Log Abelian Varieties over a Log Point

Heer Zhao

Received: October 25, 2015 Revised: August 26, 2016

Communicated by Takeshi Saito

Abstract. We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincaré complete reducibility theorem for log abelian varieties, and the semisimplicity of the endomorphism algebras of log abelian varieties.

2010 Mathematics Subject Classification: Primary 14D06; Secondary 14K99, 11G99. Keywords and Phrases: log abelian varieties with constant degeneration, endomorphism algebras, Poincaré complete reducibility theorem, dual short exact sequence for isogenies.

1 Introduction

As stated in [KKN08a], degenerating abelian varieties can not preserve group structure, properness, and smoothness at the same time. Log abelian variety is a construction aimed to make the impossible possible in the world of log geometry. The idea dates back to Kato’s construction of log Tate curve in [Kat89, Sec. 2.2], in which he also conjectured the existence of a general theory of log abelian varieties. The theory finally comes true in [KKN08b] and [KKN08a]. Log abelian varieties are defined as certain sheaves in the classical étale topology in [KKN08a], however the log flat topology is needed for studying some problems, for example finite group subobjects of log abelian varieties, l-adic realisations of log abelian varieties, logarithmic Dieudonné theory of log abelian varieties and so on. In section 2, we prove that various classical étale sheaves from [KKN08a] are also sheaves for the log flat topology, in particular we prove that (weak) log abelian varieties with constant degeneration are sheaves for the log flat topology, see Theorem 2.1. We compute the first direct image sheaves of étale locally finite rank free constant sheaves, for changing to the log flat site from the classical étale site, in Lemma 2.4. This lemma can be considered as a supplement or generalisation of [Kat91, Thm. 4.1]. We also reformulate some results from [KKN08a, §2, §3 and §7] in the context of the log flat topology. In section 3, we focus on the case that the base is a log point. In this case, a log abelian variety is automatically a log abelian variety with constant degeneration. And only in this case, log abelian variety is the counterpart of abelian variety. While for general base, log abelian variety corresponds to abelian scheme. Now one may wonder if various results for abelian variety also hold for log abelian variety. We study isogenies and general homomorphisms between log abelian varieties over a log point. More precisely, we give several equivalent characterisations of isogeny in Proposition 3.3, and prove the dual short exact sequence in Theorem 3.1, Poincaré complete reducibility theorem for log abelian varieties in Theorem 3.2, and the finiteness of homomorphism group of log abelian varieties in Theorem 3.4, Corollary 3.3, Corollary 3.4, and Corollary 3.5.

Acknowledgement

I am grateful to Professor Kazuya Kato for sending me a copy of the paper [KKN15] which had been accepted but not yet published when the author started to work on this paper. I thank Professor Chikara Nakayama for very helpful communications concerning the paper [KKN15]. I thank Professor Qing Liu for telling me the reference [Bri15, Rem. 5.4.7. (iii)] about semi-abelian varieties. Part of this work was done during the author’s informal stay at Professor Gebhard Böckle’s Arbeitsgruppe, and I thank him for his hospitality and kindness. I would like to thank the anonymous referee, whose feedback has greatly improved this article. Part of this work has been supported by SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”.

2 Log abelian varieties with constant degeneration in the log flat topology

When dealing with finite subgroup schemes of abelian varieties, one needs to work with the flat topology. Similarly, the log flat topology is needed in the study of log finite group subobjects of log abelian varieties. However, log abelian varieties in [KKN08a] are defined in the classical étale topology. In this section, we are going to reformulate some results from [KKN08a, §2, §3 and §7], which are formulated in the context of classical étale topology, in the context of log flat topology. Throughout this section, let S be any fs log scheme with its underlying scheme locally noetherian, and (fs/S) be the category of fs log schemes over S. The log schemes in this section will always be fs log schemes unless otherwise stated.

Let S_{Ét}^{cl} (resp. S_{fl}^{cl}, resp. S_{Ét}^{log}, resp. S_{fl}^{log}) be the classical étale site (resp. classical flat site, resp. log étale site, resp. log flat site)^{1} associated to the category (fs/S), and let δ = m ∘ ε_{fl} : S_{fl}^{log} ⟶^{ε_{fl}} S_{fl}^{cl} ⟶^{m} S_{Ét}^{cl} be the canonical map of sites. For any inclusion F ⊂ G of sheaves on S_{fl}^{log}, we denote by G/F the quotient sheaf in the category of sheaves on S_{fl}^{log} by convention, unless otherwise stated. We start with the following lemma, which relates the Hom sheaves in the classical étale topology to the Hom sheaves in the log flat topology. Although this lemma is somehow trivial, we still formulate it due to its extensive use in this paper.

Lemma 2.1. Let F, G be two sheaves on S_{Ét}^{cl} which are also sheaves on S_{fl}^{log}. Then we have Hom_{S_{Ét}^{cl}}(F, G) = Hom_{S_{fl}^{log}}(F, G), in particular Hom_{S_{Ét}^{cl}}(F, G) is

a sheaf on S_{fl}^{log}.

Proof. This is clear.

Now we recall some definitions from [KKN08a]. Let G be a commutative group scheme over the underlying scheme of S which is an extension of an abelian scheme B by a torus T. Let X be the character group of T which is a locally constant sheaf of finite generated free ℤ-modules for the classical étale topology. The sheaf 𝔾_{m,log} on S_{Ét}^{cl} is defined by

𝔾_{m,log}(U) = Γ(U, M_{U}^{gp}),

the sheaf T_{log} on S_{Ét}^{cl} is defined by

T_{log} := Hom_{S_{Ét}^{cl}}(X, 𝔾_{m,log}),

and the sheaf G_{log} is defined as the push-out of T_{log} ← T → G in the category of sheaves on S_{Ét}^{cl}, see [KKN08a, 2.1].

Proposition 2.1. The sheaves 𝔾_{m,log}, X, T_{log} and G_{log} on S_{Ét}^{cl} are also sheaves for the log flat topology. Moreover, T_{log} can be alternatively defined as

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}),

and G_{log} can be alternatively defined as the push-out of T_{log} ← T → G in the category of sheaves on S_{fl}^{log}.

