}^{n}) = 0 (i > 0, ∀j)

It follows

H^{i}(P_{0}, L_{0}^{n}) ≃ H^{i}(⨁ W_{∗}^{Y -inv}) ≃ H^{i}(  ⨁_{σ∈Del^{(g−i)} mod Y} H^{0}(V (σ), L_{0}^{n})).

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Let Star (x) be the union of σ ∈ Del such that x ∈ σ and Star (x)^{0} the relative interior of Star (x). Then it is easy to see

H^{i}(⨁ W_{∗}^{Y -inv}) ≃ ⨁_{x∈Xmod nY} H^{i}(Star (\frac{x}{n})^{0}, k)

We see H^{i}(Star (x)^{0}, k) ≃ H^{i}(⨁_{x∈σ} H^{0}(σ, k), ∂_{g−∗}). Therefore we have H^{0}(Star (x)^{0}, k) = k and H^{i}(Star (x)^{0}, k) = 0 for i > 0. Hence H^{i}(P_{0}, L_{0}^{n}) = 0 for i > 0, while H^{0}(P_{0}, L_{0}^{n}) = ker(∂_{g})^{Y -inv}.

By the above proof, θ ∈ H^{0}(P_{0}, L_{0}^{n}) is an (apparently infinitely many) linear combination of η_{n}(x). S_{y} acts on R̃ by S_{y}^{∗}(ā(x)ξ_{x}ϑ) = ā(x + y)ξ_{x+y}ϑ. We note that θ ∈ H^{0}(P_{0}, L_{0}^{n}) if and only if S_{y}^{∗}(θϑ^{n}) = θϑ^{n}. Theorem follows from the formula S_{y}^{∗}(η_{n}(x)ϑ^{n}) = b_{0}(y, x)a_{0}(y)^{n}η_{n}(x + ny)ϑ^{n}. Therefore h^{0}(P_{0}, L_{0}^{n}) = |X/Y |n^{g}.

Remark 3.10. Since G_{0} is irreducible, the extension of L_{η} to G is unique. Therefore Γ(G_{η}, L_{η}^{n}) = Γ(G, L^{n}) ⊗ k(η). Hence Γ(G_{η}, L_{η}^{n}) is identified with the k(η) vector subspace of Γ(G_{for}, L_{for}^{n}) ⊗ k(η) consisting of Fourier series s = ∑_{x∈X} σ_{x}(s)w^{x} such that

σ_{x+ny}(s) = a(y)^{n}b(y, x)σ_{x}(s), σ_{x}(s) ∈ k(η) (∀x ∈ X, ∀y ∈ Y)

By Theorem 3.9 we see that Γ(P, L^{n}) ⊗ k = Γ(P_{0}, L_{0}^{n}). Hence we easily see

Γ(P, L^{n}) = \{∑_{x∈X} c(x)η_{n}(x); \begin{matrix}c(x + ny) = b̄(y, x)ā(y)^{n}c(x) \\ c(x) ∈ R, ∀x ∈ X, ∀y ∈ Y \end{matrix}\}

Hence we have Γ(P_{η}, L_{η}^{n}) = Γ(G_{η}, L_{η}^{n}) and therefore G_{η} ≃ P_{η}.

We note that G is the semi-abelian scheme we started from, while P is the projective scheme we constructed with the degeneration data of G.

4. General case

4.1. Fourier expansions in the general case. In this section we will discuss the general case where the abelian part A is nontrivial. We will prove the theorems similar to the totally degenerate case.

We use freely the notation in Section 2. We assume that T_{0} := T ⊗ k is a split k-torus. Then T is a split S-torus by [SGA3, exp. IX §3]. See also [FC90, 2.2]. Let X := Hom(T, G_{m}), Y = Hom(T^{t}, G_{m}). The ample cubical sheaf L on a semi-abelian scheme G induces a sheaf L_{for} on G_{for}. This is descended to an ample cubical sheaf M_{for} on A_{for}, which is algebraized into an ample cubical sheaf M on the algebraisation A of A_{for}. The sheaf π_{∗ for}(O_{G_{for}}) is the I-adic direct sum of T_{for}-eigenspaces O_{G_{for}}^{χ}

π_{∗ for}(O_{G_{for}}) = ⊕_{χ∈X} O_{G_{for}}^{χ}

where O_{G_{for}}^{χ} = {f ∈ O_{G_{for}}; f(tx) = χ(t)f(x) ∀ functorial points t ∈ T_{for}, x ∈ G_{for}}.

Let G̃ be the algebraisation of G_{for}, 0 → T → G̃ →^{π} A → 0 the Raynaud sequence and L̃:= π^{∗}(M). It is clear that L̃_{for} = π_{for}^{∗}(M_{for}) = L_{for}.

Since T acts on G̃, we have OG̃ = ⊕_{χ∈X} O^{χ} with O^{χ} the subsheaf of OG̃ consisting of χ-eigenfunctions. Then π_{∗}(OG̃) = ⊕_{χ∈X} O_{χ} where O_{χ} is the subsheaf of π_{∗}(OG̃) such that O^{χ} = π^{·}O_{χ}, the sheaf-theoretic pull back of O_{χ}. Then O_{χ} ∈ A^{t}.

By definition we have O^{χ}O^{χ′} = O^{χ+χ′}, O_{χ}O_{χ′} = O_{χ+χ′}. If we choose isomorphisms O^{χ} ≃ O_{χ} ≃ O_{A} on an affine open U of A for a basis χ of X, we have the isomorphisms O^{χ} ≃ O_{χ} ≃ O_{U} for any χ ∈ X. Hence there exists an open affine covering {U} of A such that O^{χ} ≃ O_{χ} ≃ O_{U} for any χ ∈ X on each U ∈ {U}.

Next we prove that G̃ is a T -bundle over A in the Zariski topology. Let U := Spec B ∈ {U} be an affine open subset of A. Since π is an affine morphism, W := π^{−1}(U) = Spec C is affine. Then C = Γ(W, OG̃) = Γ(U, π_{∗}(OG̃)) = ⊕_{χ∈X} Γ(U, O_{χ}). Since O_{χ} ≃ O_{U} on U, Γ(U, O_{χ}) hence Γ(U, O^{χ}) is a B-module of rank one. Let w_{U}^{χ} be a generator of Γ(U, O^{χ}) such that Γ(U, O^{χ}) = Bw_{U}^{χ}. By O^{χ}O^{χ} = O^{χ+χ′}, we may assume w_{U}^{χ}w_{U}^{χ′} = w_{U}^{χ+χ′}, C = B[w_{U}^{χ}; χ ∈ X]. Therefore W is an S-split torus U ×T over U. It follows that G̃ is a T -bundle over A in the Zariski topology. Therefore there exist an affine covering {U} of A and coordinates w_{U}^{χ} of T such that

O_{G̃|π^{−1}(U)} = ⊕_{χ∈X} O_{U}^{χ} = ⊕_{χ∈X} O_{U}w_{U}^{χ}, O_{U}^{χ} = O_{U}w^{χ} ≃ O_{χ|U} (χ ∈ X).

For different U and V w_{U}^{χ} and w_{V}^{χ} are related by w_{U}^{χ} = g_{UV}^{χ} w_{V}^{χ} where g_{UV}^{χ} ∈ O_{U∩V}^{∗} with g_{UV}^{χ} g_{V W}^{χ} = g_{UW}^{χ}. The cocycle {g_{UV}^{χ} } ∈ H^{1}(A, O_{A}^{∗}) represents the sheaf O_{χ}.

4.2. Fourier coefficients. We define a subsheaf of M_{for}^{χ} of L_{for} by

M_{for}^{χ} = {f ∈ L_{for}; f(tx) = χ(t)f(x) ∀ functorial points t ∈ T_{for}, x ∈ G_{for}}

Since L_{for} is descended to M_{for}, we have π_{for}^{·}M_{for} = M_{for}^{0}, the subspace of L_{for} consisting of T -invariant sections. Then we have π_{∗ for}(L_{for}) = ⊕_{χ∈X}M_{for}^{χ}, M_{for}^{χ} = M_{for}^{0}⊗O_{for}^{χ}. Let M_{χ} := M⊗O_{χ}. Then M_{χ} is ample on A with M_{for}^{χ} = π_{for}^{·}M_{χ,for}. We also have Γ(G_{for}, M_{for}^{χ}) = Γ(A_{for}, M_{χ,for}) = Γ(A, M_{χ}).

By the sign change of c and c^{t} following [FC90, p.43] we have cϕ = λ_{A}c^{t} where ϕ is the injection of Y into X and λ_{A} = λ_{A}(M). This means that T_{c^{t}(y)}^{∗}(M) ⊗ M^{−1} ≃ O_{ϕ(y)}, which we denote simply by O_{y}. As rigidified sheaves M and O_{χ} at 0 ∈ A we have T_{c^{t}(y)}^{∗}(M) = M ⊗ O_{y} ⊗ M(c^{t}(y)) and T_{c^{t}(y)}^{∗}(M_{χ}) = M_{χ+y} ⊗M_{χ}(c^{t}(y)). By the isomorphism Γ(G_{for}, M_{for}^{χ}) ≃ Γ(A, M_{χ}), we have an R-homomorphism

σ_{χ} : Γ(G, L) ⊂ Γ(G_{for}, L_{for}) ≃ ⊕_{μ∈X}Γ(G_{for}, M_{for}^{μ}) → Γ(A, M_{χ}).

Two homomorphisms σ_{χ} and σ_{χ+y} will be related as follows. Since T_{c^{t}(y)}^{∗}(M_{χ}) = M_{χ+y} ⊗ M_{χ}(c^{t}(y)), we obtain a composition map

T_{c^{t}(y)}^{∗} ∘ σ_{χ} : Γ(G, L) → T_{c^{t}(y)}^{∗}(Γ(A, M_{χ})) = Γ(A, M_{χ+y}) ⊗ M_{χ}(c^{t}(y)).

For instance, for a given element m ∈ M_{χ}(c^{t}(y))^{−1}, we have a multiplication map

m · T_{c^{t}(y)}^{∗} ∘ σ_{χ} : Γ(G, L) → Γ(A, M_{χ+y}).

Theorem 4.3 asserts that there exists m(y, χ) ∈ M_{χ}(c^{t}(y))^{−1} ⊗ k(η) such that σ_{χ+y} = m(y, χ)·T_{c^{t}(y)}^{∗}∘σ_{χ} (y ∈ Y, χ ∈ X). The constant m(y, χ) will be written as a product m(y, χ) = τ(y, χ)ψ(y) where τ(y, χ) ∈ O_{χ}(c^{t}(y))^{−1} ⊗ k(η) ≃ k(η) and ψ(y) ∈ M(c^{t}(y))^{−1} ⊗ k(η) ≃ k(η). See [FC90, pp. 43-44].

Theorem 4.3. [FC90] The collection of R-module homomorphisms σ_{x} : Γ(G, L) → Γ(A, M_{x}) have the following properties

1. σ_{x} ≠ 0 (∀x ∈ X)

2. There exist a cubic trivialization ψ : 1_{Y} → (c^{t})^{∗}M_{η}^{−1} and a trivialization of biextension (Poincaré torsor) τ : 1_{Y ×X} → (c^{t} × c)^{∗}P_{η}^{−1} such that

σ_{x+y}(s) = ψ(y)τ(y, x)T_{c^{t}(y)}^{∗}(σ_{x}(s)) (∀s ∈ Γ(G, L), ∀y ∈ Y, ∀x ∈ X),

In particular, ψ(0) = 1, τ(0, x) = τ(y, 0) = 1.

3. τ(y, x) is bilinear in y and x,

4. τ(y, z) = τ(z, y) = ψ(y + z)ψ(y)^{−1}ψ(z)^{−1} (∀y, z ∈ Y)

5. τ(y, y) ∈ I if y ≠ 0,

6. For every n ≥ 0, ψ(y) ∈ I^{n} for all but finitely many y ∈ Y,

7. Γ(G_{η}, L_{η}) is identified with the k(η) vector subspace of Fourier series θ satisfying σ_{x+y}(θ) = ψ(y)τ(y, x)T_{c^{t}(y)}^{∗}(σ_{x}(θ)) with σ_{x}(θ) ∈ Γ(A_{η}, M_{x,η}).

4.4. Extension of τ and ψ. In the previous paragraphs, we have c : X → A^{t}, c^{t} : Y → A, and ϕ : Y → X such that λ_{A}c^{t} = cϕ. By the sign change of c and c^{t} we have c(χ) = O_{χ}, T_{c^{t}(y)}^{∗}(O_{χ}) ≃ O_{χ+y} and T_{c^{t}(y)}^{∗}(M) ≃ M_{y} for any χ ∈ X and y ∈ Y.

Now we extend the homomorphism c^{t} : Y → A to c_{X}^{t} : X → A so that λ_{A}c_{X}^{t} = c on X, which extends the relation λ_{A} · c^{t} = c · ϕ on Y. In fact, this is possible if we take a finite base change of S because A is divisible. It is also easy to extend the functions τ and ψ respectively to X × X and X by taking a finite base change of S if necessary so that the following are true;

1. τ is symmetric bilinear on X × X,

2. ψ(x + y) = ψ(x)ψ(y)τ(x, y) (x, y ∈ X).

This also implies that there are a cubical trivialization ψ : 1_{X} → (c_{X}^{t})^{∗}M_{η}^{−1} and a trivialization of the biextension (an isomorphism of the biextensions) τ : 1_{X×X} ≃ (c^{t} × c)^{∗}P_{η}^{−1} on X × X. In what follows we denote c_{X}^{t} simply by c^{t}.

4.5. A rigidification. Let P be the Poincaré sheaf on A × A^{t}, which is by definition the universal invertible sheaf for the functor Pic^{0} on A. Let m : A × A → A be the multiplication of A, p_{13} and p_{23} the projections of A × A × A^{t}. By the theorem of cube there is an (let us say, canonical) isomorphism Φ : (m × 1)^{∗}P ≃ p_{13}^{∗}P ⊗ p_{23}^{∗}P. By this we have a canonical isomorphism Φ_{a,χ} := Φ_{|A×a×χ} : T_{a}^{∗}(O_{χ}) = O_{χ} ⊗ O_{χ}(a) for any χ ∈ X and for any functorial point a of A. By the theorem of the cube we see Φ_{a,χ}T_{a}^{∗}(Φ_{a′,χ}) = Φ_{a+a′,χ}. Let ϕ_{a,χ} := Φ_{a,χ}T_{a}^{∗}, though it is rather formal. Hence ϕ_{0,χ} = ϕ_{a,0} = 1 and

ϕ_{a,χ+χ′} = ϕ_{a,χ} ⊗ ϕ_{a,χ′}, ϕ_{a+a′,χ} = ϕ_{a,χ}ϕ_{a′,χ}.

Let Λ(M) := m^{∗}(M)⊗p_{1}^{∗}(M)^{−1}⊗p_{2}^{∗}(M)^{−1}. With the notation in [MB85, p. 14] Λ(M) = D_{2}(M). By the theorem of square (1_{A} × λ_{A})^{∗}(P) ≃ Λ(M) as rigidified sheaves. We identify both sheaves via this (canonical) isomorphism. This induces the canonical isomorphism ρ_{y,0} : T_{c^{t}(y)}^{∗}(M) = M ⊗ M(c^{t}(y)) ⊗ O_{y} as well as the canonical isomorphisms ρ_{y,x} = ρ_{y,0} ⊗ Φ_{c^{t}(y),x} : T_{c^{t}(y)}^{∗}(M_{x}) = T_{c^{t}(y)}^{∗}(M) ⊗ T_{c^{t}(y)}^{∗}(O_{x}) = M_{y} ⊗ M(c^{t}(y)) ⊗ O_{x} ⊗ O_{x}(c^{t}(y)).

Definition 4.6. Modifying [Chai85, p.102] slightly, we define

R̃ : = the R -algebra generated by ψ(x)M_{x}ϑ (∀x ∈ X)

where ϑ is an indeterminate with deg(ϑ) = 1, while deg(R) = deg(M_{x}) = 0. Let Q̃ := Proj(R̃) and L̃:= OQ̃(1). Let R̃^{∗} := R̃⊗O_{A}. We note Proj(R̃) ≃

Proj(R̃^{∗}) by [EGA, Chap. II, (2.4.7)]. Let I(M_{x}) := ψ(x)M_{x} ⊂ M_{x}⊗_{R}k(η) and ρ_{y} := ρ_{y,x} on I(M_{x}). We define the action S_{y}^{∗} (y ∈ X) on R̃ as follows

S_{y}^{∗}(σ_{x}ψ(x)ϑ) = ρ_{y}T_{c^{t}(y)}^{∗}(σ_{x})ψ(x)ψ(y)τ(y, x)ϑ
= ρ_{y}T_{c^{t}(y)}^{∗}(σ_{x})ψ(x + y)ϑ ∈ I(M_{x+y})ϑ

where σ_{x}ψ(x) ∈ I(M_{x}). S_{y}^{∗} is well-defined as an endomorphism of the graded algebra R̃. Thus we have an endomorphism S_{y} of (Q̃, L̃). In what follows since we always consider the canonical isomorphisms, we omit Φ_{a,χ} and ρ_{y} if no confusion is possible.

Remark 4.7. Let R[M] := R ⊕ ⊕_{k≥1} M^{k} and S(M) = R[M] ⊗ O_{A}. It is well known that Proj(R[M]) ≃ Proj(S(M)) by [EGA, Chap. II, (2.4.7)].

Since M is an invertible sheaf on A, it is clear that A ≃ Proj(S(M)). Since R̃ ⊗ O_{A} is an O_{A}-algebra, we have a morphism π : Q̃ → A by the isomorphism Q̃ ≃ Proj(R̃ ⊗ O_{A}). Then S_{y} induces a translation of A_{η} by c^{t}(y) because

S_{y}^{∗}(Proj(R[M]) ⊗ k(η) = Proj(R[T_{c^{t}(y)}^{∗}(M)ψ(y)] ⊗ k(η))
= Proj(O_{A}[ψ(y)M_{y}] ⊗ k(η)) ≃ A_{η}.

Since A is the Néron model of A_{η}, S_{y} induces a translation of A. It is clear that (Q̃, L̃) ⊗ k(η) = Proj(R[⊕_{y∈X}ψ(y)M_{y}ϑ] ⊗ k(η)) ≃ (G̃, L̃) ⊗ k(η).

Definition 4.8. By a suitable finite base change of S if necessary we can construct a projective flat scheme (P, L) over S with P_{0} reduced. Let I the maximal ideal of R, s a generator of I and val_{s} the valuation of R with val_{s}(s) = 1. As in 4.4, by a finite base change of S if necessary, we choose and fix the extensions ψ, τ and c^{t} respectively to X, X × X and X so that Theorem 4.3 (3) and (4) are true for ∀y, z ∈ X and λ_{A} · c^{t} = c on X. For x, y ∈ X we define

B(x, y) = val_{s}(τ(x, y)), dA(α, x) = B(α, x) + r(x)/2
A(x) = val_{s}(ψ(x)) = B(x, x)/2 + r(x)/2
τ̄(x, y) = s^{−B(x,y)}τ(x, y), ψ̄(x) = s^{−A(x)}ψ(x)
τ̄_{0}(x, y) = τ̄(x, y) ⊗ k, ψ̄_{0}(x) = ψ̄(x) ⊗ k

where there exists an r ∈ Hom_{Z}(X, Z) as above. Since B is positive definite on X × X we have a Delaunay decomposition Del_{B} of X_{R}. We denote a Delaunay cell by σ, τ and so on, and the centre of σ by α(σ).

We choose an affine U of A, the generators w_{U}^{χ} of O_{U}^{χ} and ϑ_{U} of M_{U}ϑ. Then we define ξ_{x} = s^{A(x)}w_{U}^{x}, ξ_{x,c} = ξ_{x+c}/ξ_{c} and ζ_{x,c} = s^{B(α(σ),x)+r(x)/2}w_{U}^{x} if x + c ∈ C(c, σ) and c ∈ σ ∩ X. Let R̃_{U}^{∗} := Γ(O_{U})[ψ(x)w_{U}^{x}ϑ_{U}; x ∈ X]. Then Q̃ is covered with Proj R̃_{U}^{∗} (≃ Q̃×_{A} U), U affine open subsets of A. In terms of ζ_{x,c}, we can describe the structure of (Q̃, L̃) and its normalization (P̃, L̃) over U as in the totally degenerate case.

Theorem 4.9. Let A_{0} := A ⊗ Spec k, g′ = rank X, g′′ = dim A_{0} and g = g′ + g′′. Suppose that dA(α(σ)) ∈ X^{∗} for any g′-dimensional Delaunay cell σ ∈ Del_{B}. Then A_{0} is an abelian variety over k and P̃_{0} is a g-dimensional reduced scheme of locally of finite type over k. P̃_{0} is covered with affine A_{0}-schemes U_{0}(c) = Spec R_{0}(c) of finite type over A_{0} and the following holds.

1. R_{0}(c) is an O_{A_{0}}-module with the basis {ζ̄_{x,c}; x ∈ X}. If x_{1}, ⋯, x_{m} are cellmates (resp. otherwise) the multiplication of them is defined by

ζ̄_{x_{1},c} ⋯ ζ̄_{x_{m},c} = ζ̄_{x_{1}+···+x_{m},c} (resp. 0)

2. Let T_{0} := Spec k[w^{x}; x ∈ X]. Any irreducible component of P̃_{0} contains a T_{0}-torsor over A_{0} whose class is given by c_{0} := c ⊗ k ∈ Hom(X, A_{0}^{t}).

3. P̃_{0} has a stratification by algebraic torus-torsors O(σ), i.e. an extension of A_{0} by a torus T_{0}(σ) (σ ∈ Del_{B}) where

T_{0}(σ) = Spec k[ζ̄_{x,c}^{±1}; x ∈ C(c, σ) ∩ X, c ∈ σ ∩ X].

The extension class of O(σ) is given by the restriction of c_{0} to the Z-submodule of X spanned by all x ∈ X such that x + c ∈ C(c, σ), c ∈ σ ∩ X.

4. S_{y} (y ∈ Y) transforms O(σ + y) into O(σ) where S_{y} induces a translation of A_{0} by c_{0}^{t}(y) := c^{t}(y) ⊗ k ∈ A_{0}, while it induces a morphism of fibres T_{0}(σ + y) into T_{0}(σ) as S_{y}^{∗}(ζ̄_{x,c}) = τ̄_{0}(x, y)T_{c^{t}(y)}^{∗}(ζ̄_{x,c+y}) (c ∈ σ).

2

Proof. (1) and (2) are clear. Any irreducible component of P̃_{0} contains (a copy of) G̃_{0}, a T_{0}-torsor over A_{0}, whose extension class is given by c_{0}. See 2.2. For a Delaunay cell σ O(σ) is also given by the push-out of G_{0} to the subtorus T_{0}(σ) of T_{0}, therefore its extension class is given by the restriction of c_{0} to the Z-submodule spanned by x ∈ X such that x + c ∈ C(c, σ), c ∈ σ ∩ X. This proves (3).

Choosing w_{U}^{χ} as in Definition 4.8 we can prove (4) on a dense open subset U_{0} ∩ S_{y}(U_{0}) of A_{0}. In fact, by a formal computation we have

S_{y}^{∗}(ξ_{x,c}) = ψ(x + c + y)ψ(c)ψ(c + y)^{−1}ψ(x + c)^{−1}T_{c^{t}(y)}^{∗}(ξ̄_{x,c+y})
= τ̄(x, y)T_{c^{t}(y)}^{∗}(ξ_{x,c+y})

which proves the formula (4). See also [Chai85, pp.76-77].

2If w_{U}^{χ} are defined on an open set U, then (4) is true only over U_{0} ∩ S_{y}(U_{0}) where U_{0} := U ⊗ k.

Theorem 4.10. Let (P_{for}, L_{for}) be the quotient formal scheme of (P̃_{for}, L̃_{for}) by the periods Y. Then there exists a flat projective scheme (P, L) over Spec R which algebraizes (P_{for}, L_{for}). Let (P_{0}, L_{0}) be the closed fibre of (P, L). Then

1. (P_{0}, L_{0}) has the stratification by (the images of) O(σ) (σ ∈ Del_{B} mod Y)

2. h^{0}(P_{0}, L_{0}^{n}) = |X/Y | deg(M_{0})n^{g}, h^{i}(P_{0}, L_{0}^{n}) = 0 (i > 0)

3. For any affine open subset U of A such that G ×_{A} U ≃ U × T, we define w^{x} := w_{U}^{x} and η_{n}(x) := ζ_{x−nc,c}ξ_{c}^{n} if \frac{x}{n} ∈ σ ∩ \frac{X}{n} for a Delaunay cell σ and c ∈ σ ∩ X as in Theorem 3.9. Then we have

Γ(P_{0}, L_{0}^{N}) = \{∑_{x∈X} σ_{x}η_{n}(x); \begin{matrix}σ_{x+ny} = τ̄_{0}(y, x)ψ̄_{0}(y)^{n}T_{c^{t}(ny)}^{∗}(σ_{x}) \\ σ_{x} ∈ Γ(A_{0}, M_{0}^{n} ⊗ O_{x}), ∀x ∈ X, ∀y ∈ Y \end{matrix} \}

where the right hand side is understood as those sections written over U_{0}.

Proof. Let V (σ) be the closure of O(σ). V (σ) is a torus embedding bundle over A_{0}. In order to compute H^{i}(V (σ), L_{0}^{n}) we use the spectral sequence E_{2}^{p,q} = H^{p}(A_{0}, R^{q}π(σ)_{∗}L_{0}^{n}) where π(σ) is the projection of V (σ) onto A_{0}. By the same reason as in Theorem 3.9 we see R^{q}π(σ)_{∗}(L_{0}^{n}) = 0 for q, n > 0 and R^{0}π(σ)_{∗}(L_{0}^{n}) ≃ ⊕_{\frac{x}{n} ∈σ∩ \frac{X}{n}} M_{0}^{n} ⊗ O_{x}. Hence H^{p}(V (σ), L_{0}^{n}) = H^{p}(A_{0}, R^{0}π(σ)_{∗}L_{0}^{n}) = 0 for p > 0. Moreover

H^{0}(V (σ), L_{0}^{n}) ≃ ⊕_{\frac{x}{n} ∈σ∩ \frac{X}{n}} H^{0}(M_{0}^{n} ⊗ O_{x})

= \{ ∑_{\frac{x}{n} ∈σ∩ \frac{X}{n}} c(x)η_{n}(x); c(x) ∈ H^{0}(A_{0}, M_{0}^{n} ⊗ O_{x})\}

where the right hand side is understood as the expressions over U_{0} of those sections. The rest of the proof is the same as in Theorem 3.9.

Remark 4.11. By choosing a trivialization of L̃_{T_{for}}^{n} induced from that of L̃_{T_{for}} we see that Γ(P_{η}, L_{η}^{n}) is identified with the k(η) vector subspace of Γ(G_{for}, L_{for}^{n}) ⊗ k(η) consisting of Fourier series s = ∑_{x∈X} σ_{x}(s) such that

σ_{x+ny}(s) = τ(y, x)ψ(y)^{n}T_{c^{t}(ny)}^{∗}(σ_{x}(s)),
σ_{x}(s) ∈ Γ(A, M^{n} ⊗ O_{x}) ⊗ k(η) (∀x ∈ X, ∀y ∈ Y)

Hence we have Γ(P_{η}, L_{η}^{n}) = Γ(G_{η}, L_{η}^{n}) for any n > 0. It follows that P_{η} ≃ G_{η}.

4.12. The group scheme G. By choosing a suitable base change of S we assume dA(α(σ)) ∈ Hom(X, Z) for any σ ∈ Del_{B}. Then a closed fibre P_{0} of P is reduced. Now we prove that G is realized as an open subscheme of P. By the assumption we choose (and fix) a maximal dimensional Delaunay cell σ ∈ Del(0) and define w^{x} := ζ_{x,0} of O(σ) (x ∈ C(σ, 0)). This is equivalent to assuming the condition B(α(σ), x) + r(x)/2 = 0 (∀x ∈ X) (in fact, we may assume after taking a finite base change of S). By choosing ζ_{x,0} (instead of w^{x}) as the coordinates of T in the construction of (P, L), we obtain the same (P, L). Hence we have an open subscheme G̃(σ) ⊂ P̃ such that G̃(σ)_{0} = O(σ)_{0}, a torus torsor over A_{0}. We now imitate the argument of [Mumford72, p. 297-p. 303]. Let S_{n} = Spec (R/I^{n}), G̃^{†} := ∪_{y∈Y} S_{y}(G̃(σ)) ⊂ P̃ and G(σ)_{n} := (G̃^{†} ×_{S} S_{n})/Y ⊂ P_{n}. Clearly G(σ)_{n} ≃ G̃(σ) ×_{S} S_{n}.

Let B̃(σ) := P̃ \ G̃^{†} and make it into a reduced closed subscheme. Then B(σ)_{for}, the formal completion of B̃(σ)/Y, is algebraized into a reduced closed subscheme B(σ) of P. We let G(σ) := P \ B(σ). Then by the argument in [Mumford72, 4.1] G(σ) is smooth and a group scheme over S with G(σ)_{η} ≃ P_{η}. It is clear that G(σ)_{0} is connected and semi-abelian. Moreover

(13) G(σ)_{for} ≃ lim_{n} G(σ)_{n} ≃ lim_{n} G̃(σ)_{n} ≃ G̃(σ)_{for}.

In order to construct (P, L) we started from a semi-abelian scheme (G, L), and then constructed (G̃, L̃), where G_{for} ≃ G̃_{for} and G̃(σ) = G̃ by the definition. It follows that G(σ)_{for} ≃ G_{for}. By the universal property of G there is a group scheme morphism ρ : G(σ) → G extending the isomorphism G(σ)_{η} ≃ P_{η} ≃ G_{η}, which keeps the formal isomorphism G(σ)_{for} ≃ G_{for} unchanged. It follows that ρ is an isomorphism. Hence G(σ) ≃ G. This proves G ⊂ P.

4.13. The group scheme G^{♯}. Let G̃^{♯} := ∪_{x∈X}S_{x}(G̃(σ)) ⊂ P̃. Let B̃^{♯} := P̃ \ G̃^{♯} and make it into a reduced closed subscheme. Then B_{for}^{♯}, the formal completion of B̃^{♯}/Y, is algebraized into a reduced closed subscheme B^{♯} of P. We let G^{♯} := P \ B^{♯}. In the same manner as before we see G^{♯} is a semi-abelian group scheme with G_{η}^{♯} ≃ P_{η} such that G is an open group subscheme of G^{♯}.

Let e be the unit section of G and e(x) := S_{x}(e) ∈ G^{♯}(S) (x ∈ X). We define ẽ and ẽ(x) ∈ G̃^{♯}(S) by

ẽ:ζ_{z,0} = 1, (∀z ∈ C(0, σ)), ζ_{z,0} = 0 (∀z ∉ C(0, σ))
ẽ(x) :ζ_{z,−x} = τ̄(z, x) (∀z ∈ C(0, σ)), ζ_{z,−x} = 0 (∀z ∉ C(0, σ))

plus the sections c^{t}(0) (resp. c^{t}(x)) of A. By definition e_{for} = ẽ_{for} and e(x)_{for} = ẽ(x)_{for} as the formal sections of P_{for} = P̃_{for}/Y. Since S_{x}(U_{−x}) = U_{0} and both S_{x}, T_{ẽ(x)} are translations by c^{t}(x) in the direction of A, it follows from S_{x}^{∗}(ζ_{z,−x}) = T_{c^{t}(x)}^{∗}(ζ_{z,0})τ̄(z, x) that S_{x} = T_{ẽ(x)} on G̃^{♯}. Since S_{x}S_{y} = S_{y}S_{x} (∀y ∈ Y), S_{x} is descended to an action on P, which we again denote by S_{x}. On the other hand since e(x) ∈ G^{♯}(S), the translation T_{e(x)} of G^{♯} by e(x) is well-defined. Since S_{x,for} = T_{ẽ(x),for} = T_{e(x),for} on G_{for}^{♯}, we have S_{x} = T_{e(x)} on G^{♯}.

Lemma 4.14. The flat closure K_{S}^{♯}(L_{η}) of K(L_{η}) in G^{♯} is finite over S.

Proof. In fact, K_{S}^{♯}(L_{η}) is quasi-finite and there is the maximal flat finite subscheme K_{S}^{♯}(L_{η})^{f} of K_{S}^{♯}(L_{η}) and a flat quasi-finite subscheme Z such that K_{S}^{♯}(L_{η}) = K_{S}^{♯}(L_{η})^{f} ⨿ Z where Z ∩ G_{0}^{♯} = ∅. By definition K_{S}(L_{η}) ⊂ K_{S}^{♯}(L_{η}), K_{S}(L_{η})^{f} ⊂ K_{S}^{♯}(L_{η})^{f} and K_{S}(L_{η})^{fn} = K_{S}^{♯}(L_{η})^{fn} := K_{S}^{♯}(L_{η})^{f} ∩ G. Moreover K_{S}(L_{η})^{m} = K_{S}^{♯}(L_{η})^{m}, the maximal flat finite subgroup S-scheme of K_{S}^{♯}(L_{η})^{fn} such that its intersection with G_{0} is a subgroup scheme of T_{0}. See Subsection 2.3. Hence we have ord K_{S}^{♯}(L_{η})^{m} = d_{t}, ord K_{S}^{♯}(L_{η})^{fn} = d_{a}^{2}d_{t}, and ord_{k(η)} K(L_{η}) = d_{a}^{2}d_{t}^{2} by Subsection 2.4.

Let H^{∗} be the étale finite flat subgroup scheme of G^{♯} consisting of e(x) (x ∈ X/Y). Since T_{e(x)}^{∗}(L) = S_{x}^{∗}(L) ≃ L by Subsection 4.13, H^{∗} is a subgroup scheme of K_{S}^{♯}(L_{η}). Since H^{∗} is étale finite, the subscheme

H^{∗} · K_{S}^{♯}(L_{η})^{fn} := ∪_{x∈X/Y} S_{x} · K_{S}^{♯}(L_{η})^{fn} ≃ H^{∗} ⊕ K_{S}^{♯}(L_{η})^{fn}

is a finite flat subgroup scheme of K_{S}^{♯}(L_{η}) of order d_{a}^{2}d_{t}^{2}. This shows that Z = ∅, K_{S}^{♯}(L_{η}) = K_{S}^{♯}(L_{η})^{f} = H^{∗} · K_{S}^{♯}(L_{η})^{fn} = H^{∗} ⊕ K_{S}^{♯}(L_{η})^{fn}. Hence K_{S}^{♯}(L_{η}) is finite.

Definition 4.15. Let L^{tsr} be the G_{m}-torsor on G^{♯} associated with the invertible sheaf L_{G^{♯}}. Let G_{S}^{♯}(L_{η}) := L_{|K_{S}^{♯}(L_{η})}^{tsr} and e_{S}^{♯} an extension of e^{L_{η}}. By

[MB85, IV, 7.1 (ii)] G_{S}^{♯}(L_{η}) is a group scheme over S. See [MB85, I, 3.1, 3.2] for the definition of the group scheme structure of G_{S}^{♯}(L_{η}). By the natural projection of L^{tsr} onto G^{♯}, it is a central extension of K_{S}^{♯}(L_{η}) by G_{m,S} with e_{S}^{♯} the commutator form as is given in Subsection 2.4. The bilinear form e_{S}^{♯} on K_{S}^{♯}(L_{η}) is nondegenerate alternating by [MB85, IV, 2.4] and by Lemma 4.14. We note G^{♯} = A^{[1]} with the notation of [MB85, IV, 7.1].

By [MB85, I, 3.1, 3.2] G_{S}^{♯}(L_{η}) acts on Γ(G^{♯}, L) in the natural manner. Moreover Γ(G^{♯}, L) is an irreducible G_{S}^{♯}(L_{η})-module by [MB85, V, 2.4.2; VI, 1.4.7].

5. (Q, L)

5.1. The totally degenerate case. We consider the totally degenerate case.

From Section 3 we recall

R̃ := R[a(x)w^{x}ϑ; x ∈ X] ≃ R[ξ_{x}ϑ; x ∈ X]
ξ_{x} := s^{B(x,x)/2+r(x)/2}w^{x}, ξ_{x,c} := ξ_{x+c}/ξ_{c}
S_{y}^{∗}(a(x)w^{x}ϑ) = a(x + y)w^{x+y}ϑ

Content-Length: 107348 X-Lines: 3072 Let Q̃ := Proj R̃ and L̃:= OQ̃(1).