Proof. The statement for 𝔾_{m,log} is just [Kat91, Thm. 3.2], see also [Niz08, Cor. 2.22]. Being representable by a group scheme, X is a sheaf on S_{fl}^{log} by [Kat91, Thm. 3.1] and [KKN15, Thm. 5.2]. It follows then T_{log} = Hom_{S_{Ét}^{cl}}(X, 𝔾_{m,log}) = Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}) is also a sheaf on S_{fl}^{log}. By its definition G_{log} fits into a short exact sequence 0 → T_{log} → G_{log} → B → 0 of sheaves on S_{Ét}^{cl}. Consider the following commutative diagram

1Here we are following the terminology from [Kat91]. Note that in [KKN15] S_{Ét}^{cl} is called the strict étale site, while S_{Ét}^{log} and S_{fl}^{log} are called the Kummer log étale site and the Kummer log flat site respectively.

Documenta Mathematica 22 (2017) 505–550

\xymatrix{0 \ar[r] & T_{log} \ar[r] \ar[d]^{=} & G_{log} \ar[r] \ar[d] & B \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & T_{log} \ar[r] & δ_{∗}δ^{∗}G_{log} \ar[r] & B \ar[r] & R^{1}δ_{∗}T_{log}}

with exact rows in the category of sheaves on S_{Ét}^{cl}, where the vertical maps come from the adjunction (δ^{∗}, δ_{∗}). The sheaf R^{1}δ_{∗}T_{log} is zero by Kato’s logarithmic Hilbert 90, see [Kat91, Cor. 5.2] or [Niz08, Thm. 3.20]. It follows that the canonical map G_{log} → δ_{∗}δ^{∗}G_{log} is an isomorphism, whence G_{log} is a sheaf on S_{fl}^{log}. Since G_{log}, as a push-out of T_{log} ← T → G in the category of sheaves on S_{Ét}^{cl}, is already a sheaf on S_{fl}^{log}, it coincides with the push-out of T_{log} ← T → G in the category of sheaves on S_{fl}^{log}.

Proposition 2.2. We have canonical isomorphisms

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) ≅ T_{log}/T ≅ G_{log}/G.

Proof. By Proposition 2.1, G_{log} is the push-out of T_{log} ← T → G in the category of sheaves on S_{fl}^{log}, so we get a commutative diagram

[figure: 0; T; G; B; 0; 0; T_{log}; G_{log}; B; 0]

with exact rows. Then the isomorphism T_{log}/T ≅ G_{log}/G follows. Applying the functor Hom_{S_{fl}^{log}}(X, −) to the short exact sequence

0 → 𝔾_{m} → 𝔾_{m,log} → 𝔾_{m,log}/𝔾_{m} → 0,

we get a long exact sequence

0 → T → T_{log} → Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) → Ext_{S_{fl}^{log}}(X, 𝔾_{m})

of sheaves on S_{fl}^{log}. Since X is classical étale locally represented by a finite rank free abelian group, the sheaf Ext_{S_{fl}^{log}}(X, 𝔾_{m}) is zero. It follows that the sheaf Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) is canonically isomorphic to T_{log}/T.

It is obvious that the association of G_{log} to G is functorial in G. Hence we have a natural map Hom_{S_{fl}^{log}}(G, G′) → Hom_{S_{fl}^{log}}(G_{log}, G′_{log}), where G′ is another commutative group scheme which is an extension of an abelian scheme by a torus over the underlying scheme of S. The following proposition describes some properties of this map.

Proposition 2.3. (1) The association of G_{log} to G is functorial in G.

(2) The canonical map Hom_{S_{fl}^{log}}(G, G′) → Hom_{S_{fl}^{log}}(G_{log}, G′_{log}) is an isomorphism.

(3) For a group scheme H of multiplicative type with character group X_{H} over the underlying scheme of S, let H_{log} denote Hom_{S_{fl}^{log}}(X_{H}, 𝔾_{m,log}). Let 0 → H′ → H → H′′ → 0 be a short exact sequence of group schemes of multiplicative type over the underlying scheme of S such that their character groups are étale locally finite rank constant sheaves, then the sequences

0 → H′_{log} → H_{log} → H′′_{log} → 0

and

0 → Hom_{S_{fl}^{log}}(X_{H′}, 𝔾_{m,log}/𝔾_{m}) → Hom_{S_{fl}^{log}}(X_{H}, 𝔾_{m,log}/𝔾_{m})
→ Hom_{S_{fl}^{log}}(X_{H′′}, 𝔾_{m,log}/𝔾_{m}) → 0

are both exact.

(4) If G → G′ is injective, so is G_{log} → G′_{log}.

(5) If G → G′ is surjective, so is G_{log} → G′_{log}.

(6) Let 0 → G′ → G → G′′ → 0 be a short exact sequence of semi-abelian schemes over the underlying scheme of S, such that G′ (resp. G, resp. G′′) is an extension of an abelian scheme B′ (resp. B, resp. B′′) by a torus T′ (resp. T, resp. T′′). Then we have a short exact sequence 0 → G′_{log} → G_{log} → G′′_{log} → 0.

Proof. Part (1) is clear. The isomorphism of part (2) follows from [KKN08a, Prop. 2.5]. We prove part (3). Since we have a long exact sequence

0 → H′_{log} → H_{log} → H′′_{log} → Ext_{S_{fl}^{log}}(X_{H′}, 𝔾_{m,log}),

it suffices to show Ext_{S_{fl}^{log}}(X_{H′}, 𝔾_{m,log}) = 0. Since Ext_{S_{fl}^{log}}(ℤ, 𝔾_{m,log}) = 0, we are further reduced to show Ext_{S_{fl}^{log}}(ℤ/nℤ, 𝔾_{m,log}) = 0 for any positive

integer n. The short exact sequence 0 → ℤ ⟶^{n} ℤ → ℤ/nℤ → 0 gives rise to a long exact sequence 0 → Hom_{S_{fl}^{log}}(ℤ/nℤ, 𝔾_{m,log}) → 𝔾_{m,log} ⟶^{n} 𝔾_{m,log} → Ext_{S_{fl}^{log}}(ℤ/nℤ, 𝔾_{m,log}) → 0. Since 𝔾_{m,log} ⟶^{n} 𝔾_{m,log} is surjective, the sheaf Ext_{S_{fl}^{log}}(ℤ/nℤ, 𝔾_{m,log}) must be zero. The other short exact sequence is proved similarly.