Lemma 5.2. 1. If ξ_{x_{1}} ⋯ ξ_{x_{m}}s^{α} = 1 for some x_{i} ∈ X and α ≥ 0, then

x_{i} = α = 0.

2. If B(a, b) > 0, then ξ_{a+b} is divisible by sξ_{a}ξ_{b}. If B(a, b) < 0, then ξ_{a}ξ_{b} is divisible by sξ_{a+b}.

3. For any x ∈ X there exist x_{i} ∈ Star (0) ∩ X with x = x_{1} + ⋯ + x_{m} such that ξ_{x} is a product of ξ_{x_{1}} ⋯ ξ_{x_{m}} and some power of s.

4. If x ∉ Star (0), then ξ_{x} is divisible by s.

Proof. We will prove (1). By assumption x_{1} + ⋯ + x_{m} = 0. Hence

B(x_{1}, x_{1}) + ⋯ + B(x_{m}, x_{m}) + 2α = 0

from which it follows x_{i} = α = 0. This proves (1).

If B(a, b) > 0, then B(a + b, a + b) = B(a, a) + B(b, b) + 2B(a, b) > B(a, a) + B(b, b). Hence ξ_{a+b} is divisible by sξ_{a}ξ_{b}. This proves (2).

We prove (3) by the induction on B(x, x). If x ∈ C(0, σ) ∩ X \ σ for some σ ∈ Del(0), then we can choose a ∈ σ ∩ X such that B(x − a, a) > 0 by Lemma 1.4. Hence ξ_{x} is divisible by sξ_{x−a}ξ_{a} and B(x, x) > B(x − a, x − a) + B(a, a) > B(x − a, x − a). Hence by the induction hypothesis, ξ_{x−a} is a product of s and ξ_{a_{i}}, a_{i} ∈ Star (0)∩X. (4) follows from the proof of (3).

Theorem 5.3. Let (Q̃, L̃) = (Proj R̃, O_{Proj R̃}(1)). Then

1. Q̃ is covered with open affine subschemes W (c) := Spec S(c) where

S(c) := R[ξ_{x,c}; x ∈ X] (c ∈ X).

2. The coordinate ring S(c) of W (c) is an R-algebra of finite type generated by ξ_{x,c} (x ∈ Star (0) ∩ X). All the ring S(c) are isomorphic to each other.

3. Let x_{i}, y_{j} ∈ X (i ∈ I, j ∈ J). If p := ∑_{i∈I} A(x_{i}) − ∑_{j∈J} A(y_{j}) ≥ 0, then

∏_{i∈I} ξ_{x_{i},c} = s^{p} ∏_{j∈J} ξ_{y_{j},c}

The assertion (2) follows from Lemma 5.2.

Definition 5.4. We set

v(x) = min{val_{s}(ξ_{x_{1}} ⋯ ξ_{x_{m}}); x = x_{1} + ⋯ + x_{m}, x_{i} ∈ X, m ≥ 1}

Then we define ξ(x) = s^{v(x)}w^{x} ∈ Γ(W (0), OQ̃). Let v(x, c) = v(x) + B(x, c), ξ(x, c) = s^{v(x)+B(x,c)}w^{x} and ξ̄(x, c) := ξ(x, c) ⊗ k. In particular v(x) = v(x, 0) and ξ(x) = ξ(x, 0). We note that S_{d}^{∗}(ξ(x, c)) = τ̄(x, d)ξ(x, c + d). It is clear that

Γ(W (c), OQ̃) ⊗ k = ⊕_{x∈X} k · ξ(x, c)

Lemma 5.5. Let σ ∈ Del(0) and define Semi(0, σ) to be the semigroup generated by σ ∩ X. Let ξ̄_{x} := ξ_{x} ⊗ k ∈ S(0) ⊗ k (x ∈ X).

1. If x ∉ Star (0) ∩ X, then ξ̄_{x} = 0.

2. If x_{1}, ⋯, x_{m} ∈ Star (0) are not cellmates, then the product ξ̄_{x_{1}} ⋯ ξ̄_{x_{m}} is either zero or nilpotent.

3. If x_{1}, ⋯, x_{m} ∈ Star (0) are not cellmates and if x := x_{1} + ⋯ + x_{m} ∈ Semi(0, σ) for some σ ∈ Del(0), then ξ̄_{x_{1}} ⋯ ξ̄_{x_{m}} is zero.

4. If the product ξ̄_{x_{1}} ⋯ ξ̄_{x_{m}} is nonzero, then B(x_{i}, x_{j}) ≥ 0 for any i, j.

Proof. (1) follows from Lemma 5.2 (4). Next we prove (2).

Let x = x_{1}+⋯+x_{m} and σ ∈ Del^{max}(0) a maximal-dimensional Delaunay cell containing x, and α(σ) the centre of σ. Then there exist some n ∈ Z_{+}, n_{i} ∈ Z_{0} and an a_{i} ∈ σ ∩ X such that nx = n_{1}a_{1} + ⋯ + n_{r}a_{r}. We also have

n ∑_{i=1}^{m} B(x_{i}, x_{i}) ≥ 2nB(α(σ), ∑_{i=1}^{m} x_{i}) = 2B(α(σ), nx)

= 2 ∑_{i=1}^{r} n_{i}B(α(σ), a_{i}) = ∑_{i=1}^{r} n_{i}B(a_{i}, a_{i})

Since x_{i} are not cellmates, there is at least an x_{i} such that x_{i} ∉ σ, hence B(x_{i}, x_{i}) > 2B(α(σ), x_{i}). Therefore the above inequality is strict. This implies that (ξ̄_{x_{1}} ⋯ ξ̄_{x_{m}})^{n} is zero in O_{Q_{0}}. This proves (2).

Suppose x := x_{1} + ⋯ + x_{m} ∈ Semi(0, σ). Then we see

2B(α(σ), x) = ∑_{i=1}^{m} 2B(α(σ), x_{i}) = ∑_{i=1}^{m} B(x_{i}, x_{i}).

Since x_{i} are not cellmates, there is an x_{i} such that 2B(α(σ), x_{i}) < B(x_{i}, x_{i}). Therefore ξ_{x_{1}} ⋯ ξ_{x_{m}} is divisible by s · s^{2B(α(σ),x)}w^{x}. On the other hand, since x ∈ Semi(0, σ), there are a_{i} ∈ σ ∩ X such that x = ∑_{j=1}^{r} n_{i}a_{j}. Then s^{2B(α(σ),x)}w^{x} = ∏ ξ_{a_{j}}^{n_{j}}. This proves (3). (4) is clear from Lemma 5.2 (2).

Corollary 5.6. Let σ be a Delaunay cell with 0 ∈ σ, and Semi(0, σ) the semi-group generated by σ ∩ X. If x ∈ Semi(0, σ), then ξ̄(x, c) is a product of ξ̄_{a,c} (a ∈ σ ∩ X), which is not nilpotent. If x ∈ C(0, σ) ∩ X \ Semi(0, σ), then ξ̄(x, c) is nonzero and nilpotent.

Theorem 5.7. Let (Q̃_{0}, L̃_{0}) be the closed fibre of (Q̃, L̃). Then

1. Q̃_{0} is covered with finitely many open affine subschemes of finite type

W_{0}(c) := W (c) ⊗ k = Spec k[ξ̄_{x,c}; x ∈ X] (c ∈ X).

2. The coordinate ring S_{0}(c) := k[ξ̄_{x,c}; x ∈ X] of W_{0}(c) is a k-algebra of finite type possibly with nilpotent elements. Let x_{i}, y_{j} ∈ X (i ∈ I, j ∈ J). Then

∏_{i∈I} ξ̄_{x_{i},c} = \begin{cases}∏_{j∈J} ξ̄_{y_{j},c} & if ∑_{i∈I} A(x_{i}) = ∑_{j∈J} A(y_{j}) \\ 0 & if ∑_{i∈I} A(x_{i}) > ∑_{j∈J} A(y_{j}) \end{cases}

3. S_{0}(c) is generated by ξ̄_{x,c} (x ∈ Star (0) ∩ X).

4. Let S_{0}^{red}(c) be the coordinate ring of W_{0}(c)_{red}. Then S_{0}^{red}(c) is generated by ξ̄_{x,c} (x ∈ Star (0) ∩ X) as a k-algebra. It is naturally a ksubmodule of R_{0}(c) generated by ξ̄(x, c) for x ∈ Semi(0, σ), σ ∈ Del(0). The product of ξ̄_{x_{i},c} (1 ≤ i ≤ m) in S_{0}^{red}(c) is nonzero if and only if x_{i} ∈ Star (0) ∩ X are cellmates.

Proof. The assertions (1) and (2) are clear from Theorem 5.3. (3) follows from Lemma 5.2. (4) follows from Corollary 5.6. See also Theorem 3.7.

The construction of the quotient (Q, L) := (Q̃, L̃)/Y is quite similar to [Mumford72]. See also [AN96] and Theorem 3.5.

Theorem 5.8. The quotient (Q̃_{for}, L̃_{for})/Y is a flat projective formal Sscheme. There exists a flat projective S-scheme (Q, L) such that (Q_{for}, L_{for}) ≃ (Q̃_{for}, L̃_{for})/Y. Moreover (P_{η}, L_{η}) ≃ (Q_{η}, L_{η}) ≃ (G_{η}, L_{η}) and (P, L) is the normalization of (Q, L).

Proof. By the definition of P and Q P_{for} is finite over Q_{for} hence there is a finite surjective morphism f : P → Q. Hence Γ(Q_{η}, L_{η}^{n}) ⊂ Γ(P_{η}, L_{η}^{n}) = Γ(G_{η}, L_{η}^{n}) by Remark 4.11. It is clear from Remark 4.11 and the definition of Q̃ that any element of Γ(G_{η}, L_{η}^{n}) has some R-multiple contained in Γ(Q̃_{for}, L̃_{for}^{n})^{Y −inv}. Therefore

Γ(G_{η}, L_{η}^{n}) ⊂ Γ(Q̃_{for}, L̃_{for}^{n})^{Y −inv} ⊗ k(η) = Γ(Q, L^{n}) ⊗ k(η) ⊂ Γ(Q_{η}, L_{η}^{n})

It follows Γ(P_{η}, L_{η}^{n}) = Γ(Q_{η}, L_{η}^{n}) hence (P_{η}, L_{η}) ≃ (Q_{η}, L_{η}) ≃ (G_{η}, L_{η}). Hence P is the normalization of Q. We note that this proves Theorem 0.1.

Definition 5.9. We call the closed fibre (Q_{0}, L_{0}) of (Q, L) a projectively stable quasi-abelian scheme over k := R/I.

The Delaunay decomposition describes the closed fibre by the following

Theorem 5.10. Let Q_{0} be the closed fibre of the flat family (Q, L) and Del the Delaunay decomposition corresponding to (Q, L). Let σ,τ be Delaunay cells in Del.

1. For each σ there exists a T -invariant subscheme O_{Q}(σ) of the closed fibre Q_{0} which is a torus of dimension dim σ over k such that Q_{0} is the disjoint union of O_{Q}(σ)

2. σ ⊂ τ iff O_{Q}(σ) is contained in O_{Q}(τ),

3. For each σ there exists a unique closed reduced subscheme Z(σ) which is the closure of O_{Q}(σ) and has dimension dim σ over k, where the scheme structure of Z(σ) is defined by

Z(σ) ∩ W (c) = Spec k[ξ̄_{x−c,c}; x ∈ σ ∩ X] (c ∈ σ ∩ X)

4. σ ⊂ τ iff Z(σ) is contained in Z(τ),

5. Q_{0} = ⋃_{σ∈Del mod Y} Z(σ)

The general case where the abelian part A_{0} is nontrivial is discussed in a manner similar to Section 4. We omit the details.

Definition 5.11. With the notation in Definition 4.15 we define

K(Q, L) = K(P, L) = K_{S}^{♯}(L_{η})
G(Q, L) = G(P, L) = G_{S}^{♯}(L_{η})
K(G_{η}, L_{η}) = K(L_{η}) = K(Q, L) ⊗ k(η)
G(G_{η}, L_{η}) = G(L_{η}) = G(Q, L) ⊗ k(η)
K(Q_{0}, L_{0}) = K(P_{0}, L_{0}) := K(Q, L) ⊗ k,
G(Q_{0}, L_{0}) = G(P_{0}, L_{0}) = G(Q, L) ⊗ k

We call K(Q, L) (resp. G(Q, L)) the abelian Heisenberg group scheme (resp. the Heisenberg group scheme) of (Q, L). We call K(Q_{0}, L_{0}) (resp. G(Q_{0}, L_{0})) the abelian Heisenberg group scheme (resp. the Heisenberg group scheme) of (Q_{0}, L_{0}). We will see in Lemma 5.14 that K(Q_{0}, L_{0}) is independent of the choice of (Q, L). Therefore by Definition 4.15 and by [MB85, I, 3.1, 3.2] G(Q_{0}, L_{0}) as well as its group scheme structure is uniquely determined by (Q_{0}, L_{0}).

Lemma 5.12. Suppose G(σ) = G ⊂ P for some σ ∈ Del^{max}(0). Let H^{∗} be the subgroup consisting of all e(x) (x ∈ X/Y) in Lemma 4.14 and G^{♯} := H^{∗} · G. Then

1. Γ(Q, L) = Γ(P, L) = Γ(G^{♯}, L),

2. Γ(Q, L) is an irreducible G(Q, L)-module of weight one, that is by definition, any G(Q, L)-submodule of Γ(Q, L) of weight one is of the form JΓ(Q, L) for some ideal J of R,

3. Γ(Q, L) ⊗ k is an irreducible G(Q_{0}, L_{0})-module of weight one.

Proof. Since K_{S}^{♯}(L_{η}) is finite over S by Lemma 4.14, Γ(G^{♯}, L) is an irreducible G_{S}^{♯}(L_{η})-module by [MB85, V, 2.4.2; VI, 1.4.7]. To fix an embedding G = G(σ) ⊂ P, we choose σ with B(α(σ), x)+r(x)/2 = 0 for any x ∈ X. See 4.12. Since Γ(P, L) is R-free, we have Γ(P, L) ↪ Γ(P_{η}, L_{η}) = Γ(G_{η}^{♯}, L_{η}) = Γ(G_{η}, L_{η}). Hence Γ(P, L) ⊂ Γ(G^{♯}, L). Since both Γ(P, L) and Γ(G^{♯}, L) are G(P, L)-modules, there is an ideal J of R such that Γ(P, L) = JΓ(G^{♯}, L) by [MB85, V, 2.4] because Γ(G^{♯}, L) is an irreducible G(P, L)-module. If J ≠ R, then the image of Γ(P, L) ⊗ k in Γ(G^{♯}, L) ⊗ k is zero. However if x ∈ Semi(0, σ) ∩ X, then η_{1}(x) = ξ_{x} = w^{x} and θ ⊗ k = σ_{0}w^{x} + ⋯ is nontrivial if we choose a nontrivial σ_{0} ∈ Γ(A_{0}, M_{0}). Hence J = R and Γ(P, L) = Γ(G^{♯}, L). Since P is the normalization of Q, we have Γ(Q, L) ⊂ Γ(P, L). By the same reason as before there exists an ideal J of R such that Γ(Q, L) = JΓ(P, L). By Theorems 3.9 and 4.10 we have a basis of Γ(P, L) very explicitly. Any element of Γ(P, L) is easily seen to belong to Γ(Q, L). Thus as before we see J = R, Γ(Q, L) = Γ(P, L). (2) follows from (1) and [MB85, V, 2.4.2; VI, 1.4.7]. (3) is a corollary to (2).

Definition 5.13. Let k = R/I, I maximal. Let (Q_{0}, L_{0}) be a projective stable quasi-abelian k-scheme. We call a k-submodule V of Γ(Q_{0}, L_{0}) the theta k-module of (Q_{0}, L_{0}) if V = Γ(Q, L)⊗k. We note that by Theorem 4.10 Γ(Q, L) ⊗ k is the unique submodule of Γ(Q_{0}, L_{0}) generated by a sum of monomials ξ_{a} with Γ(A_{0}, M_{a}) coefficients (a ∈ X) invariant under the action of S_{y} (y ∈ Y). This is restated as

(14)  V = Γ(Q_{0}, L_{0}) ∩ (⊕_{a∈X}Γ(A_{0}, M_{a})ξ_{a})

Lemma 5.14. We define a morphism λ(L_{0}) : G_{0}^{♯} → Pic^{0}(Q_{0}) by

λ(L_{0})(a) = T_{a}^{∗}(L_{0}) ⊗ L_{0}^{−1}

for any U-valued point a of G_{0}^{♯}, U any k-scheme. Then K(Q_{0}, L_{0}) = ker λ(L_{0}).

Proof. First we recall Pic^{0}(Q_{0}) is a k-scheme by [FGA, 232, Corollaire 6.6]. By Definition 5.11 and the proof of Lemma 4.14 K(Q_{0}, L_{0}) := K_{S}^{♯}(L_{η})_{0} = H_{0}^{∗} ⊕ K_{S}^{♯}(L_{η})_{0}^{fn} and moreover the sequence

1 → K_{S}^{♯}(L_{η})_{0}^{m} → K_{S}^{♯}(L_{η})_{0}^{fn} → K(A_{0}, M_{0}) → 0 (15)

is exact. We note G_{0}^{♯} = ∪_{x∈X/Y} S_{x}(G_{0}) ≃ H_{0}^{∗} ⊕ G_{0}, and S_{x} = T_{e(x)_{0}} on G_{0}^{♯} by Subsection 4.13.

Let U be any connected k-scheme. Suppose a ∈ G_{0}^{♯}(U) such that T_{a}^{∗}(L_{0}) ≃ L_{0}. Then there exists x ∈ (X/Y)(U) and a = b·e(x)_{0,U} for some b ∈ G_{0}(U). Since T_{e(x)}^{∗}(L_{0,U}) = S_{x}^{∗}(L_{0,U}) ≃ L_{0,U}, we see T_{b}^{∗}(L_{0,U}) ≃ L_{0,U}. Since L_{0|G_{0,U}} ≃ π_{0,U}^{∗}(M_{0,U}), we have T_{π_{0,U} (b)}^{∗}(M_{0,U}) ≃ M_{0,U}, hence π_{0,U}(b) ∈

K(A_{0}, M_{0})(U). Hence there is c ∈ K_{S}^{♯}(L_{η})_{0}^{fn}(U) such that π_{0,U}(c) = π_{0,U}(b) and T_{c}^{∗}(L_{0,U}) ≃ L_{0,U}. Thus d := b · c^{−1} ∈ T_{0}(U) satisfies T_{d}^{∗}(L_{0,U}) ≃ L_{0,U} where T_{0} = G_{0}^{m} ⊂ G_{0}. It follows that there exists α ∈ Hom(X, G_{m})(U) such that T_{d}^{∗}(w^{x}) = α(x)w^{x}. T_{d} induces an endomorphism of Γ(Q_{0,U}, L_{0,U}), which is spanned by Fourier series ∑_{y∈Y} σ_{x+y}η_{1}(x + y) in view of Lemmas 4.10 (3) and 5.12. Hence T_{d}^{∗}(η_{1}(x + y))/η_{1}(x + y) = T_{d}^{∗}(η_{1}(x))/η_{1}(x) because T_{d}^{∗}(σ_{x}) = σ_{x}, therefore α(y) = 1 (∀y ∈ Y (U)). It follows that α ∈ Hom(X/Y, G_{m})(U) = (X/Y)^{∨}(U), that is, d ∈ K_{S}^{♯}(L_{η})_{0}^{m}(U) (≃ (X/Y)^{∨}(U)). This proves Lemma.

6. Projective embedding

Definition 6.1. Let k be an algebraically closed field and (Q_{0}, L_{0}) be a PSQAS over k. Then we define

e_{min}(K(Q_{0}, L_{0})) = max{n > 0; ord ker(n · id_{K(Q_{0},L_{0})}) ≥ n^{2g}}
e_{max}(K(Q_{0}, L_{0})) = min{n > 0; n · id_{K(Q_{0},L_{0})} = 0}

If the order of K(Q_{0}, L_{0}) and the characteristic of k are coprime, then K(Q_{0}, L_{0}) ≃ ⊕_{i=1}^{g}(Z/e_{i}Z)^{⊕2} for some positive integers e_{i} with e_{i}|e_{i+1}. From this it follows that e_{min}(K(Q_{0}, L_{0})) = e_{1} and e_{max}(K(Q_{0}, L_{0})) = e_{g}. See Definition 7.5.

Theorem 6.2. Let A be an abelian variety over an algebraically closed field k, L an ample invertible sheaf on A. If e_{min}(K(A, L)) ≥ 3, L is very ample.

Proof. This is a corollary to [Mumford74, p.231, Theorem 3]. By the assumption K(A, L) ⊃ A_{n}, all the n-division points of A for some n ≥ 3. By [Mumford74, p.231, Theorem 3] there exists an invertible sheaf M on A such that L = M^{n}. Since L is ample, M is ample. By Lefschetz’s theorem, L is very ample because n ≥ 3.

Theorem 6.3. Let (Q_{0}, L_{0}) be a polarized projectively stable quasi-abelian scheme over an algebraically closed field k, and K(Q_{0}, L_{0}) the abelian Heisenberg group. If e_{min}(K(Q_{0}, L_{0})) ≥ 3, then Γ(Q, L)⊗k is very ample, a fortiori L_{0} is very ample.

Proof. We consider first the totally degenerate case. By Theorem 5.12 Γ(Q, L) = Γ(P, L). Let V := Γ(Q, L) and V^{0} := Γ(Q, L) ⊗ k.

We define

θ(x) := ∑_{y∈Y} a_{0}(x + y)ξ_{x+y} for x ∈ X/Y (16)

We note θ(x + z)ϑ = S_{z}^{∗}(θ(x)ϑ) (∀z ∈ X). Hence it suffices to prove the assertion on W_{0}(c) := W (c) ⊗ k for c ∈ Star (0) where Star (0) is the union of Delaunay cells containing 0.

It follows from Lemma 5.5 (1) that on W_{0}(c)

(17)  θ(x + c)/ξ_{c} = ∑_{y∈Y} a_{0}(x + y + c)ξ_{x+y+c}/ξ_{c}

= ∑_{y∈(−x+Star (0))∩Y} a_{0}(x + y + c)ξ_{x+y,c} (18)

By the assumption Y ⊂ nX for n ≥ 3. Hence Star (c) ∩ (y + Star (c)) = ∅ for y ∈ Y by Lemma 1.2. It follows that (θ(x)/ξ_{c}) ⊗ k consists of at most a single term. Therefore

(19) θ(x + c)/ξ_{c} = ∑_{y∈(−x+Star (0))∩Y} a_{0}(x + y + c)ξ_{x+y,c}

(20) = \begin{cases}a_{0}(x + y + c)ξ_{x+y,c} & if ∃y ∈ Y ∩ (−x + Star (0)) \\ 0 & otherwise \end{cases}

By Theorem 5.10 Q_{0} is a disjoint union of the torus orbits O_{Q}(σ) ≃ Spec k[ξ_{z,c}^{±}; z + c ∈ σ ∩ X] (σ ∈ Del(c), c ∈ X/Y). For any point of O_{Q}(σ), θ(x + c)/ξ_{c} = a_{0}(x + y + c)ξ_{x+y,c} is nonvanishing if {x + y + c} = (x + c + Y) ∩ (σ ∩ X). Therefore the linear system V^{0} is base point free.

Let ϕ_{V^{0}} be the morphism defined by V^{0}. Suppose that we are given two distinct points q and q′ of (Q_{0})_{red}. Suppose that q ∈ O_{Q}(σ) and q′ ∈ O_{Q}(σ′) and σ ≠ σ′ mod Y. Then choose c ∈ σ ∩ X \ (σ′ + Y). Then θ(c) is nonvanishing at q while θ(c) is zero at q′. Hence ϕ_{V^{0}}(q) ≠ ϕ_{V^{0}}(q′). If q, q′ ∈ O_{Q}(σ), then choose z + c, c ∈ σ ∩ X such that ξ_{z,c}(q) ≠ ξ_{z,c}(q′). Since (θ(c)/ξ_{c})_{|W_{0}(c)} = a_{0}(c) ∈ k is nonzero, this implies that (θ(z + c)/θ(c))(q) ≠ (θ(z + c)/θ(c))(q′). Thus ϕ_{V^{0}} separates two distinct points of W_{0}(c)_{red}.

The ring S_{0}(c) is generated by ξ_{x,c} (x ∈ Star (0) ∩ X), while ξ_{x,c} = 0 in S_{0}(c) by Lemma 5.5 if x ∉ Star (0). Hence ϕ_{V^{0}} is an immersion locally on W_{0}(c) by Theorem 5.7 (3) and Equation (20). This proves Theorem in the totally degenerate case.

The general case easily follows from the above and Theorem 6.2. With the notation in Lemma 4.14 and Definition 5.11 let K(Q, L) := K(P, L) = K_{S}^{♯}(L_{η}) and let H^{∗} be an étale subgroup scheme of K(Q, L) generated by e(x) := S_{x}(e) (x ∈ X/Y) and K_{ab} := (H^{∗} ⊕ K_{S}(L_{η})^{m})^{⊥}. Let e_{S}^{♯} : K_{S}^{♯}(L_{η}) × K_{S}^{♯}(L_{η}) → G_{m,S} be the extension of e^{L_{η}}. Hence e_{S}^{♯} is also an extension of e_{S}^{L_{η}}. By [MB85, IV, 2.4] and Lemma 4.14 e_{S}^{♯} is a perfect pairing. Since X(K_{S}(L_{η})^{m}) ≃ H^{∗}, we see that K(Q, L) ≃ (H^{∗} ⊕ K_{S}(L_{η})^{m}) ⊕ K_{ab}. It is clear that K_{ab} ≃ K_{S}(L_{η})^{ab} ≃ K(A, M), hence K_{ab} ⊗ k ≃ K(A_{0}, M_{0}). Hence e_{min}(K(A_{0}, M_{0})) ≥ 3 and Y ⊂ nX for some n ≥ 3. Hence we can apply Theorem 6.2 and Theorem in the totally degenerate case.

We need only to consider W_{0}(0). We take θ = ∑_{y∈Y} σ_{y}ξ_{y} ∈ Γ(Q_{0}, L_{0}). Then by the assumption and by the same argument we have θ_{|W (0)} = σ_{0} ∈ Γ(A_{0}, M_{0}). By Theorem 6.2 there is σ_{0} ∈ Γ(A_{0}, M_{0}) nonvanishing at any given point of A_{0}. This shows that the linear system V^{0} is base point free. Suppose that we are given two distinct points q and q′ of W_{0}(0). If moreover their images in A are different, then |M_{0}|_{A} separates these points. Hence V^{0} separates q and q′. Suppose their images in A are the same, say a. For any point a of A, we choose an affine open subset U of A and coordinates w_{U}^{x} (x ∈ X) of the split U-torus G̃_{U} := G̃ ×_{A} U.

Then by Theorem 4.10, Lemma 5.12 and by the totally degenerate case a general global section θ ∈ Γ(Q_{0}, L_{0}) is given on W_{0}(0) by θ_{|W_{0}(0)} = ∑_{x∈Star (0)∩X} σ_{x}ξ_{x}. We have a basis of Γ(Q_{0}, L_{0}) whose restriction to W_{0}(0) consists of at most a single term as in Equation (20) by Lemma 1.2. Since M_{x} is very ample, we can choose a section σ_{x} ∈ Γ(A, M_{x}) for any x ∈ Star (0) ∩ X such that σ_{x} is nonvanishing at a. Therefore V^{0} is base point free. Then by the argument as in the totally degenerate case, we can separate the points q and q′. We can prove similarly that ϕ_{V^{0}} is an immersion. This completes the proof.

7. Central extension

Definition 7.1. Let R be an integral domain, k(R) its fraction field and S = Spec R. Let K be a finite constant reduced group scheme over S. A group S-scheme G is said to be a central extension of K by G_{m,S} if G_{m,S} is contained in the center of G and the quotient of G by G_{m,S} is K. Hence we have an exact sequence of group S-schemes

1 → G_{m,S} →^{i} G →^{τ} K → 0

A central extension G is said to be totally split if K(R) ≃ K(k(R)) ≃ K(\overline{k(R)}) and τ(R) : G(R) → K(R) is surjective.

Each section P_{i} ∈ K(R) is a subscheme of K isomorphic to S. There is an irreducible component G_{i} of G mapped onto P_{i}. Although G_{i} may have no sections over S, by taking a finite base change if necesary, there is a section Q_{i} of G such that P_{i} = τ(Q_{i}) and G_{i} = G_{m,S} · Q_{i} ≃ G_{m,S}. Thus any central extension G of K by G_{m,S} is totally split by a finite base change.

Now we recall a well known lemma.

Lemma 7.2. The set of equivalence classes of totally split central extensions is the cohomology group H^{2}(K(R), G_{m,S}(R)) = H^{2}(K(R), R^{∗}).

Proof. Suppose that G is totally split. Then G(R) = {α·Q_{i}; α ∈ G_{m,S}(R), P_{i} ∈ K(R)}. We denote α · Q_{i} by (α, k) where P_{i} = k ∈ K(R). Then in the same manner as in [Mumford74, p. 221] we can express the twisted group law of G as a two cocycle {f(k, k′); k, k′ ∈ K(R)}

(α, k) · (α′, k′) = (α · α′ · f(k, k′), k + k′) (21)

where α, α′ are R-valued points of G_{m,S}, k, k′ are R-valued points of K and K is written additively. f(·, ·) satisfies the cocycle condition

(22)  f(λ + ν, μ)f(λ, ν) = f(λ, ν + μ)f(ν, μ)

If another cocycle h describes an isomorphic cohomologous to f, that is, h/f is a coboundary in the sense that

(23)  h(λ, ν) = f(λ, ν)g(λ + ν)g(λ)^{−1}g(ν)^{−1}

for some g where λ, ν ∈ K(R).

Remark 7.3. The commutator form e_{G}(λ, ν) for G is given by the formula [Mumford74, p. 222]

(24)  e_{G}(λ, ν) = f(λ, ν)/f(ν, λ)

even if f(0, 0) ≠ 1. It is clear e_{G}(λ, ν) = h(λ, ν)/h(ν, λ) if h is cohomologous to f.

Conversely we prove

Lemma 7.4. Let G and G′ be totally split central extensions of K by G_{m,S}. Suppose that their commutator forms are equal. Then G ≃ G′ after a suitable finite base change.

Proof. Let f and h be two-cocycles associated with the extensions G and G′ respectively. Then we prove that after a suitable finite base change there exists g satisfying (23) if e_{G} = e_{G′}. Since H^{2}(K(R), G_{m,S}(R)) is a group, it suffices to prove that if a cocycle f determine a central extension G with its commutator form trivial, then the pull back of G by a finite base change is trivial, that is, the pull back of f is a coboundary. We prove the assertion by induction on the minimal number of generators of K. Now we write the group law of G_{m,S} additively. Thus our conditions on f are

(25)  f(λ + ν, μ) + f(λ, ν) = f(λ, ν + μ) + f(ν, μ)
(26)  f(λ, ν) = f(ν, λ)

where f(λ, ν) ∈ G_{m}(R) = R^{∗}, λ, ν ∈ K(R). First we may assume f(0, 0) = 0. In fact, otherwise, let g(λ) = −f(0, 0) and h(λ, ν) := f(λ, ν) − g(λ + ν) + g(λ) + g(ν). Then h(0, 0) = 0, h(λ, ν) = h(ν, λ) and h is cohomologous to f. Now we assume f(0, 0) = 0. The condition (25) implies f(λ, 0) + f(λ, 0) = f(λ, 0) + f(0, 0) by setting ν = 0. Hence f(λ, 0) = 0 for any λ ∈ K(R).

Let N = ord K := rank_{R} B. Hence by [OT70] Nλ = 0 for any λ ∈ K(R). We choose and fix any λ. We define (formally at this moment)

(27)  g(λ) = −(1/N) ∑_{i=0}^{N−1} f(λ, iλ)

(28)  g(kλ) = kg(λ) + ∑_{i=0}^{k−1} f(λ, iλ)

First we note g((k + N)λ) = g(kλ). We see readily f(λ, kλ) = g((k + 1)λ) − g(kλ) − g(λ) for any k. Next by induction on ℓ we prove

(29)  f(ℓλ, kλ) = g((ℓ + k)λ) − g(ℓλ) − g(kλ)

In fact, if ℓ = 1, clear. Assume (29) for any k ∈ Z and 1 ≤ ℓ ≤ n. Since

f((n + 1)λ, kλ) = −f(λ, nλ) + f(nλ, (k + 1)λ) + f(λ, kλ)

we have f((n + 1)λ, kλ) = g((n + k + 1)λ) − g((n + 1)λ) − g(kλ). Therefore if K(R) is generated by λ and if R contains g(λ) defined by (27), then the pull back of f is cohomologous to zero.

Next we consider the case where K(R) = H(R) ⊕ Zλ for some λ ∈ K(R). For ν, μ ∈ H(R) and k, ℓ ∈ Z, we write ν + kλ ∈ K(R) and μ + ℓλ ∈ K(R). We define

(30)  g(ρ) = −(1/N) ∑_{i=0}^{N−1} f(ρ, iρ)
(31)  g(ν + kλ) = f(ν, kλ) + g(ν) + g(kλ)

for ρ ∈ H(R) or ρ ∈ Zλ, and ν ∈ H(R), k ∈ Z. By induction hypothesis on H we have f(μ, ν) = g(μ + ν) − g(μ) − g(ν) for μ, ν ∈ H(R). It follows from the relation (25), the definition (31) and symmetry of f that

f(μ + kλ, ν) = −f(kλ, μ) + f(kλ, ν + μ) + f(μ, ν)
= g(ν + μ + kλ) − g(ν) − g(μ + kλ)

Therefore by symmetry of f and the relation (25)

f(μ + kλ, ν + ℓλ) = −f(kλ, μ) + f(kλ, ℓλ + μ + ν) + f(μ, ν + ℓλ)

we have f(μ + kλ, ν + ℓλ) = g(μ + ν + (k + ℓ)λ) − g(μ + kλ) − g(ν + ℓλ).

Now we switch into the first multiplicative cocycle f(λ, ν). Let R′ be the ring obtained by adding to R N-th roots of ∏_{i=0}^{N−1} f(ρ, iρ) for all ρ ∈ K(R). Clearly R′ is finite over R. Then by the above argument the pull back to Spec R′ of f becomes cohomologous to zero. This completes the proof.

Definition 7.5. Let N and M be a pair of positive integers such that N is divisible by M, and ζ_{M} a primitive M-th root of unity and O_{M} := Z[ζ_{M}, 1/M]. For simplicity we write O = O_{M} if there is no fear of confusion.

Let H be a finite abelian group of order N such that e_{max}(H), the maximal order of elements in H, is equal to M. Now by abuse of notation we regard H as a constant affine finite abelian group O-scheme of order N. Let H^{∨} := Hom_{O}(H, G_{m,O}) be the Cartier dual of H over O. We set K := K(H) = H ⊕ H^{∨} and define a bimultiplicative (we say simply a bilinear) form e_{K} : K × K → G_{m,O} by

e_{K}(z ⊕ α, w ⊕ β) = β(z)α(w)^{−1} (32)

where z, w ∈ H, α, β ∈ H^{∨}. We denote e_{K} and K by ℓ_{H} and K(H) when it is necessary to emphasize dependence on H. However we note that H is a maximally isotropic subgroup of K with respect to e_{K}, which is uniquely determined by this property up to isomorphism.

Let μ_{M} := Spec O[x]/(x^{M} − 1) be the group scheme of all M-th roots of unity. We define group O-schemes G(K) and G(K) by

(33)  G(K) := {(a, z, α); a ∈ G_{m,O}, z ∈ H, α ∈ H^{∨}}
(34)  G(K) := {(a, z, α); a ∈ μ_{M}, z ∈ H, α ∈ H^{∨}}

endowed with a group scheme structure

(a, z, α) · (b, w, β) = (abβ(z), z + w, α + β) (35)

where a, b ∈ G_{m,O} or a, b ∈ μ_{M}, z, w ∈ H and α, β ∈ H^{∨}.