We prove part (4). Since G → G′ is injective, then the corresponding map T → T′ on the torus parts is also injective and the corresponding map X′ → X on the character groups is surjective. It follows that the induced map

G_{log}/G = Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) → Hom_{S_{fl}^{log}}(X′, 𝔾_{m,log}/𝔾_{m}) = G′_{log}/G′

is injective. Hence G_{log} → G′_{log} is injective.

Now we prove part (5). Let f denote the map G → G′. Consider the torus and abelian variety decomposition of f

\xymatrix{0 \ar[r] & T \ar[r] \ar[d]^{f_{t}} & G \ar[r] \ar[d]^{f} & B \ar[r] \ar[d]^{f_{ab}} & 0 \\ 0 \ar[r] & T′ \ar[r] & G′ \ar[r] & B′ \ar[r] & 0.}

We first show that f_{t} is surjective. Assume that the underlying scheme of S is a point. The snake lemma gives an exact sequence Ker(f_{ab}) → Coker(f_{t}) → 0. Since Coker(f_{t}) is a torus and the reduced neutral component of Ker(f_{ab}) is an abelian variety by [Bri15, Lem. 3.3.7], we must have Coker(f_{t}) = 0. Hence f_{t} is surjective. In the general case, f_{t} is fiberwise surjective, hence it is also set-theoretically surjective. The fibers of f_{t} over S are all flat, hence f_{t} is flat by the fiberwise criterion of flatness, see [Gro66, Cor. 11.3.11]. Then f_{t} is faithfully flat, hence it is surjective. Then we get a short exact sequence 0 → X′ → X → X/X′ → 0 of étale locally constant sheaves. Applying the functor Hom_{S_{fl}^{log}}(−, 𝔾_{m,log}/𝔾_{m}) to this short exact sequence, we get a long exact sequence

→ G_{log}/G → G′_{log}/G′ → Ext_{S_{fl}^{log}}(X/X′, 𝔾_{m,log}/𝔾_{m}).

Let Z_{tor} be the torsion part of X/X′, and let n be a positive integer such that nZ_{tor} = 0. Since the multiplication-by-n map on 𝔾_{m,log}/𝔾_{m} is an isomorphism, we get that the sheaf Ext_{S_{fl}^{log}}(Z_{tor}, 𝔾_{m,log}/𝔾_{m}) is zero. The torsion-free nature of (X/X′)/Z_{tor} implies Ext_{S_{fl}^{log}}((X/X′)/Z_{tor}, 𝔾_{m,log}/𝔾_{m}) = 0, hence Ext_{S_{fl}^{log}}(X/X′, 𝔾_{m,log}/𝔾_{m}) = 0. It follows that G_{log}/G → G′_{log}/G′ is surjective, hence G_{log} → G′_{log} is surjective.

At last, we prove part (6). Consider the following commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & G′ \ar[r] \ar[d] & G′_{log} \ar[r] \ar[d] & Hom_{S_{fl}^{log}}(X′, 𝔾̄_{m,log}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & G \ar[r] \ar[d] & G_{log} \ar[r] \ar[d] & Hom_{S_{fl}^{log}}(X, 𝔾̄_{m,log}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & G′′ \ar[r] \ar[d] & G′′_{log} \ar[r] \ar[d] & Hom_{S_{fl}^{log}}(X′′, 𝔾̄_{m,log}) \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

with the first column and all rows exact, where 𝔾̄_{m,log} denotes 𝔾_{m,log}/𝔾_{m}. The maps G′ → G → G′′ induce T′ → T → T′′, furthermore X′ ← X ← X′′, lastly the third column of the diagram. Although 0 → X′′ → X → X′ → 0 is not necessarily exact, it gives two exact sequences 0 → Z → X → X′ → 0 and 0 → X′′ → Z → Z/X′′ → 0, where Z := Ker(X → X′) is étale locally a finite rank free constant sheaf and Z/X′′ is étale locally a finite torsion constant sheaf. By part (3), we get two short exact sequences

0 → Hom_{S_{fl}^{log}}(X′, 𝔾̄_{m,log}) → Hom_{S_{fl}^{log}}(X, 𝔾̄_{m,log}) → Hom_{S_{fl}^{log}}(Z, 𝔾̄_{m,log}) → 0

and

0 →Hom_{S_{fl}^{log}}(Z/X′′, 𝔾̄_{m,log}) → Hom_{S_{fl}^{log}}(Z, 𝔾̄_{m,log})
→Hom_{S_{fl}^{log}}(X′′, 𝔾̄_{m,log}) → 0.

But Hom_{S_{fl}^{log}}(Z/X′′, 𝔾̄_{m,log}) = 0, it follows that the third column of the diagram is exact. So is the middle column.

Recall that in [KKN08a, Def. 2.2], a log 1-motive M over S_{Ét}^{cl} is defined as a two-term complex [Y ⟶^{u} G_{log}] in the category of sheaves on S_{Ét}^{cl}, with the degree −1 term Y an étale locally constant sheaf of finitely generated free abelian groups and the degree 0 term G_{log} as above. Since both Y and G_{log} are sheaves on S_{fl}^{log}, M can also be defined as a two-term complex [Y ⟶^{u} G_{log}] in the category of sheaves on S_{fl}^{log}. Parallel to [KKN08a, 2.3], we have a natural pairing

<, >: X × Y → X × (G_{log}/G) = X × Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) → 𝔾_{m,log}/𝔾_{m}. (2.1)

It is clear that our pairing is induced from the one of [KKN08a, 2.3]. By our convention, T_{log}/T denotes the quotient in the category of sheaves on S_{fl}^{log}. For the quotient of T ⊂ T_{log} in the category of sheaves on S_{Ét}^{cl}, we use the notation (T_{log}/T)_{S_{Ét}^{cl}}. Now we assume that the pairing (2.1) is admissible (see [KKN08a, 7.1] for the definition of admissibility), in other words the log 1-motive M is admissible. Recall that in [KKN08a, 3.1], the subgroup sheaf Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)} of the sheaf Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) on S_{Ét}^{cl} is defined by

Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)}(U) :=
{φ ∈ Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})(U) | for every u ∈ U and x ∈ Xū,
there exist y_{u,x}, y′_{u,x} ∈ Yū such that < x, y_{u,x} > |φū(x)| < x, y′_{u,x} >}.