An automorphism ϕ of G(K) as a group O-scheme is said to be a central extension automorphism if

ϕ(ah) = aϕ(h) for a ∈ G_{m,O}, h ∈ G(K) (36)

A level subgroup scheme of G(K) (resp. G(K)) is by definition a subgroup scheme B of G(K) (resp. G(K)) whose natural image in K(H) is a maximal isotropic subgroup scheme with respect to e_{K} isomorphic to B. Let H̃ := {(1, z, 0); z ∈ H} and H̃^{∨} := {(1, 0, α); α ∈ H^{∨}}. It is clear that H̃ and H̃^{∨} are level subgroup schemes of G(K) and G(K).

Let V (K) be the group algebra O[H^{∨}] of H^{∨} over O, and an O-basis v(χ) (χ ∈ H^{∨}) of V (K). The group scheme G(K) as well as G(K) acts on V (K) by

U(K)(a, z, α)(v(χ)) = aχ(z)v(χ + α). (37) where a ∈ μ_{M} or a ∈ G_{m,O}, z ∈ K and α ∈ H^{∨}. We define subgroup O-schemes Ḡ(K) and Ḡ(K) of an algebraic group O-scheme GL(V (K)) by

(38)  Ḡ(K) : = {U(K)(g); g ∈ G(K)}
(39)  Ḡ(K) : = {U(K)(g); g ∈ G(K)}

Lemma 7.6. Let ψ : G(K) → K be the natural surjection. Let S be a subgroup O-scheme of G(K) containing two level subgroup O-schemes H̃_{1} and H̃_{2} with ψ(H̃_{1}) ∩ ψ(H̃_{2}) trivial. Then G(K) ⊂ S. In particular, G(K) is the unique minimal subgroup O-scheme among those subgroup O-schemes S with the above property.

Proof. Let H_{i} := ψ(H̃_{i}). By definition H_{i} ≃ H̃_{i}. Let R be any ring over O. It follows that there exists a homomorphism f_{i} : H_{i} → G_{m} such that

H̃_{i}(R) = {(f_{i}(z, α), z, α); (z, α) ∈ H_{i}(R)} (40)

Since e_{max}(K) = M, we have f_{i}(z, α) ∈ μ_{M}(R) for (z, α) ∈ H_{i}(R). Since H_{1} ∩ H_{2} = {1} and H_{i} are maximally isotropic, we have K ≃ H_{1} ⊕ H_{2} and H_{2} ≃ Hom(H_{1}, G_{m}). It follows that ζ_{M} ∈ e_{K}(H_{1}, H_{2}) = [H̃_{1}(R), H̃_{2}(R)], hence (f_{i}(z, α), 0, 0) ∈ S(R). Therefore G(K)(R) ⊂ S(R). This proves Lemma.

Lemma 7.7. Let ϕ be a central extension automorphism of G(K). Assume that it induces the identity of K. Then ϕ(G(K)) = G(K).

Proof. We see that ϕ is written as

ϕ : (a, u) ↦ (ϕ(a, u), u) for a ∈ G_{m,O} and u ∈ K (41)

Then ϕ is subject to the relation

ϕ(ab, u + v) = ϕ(a, u)ϕ(b, v), ϕ(a, u) = aϕ(1, u) (42)

Hence ϕ(1,?) ∈ Hom_{O}(K, G_{m,O}) = Hom_{O}(K, μ_{M}). It follows ϕ(G(K)) = G(K). This Lemma also follows from Lemma 7.6.

Lemma 7.8. Let T be an O-scheme, and G a central extension of K_{T} by G_{m,T},

1 → G_{m,T} → G → K_{T} → 0

Suppose that there exists an open affine covering U_{i} := Spec A_{i} of T such that G_{U_{i}} ≃ G(K)_{U_{i}}. Then there exists an O-scheme W finite over T such that G_{W} ≃ G(K)_{W}.

Proof. Let ρ_{i} : G_{U_{i}} ≃ G(K)_{U_{i}} be the given isomorphisms and U_{ij} = U_{i} ∩ U_{j}. There is a natural isomorphism g_{ij} : (G_{U_{j}})_{U_{ij}} ≃ (G_{U_{i}})_{U_{ij}}. Then h_{ij} := ρ_{i} ·g_{ij} · (ρ_{j})^{−1} is an isomorphism of (G(K)_{U_{j}})_{U_{ij}} and (G(K)_{U_{i}})_{U_{ij}}. Therefore G is obtained by gluing G(K)_{U_{i}} by a one-cocycle h_{ij}. By the proof of Lemma 7.7 we see h_{ij} ∈ H_{Zar}^{1}(T, Hom(K, μ_{M})). We prove that h_{ij} is cohomologous zero by a suitable finite base change. Since K is finite abelian, it suffices to prove that any element θ ∈ H_{Zar}^{1}(T, μ_{M}) is cohomologous zero by a suitable finite base change. The cocycle θ determines a G_{m}-torsor F_{θ} on T of order at most M. Let (F_{θ})_{U_{i}} = {(ζ_{i}, z); ζ_{i} ∈ G_{m}, z ∈ U_{i}}. We may assume that F_{θ} = ∪_{i} (F_{θ})_{U_{i}} is obtained by gluing together by the relations ζ_{i} = θ_{ij}ζ_{j}. Let W := ∪{(ζ_{i}, z) ∈ (F_{θ})_{U_{i}}; ζ_{i}^{M} = 1}. Since θ_{ij}^{M} = 1, W is a well-defined subscheme of F_{θ} finite over T. The pull back of θ to W is trivial. This completes the proof.

Remark 7.9. Lemma 7.2 and Lemma 7.4 are true over any algebraically closed field k by [MB81]. Lemma 7.7 is clearly true over any algebraically closed field k.

Definition 7.10. Let Spec k be a point of Spec O. Any G(K) ⊗ k-module V is of weight one if every a ∈ G_{m,k} of G(K) ⊗ k acts on V as scalar multiplication a · id_{V}. In this case we say also that the action of G(K) on V is of weight one.

Lemma 7.11. Let Spec k be a point of Spec O. Then V (K) ⊗ k is an irreducible G(K) ⊗ k-module of weight one, unique up to equivalence. Any finite dimensional G(K)-module of weight one over k is a direct sum of copies of V (K) ⊗ k.

Proof. We imitate the argument in [Mumford66]. Let V be a nontrivial G(K)-k-module of weight one. Let V (χ) be the maximal k-submodule of V such that H acts on V (χ) by a character χ ∈ H^{∨}. By definition V (0) = V^{H}, the submodule of H-invariants. Since (1, 0, −χ) · V (χ) = V (0), V (χ) ≠ 0 if and only if V (0) ≠ 0. Hence V (0) ≠ 0. Then we set v(χ, w) := (1, 0, χ) · w ∈ V (χ) for w ∈ V (0) = V^{H}. Let V_{0} be a k-submodule of V spanned by v(χ, w) (∀χ ∈ H, w ∈ V (0)). We see

(a, z, α) · v(χ, w) = (a, 0, α) · (1, z, 0) · v(χ, w)
= aχ(z)v(χ + α, w)

This proves that V_{0} is a G(K)-module. Since V/V_{0} is a G(K)-module with (V/V_{0})^{H} = 0, we have V = V_{0}. Hence V ≃ V (K) ⊗_{k} V (0) ≃ (V (K) ⊗ k)^{⊕ dim_{k} V/ dim_{k} V_{0}} as G(K) ⊗ k modules.

Definition 7.12. Let R be a complete discrete valuation ring over O with k = R/I the residue field and S := Spec R. Suppose we are given a PSQAS (Q, L), a SQAV (P, L) and a semi-abelian scheme (G, L) over S such that (P, L) is the normalization of (Q, L) and G is an open subscheme of (P, L). In what follows we need only a pair (Q, L) and (G, L). We define G(Q, L) as follows. Let L^{tsr} be the G_{m}-torsor on G associated with the invertible sheaf L. Then we define G(Q, L) to be (L^{tsr})_{|K(Q,L)}. Since L is a cubical sheaf on G, G(Q, L) is shown to be a group scheme [MB81, 3.2]. The natural projection from L^{tsr} to G makes G(Q, L) a central extension of K(Q, L) by G_{m,S}

1 → G_{m,S} → G(Q, L) → K(Q, L) → 0 (43)

with commutator form e_{S}^{♯}.

Remark 7.13. Assume that the order of K(Q, L) and the characteristic of k are coprime. Then K(Q, L) is a reduced flat finite group S-scheme, étale over S. Hence by a finite base change of S we may assume that K(Q, L) is a constant group scheme. The symplectic group scheme (K(Q, L), e_{S}^{♯}) is then isomorphic to (K_{S}, (e_{K})_{S}) = (K(H)_{S}, ℓ_{H,S}) for some finite group O-scheme K = K(H). By taking a further finite base change if necessary, G(Q, L) is a totally split central extension of K(Q, L) by G_{m,S}. By Lemma 7.4 G(Q, L) ≃ G(K)_{S} after a finite base change.

Definition 7.14. We choose a central extension isomorphism ρ : G(K)_{S} ≃ G(Q, L) and define G(Q, L) to be the image of G(K)_{S} by ρ. G(Q, L) is a central extension of K(Q, L) by μ_{M}. By Lemma 7.7 G(Q, L) is independent of the choice of ρ.

We also define G(Q_{η}, L_{η}) := G(Q, L) ⊗ k(η), and G(Q_{0}, L_{0}) := G(Q, L) ⊗ k. We note that both G(Q, L) and G(Q, L) are defined only after a base change because G(Q, L) is defined by choosing an open subgroup scheme G^{♯} of P. They act on Γ(Q, L) and Γ(Q, L) is irreducible by these actions in view of Lemma 7.11.

8. Kempf-Stability

8.1. Hilbert points. Let (Q_{0}, L_{0}) be a projectively stable quasi-abelian scheme over k. Suppose that V^{0} := Γ(Q, L) ⊗ k is very ample. Hence (Q_{0}, L_{0}) is a closed subscheme of the projective space (P, O_{P}(1)) where P = P(V^{0}). Let I be the ideal of O_{P} defining (Q_{0}, L_{0}). Then by Serre vanishing theorem there exists a sufficiently large n_{0} such that H^{1}(P, I ⊗ O_{P}(n)) = 0 and H^{q}(Q_{0}, L_{0}^{n}) = 0 for n ≥ n_{0} and q ≥ 1. Hence we have an epimorphism

ϕ_{n} : S^{n}Γ(Q, L) ⊗ k → Γ(Q_{0}, L_{0}^{n}) (44)

The epimorphism ϕ_{n} determines a point of the Grassman variety. Let n(g) := n^{g} deg(L_{0}). By taking the Plücker coordinates we obtain a point

∧^{n(g)}ϕ_{n} of the projective space P( ∧^{n(g)}S^{n}Γ(Q, L) ⊗ k)

∧^{n(g)}ϕ_{n} : ∧^{n(g)}S^{n}Γ(Q, L) ⊗ k → ∧^{n(g)}Γ(Q_{0}, L_{0}^{n}) ≃ k. (45)

We call ∧^{n(g)}ϕ_{n} the n-th normalized Hilbert point of (Q_{0}, L_{0}), which we denote by hilb_{n}(Q_{0}, L_{0}). If Γ(Q_{0}, L_{0}) = Γ(Q, L)⊗k, for instance if Q_{0} = P_{0}, then hilb_{n}(Q_{0}, L_{0}) is just the n-th Hilbert point of (Q_{0}, L_{0}) in the usual sense.

We say that hilb_{n}(Q_{0}, L_{0}) is Kempf-stable if it has a closed SL_{±}(V^{0})-orbit. By Lemma 7.11 the following is a corollary to [Kempf78, Corollary 5.1].

Theorem 8.2. Let (Q_{0}, L_{0}) be a polarized projectively stable quasi-abelian scheme over an algebraically closed field k. Suppose that the characteristic of k and deg L_{0} := (L_{η}^{g})/g! are coprime. If Γ(Q, L) ⊗ k is very ample, then hilb_{n}(Q_{0}, L_{0}) is Kempf-stable for all large n. In particular it is Mumfordsemistable.

Proof. Let V^{0} := Γ(Q, L) ⊗ k. Let SL_{±}(V^{0}) be a subgroup of GL(V^{0}) consisting of elements with determinant ±1. We note that closedness of the orbits for the actions of SL or SL_{±} are equivalent to each other because [SL_{±}(V^{0}); SL(V^{0})] is finite. G(Q_{0}, L_{0}) acts on Γ(Q_{0}, L_{0}) keeping V^{0} stable, hence hilb_{n}(Q_{0}, L_{0}) is G(Q_{0}, L_{0})-invariant. Since V^{0} is an irreducible G(Q_{0}, L_{0}) (≃ G(K) ⊗ k)-module by Lemma 7.11, G(Q_{0}, L_{0}) is contained in no parabolic subgroup of SL_{±}(V^{0}). By applying [Kempf78, Corollary 5.1] to SL_{±}(V^{0}), we see that hilb_{n}(Q_{0}, L_{0}) has a closed SL_{±}(V^{0}) orbit. They are semistable in the sense of Mumford by [Seshadri77, p. 252, Proposition 6 (1)].

We remark that Theorem 8.2 is true without the assumption on deg L_{0} by [MB85, V, 2.5.5].

Corollary 8.3. Let (P_{0}, L_{0}) be a polarized torically stable quasi-abelian variety over an algebraically closed field k. Suppose that the characteristic of k and deg L_{0} are coprime. If L_{0} is very ample, then hilb_{n}(P_{0}, L_{0}) is Kempf-stable for all large n.

The (normalized) Hilbert points hilb_{n}(Q_{0}, L_{0}) are not necessarily properly stable, for instance a 3-gon of rational curves. We note P_{0} = Q_{0} by Theorem 1.6 if g ≤ 4. Corollary 8.3 is an immediate consequence of Theorem 8.2 because (P_{0}, L_{0}) is very ample only if P_{0} = Q_{0}.

Remark 8.4. There are various notions of stability, which differ one from another. Since the situation is rather confusing, we explain here the relation between them so far as the present article is concerned ([Kempf78, p. 314], [MFK94, p. 36], [Seshadri77, p. 252]). Semi-stability in all these literatures is equivalent to each other by [Seshadri77, Proposition 6 (1)] as well as Properstability. Mumford-stability is strictly stronger than Kempf-stability, which is defined to be closedness of the orbit of the point. Kempf-stability is strictly stronger than semi-stability.

Let V = Zx_{0} + Zx_{1} + Zx_{2}. Let P(S^{3}(V^{∨})) be the projective space of ternary cubic forms on V. Let U_{node} be an open affine subscheme of P(S^{3}(V^{∨})) consisting of cubic curves with at worst nodal singularities. The action of SL(3) on U_{node} is not closed because cubic curves with exactly one or two nodes are only semi-stable, their orbits are not closed. Hence cubic curves with exactly one or two nodes are not Kempf-stable. However a 3-gon of rational curves C_{0} – a union of three lines with nodal singularities only – is Kempf-stable by Theorem 8.2. It is not Mumford-stable but Mumfordsemistable. This example shows in particular that Kempf-stability is not an open condition. We also note that by Theorem 11.6 (C, O_{C}(n)) is not Kempf-stable for any n ≥ 2 if C is a 3-gon.

9. Level G(K)-structures

9.1. The ring O_{M}. We keep the same notation in Section 7. Let N and M be a pair of positive integers such that N is divisible by M, and ζ_{M} a primitive M-th root of unity and O = O_{M} := Z[ζ_{M}, 1/M].

Let H be a constant affine finite abelian group O-scheme of order N with e_{max}(H) = M. Let (K, e_{K}) := (H ⊕ H^{∨}, ℓ_{H}) be the constant symplectic finite abelian group O-scheme of rank N^{2} defined in Section 7.

Definition 9.2. Let Spec k be a (not necessarily closed) point of Spec O. A triple (Q_{0}, L_{0}, V_{0}) is called a g-dimensional K-symplectic projectively stable quasi-abelian scheme over k or a K-symplectic PSQAS over k if

(i) (Q_{0}, L_{0}) is a g-dimensional projectively stable quasi-abelian scheme over k, that is, a closed fibre of some (Q, L) in Theorem 5.8,

(ii) (K(Q_{0}, L_{0}), e_{0}^{♯}) ⊗ k̄ ≃ (K, e_{K}) ⊗ k̄. (iii) V_{0} is a G(Q_{0}, L_{0})-submodule of H^{0}(Q_{0}, L_{0}) such that V_{0} ⊗ k̄ is the

theta k̄-module of (Q_{0}, L_{0}) ⊗ k̄ where k̄ is the algebraic closure of k. See Definition 5.13 for theta modules. By definition We note that (ii) implies a central extension isomorphism G(Q_{0}, L_{0}) ⊗ k̄ ≃ G(K) ⊗ k̄.

If the following conjecture for n = 1 is affirmatively solved, the condition (iii) as well as V_{0} in the triple is unnecessary

Conjecture: H^{q}(Q_{0}, L_{0}^{n}) = 0 for q, n > 0.

9.3. Separable polarizations. Let Spec k be a point of Spec O. Hence the characteristic of k := R/I is prime to N. Therefore the condition (ii) in Definition 9.2 implies that d := deg L_{η}/(g!) is prime to the characteristic of k := R/I because d = \sqrt{ord K} = N, hence the polarizations L_{η} and L_{0} are separable. By the discussion in Subsection 2.3 d_{t} := ord K(L_{η})^{m} and d_{a} = ord K(M) are prime to the characteristic of k. In particular, M is also a separable polarization of the abelian scheme A, the abelian part of the Raynaud sequence of G.

Suppose e_{min}(K) ≥ 3 in what follows.

Definition 9.4. Let Spec k be a point of Spec O, (Z, L, V) a K-symplectic PSQAS over k. A level G(K)-structure (ϕ, ρ) on (Z, L, V) is a pair of a closed k-immersion ϕ : Z → P(V (K) ⊗ k) and a central extension isomorphism ρ : G(K) ⊗_{O} k → G(Z, L) which induce ρ-equivariant isomorphisms

(i) ϕ^{∗}(O_{P(V (K)⊗k)}(1)) ≃ L, (ii) ϕ^{∗} : H^{0}(O_{P(V (K))}(1)) ⊗ k ≃ V (K) ⊗ k ≃ V for some action of G(K) on V (K) of weight one (possibly different from U(K)).

The level G(K)-structure (ϕ, ρ) is called a rigid G(K)-structure if in addition to (i) and (ii) the following is true, (iii) ρ = G(ϕ^{∗}) · (U(K) ⊗_{O} k) where G(ϕ^{∗})(g) = (ϕ^{∗})g(ϕ^{∗})^{−1} for any g ∈ Ḡ(K), that is, the action of G(K) on V (K) is just U(K).

Evidently L and V are uniquely determined by ϕ. We denote the sextuplet (Z, L, G(Z, L), V, ϕ, ρ) by (Z, ϕ, ρ)_{LEV}. If moreover (iii) is true, we denote it by (Z, ϕ, ρ)_{RIG}.

Definition 9.5. Let (Z_{i}, L_{i}, G(Z_{i}, L_{i}), V_{i}, ϕ_{i}, ρ_{i}) be k-PSQAS’s with level G(K)-structures (i = 1, 2). They are defined to be isomorphic as k-PSQAS’s with level G(K)-structures if there is a k-isomorphism f : Z_{1} ≃ Z_{2} such that (i) L_{1} ≃ f^{∗}L_{2}, (ii) V_{1} = f^{∗}V_{2} as k-linear subspaces of H^{0}(Z_{1}, L_{1}), (iii) ρ_{1} = G(f^{∗}) · ρ_{2} where G(f^{∗})(g) = f^{∗}g(f^{∗})^{−1} for any g ∈ G(Z_{2}, L_{2}).

In this case we write (Z_{1}, ϕ_{1}, ρ_{1})_{LEV} ≃ (Z_{2}, ϕ_{2}, ρ_{2})_{LEV}. By (iii) we have G(Z_{1}, L_{1}) = G(f^{∗})G(Z_{2}, L_{2}). We define (Z_{i}, ϕ_{i}, ρ_{i})_{RIG} to be isomorphic if so are (Z_{i}, ϕ_{i}, ρ_{i})_{LEV}. For any PSQAS (Z, ϕ, ρ)_{RIG}, we see

(Z, ϕ, ρ)_{RIG} ≃ (ϕ(Z), i, U(K))_{RIG} (46)

where i : ϕ(Z) → P(V (K) ⊗ k) is the natural inclusion map.

Lemma 9.6. Let Spec k be a point of Spec O. Let (Z_{i}, ϕ_{i}, ρ_{i})_{RIG} be k-PSQAS’s with rigid G(K)-structures (i = 1, 2). Then the following are equivalent;

1. (Z_{1}, ϕ_{1}, ρ_{1})_{RIG} ≃ (Z_{2}, ϕ_{2}, ρ_{2})_{RIG}

2. there is a k-isomorphism f : Z_{1} ≃ Z_{2} with ϕ_{1} = ϕ_{2} · f.

Proof. First we prove that (2) implies (1). Definition 9.5 (ii) is clear from the uniqueness of theta modules. (i) follows from it. See Definition 5.13 for theta modules. Definition 9.5 (ii) is clear.

Next we prove that (1) implies (2). By Definition 9.5 (i) and by the very-ampleness of V_{i} there is an h ∈ GL(V (K) ⊗ k) such that ϕ_{1}^{∗} · h = f^{∗} · ϕ_{2}^{∗} ∈ Hom(V (K) ⊗ k, V_{1}). Hence G(ϕ_{1}^{∗})G(h) = G(f^{∗})G(ϕ_{2}^{∗}). It follows from Definition 9.5 (ii) that G(h) is the identity, i.e., h · g = g · h for any g ∈ Ḡ(K). By Schur’s lemma, h is a scalar multiple of the identity. Hence ϕ_{1} = ϕ_{2} · f.

Corollary 9.7. If e_{min}(K) ≥ 3, then Aut ((Z, ϕ, ρ)_{LEV}) is trivial.

Lemma 9.8. Let Spec k be a point of Spec O. Let (Z, L, V) be a Ksymplectic PSQAS over k. For any level G(K)-structure (ϕ, ρ) on (Z, L, V) there exists a unique rigid G(K)-structure (ϕ(ρ), ρ) such that

(Z, ϕ(ρ), ρ)_{LEV} ≃ (Z, ϕ, ρ)_{LEV} (47)

Proof. We choose and fix a symplectic isomorphism σ : (K, e_{K}) ⊗ k ≃ (K(Q_{0}, L_{0}), e_{0}^{♯}) and a central extension isomorphism ρ : G(K)⊗k ≃ G(Q_{0}, L_{0}) in a compatible way by Remark 7.9. See Definition 4.15 for e^{♯}. By Lemma 7.11 there is a k-isomorphism ϕ(ρ)^{∗} : V (K) ⊗ k → V = Γ(Q, L) ⊗ k such that

U(ρ(g))(ϕ(ρ)^{∗}(w)) = ϕ(ρ)^{∗}U(K)(g)(w) (∀g ∈ G(K), ∀w ∈ V (K)) (48)

where U is the action of G(Q_{0}, L_{0}) on V. Hence ρ = G(ϕ(ρ)^{∗})U(K). By Theorem 6.3 ϕ(ρ) is a closed k-immersion ϕ(ρ) : Z → P(V (K) ⊗_{O} k).

Uniqueness of ϕ follows from Lemma 9.6.

Remark 9.9. Let (Z, ϕ, ρ)_{LEV} be a k-PSQAS with a level G(K)-structure. The stabilizer group Stab(ϕ(ρ)(Z)) in GL(V (K) ⊗ k) contains Ḡ(K) ⊗ k. There is a unique h ∈ GL(V (K)⊗k) such that ϕ = h·ϕ(ρ). Then Stab(ϕ(Z)) contains h(Ḡ(K) ⊗ k)h^{−1}. Thus we see

G(ϕ(Z), O_{P(V (K))}(1)_{ϕ(Z)}) = h(Ḡ(K) ⊗ k)h^{−1} (49)

Lemma 9.10. Let T be an irreducible reduced O-scheme, π : (A, L) → T a polarized abelian T -scheme, A^{t} := Pic^{0}(A/T) and λ(L) : A → A^{t} the polarization morphism. Assume that L_{s} is a separable polarization for any geometric point s ∈ T and assume that (ker λ(L_{s}), e^{L_{s}}) ≃ (K, e_{K}) ⊗ k(s) for a geometric point s of T. Then there exists a finite étale covering f : T^{∗} → T and a constant finite reduced symplectic abelian group O-scheme K such that (ker λ(L_{T^{∗}}), (e^{L})_{T^{∗}}) ≃ (K, e_{K})_{T^{∗}} where e^{L} is the Weil pairing on ker λ(L).

Proof. We have an exact sequence of group schemes

0 → ker λ(L) → A →^{λ(L)} A^{t} → 0.

By the assumption λ(L)_{s} = λ(L_{s}) is étale for any geometric point s of T. Hence λ(L) is étale. Therefore ker λ(L) is étale and finite over T. Let T′ be one of the irreducible components of ker λ(L) with T′ ≄ T. Then ker λ(L)_{T′} has a new section over T′. By repeating the same argument we see that there exists an irreducible reduced O-scheme T^{∗} étale and finite over T such that H := ker λ(L)_{T^{∗}} is a disjoint union of copies of T^{∗}. The multiplication morphism of H as a group scheme as well as its inversion maps each connected component of H × H into a connected component of H. Since T^{∗} is reduced, it follows that H is a trivial group T^{∗}-scheme, say H = K_{T^{∗}} for some finite abelian group K (regarded as a reduced group O-scheme). The Weil pairing (e^{L})_{T^{∗}} is a symplectic bilinear form on H × H with values in (μ_{M})_{T^{∗}}, which is constant on each component of H × H because T^{∗} is reduced. Hence there exists a symplectic form e_{K} on K such that (H, e_{T^{∗}}^{L}) ≃ (K, e_{K})_{T^{∗}}. This proves Lemma.

Lemma 9.11. Let T be an irreducible reduced O-scheme, π : (A, L) → T a polarized abelian T -scheme. Suppose that there exists a constant finite reduced symplectic abelian group O-scheme K such that (ker λ(L), e^{L}) ≃ (K, e_{K})_{T}. Then after a suitable finite base change of T if necessary G(A, L) ≃ G(K)_{T}.

Proof. Let G(A, L) = L_{|K(A,L)}^{tsr} be the Heisenberg group of (A, L), a central extension of K(A, L) := ker λ(L) by G_{m,T} with commutator form e^{L}. By the assumption there exists a constant subgroup scheme H of K such that (K, e_{K}) ≃ (K(H), ℓ_{H}).

We prove that after a finite base change of T there are an open affine covering U_{i} of T and isomorphisms ρ_{i} : G(K)_{U_{i}} ≃ G(A, L)_{U_{i}} as group U_{i}- schemes.

In fact, let U_{i} := Spec A_{i} be an affine covering of T and k(η) the field of a generic point of T. We note k(η) = k(A_{i}), the filed of fractions of A_{i}. By choosing a finite extension k(η′) of k(η) for k(η) and integral closures of A_{i} in k(η′) for A_{i}, we may assume from the start that G(A, L)_{U_{i}} is a totally split central extension and described by two cocycles f_{i}(λ, ν) (λ, ν ∈ K(A_{i})).

Hence by the argument in the proof of Lemma 7.4, we can construct a field k(η′) by adding to k(η) N-th roots of some elements of k(η) as in Equation (30). Then by taking integral closures A′_{i} of A_{i} in k(η′) we may assume that G(K)_{U′_{i}} ≃ G(A, L)_{U′_{i}} where U′_{i} := Spec A′_{i}.

The affine open sets U′_{i} form an open covering of the normalization T′ of T in k(η′). Therefore by Lemma 7.8 by taking a finite base change W of T′ we infer that G(K)_{W} ≃ G(A, L)_{W} = G(A_{W} , L_{W}).

Remark 9.12. By Lemma 7.7 and Lemma 9.11, after a finite base change we have a subgroup W -scheme G(A_{W} , L_{W}) of G(A_{W} , L_{W}), a central extension of K_{W} by (μ_{M})_{W}.

Lemma 9.13. Let T be an irreducible reduced O-scheme, π : (A, L) → T a polarized abelian T -scheme. Suppose that there exists a constant reduced symplectic finite abelian group O-scheme K such that (ker λ(L), e^{L}) ≃ (K, e_{K})_{T}. Then after a suitable finite base change of T if necessary there exist an invertible O_{T} -module M with trivial G(K)_{T} -action and an isomorphism ρ^{−1} : G(A, L) ≃ G(K)_{T} such that π_{∗}(L) is ρ-equivariant isomorphic to M ⊗_{O} V (K).

Proof. We may assume (K, e_{K}) = (K(H), ℓ_{H}) for some contant reduced finite group O-scheme H. By Lemma 9.11 we may assume that G(A, L) ≃ G(K)_{T}. Let s be a closed point of T, U an affine open subset of T with s ∈ U. It follows that through ρ π_{∗}(L) is a G(K)_{T} -module of weight one, that is, the centre of G(K)_{T} acting on π_{∗}(L) by scalar multiplication. Since rank π_{∗}(L) = ord H, we have by Lemma 7.11 π_{∗}(L) ⊗ k(s) ≃ V (K) ⊗_{O} k(s) as G(K)_{T} ⊗ k(s)-modules for any closed point s ∈ T. Since any eigenvalue of the action of H_{T} (T) on π_{∗}(L) ⊗ k(s) belongs to Γ(T, O_{T}), the action of the subgroup scheme H_{T} on π_{∗}(L) is diagonalized with respect to some O_{T} - module basis of π_{∗}(L) locally at s. By the proof of Lemma 7.11 this implies that there exists an open affine covering {U_{i}} of T such that π_{∗}(L)⊗_{O_{T}} O_{U_{i}} ≃ O_{U_{i}} ⊗_{O} V (K) as G(K)_{U_{i}}-O_{U_{i}}-modules. Let U(K)_{i} = 1_{O_{U_{i}}} ⊗ U(K) be the action of G(K)_{U_{i}} on O_{U_{i}} ⊗_{O} V (K). It follows that there is a one-cocycle ℓ_{jk} ∈ H^{1}({U_{jk}}, GL(O_{T} ⊗_{O} V (K))) such that

U(K)_{j}(g)ℓ_{jk} = ℓ_{jk}U(K)_{k}(g) (∀g ∈ G(K)_{T} (T))

on U_{jk} := U_{j} ∩ U_{k}. Hence U(K)(g)ℓ_{jk} = ℓ_{jk}U(K)(g). Therefore ℓ_{jk} is a scalar matrix by Schur’s lemma. Hence ℓ_{jk} ∈ H^{1}({U_{jk}}, O_{T}^{∗}), which defines an invertible O_{T} -module M. This proves Lemma.

Corollary 9.14. Let anything be as in Lemma 9.13. Suppose e_{min}(K) ≥ 3. Then by taking a suitable finite base change of T if necessary there exist a closed T -immersion ϕ : A → P(V (K) ⊗_{O} O_{T}), a central extension isomorphism ρ : G(K)_{T} → G(A, L) and an invertible O_{T} -module M with trivial G(K)_{T} -action such that 1. L ≃ ϕ^{∗}(M ⊗_{O} O_{P(V (K))}(1)),

2. π_{∗}(L) ≃ M ⊗_{O} V (K),

where the isomorphisms (1) and (2) are ρ-equivariant and Γ(O_{P(V (K))}(1)) is identified with V (K).

Proof. Since e_{min}(K) ≥ 3, L is relatively very ample by Theorem 6.3. Hence any choice of O_{T} -module basis of π_{∗}(L) defines a projective embedding of A over T. Corollary follows from Lemma 9.13.

Corollary 9.14 motivates the following definition.

Definition 9.15. Let T be a noetherian O_{M}-scheme. Then a sextuplet (Q, L, G(Q, L), V, ϕ, ρ) is called a projectively stable quasi-abelian T -scheme of relative dimension g with a level G(K)-structure if

(i) Q is a proper flat T -scheme with a relatively ample invertible sheaf L, (ii) ϕ is a closed T -immersion of Q into P(V (K) ⊗_{O} O_{T}), (iii) G(Q, L) is a finite flat reduced group T -scheme acting on (Q, L) (iv) ρ : G(K)_{T} → G(Q, L) is an isomorphism of group T -schemes

(v) V is a locally O_{T} -free subsheaf of π_{∗}(L) (vi) L ≃ ϕ^{∗}(O_{P(V (K))}(1)⊗_{O} M) and V ≃ π_{∗}(ϕ^{∗}H^{0}(O_{P(V (K))}(1)))⊗_{O} M are

ρ-equivariant isomorphisms for some action of G(K)_{T} on V (K)_{T} (vii) for any geometric point s of T, (Q_{s}, ϕ_{s}, ρ_{s}) is a PSQAS of dimension g

over k(s) with a level G(K)-structure,

where M is some invertible O_{T} -module with trivial G(K)_{T} -action and π : Q → T is the structure morphism.

We denote (Q, L, G(Q, L), V, ϕ, ρ) by (Q, ϕ, ρ)_{LEV} for brevity. If any (Q_{s}, ϕ_{s}, ρ_{s}) in (vii) is a PSQAS over k(s) with a rigid G(K)-structure, then we call (Q, ϕ, ρ) a T -PSQAS with a rigid G(K)-structure and denote it by (Q, ϕ, ρ)_{RIG}.

We note V ≃ V (K)⊗_{O}M by (vi). (vii) implies G(Q_{s}, L_{s}) = G(Q, L)⊗k(s) for any point s ∈ T.

Definition 9.16. Let (Q_{i}, ϕ_{i}, ρ_{i})_{LEV} := (Q_{i}, L_{i}, G(Q_{i}, L_{i}), V_{i}, ϕ_{i}, ρ_{i}) (i = 1, 2) be T -PSQAS’s with level G(K)-structures. We define them to be isomorphic if there exist a T -isomorphism f : Q_{1} → Q_{2} and an invertible O_{T} -module M with trivial G(K)-action such that

(i) L_{1} ≃ f^{∗}L_{2} ⊗_{O} M is a G(K)-equivariant isomorphism, (ii) V_{1} = f^{∗}(V_{2} ⊗_{O} M) as G(K)-O_{T} -submodules of π_{∗}(L_{1}), (iii) ρ_{1} = G(f^{∗}) · ρ_{2}.

If (Q_{i}, ϕ_{i}, ρ_{i}) are rigid G(K)-structures, then (iii) is equivalent to the following

(iv) ϕ_{1} = ϕ_{2} · f

which implies (i) and (ii) and that M in (i) and (ii) is necessarily trivial.

The above equivalence of (iii) and (iv) is proved in the same manner as in Lemma 9.6.

Definition 9.17. We define the functor SQ_{g,K} as follows. For any noetherian O_{N,M}-scheme T, we set

SQ_{g,K}(T) = the set of projectively stable quasi-abelian

T -schemes (Q, ϕ, ρ)_{LEV} of relative dimension g with level G(K) -structures modulo T -isom

SQ_{g,K}^{RIG}(T) = the set of projectively stable quasi-abelian

T -schemes (Q, ϕ, ρ)_{RIG} of relative dimension g with rigid G(K) -structures modulo T -isom .

10. Stable reduction theorem – summary

10.1. The rigid G(K)-structure we start from. Let R be a complete discrete valuation ring with the fraction field k(η) and k the residue field. Let (G_{η}, L_{η}) be a polarized abelian variety over k(η) with separable polarization L_{η} and K(L_{η}) := ker λ(L_{η}). Let e^{L_{η}} be the Weil pairing of K(L_{η}). Then after a suitable finite ramified cover of Spec R we may assume that K(L_{η})(k(η)) = K(L_{η})(\overline{k(η)}). Hence there exists a finite symplectic abelian group (K, e_{K}) (regarded as a group Z-scheme) such that (K(L_{η}), e^{L_{η}}) ≃ (K, e_{K}) ⊗ k(η).

By Lemma 7.4 or Remark 7.9, after a suitable finite ramified cover of Spec R we may assume that G(K)⊗k(η) ≃ G(G_{η}, L_{η}) and ρ_{η} : G(K)⊗k(η) ≃ G(G_{η}, L_{η}).

Then Γ(G_{η}, L_{η}) is a unique irreducible G(G_{η}, L_{η})-module of weight one by Lemma 7.11, hence so as a G(G_{η}, L_{η})-module because we may of course assume ζ_{M} ∈ R. By Lemma 9.6 (G_{η}, L_{η}, Γ(G_{η}, L_{η})) has a unique rigid G(K)-structure for the above ρ_{η}.

We summarize the previous results in the separably polarized case as follows.