Here, ū denotes a classical étale geometric point above u, and for a, b ∈ (M_{U}^{gp}/O_{U}^{×})ū, a|b means a^{−1}b ∈ (M_{U}/O_{U}^{×})ū. It is natural to define the analogue of Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)} in the log flat topology. We need the following lemma first.

Lemma 2.2. Let δ : S_{fl}^{log} → S_{Ét}^{cl} be the canonical map between these two sites.

(1) δ_{∗}(𝔾_{m,log}/𝔾_{m}) = (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ.

(2) Let H be a commutative group scheme over the underlying scheme of S with connected fibres. Then Hom_{S_{fl}^{log}}(H, 𝔾_{m,log}/𝔾_{m}) = 0.

Proof. We denote the sheaf 𝔾_{m,log}/𝔾_{m} on S_{fl}^{log} by 𝔾̄_{m,log}. For any positive integer n, we have the following commutative diagram

[figure: 0; 0; 0; ℤ/n(1); 𝔾_{m}; n; 𝔾_{m}; 0; 0; ℤ/n(1); 𝔾_{m,log}; n; 𝔾_{m,log}; 0; 𝔾̄_{m,log}; n; ≅; 𝔾̄_{m,log}; 0; 0]

with exact rows and columns, where ℤ/n(1) denotes the group scheme of n-th roots of unity. Applying the functor ε_{fl∗} to the above diagram, we get a new commutative diagram

[figure: 0; ℤ/n(1); 𝔾_{m}; n; 𝔾_{m}; 0; ℤ/n(1); 𝔾_{m,log}; n; α; 𝔾_{m,log}; γ; ε_{fl∗}𝔾̄_{m,log}; n; ≅; β; ε_{fl∗}𝔾̄_{m,log}; δ; 𝔾_{m}; n; 𝔾_{m}; R^{1}ε_{fl∗}ℤ/n(1); η; R^{1}ε_{fl∗}𝔾_{m}; n; R^{1}ε_{fl∗}𝔾_{m}; 𝔾_{m,log}; n; 𝔾_{m,log}; θ; R^{1}ε_{fl∗}ℤ/n(1); R^{1}ε_{fl∗}𝔾_{m,log}]

with exact rows and columns. Since the map 𝔾_{m} ⟶^{n} 𝔾_{m} is surjective and R^{1}ε_{fl∗}𝔾_{m,log} = 0, we get a new commutative diagram

[figure: G ⊗_{ℤ} ℤ/n; θ̄; ≅; 0; G; n; G; \frac{1}{n} γ; ξ; ω; R^{1}ε_{fl∗}ℤ/n(1); η; 0; 0; G; ᾱ; ε_{fl∗}𝔾̄_{m,log}; β; R^{1}ε_{fl∗}𝔾_{m}; 0]

with exact rows, where G denotes (𝔾_{m,log}/𝔾_{m})_{S_{fl}^{cl}}, ᾱ (resp. θ̄) is the canonical map induced by α (resp. θ), ω is the canonical projection map and ξ is the unique map guaranteed by nβ ∘ (\frac{1}{n}γ) = δ ∘ (n(\frac{1}{n}γ)) = δ ∘ γ = 0. Taking colimit of the above diagram with respect to n, we get a commutative diagram

[figure: 0; G; G ⊗_{ℤ} ℚ; G ⊗_{ℤ} ℚ/ℤ; 0; 0; G; ᾱ; ε_{fl∗}𝔾̄_{m,log}; β; R^{1}ε_{fl∗}𝔾_{m}; 0]

with exact rows. Since the map G ⊗_{ℤ} ℚ/ℤ → R^{1}ε_{fl∗}𝔾_{m} is an isomorphism by Kato’s theorem [Kat91, Thm. 4.1] (see also [Niz08, Thm. 3.12]), we get G ⊗_{ℤ} ℚ ≅ ε_{fl∗}(𝔾_{m,log}/𝔾_{m}). Then

δ_{∗}(𝔾_{m,log}/𝔾_{m}) = m_{∗}ε_{fl∗}(𝔾_{m,log}/𝔾_{m}) = m_{∗}(G ⊗_{ℤ} ℚ)
= (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ,

where the last equality follows from the following fact: for any U ∈ (fs/S), the sheaf M_{U}^{gp}/O_{U}^{×} on the small étale site of U is constructible. This proves part (1). Now we prove part (2) which corresponds to [KKN08a, Lem. 6.1.1]. We have

Hom_{S_{fl}^{log}}(H, 𝔾̄_{m,log}) = Hom_{S_{Ét}^{cl}}(H, δ_{∗}𝔾̄_{m,log})
= Hom_{S_{Ét}^{cl}}(H, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ).

By the same argument of the proof of [KKN08a, Lem. 6.1.1], we have

Hom_{S_{Ét}^{cl}}(H, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ) = 0.

Hence part (2) is proved.

Now we define the analogue of Hom_{S_{Ét}^{log}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)}. It is the sub-

group sheaf Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} of the sheaf Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})

on S_{fl}^{log} given by

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)}(U) :=
{φ ∈ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})(U) | after pushing forward to U_{Ét}^{cl},
for every u ∈ U and x ∈ Xū, there exist y_{u,x}, y′_{u,x} ∈ Yū such that
< x, y_{u,x} > |φū(x)| < x, y′_{u,x} >}.

Here ū still denotes a classical étale geometric point above u. Let F := δ_{∗}(𝔾_{m,log}/𝔾_{m}) = (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ with δ the canonical map U_{fl}^{log} → U_{Ét}^{cl}.