Theorem 10.2. Let R be a complete discrete valuation ring with the fraction field k(η) and k the residue field. Let (G_{η}, L_{η}) be a polarized abelian variety over k(η) and K(L_{η}) := ker λ(L_{η}). Assume that the charateristic of k and the order of K(L_{η}) are coprime. Then after a suitable finite base change if necessary, there exist flat projective schemes (P, L) and (Q, L), semiabelian group schemes G and G^{♯}, the flat closure K(Q, L) of K(L_{η}) in G^{♯}, and the finite and infinite Heisenberg group schemes G(Q, L) and G(Q, L) of (Q, L), all of these being defined over S, such that

1. L is relatively ample on P and Q,

2. P is reduced over S

3. (G, L) and (G^{♯}, L) are open subschemes of (P, L),

4. G^{♯} = K(Q, L) · G,

5. (G_{η}, L_{η}) ≃ (G_{η}^{♯}, L_{η}) ≃ (P_{η}, L_{η}) ≃ (Q_{η}, L_{η}), 6. (K(Q, L), e^{♯}) is a constant finite flat reduced subgroup scheme of G^{♯}

with a constant symplectic form e^{♯} extending (K(L_{η}), e^{L_{η}})

7. G(Q, L) and G(Q, L) are respectively totally split central extensions of K(Q, L) by μ_{M,S} and G_{m,S} with commutator form e^{♯}

8. Γ(G^{♯}, L) ≃ Γ(P, L) ≃ Γ(Q, L) and they are irreducible G(Q, L)-module

of weight one, unique up to equivalence where M = e_{max}(K(Q_{η}, L_{η})).

Proof. By the assumption L_{η} is a separable polarization, hence we can start from Subsection 10.1. (1) and (5) follow from Theorems 3.5, 4.10, 5.8, Remark 3.10 and Subsection 4.13. (2) follows from [AN96] and Definition 3.6. (3) and (4) follow from Subsections 4.12, and 4.13. (6) follows from Lemma 4.14 and Definition 5.11. (7) follows from Subsection 2.4, Definitions 5.11, 7.12, 7.14 and Lemma 7.4. (8) follows from Lemma 5.12.

As a corollary to the proof of Theorem 10.2 we obtain,

Corollary 10.3. Under the same assumptions and notation as in Theorem 10.2 there exists a finite symplectic abelian group Z-scheme (K, e_{K}) such that

1. (K(Q, L), e^{♯}) ≃ (K, e_{K})_{S},

2. G(Q, L) ≃ G(K)_{S} and G(Q, L) ≃ G(K)_{S}

3. Γ(Q, L) ≃ V (K) ⊗_{O_{M}} R where M = e_{max}(K). All the isomorphisms in (1) and (2) are compatible and coincide with those given in Subsection 10.1 over k(η). The isomorphism in (3) is (2)-equivariant with respect to some action of G(K)_{S} on V (K) ⊗_{O_{M}} R.

The following is the stable reduction theorem we need in Sections 11 and 12.

Theorem 10.4. Let (K, e_{K}) be a finite reduced symplectic abelian group Zscheme with M = e_{max}(K) and e_{min}(K) ≥ 3. Let R be a complete discrete valuation ring over O_{M} with the fraction field k(η) and k the residue field. Assume that M and the charateristic of k are coprime. For a given abelian variety (G_{η}, ϕ_{η}, ρ_{η})_{RIG} over k(η) with a rigid G(K)-structure, by a finite base change of Spec R if neccesary, there exists a PSQAS (Q, ϕ, ρ)_{RIG} over Spec R with a rigid G(K)-structure extending (G_{η}, ϕ_{η}, ρ_{η})_{RIG}.

Proof. Since e_{min}(K) ≥ 3, L_{η} is very ample by Theorem 6.2. By assumption we have a rigid G(K)-structure (ϕ_{η}, ρ_{η}) on (G_{η}, L_{η}, Γ(G_{η}, L_{η})). By Theorem 10.2 and Corollary 10.3 we have an S-PSQAS (Q, L, Γ(Q, L)), an isomorphism ρ : G(K)_{S} ≃ G(Q, L) extending ρ_{η} and a ρ-equivariant isomorphism ψ^{∗} : V (K) ⊗_{O} R ≃ Γ(Q, L) with respect to some action of G(K)_{S} on V (K) ⊗_{O} R. It follows from Lemma 9.8 that there exists a rigid G(K)-structure (ϕ, ρ), uniquely determined for a given ρ, on an S-PSQAS (Q, L, Γ(Q, L)) such that (Q, ϕ, ρ)_{LEV} ≃ (Q, ψ, ρ)_{LEV}. By the uniqueness of a rigid G(K)-structure for ρ_{η}, we have ϕ ⊗ k(η) = ϕ_{η}.

11. The scheme SQ_{g,K}

11.1. Z_{g,K} and H_{g,K}. Let O = O_{M} and let (K, e_{K}) be a constant reduced finite symplectic abelian group O-scheme with e_{min}(K) ≥ 3, M = e_{max}(K) and N := \sqrt{ord_{O} K}. Let P (n) := n^{g}N, S_{O} := Spec O and let H_{g,K} := Hilb_{P}^{P (n)} be the Hilbert scheme parametrizing all projective subschemes of P := P(V (K)) with their Hilbert polynomial P (n), (Z_{g,K}, L_{g,K}) the universal subscheme over H_{g,K} and π : Z_{g,K} → H_{g,K} the natural morphism. Let i : (Z_{g,K}, L_{g,K}) → P(V (K)) ×_{S_{O}} H_{g,K} be the natural (given) closed immersion of the universal subscheme Z_{g,K} over H_{g,K}. We remark that H_{g,K} and Z_{g,K} are O-schemes by [FGA, 221, Théorème 3.1]. In this section we will define an O-subscheme SQ_{g,K} of H_{g,K} which ought to be the moduli scheme. See [MFK94, Proposition 7.3, pp.132-134].

Since e_{min}(K) ≥ 3, any K-symplectic PSQAS (Q_{0}, L_{0}, V_{0}) over k is a closed point of H_{g,K} by choosing any isomorphism V_{0} ≃ V (K) ⊗_{O} k in view of Theorem 6.3.

Let U be the maximal subscheme of H_{g,K} such that π is smooth, which is an open O-subscheme of H_{g,K} by [EGA, IV, Corollaire 6.8.7]. Let U_{1} be the union of all connected components (as O-subschemes) of U containing at least a geometric point s of U, over which a fibre of π is an abelian variety with ker λ(L_{g,K,s}) ≃ K ⊗_{O} k(s). Let Z_{1} := Z_{g,K} ×_{H_{g,K}} U_{1}. By the base change U_{2} of U_{1} we may assume Z_{2} := Z_{1} ×_{U_{1}} U_{2} has a section e over U_{2}. For instance choose U_{2} = Z_{1}. Then by [MFK94, Theorem 6.14] Z_{2} is an abelian scheme over U_{2} with e unit section. It follows from Lemma 9.10 that any geometric fibre of Z_{2}, a fortiori, any geometric fibre of Z_{1} is an abelian variety with ker λ(L_{g,K,s}) ≃ K ⊗_{O} k(s).

Let

U_{3} := U_{2}^{Ḡ(K)} = U_{2}^{Ḡ(K)} = {s ∈ U_{2}; g(s) = s for ∀g ∈ Ḡ(K)} (50)

It is clear that U_{3} is an O-scheme. Let (Z_{3}, L_{3}) := (Z, L) ×_{H_{g,K}} U_{3}. The relative polarized-Aut-scheme Aut_{U_{3}}(Z_{3}, L_{3}) is a closed subgroup O-scheme of PGL(V (K))_{U_{3}} finite over U_{3} because L_{3} is very ample. Meanwhile the relative Aut-scheme Aut_{U_{3}}^{0}(Z_{3}) is quasi-projective. Let

(51) Aut_{U_{3}}^{0}(Z_{3}, L_{3}) := Aut_{U_{3}}(Z_{3}, L_{3}) ∩ Aut_{U_{3}}^{0}(Z_{3})

It is an open and closed normal subgroup O-scheme of Aut_{U_{3}}(Z_{3}, L_{3}). By the definition of U_{3}, Ḡ(K)_{U_{3}} is a subgroup scheme of Aut_{U_{3}}(Z_{3}, L_{3}). Let

U_{4} := {s ∈ U_{3}; Ḡ(K) ⊗ k(s) ⊂ Aut_{U_{3}}^{0}(Z_{3}, L_{3}) ⊗ k(s)} (52)

Definition 11.2. Let U_{g,K} = U_{1}, and let W_{g,K} be the closure of (U_{g,K})_{red} in H_{g,K} with reduced structure. Let A_{g,K} := U_{4}. By (52) A_{g,K} is an Osubscheme of H_{g,K}.

We define SQ_{g,K} to be the closure of A_{g,K} in H_{g,K}, i.e., the minimal (reduced) closed O-subscheme of H_{g,K} containing (A_{g,K})_{red}. Let Z_{g,K}^{SQ} := Z_{g,K} ×_{H_{g,K}} SQ_{g,K}, and let π_{g,K} : Z_{g,K}^{SQ} → SQ_{g,K} be the natural projection.

We note that for any abelian scheme (A, L) ∈ U_{g,K}(k) over a closed field k, there is by Lemma 9.8 a closed immersion ϕ : A → P(V (K) ⊗ k) such that (ϕ(A), O_{P}(1)_{|ϕ(A)}) ∈ A_{g,K}(k).

11.3. The level structure over A_{g,K}. There is a natural isomorphism

ρ_{g,K} : G(K)_{A_{g,K}} ≃ G((Z_{g,K}, L_{g,K}) ×_{H_{g,K}} A_{g,K}) (53)

In fact, any element h of Ḡ(K) ⊗ k(s) is a translation T_{a(s)} by some point a(s) of an abelian variety Z_{s} for any s ∈ U_{4} because h ∈ Aut_{U_{3}}^{0}(Z_{3}). Since h keeps L_{s} invariant, a(s) belongs to K(Z_{s}, L_{s}) = ker λ(L_{g,K,s}). Therefore

G((Z_{g,K}, L_{g,K}) ×_{H_{g,K}} A_{g,K}) ⊗ k(s) = Ḡ(K) ⊗_{O} k(s) (54)

for any geometric point s of A_{g,K} because both group k(s)-schemes have order N^{2}M. The isomorphism U(K) : G(K) → Ḡ(K) induces a natural morphism ρ_{g,K} of group O-schemes in (53), which is an isomorphism by the fibrewise isomorphism (54).

Theorem 11.4. Assume e_{min}(K) ≥ 3. Then SQ_{g,K} is a projective Osubscheme of W_{g,K} pointwise fixed by Ḡ(K) such that for any geometric point Spec k of Spec O,

1. A_{g,K}(k) = \{(Z, L) ∈ W_{g,K}(k); \begin{matrix}(Z, i, U(K)) is an abelian variety \\ over k with a rigid G(K)-structure \end{matrix} \}

2. SQ_{g,K}(k) = \{(Z, L) ∈ W_{g,K}(k); \begin{matrix}(Z, i, U(K)) is a PSQAS over k \\ with a rigid G(K)-structure \end{matrix} \}

where i is the natural inclusion of Z into P(V (K) ⊗ k).

Proof. By Equality (54) A_{g,K} is an O-subscheme of W_{g,K} such that any (Z, L) ∈ A_{g,K}(k) is an abelian variety with G(Z, L) = i^{∗}Ḡ(K), hence (Z, i, U(K)) is an abelian variety with a rigid G(K)-structure. Conversely if (Z, i, U(K)) is an abelian variety with a rigid G(K)-structure and (Z, L) ∈ H_{g,K}, then (Z, L) is fixed by the action of Ḡ(K). Hence (Z, L) ∈ U_{3}. Any automorphism of Z induced from Ḡ(K) is a translation by the definition of rigid G(K)-structures, hence (Z, L) ∈ U_{4} = A_{g,K}. This proves (1).

A_{g,K}(k) is pointwise fixed by the action of Ḡ(K) on H_{g,K} because i^{∗}Ḡ(K) = G(Z, L) ⊂ Aut(Z, L). It follows from it that SQ_{g,K} is an O-subscheme of W_{g,K} pointwise fixed by Ḡ(K). It remains to prove (2).

Let R be a complete discrete valuation ring with fraction field k(η), η the generic point of S := Spec R. Suppose that we are given a flat Ssubscheme (Z, L) of (Z_{g,K} ×_{H_{g,K}} SQ_{g,K})_{S} such that (Z_{η}, L_{η}) is a polarized abelian scheme. Let i be the inclusion immersion of (Z, L) into P(V (K))_{S}, G(Z, L) := i^{∗}(Ḡ(K)_{S}) and χ := G(i^{∗}) · (U(K)_{S}). Then (i, χ) is a rigid G(K)-structure on (Z, L, i^{∗}V (K)_{S}) over S.

In view of Lemma 7.4, Lemma 9.8 and Theorem 10.4, by choosing a suitable finite ramified cover of S if necessary we may assume that we have a projective flat family (W, M) with a rigid G(K)-structure (ϕ, ρ) such that (W_{η}, M_{η}) ≃ (Z_{η}, L_{η}) and (W_{0}, M_{0}) is a PSQAS. To be more precise, (i) ϕ : (W, M) → P(V (K))_{S} is a closed S-immersion (ii) ρ : G(K)_{S} ≃ G(W, M) is an isomorphism with ρ = G(ϕ^{∗})(U(K)_{S}) (iii) (W_{η}, ϕ_{η}, ρ_{η})_{LEV} ≃ (Z_{η}, i_{η}, U(K)_{η})_{LEV}, (iv) (W_{0}, ϕ_{0}, ρ_{0}) is a PSQAS with a rigid G(K)-structure

Hence there is by Lemma 9.6 a k(η)-isomorphism f : Z_{η} → W_{η} such that i_{η} = ϕ_{η} · f, that is, i_{η}(Z_{η}) = ϕ_{η}(W_{η}). Since i and ϕ are closed Simmersions respectively, they induce natural morphisms Hilb(i) and Hilb(ϕ) from Spec R into SQ_{g,K} by the universal property of H_{g,K}. It follows from i_{η} = ϕ_{η} · f that we have Hilb(i_{η}) = Hilb(ϕ_{η}) : Spec k(η) → SQ_{g,K}. Since SQ_{g,K} is projective (separated), we have Hilb(i) = Hilb(ϕ) [EGA, II, 7.2.3]. This implies that i(Z) = ϕ(W), a fortiori, i_{0}(Z_{0}) = ϕ_{0}(W_{0}). Hence (Z_{0}, L_{0}) ≃ (W_{0}, M_{0}), and (Z_{0}, i_{0}, U(K)) is a k-PSQAS with a rigid G(K)-structure. In fact, G(Z_{0}, L_{0}) = Ḡ(K). This proves (2).

Definition 11.5. Now we define the relative n-th Hilbert point hilb_{n}. Let π : Z_{g,K} → H_{g,K} be the natural morphism, and let

V := O_{H_{g,K}} ⊗_{O} V (K), V_{n} := π_{∗}(O_{Z_{g,K}}(n)).

We note that V_{n} is locally free of rank n(g) := n^{g}N for sufficiently large n ≥ n_{0}. Let ϕ_{n} : S^{n}V → V_{n} be the natural epimorphism for n ≥ n_{0}. Thus we

have a morphism hilb_{n} := ∧^{n(g)}ϕ_{n} of H_{g,K} into the projective space P_{large} :=

P( ∧^{n(g)}S^{n}V (K)). For any large n ≥ n_{0} hilb_{n} is a closed O-immersion of H_{g,K} into P_{large}. For a geometric fibre (Z, L) of π we call hilb_{n}(Z, L) the n-th Hilbert point of (Z, L).

Theorem 11.6. Let Spec k be a geometric point of Spec O and (Z, L) ∈ H_{g,K}(k). Suppose e_{min}(K) ≥ 3 and that (Z, L) is smoothable into an abelian variety (A, M) with ker λ(M) ≃ K ⊗ k. Then the following are equivalent.

1. (Z, L) is a K-symplectic PSQAS over k,

2. Aut(Z, L) contains a subgroup of SL_{±}(V (K)⊗k) conjugate to Ḡ(K)⊗k,

3. the normalized Hilbert points hilb_{n}(Z, L) are Kempf-stable for any large n ≥ n_{0}.

Proof. (2) follows from (1) by Lemma 9.8. (3) follows from (2) by [Kempf78, Corollary 5.1]. It remains to prove that (3) implies (1). We choose and fix a large n so that hilb_{n} may be a closed immersion of W_{g,K}. Suppose (3) and that (Z, L) is smoothable into an abelian variety with ker λ ≃ K ⊕ K^{∨}. It follows (Z, L) ∈ W_{g,K}(k). By (3) hilb_{n}(Z, L) is Mumford-semistable. By [Seshadri77, p. 269, Remark 8] there is a categorical quotient

W̄_{g,K} := W_{g,K} ∩ hilb_{n}^{−1}(P_{large}(SemiStable))/ SL(V (K))

which is a projective O-scheme. By Lemma 9.8 and Theorem 10.4 W̄_{g,K} is set-theoretically SL(V (K) ⊗ k)-orbits of PSQAS’s modulo some relation between orbit closures. By [Seshadri77, p. 269, Theorem 4] the closure of the SL(V (K) ⊗ k)-orbit of hilb_{n}(Z, L) intersects the SL(V (K) ⊗ k)-orbit of a PSQAS hilb_{n}(W, M). By the assumption (3) the orbit of hilb_{n}(Z, L) is closed. Therefore the orbit of hilb_{n}(Z, L) is that of hilb_{n}(W, M), hence (Z, L) ≃ (W, M) is a PSQAS.

12. Reduced-fine-moduli schemes

Definition 12.1. Let O_{M} := Z[ζ_{M}, 1/M]. When no confusion is possible, we write O = O_{M}. For a contravariant functor F over O a reduced Oscheme W is said to be a reduced-fine-moduli scheme over O of F or we say that F is reductively-represented over O by W if the following conditions are satisfied;

(a) f_{W} (T) : F (T) → Hom_{S}(T, W) is a bijection for a reduced O-scheme T (b) f_{W} (T) · F (f) = Hom(f, W) · f_{W} (U) for an O-morphism f : T → U of

reduced O-schemes T, U

It is easy to see that a reduced-fine-moduli scheme is unique if there exists.

Theorem 12.2. If e_{min}(K) ≥ 3, then SQ_{g,K} is reductively-represented by a projective scheme SQ_{g,K} over Z[ζ_{M}, 1/M] where M = e_{max}(K).