For a, b ∈ (M_{U}^{gp}/O_{U}^{×})ū ⊗_{ℤ} ℚ, a|b means a^{−1}b = α ⊗_{ℤ} r for some α ∈ (M_{U}/O_{U}^{×})ū and r ∈ ℚ. Remark 2.1. In [KKN08a, 7.1], admissibility and non-degeneracy are defined for pairings into (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} in the classical étale site on (fs/S). We can define admissibility and non-degeneracy for pairings into 𝔾_{m,log}/𝔾_{m} on the log flat site in the same way. Since both X and Y are classical étale locally constant sheaves of finite rank free abelian groups, the definitions of admissibility and non-degeneracy are independent of the choice of the topology. The next lemma compares the sheaf Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)} on S_{Ét}^{cl}

with the sheaf Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} on S_{fl}^{log}.

Lemma 2.3. Let X, Y be two free abelian groups of finite rank, <, >: X × Y → (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} an admissible pairing on S_{Ét}^{cl}. Let

Q_{cl} := Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)}, Q := Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)},

and δ : S_{fl}^{log} → S_{Ét}^{cl} the canonical map between these two sites. Then we have Q = δ^{∗}Q_{cl} and δ_{∗}Q = Q_{cl} ⊗_{ℤ} ℚ.

Log Abelian Varieties over a Log Point 515

Proof. Denote 𝔾_{m,log}/𝔾_{m} by 𝔾̄_{m,log}. We have

δ^{∗}(𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} = 𝔾̄_{m,log},

and

δ_{∗}(𝔾̄_{m,log}) = (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ

by part (1) of Lemma 2.2, hence

δ^{∗}Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) = Hom_{S_{fl}^{log}}(X, 𝔾̄_{m,log}),

and

δ_{∗}Hom_{S_{fl}^{log}}(X, 𝔾̄_{m,log}) = Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) ⊗_{ℤ} ℚ.

Then by the definition of Q and Q_{cl}, we get Q = δ^{∗}Q_{cl} and δ_{∗}Q = Q_{cl}⊗_{ℤ}ℚ. Recall that in [KKN08a, 3.2, Thm. 7.3], G_{log}^{(Y)} ⊂ G_{log} (resp. T_{log}^{(Y)} ⊂ T_{log}) on S_{Ét}^{cl} is defined to be the inverse image of Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)} under the map

G_{log} → (G_{log}/G)_{S_{Ét}^{cl}} ≅ Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})
(resp. T_{log} → (T_{log}/T)_{S_{Ét}^{cl}} ≅ Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})).

We could also consider the inverse image sheaf of Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} under the map

G_{log} → G_{log}/G ≅ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})
(resp. T_{log} → T_{log}/T ≅ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})).

The following proposition states that these two constructions coincide. Proposition 2.4. (1) The sheaf G_{log}^{(Y)} on S_{Ét}^{cl} is also a sheaf on S_{fl}^{log}.

(2) The sheaf G_{log}^{(Y)} fits into a canonical short exact sequence

0 → G → G_{log}^{(Y)} → Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} → 0  (2.2)

of sheaves on S_{fl}^{log}.

(3) The association of G_{log}^{(Y)} to a log 1-motive M = [Y → G_{log}] is functorial. Proof. Let T (resp. B) be the torus (resp. abelian scheme) part of G, then we have a short exact sequence 0 → T_{log}^{(Y)} → G_{log}^{(Y)} → B → 0 of sheaves on S_{Ét}^{cl}. To show that G_{log}^{(Y)} on S_{Ét}^{cl} is a sheaf on S_{fl}^{log}, it suffices to show that T_{log}^{(Y)} is so. By [KKN08a, 7.7], locally on S_{Ét}^{cl} the sheaf T_{log}^{(Y)} is a union of representable sheaves. Hence it is also a sheaf on S_{fl}^{log}. So part (1) is proven.

Documenta Mathematica 22 (2017) 505–550

By the definition of G_{log}^{(Y)}, we have a pullback diagram

[figure: 0; G; G_{log}^{(Y)}; Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)}; 0; 0; G; G_{log}; Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}); 0]

in the category of sheaves on S_{Ét}^{cl}. Since G, G_{log}^{(Y)} and G_{log} are all sheaves on S_{fl}^{log}, applying the functor δ^{∗} to the above commutative diagram, we get the following commutative diagram

[figure: 0; G; G_{log}^{(Y)}; δ^{∗}Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)}; 0; 0; G; G_{log}; δ^{∗}Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}); 0.]

Since we have canonical isomorphisms

δ^{∗}Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) ≅ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})

and

δ^{∗}Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)} ≅ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)},

part (2) follows. Now we prove part (3). It is enough to prove that for a given homomorphism (f_{−1}, f_{0}) : M = [Y → G_{log}] → M′ = [Y′ → G′_{log}], the composition G_{log}^{(Y)} ↪

G_{log} ⟶^{f_{0}} G′_{log} factors through G_{log}^{′(Y′)} ↪ G′_{log}. Let X and X′ be the character groups of the torus parts of G and G′ respectively, let f_{l} : X′ → X be the map induced from f_{0}, and let

f̃_{d} : Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) → Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})

be the map induced from f_{l}. By the definition of G_{log}^{(Y)} and G_{log}^{′(Y′)}, we are reduced to show the composition

Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)} ↪ Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) ⟶^{f̃_{d}}

Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})

factors through

Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y′)} ↪ Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}).

Let <, >: X × Y → (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}} (resp. <, >′: X′ × Y′ → (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}}) be the pairing associated to M (resp. M′), then we have

< f_{l}(x′), y >=< x′, f_{−1}(y) >′

for any x′ ∈ X′, y ∈ Y. For any U ∈ (fs/S),

φ ∈ Hom_{S_{Ét}^{cl}}(X, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y)}(U),

we need to show

ψ := φ ∘ f_{l} ∈ Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y′)}(U).

For every u ∈ U and every x′ ∈ X′ ; ū, there exist y_{u,x′,1}, y_{u,x′,2} ∈ Yū such that < f_{l}(x′), y_{u,x′,1} > | φū(f_{l}(x′)) | < f_{l}(x′), y_{u,x′,2} >. The relation can be rewritten as

< x′, f_{−1}(y_{u,x′,1}) >′ | ψū(x′) | < x′, f_{−1}(y_{u,x′,2}) >′,

which implies that ψ ∈ Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y′)}(U). This finishes the proof of part (3).

Remark 2.2. Clearly, the image of u : Y → G_{log} is contained in G_{log}^{(Y)}. We further assume that the pairing (2.1) is non-degenerate (see [KKN08a, 7.1] and Remark 2.1 for the definition of non-degenerate pairings), then the two maps X → Hom_{S_{fl}^{log}}(Y, 𝔾_{m,log}/𝔾_{m}) and Y → Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) associated to the pairing are both injective. Recall that in [KKN08a, Def. 3.3. (1)] (resp. [KKN15, 1.7]) a log abelian variety with constant degeneration (resp. weak log abelian variety with constant degeneration) over S is defined to be a sheaf of abelian groups on S_{Ét}^{cl} which is isomorphic to the quotient sheaf

(G_{log}^{(Y)}/Y)_{S_{Ét}^{cl}} for a pointwise polarisable (resp. non-degenerate) log 1-motive

M = [Y ⟶^{u} G_{log}]. Here a log 1-motive is said to be non-degenerate if its associated pairing (2.1) is non-degenerate. Since the polarisability implies the non-degeneracy, a log abelian variety with constant degeneration over S is in particular a weak log abelian variety with constant degeneration over S.

Theorem 2.1. Let A be a weak log abelian variety with constant degeneration over S. Suppose A = (G_{log}^{(Y)}/Y)_{S_{Ét}^{cl}} for a non-degenerate log 1-motive M =

[Y ⟶^{u} G_{log}]. Then

(1) A is a sheaf on S_{fl}^{log};

(2) A = G_{log}^{(Y)}/Y, in other words A fits into a canonical short exact sequence

0 → Y → G_{log}^{(Y)} → A → 0  (2.3)

in the category of sheaves of abelian groups on S_{fl}^{log}; (3) A fits into a canonical short exact sequence

0 → G → A → Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)}/Y → 0  (2.4)

in the category of sheaves of abelian groups on S_{fl}^{log}.

Proof. Part (2) follows from part (1). Since the log 1-motive M is nondegenerate, the map Y → Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} is injective. Then the short exact sequence in part (3) is induced from the short exact sequences (2.2) and (2.3). We are left with part (1). The proof of part (1) is similar to that for the log étale case in [KKN15, §5]. Consider the short exact sequence 0 → Y → Ã → A → 0 of [KKN15, 5.3]. Note that Ã is nothing but G_{log}^{(Y)} in our situation, however we stick to the notation Ã for the sake of coherence with [KKN15, 5.3]. The argument showing that Ã is a log étale sheaf, also shows that Ã is a log flat sheaf, since representable functors are sheaves for the log flat topology by [KKN15, Thm. 5.2]. We have the canonical map δ := m ∘ ε_{fl} : S_{fl}^{log} ⟶^{ε_{fl}} S_{fl}^{cl} ⟶^{m} S_{Ét}^{cl} of sites. Applying δ^{∗} and δ_{∗} to 0 → Y → Ã → A → 0, we get a commutative diagram

\xymatrix{0 \ar[r] & Y \ar[r] \ar[d]^{=} & Ã \ar[r] \ar[d]^{=} & A \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Y \ar[r] & Ã \ar[r] & δ_{∗}δ^{∗}A \ar[r] & R^{1}δ_{∗}Y}

with exact rows, where the vertical maps are the ones given by the adjunction (δ^{∗}, δ_{∗}). To prove that A is a sheaf for the log flat topology, it is enough to show that the canonical map A → δ_{∗}δ^{∗}A is an isomorphism. This follows from the above commutative diagram with the help of the lemma below.

Lemma 2.4. The sheaf R^{1}δ_{∗}Y is zero.

Proof. Since Y is étale locally isomorphic to a finite rank free abelian group, we are reduced to the case Y = ℤ. Note that Y is a smooth group scheme over S. The proof here is the same as the proof of [Kat91, Thm. 4.1] (see also the proof of [Niz08, Thm. 3.12]) except the very last part where the condition G being affine is used. The reason why the proof there can be generalised to our case lies in the fact that Y is étale over S. Now we start from [Kat91, the second half of page 22] or [Niz08, the second last paragraph of page 524], since these two parallel parts are the very parts needed to be modified. Let B be a strict local ring, B̂ its completion, and let α ∈ H^{1}((SpecB)_{fl}^{log}, ℤ) such that it vanishes in H^{1}((SpecB̂)_{fl}^{log}, ℤ). By fpqc descent, α is a class of a representable ℤ-torsor over SpecB such that its structure morphism is étale. Since B is a strict local ring, the torsor admits a section by [Gro67, Prop. 18.8.1], so α is zero. It follows that [Kat91, Thm. 4.1] also holds for the case G = ℤ, so R^{1}ε_{fl∗}ℤ = 0. The Leray spectral sequence gives a short exact sequence 0 → R^{1}m_{∗}ℤ → R^{1}δ_{∗}ℤ → m_{∗}R^{1}ε_{fl∗}ℤ. The sheaf R^{1}m_{∗}ℤ = 0 by [Gro68, Thm. 11.7], it follows that R^{1}δ_{∗}ℤ = 0.

Remark 2.3. Lemma 2.4 can be viewed as a generalisation of Kato’s theorem (see [Kat91, Thm. 4.1] or [Niz08, Thm. 3.12]) to étale locally constant finitely generated torsion-free group schemes. Now we give a reformulation of [KKN08a, Thm. 7.4] in the context of the log flat topology.