Proof. First we prove SQ_{g,K} ≃ SQ_{g,K}^{RIG}. Suppose (Q, ϕ, ρ)_{LEV} is a T -PSQAS with a level G(K)-structure. By Definition 9.15 (iv), we have an isomorphism ρ : G(K)_{T} → G(Q, L). Let {U_{i} = Spec R_{i}} be an open affine covering, V_{i} := V_{U_{i}} ≃ V_{i} ⊗_{O} O_{U_{i}} for some free R_{i}-module V_{i}, ϕ_{i} := ϕ_{U_{i}} and ρ_{i} := ρ_{U_{i}}. By Lemma 7.11, by choosing a finer open covering of {U_{i}} if necessary, there is a collection of A_{i} ∈ GL(V_{i}) such that ρ_{i}(g) = A_{i}G(ϕ_{i}^{∗})U(K)(g)A_{i}^{−1} (∀g ∈ G(K)_{U_{i}}(U_{i})). Let ϕ(ρ)_{i}^{∗} := A_{i}ϕ_{i}^{∗}. Then ϕ(ρ)_{i}^{∗} induces a closed U_{i}- immersion of Q_{U_{i}} into P(V (K)⊗_{O} O_{U_{i}}) such that ρ_{i}(g) = G(ϕ(ρ)_{i}^{∗})U(K)(g). On U_{i} ∩ U_{j}, we have ρ_{i} = ρ_{U_{i}∩U_{j}} = ρ_{j}, hence G(ϕ(ρ)_{i}^{∗})(g) = G(ϕ(ρ)_{j}^{∗})(g) for any g ∈ Ḡ(K). Hence by Schur’s lemma there is a scalar ℓ_{ij} ∈ GL(V (K) ⊗_{S} O_{U_{ij}}) such that ϕ(ρ)_{i}^{∗} = ℓ_{ij}ϕ(ρ)_{j}^{∗}.

It follows that there are a closed T -immersion ϕ(ρ) of Q into P(V (K))_{T} and M := {ℓ_{ij}} ∈ H^{1}(O_{T}^{∗}) such that V ≃ ϕ(ρ)^{∗}(V (K) ⊗_{O} M) and ρ = G(ϕ(ρ)^{∗})U(K). By Lemma 9.8 (Q, ϕ(ρ), ρ)_{RIG} is a unique T -PSQAS with a rigid G(K)-structure such that (Q, ϕ(ρ), ρ)_{LEV} ≃ (Q, ϕ, ρ)_{LEV} where the isomorphism is induced from the identity of Q. It follows SQ_{g,K} ≃ SQ_{g,K}^{RIG}.

It remains to prove that SQ_{g,K}^{RIG} is reductively-represented by SQ_{g,K}. Let T be a reduced O_{M}-scheme. Suppose we are given a projectively stable quasi-abelian T -scheme (Z, ϕ, ρ)_{RIG} with a rigid G(K)-structure. Then ϕ : Z → P(V (K) ⊗_{O} O_{T}) is a closed immersion of Z. Hence we have a natural T -morphism Hilb(ϕ) : T → H_{g,K} which factors through SQ_{g,K} by Theorem 11.4 (2). Moreover Z = Hilb(ϕ)^{∗}(Z_{g,K}^{SQ}) = Z_{g,K}^{SQ} ×_{SQ_{g,K}} T by the universal property of Z_{g,K}. Hence ϕ is clearly the pull back of the universal inclusion map of Z_{g,K}^{SQ} into P(V (K))_{SQ_{g,K}} by Hilb(ϕ). Since ρ = G(ϕ^{∗})U(K) by definition, it follows that the map (Z, ϕ, ρ)_{RIG} ↦ Hilb(ϕ) is a bijection between SQ_{g,K}(T) and Hom_{S}(T, SQ_{g,K}). This proves Definition 12.1 (a).

The second condition (b) in Definition 12.1 is clear. This proves that SQ_{g,K} is reductively-represented by SQ_{g,K}.

Remark 12.3. Modifying [FC90, Chapter IV, Section 6] slightly we can construct a family of PSQAS’s over the Voronoi compactification A_{g,K}^{Vor} of A_{g,K} if e_{min}(K) ≥ 3. Hence we have a natural morphism of A_{g,K}^{Vor} onto SQ_{g,K}, which is an isomorphism over A_{g,K}, the moduli scheme of nonsingular abelian varieties with level G(K)-structures. We conjecture that A_{g,K}^{Vor} ≃ SQ_{g,K}. The reduced-coarse-moduli scheme is defined in a manner similar to Definition 12.1. For smaller K the functor SQ_{g,K} is defined as in Definition 9.15 except one point that ϕ is no longer a closed immersion but ϕ^{∗} is a ρ-equivariant isomorphism of k-vector spaces.

The reduced-coarse-moduli scheme for SQ_{g,K} will be obtained by taking a quotient SQ_{g,K_{n}}/G(B_{n}) for n ≥ 3 where B_{n} and K_{n} are symplectic finite abelian groups fitting into the following exact sequece

0 → B_{n} (≃ (Z/nZ)^{2g}) → K_{n} → K → 0 (55)

One of the difficulities we encounter with in constructing the reducedcoarse-moduli scheme for SQ_{g,K} is that if SQ_{g,K_{n}} is nonnormal, then the quotient SQ_{g,K_{n}}/G(B_{n}) may depend on n, though all of these SQ_{g,K_{n}}/G(B_{n}) are naturally bijective and there are morphisms from larger n to smaller n. We would like to discuss these topics somewhere else.

13. Examples in dimension one

13.1. Cubic curves. Let V = Zx_{0}+Zx_{1}+Zx_{2}. Let P := P(S^{3}(V^{∨})) be the projective space of ternary cubic forms on V. Let U be an open subscheme of P consisting of curves with at worst a unique nodal singularity. The categorical quotient of U by SL(3) is P_{Z}^{1}, which is the coarse moduli scheme of the functor of smooth elliptic curves and a rational curve with a node. This is the moduli scheme corresponding to the j-invariant of ellitic curves. This is also regarded as the coarse moduli scheme of one-pointed stable curves of genus one. However as we saw in Section 12 and as we also see in Subsection 13.2, we have a different kind of compactification SQ_{1,(Z/3Z)^{⊕2}}.

13.2. The case of K = (Z/3Z)^{⊕2}. Let O := O_{3} = Z[ζ_{3}, 1/3]. Let H = Z/3Z and K = H ⊕ H^{∨} = (Z/3Z)^{⊕2}. The order of G(K) is 3^{3} = 27. We define χ_{i} ∈ H^{∨} by χ_{i}(1) = ζ_{3}^{i}. Let

V := O[H^{∨}] = O[(Z/3Z)^{∨}] = Ox_{0} + Ox_{1} + Ox_{2} (56)

where x_{i} := v(χ_{i}). The group Ḡ(K) (≃ G(K)) is generated by σ and τ

(57)  σ : (x_{0}, x_{1}, x_{2}) ↦ (x_{0}, ζ_{3}x_{1}, ζ_{3}^{2}x_{2})
(58)  τ : (x_{0}, x_{1}, x_{2}) ↦ (x_{1}, x_{2}, x_{0})

The group G(K) has 9 one-dimensional representations ρ_{i,j} (i, j ∈ (Z/3Z)) and two three-dimensional representations ρ_{3} and ρ′_{3} where

(59)  ρ_{i,j}(a ⊕ b) = ζ_{3}^{ia+jb}, (a, b) ∈ H ⊕ H^{∨}
(60)  ρ_{3} ≃ V, ρ′_{3} ≃ Hom_{O}(V, O)

which checks that |G(K)| = 27 = 9 · 1^{2} + 3^{2} + 3^{2}.

Let S_{n} = S^{n}V. We note V = S_{1} ≃ ρ_{3}. Then we see

S_{2} = 2ρ′_{3}, S_{3} = 2ρ_{0,0} ⊕ ⊕_{(i,j)≠(0,0)}ρ_{i,j} (61)

Let V′ = {x_{0}^{2}, x_{1}^{2}, x_{2}^{2}} and V′′ = {x_{1}x_{2}, x_{2}x_{0}, x_{0}x_{1}} be the subspaces of S_{2} spanned by those polynomials. We see V′ ≃ V′′ ≃ ρ′_{3}. The only possible nontrivial subschemes of P^{2} with nontrivial moduli defined by quadric polynomials are therefore

t_{0}x_{i}^{2} − t_{1}x_{i+1}x_{i+2} = 0 (i ∈ Z/3Z) (62)

which are clearly zero-dimensional. These subschemes are Kempf-stable by [Kempf78, Corollary 5.1].

Our case of cubic curves arises from 2ρ_{0,0} in S_{3}. Let π : (E, L) → T be any non-isotrivial cubic curve (or a one-dimensional PSQAS) with a rigid G(K)- structure over some connected O-scheme T. By the fibrewise Riemann-Roch theorem, the O_{T} -module π_{∗}(nL) is locally free of rank 3n and the natural homomorphisms π_{n} : S^{n}(π_{∗}L) → π_{∗}(nL) are surjective for any n ≥ 1. It follows that ker π_{3} is locally free of rank one. Since ker π_{3} has a G(K)-action, it is generated by a Ḡ(K)-(semi)invariant cubic polynomial with nontrivial parameters. Since ρ_{i,j} (i, j) ≠ (0, 0) has multiplicity one in S_{3}, the equation f_{E} defining E on P(V)_{T} lies in 2ρ_{0,0}. It follows that

f_{E} = A_{0}(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) − A_{1}x_{0}x_{1}x_{2} (63)

for some (A_{0}, A_{1}) ∈ P_{T}^{1} . Combining this with the subsequent observation on the curves defined by cubic polynomials with character ρ_{i,j} for (i, j) ≠ (0, 0), the above argument reproves that the universal subscheme Z := Z_{1,K}^{SQ} is given by the Hesse cubic

μ_{0}(x_{0}^{3} + x_{1}^{3} + x_{2}^{3}) − μ_{1}x_{0}x_{1}x_{2} = 0 (64)

and that SQ_{1,K} ≃ P(O[μ_{0}, μ_{1}]) ≃ P_{O}^{1}.

Let k be any closed field of char k ≠ 3. Let f := x_{0}^{3} + ζ_{3}x_{1}^{3} + ζ_{3}^{2}x_{2}^{3} and let C be a curve defined by f = 0. By the natural isomorphism G(K)/μ_{3} ≃ H ⊕ H^{∨}, we see that kf ≃ ρ_{0,1}. Let y_{i} = ζ_{9}x_{i}. Then C is isomorphic to the Fermat cubic y_{0}^{3} + y_{1}^{3} + y_{2}^{3} = 0. Via the isomorphism of C with the Fermat cubic, we see that G(C, O_{C}(1)) is generated by two elements σ and τ_{C} where

τ_{C} : (x_{0}, x_{1}, x_{2}) ↦ (ζ_{9}x_{1}, ζ_{9}x_{2}, ζ_{9}^{−2}x_{0}) (65)

Let ρ : (x_{0}, x_{1}, x_{2}) ↦ (ζ_{9}^{−2}x_{0}, ζ_{9}x_{1}, ζ_{9}x_{2}). Then ρ = τ^{−1}τ_{C}. Since ρ ∈ Aut(C, O_{C}(1)) is of order 3 and has 3 fixed points on C, it is neither a translation of C nor an endomorphism of C. This shows that ρ ∉ G(C, O_{C}(1)) and τ ∈ Ḡ(K) \ G(C, O_{C}(1)). It follows that (C, O_{C}(1)) ∈ H_{1,K}^{Ḡ(K)}(k) \ S_{1,K}(k) with the notation in Subsection 11.1. The same is true for the other ρ_{i,j} (i, j) ≠ (0, 0). This proves that any isotrivial cubic curve E with a rigid G(K)-structure is also defined by a cubic polynomial (63). This completes the proof of the universality of Z_{1,K}^{SQ} and SQ_{1,K} ≃ P(O[μ_{0}, μ_{1}]).

We see that the fibre Z_{μ} of Z is singular only for (μ_{0}, μ_{1}) = (0, 1) or (1, −3ζ_{3}^{i}) (i = 0, 1, 2). Let G^{♯} be the Néron model of Z. Then G^{♯} is a semiabelian scheme with fibres elliptic curves or extensions of a torus G_{m} by H. Let Z_{3} := ker λ(O_{Z}(1)) be the set of all flexes of the Hesse cubic Z,

(66)  Z_{3} = {(x_{0}, x_{1}, x_{2}) ∈ Z; x_{0}x_{1}x_{2} = 0}

Then Z_{3} is regarded as the set of all 3-torsion points of G^{♯}. Let e_{0} = (1, −1, 0), e_{1} = (1, −ζ_{3}, 0) and e_{2} = (−1, 0, 1). If we choose e_{0} ∈ Z_{3} as the identity of G^{♯}, then e_{1} and e_{2} generate Z_{3} (≃ K) as an abelian group. Moreover σ and τ are translations of G^{♯} by e_{1} and e_{2} respectively because e_{1} = σ(e_{0}) and e_{2} = τ(e_{0}).

13.3. The case of K = (Z/4Z)^{⊕2}. Let O := Z[ζ_{4}, 1/2]. Let H = Z/4Z and K = H ⊕ H^{∨}. We define χ_{i} ∈ H^{∨} by χ_{i}(1) = ζ_{4}^{i}. Let

V := O[H^{∨}] = Ox_{0} + Ox_{1} + Ox_{2} + Ox_{3} (67)

where x_{i} := v(χ_{i}). G(K) has 16 one-dimensional representations, 4 twodimensional representations V^{i} (i = 0, 1, 2, 3) and 2 four-dimensional representations V and its dual where we check |G| = 64 = 16 + 4 · 2^{2} + 2 · 4^{2}.

Let V′ (rresp. V′′) be the subspace of S^{2}V spanned by the pair x_{0}^{2} + x_{2}^{2}, x_{1}^{2} + x_{3}^{2} (resp. x_{1}x_{3}, x_{0}x_{2}). Then V′ ≃ V′′ ≃ V^{0} and S^{2}V = 2V^{0} + V^{1} + V^{2} + V^{3}. Let Z be the curve in P_{O}^{1} × P(V)

(68)  μ_{0}(x_{0}^{2} + x_{2}^{2})−μ_{1}x_{1}x_{3} = 0
(69)  μ_{0}(x_{1}^{2} + x_{3}^{2})−μ_{1}x_{0}x_{2} = 0

over P_{O}^{1}. We see as before that any non-isotrivial elliptic curve (or a nonisotrivial one-dimensional PSQAS) E with a rigid G(K)-structure over some connected O-scheme T is defined by the equations on P_{T}^{1} × P(V)_{T}

(70)  A_{0}(x_{0}^{2} + x_{2}^{2})−A_{1}x_{1}x_{3} = 0
(71)  A_{0}(x_{1}^{2} + x_{3}^{2})−A_{1}x_{0}x_{2} = 0

for some (A_{0}, A_{1}) ∈ P_{T}^{1} . It follows that (Z, i, U(K))_{RIG} is the universal PSQAS with a rigid G(K)-structure where i : Z → P_{O}^{1} × P(V) is the natural inclusion. In particular, SQ_{1,K} ≃ P(O[μ_{0}, μ_{1}]) ≃ P_{O}^{1}. The curves defined by V^{i} (i = 1, 2, 3) belong to H_{1,K}^{Ḡ(K)} \ S_{1,K} as in Subsection 13.2.

We define a morphism of P_{O}^{1} to P_{O}^{1} by h^{∗}μ_{0} = ν_{0}^{2} and h^{∗}μ_{1} = ζ_{4}ν_{1}^{2}. The pull back h^{∗}Z of Z has exactly 16 4-torsion points x_{0}x_{1}x_{2}x_{3} = 0 rational over P(O[ν_{0}, ν_{1}]), while it has no rational points over P(O[μ_{0}, μ_{1}]). It follows that Z is not isomorphic to S(4), the universal generalized elliptic curve with a classical level 4-structure. This will be discussed in detail in [NT98].

Notation/Terminology

a(x), b(x, y)  degeneration data, Theorem 4.3
A_{g,K}  Definition 11.2
Del, Del(P_{0}), Del(Q_{0})  Definitions 1.1, 3.6,
e_{K}  Definition 7.5
e_{S}^{L_{η}}, e^{L_{η}}  Eq.(9)
G  semi abelian scheme, Subsections 2.1, 4.12
G^{♯}  Subsection4.13
G(Q, L)  Definition 7.14
G(Q, L)  Definitions 5.11, 7.12
G(K), Ḡ(K)  Heisenberg group, Definition 7.5
G(K), Ḡ(K)  Definition 7.5
Hilbert point  Introduction, Section 8
H, K, K(H)  Definition 7.5
Kempf-stable  Introduction, Section 8
(K, e_{K})  Definition 7.5
K_{S}(L_{η}), K_{S}^{♯}(L_{η})  Subsection 2.3, Lemma 4.14
K_{S}(M)  Subsection 2.3
K-symplectic PSQAS  Definitions 9.2
K(Q, L)  Definitions 5.11
λ, λ(L_{0})  Subsection 2.2
level G(K)-structure  Definitions 9.4, 9.15, Lemma 5.14
μ_{M}  Definition 7.5
O, O_{M}  Definition 7.5
(P, L), (P_{0}, L_{0})  SQAV, Theorem 3.5
(Q, L), (Q_{0}, L_{0})  PSQAS, Theorem 5.8, Definition 5.9
(Q, ϕ, ρ)_{LEV}, (Z, ϕ, ρ)_{LEV} Definitions 9.4, 9.15
(Q, ϕ, ρ)_{RIG}, (Z, ϕ, ρ)_{RIG} Definitions 9.2, 9.4, 9.15
Raynaud sequence  Subsection 2.2
rigid G(K)-structure  Definitions 9.2, 9.4, 9.15
S_{g,K}  Definition 11.2
S_{y}  Eq.(12)
theta submodule  Definition 5.13, Eq (14)
V (K)  Definition 7.5
W_{g,K}  Subsection 11.2
ξ_{x}  Definition 3.4
ζ_{x,c}  Definition 3.4
(Z, L, V)  Definition 9.2

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Department of Mathematics, Hokkaido University,Sapporo, 060, Japan

E-mail address: nakamura@math.sci.hokudai.ac.jp