Theorem 2.2. Let [Y → G_{log}] be a log 1-motive over S of type (X, Y) (see [KKN08a, Def. 2.2]) such that the induced paring X × Y → 𝔾_{m,log}/𝔾_{m} is non-degenerate, and let [X → G_{log}^{∗}] be its dual. Let A = G_{log}^{(Y)}/Y. Then we have:

(1) Ext_{S_{fl}^{log}}(A, ℤ) ≅ Hom_{S_{fl}^{log}}(Y, ℤ);

(2) the sheaf δ_{∗}Ext_{S_{fl}^{log}}(A, 𝔾_{m}) fits into an exact sequence

0 → G^{∗} → δ_{∗}Ext_{S_{fl}^{log}}(A, 𝔾_{m}) → Hom_{S_{Ét}^{cl}}(A, R^{1}δ_{∗}𝔾_{m});

(3) Ext_{S_{fl}^{log}}(A, 𝔾_{m,log}) ≅ (G_{log}^{∗}/X)_{S_{Ét}^{cl}} ≅ G_{log}^{∗}/X;

(4) Hom_{S_{fl}^{log}}(A, ℤ) = Hom_{S_{fl}^{log}}(A, 𝔾_{m}) = Hom_{S_{fl}^{log}}(A, 𝔾_{m,log}) = 0.

Proof. By Proposition 2.1 and Theorem 2.1, the sheaves ℤ, 𝔾_{m,log} and A on S_{Ét}^{cl} are also sheaves on S_{fl}^{log}. Then part (4) follows from [KKN08a, Thm. 7.4 (4)] with the help of Lemma 2.1. Before going to the rest of the proof, we first introduce two spectral sequences. Let F_{1} (resp. F_{2}) be a sheaf on S_{Ét}^{cl} (resp. S_{fl}^{log}), then we have

δ_{∗}Hom_{S_{fl}^{log}}(δ^{∗}F_{1}, F_{2}) = Hom_{S_{Ét}^{cl}}(F_{1}, δ_{∗}F_{2}).

Let θ be the functor sending F_{2} to δ_{∗}Hom_{S_{fl}^{log}}(δ^{∗}F_{1}, F_{2}) = Hom_{S_{Ét}^{cl}}(F_{1}, δ_{∗}F_{2}), then we get two Grothendieck spectral sequences

E_{2}^{p,q} = R^{p}δ_{∗}R^{q}Hom_{S_{fl}^{log}}(δ^{∗}F_{1}, −) ⇒ R^{p+q}θ  (2.5)

and

E_{2}^{p,q} = R^{p}Hom_{S_{Ét}^{cl}}(F_{1}, −)R^{q}δ_{∗} ⇒ R^{p+q}θ.  (2.6)

These two spectral sequences give two exact sequences

0 → R^{1}δ_{∗}Hom_{S_{fl}^{log}}(δ^{∗}F_{1}, F_{2}) → R^{1}θ(F_{2}) → δ_{∗}Ext_{S_{fl}^{log}}(δ^{∗}F_{1}, F_{2})

→ R^{2}δ_{∗}Hom_{S_{fl}^{log}}(δ^{∗}F_{1}, F_{2}) (2.7)

and

0 → Ext_{S_{Ét}^{cl}}(F_{1}, δ_{∗}F_{2}) → R^{1}θ(F_{2}) → Hom_{S_{Ét}^{cl}}(F_{1}, R^{1}δ_{∗}F_{2}).  (2.8)

Let F_{1} = A, and let F_{2} be ℤ, 𝔾_{m} or 𝔾_{m,log}, part (4) together with Theorem 2.1 and exact sequence (2.7) imply

R^{1}θ(A) ≅ δ_{∗}Ext_{S_{fl}^{log}}(A, F_{2}),

so we get an exact sequence

0 → Ext_{S_{Ét}^{cl}}(A, δ_{∗}F_{2}) → δ_{∗}Ext_{S_{fl}^{log}}(A, F_{2}) → Hom_{S_{Ét}^{cl}}(A, R^{1}δ_{∗}F_{2}).

Since Ext_{S_{Ét}^{cl}}(A, 𝔾_{m}) ≅ G^{∗} by [KKN08a, Thm. 7.4 (2)], the case F_{2} = 𝔾_{m} gives part (2). Since R^{1}δℤ = 0 by Lemma 2.4 and Ext_{S_{Ét}^{cl}}(A, ℤ) ≅ Hom_{S_{fl}^{log}}(Y, ℤ) by [KKN08a, Thm. 7.4 (1)], the case F_{2} = ℤ gives part (1). The sheaf R^{1}δ_{∗}𝔾_{m,log} equals zero by Kato’s logarithmic Hilbert 90 [Kat91, Cor. 5.2]. And we have Ext_{S_{Ét}^{cl}}(A, 𝔾_{m,log}) ≅ (G_{log}^{∗}/X)_{S_{Ét}^{cl}} by [KKN08a, Thm. 7.4 (3)]. Then part (3) follows from the case F_{2} = 𝔾_{m,log}.

Let A be a weak log abelian variety with constant degeneration over S, and let M = [Y → G_{log}] be the log 1-motive of type (X, Y) defining A. Then the paring <, >: X × Y → 𝔾_{m,log}/𝔾_{m} associated to M is non-degenerate. Let M^{∗} = [X → G_{log}^{∗}] be the dual log 1-motive of M, then the pairing associated to M^{∗} is the same (up to switching the positions of X and Y) as the paring associated to M, hence it is automatically non-degenerate. If A is further a log abelian variety with constant degeneration, i.e. the log 1-motive M is pointwise polarisable, then M^{∗} is also pointwise polarisable.

Definition 2.1. Let A be a weak log abelian variety with constant degeneration (resp. log abelian variety with constant degeneration) over S. The dual weak log abelian variety with constant degeneration (resp. dual log abelian variety with constant degeneration) of A is the weak log abelian variety with constant degeneration (resp. log abelian variety with constant degeneration) G_{log}^{∗(X)}/X associated to the log 1-motive M^{∗} = [X → G_{log}^{∗}]. We denote the dual of A by A^{∗}.

Let WLAV_{S}^{CD} (resp. LAV_{S}^{CD}) denote the category of weak log abelian varieties with constant degeneration (resp. log abelian varieties with constant degeneration) over S. Then we have the following proposition.

Proposition 2.5. The association of A^{∗} to A gives rise to a contravariant functor

(−)^{∗} : WLAV_{S}^{CD} → WLAV_{S}^{CD}

which restricts to a contravariant functor

(−)^{∗} : LAV_{S}^{CD} → LAV_{S}^{CD}.

Moreover the functor is a duality functor, i.e. there is a natural isomorphism from the identity functor to (−)^{∗∗}.

Proof. This follows from [KKN15, 1.7], [KKN08a, Thm. 3.4], and the corresponding duality theory of log 1-motives over S.

Remark 2.4. Given an abelian scheme A over the underlying scheme of S, the dual abelian scheme A^{∗} can also be interpreted as Ext_{S_{fl}^{cl}}(A, 𝔾_{m}). One may wonder if something similar happens in the case of (weak) log abelian varieties with constant degeneration. Note that in the log world, 𝔾_{m,log} plays the role of 𝔾_{m} in the non-log world. Part (3) of Theorem 2.2 indicates that Ext_{S_{fl}^{log}}(A, 𝔾_{m,log}) ≅ G_{log}^{∗}/X is not A^{∗} = G_{log}^{∗(X)}/X but closely related to it.

The following is a partial reformulation of [KKN08a, Thm. 7.3].

Theorem 2.3. Let X and Y be two finitely generated free ℤ-modules, and let <, >: X × Y → 𝔾_{m,log}/𝔾_{m} be a non-degenerate pairing on S_{fl}^{log}. Let G be a commutative group scheme over the underlying scheme of S which is an extension of an abelian scheme B by a torus T over S. Assume that X is the character group of T. Let T_{log}^{(Y)} = Hom_{S_{fl}^{log}}(X, 𝔾_{m,log})^{(Y)} ⊂ T_{log} =

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}) (resp. G_{log}^{(Y)} ⊂ G_{log}) be the inverse image of

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} ⊂ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) ≅ T_{log}/T ≅ G_{log}/G.

(1) Let H be a commutative group scheme over the underlying scheme of S. Then we have

Hom_{S_{fl}^{log}}(G_{log}^{(Y)}, H) ≅ Hom_{S_{fl}^{log}}(B, H), Hom_{S_{fl}^{log}}(G_{log}^{(Y)}/G, H) = 0.

If further H satisfies the condition R^{1}δ_{∗}H = 0, we also have

Ext_{S_{fl}^{log}}(G_{log}^{(Y)}, H) ≅ Ext_{S_{fl}^{log}}(B, H)

and

Ext_{S_{fl}^{log}}(G_{log}^{(Y)}/G, H) ≅ Hom_{S_{fl}^{log}}(T, H).

In particular, since R^{1}δ_{∗}ℤ = 0, we have

Ext_{S_{fl}^{log}}(G_{log}^{(Y)}, ℤ) ≅ Ext_{S_{fl}^{log}}(B, ℤ) = 0

and

Ext_{S_{fl}^{log}}(G_{log}^{(Y)}/G, ℤ) ≅ Hom_{S_{fl}^{log}}(T, ℤ) = 0.

(2) We have Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}) ≅ X, Hom_{S_{fl}^{log}}(T_{log}^{(Y)}/T, 𝔾_{m,log}) = 0,

Ext_{S_{fl}^{log}}(T_{log}^{(Y)}/T, 𝔾_{m,log}) = 0, and δ_{∗}Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}) ⊂ R^{2}δ_{∗}X.

(3) Let G′ be another commutative group scheme over S which is an extension of an abelian scheme B′ by a torus T′ over S. Let X′ be the character group of T′. Then we have

Hom_{S_{fl}^{log}}(G, G′) ⟶^{≅} Hom_{S_{fl}^{log}}(G_{log}^{(Y)}, G′_{log})

and

Hom_{S_{fl}^{log}}(X′, X) ⊗_{ℤ} ℚ ⟶^{≅} Hom_{S_{fl}^{log}}(G_{log}^{(Y)}/G, G′_{log}/G′).

Our proof of Theorem 2.3 follows the structure of the proof [KKN08a, 7.17-7.21] of [KKN08a, Thm. 7.3]. Firstly we state a lemma which almost corresponds to [KKN08a, 7.17].

Lemma 2.5. Let the notation be as in Theorem 2.3. Then the following hold.

(1) Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, H) = 0;

(2) If R^{1}δ_{∗}H = 0, then Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, H) = 0;

(3) δ_{∗}Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}) ⊂ R^{2}δ_{∗}X.

Proof. Part (1) follows from the corresponding result in the classical étale topology of [KKN08a, 7.17], with the help of Lemma 2.1. We use the exact sequences (2.7) and (2.8) from the proof of Theorem 2.2 to investigate the sheaves Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, H) and Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}). If R^{1}δ_{∗}H =

0, then the vanishing of Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, H) and Ext_{S_{Ét}^{cl}}(T_{log}^{(Y)}, H) implies part (2) via (2.7) and (2.8). We have

Ext_{S_{Ét}^{cl}}(T_{log}^{(Y)}, 𝔾_{m,log}) = 0

by [KKN08a, 7.17] and R^{1}δ_{∗}𝔾_{m,log} = 0 by Kato’s logarithmic Hilbert 90. We also have

Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}) = Hom_{S_{Ét}^{cl}}(T_{log}^{(Y)}, 𝔾_{m,log}) ≅ X

by [KKN08a, Thm. 7.3 (2)]. Then the inclusion of part (3) follows from (2.7) and (2.8).

Remark 2.5. In the proof of part (3) of Lemma 2.5, since

Ext_{S_{Ét}^{cl}}(T_{log}^{(Y)}, 𝔾_{m,log}) = R^{1}δ_{∗}𝔾_{m,log} = 0,

we must have R^{1}δ_{∗}X = 0 by the exact sequences (2.7) and (2.8). This gives rise to an alternative proof of Lemma 2.4.

Log Abelian Varieties over a Log Point 523

Proof of Theorem 2.3: Firstly we prove part (1). The isomorphism

Hom_{S_{fl}^{log}}(G_{log}^{(Y)}, H) ≅ Hom_{S_{fl}^{log}}(B, H)

comes from the corresponding isomorphism of [KKN08a, Thm. 7.3 (1)] for the classical étale topology with the help of Lemma 2.1. The short exact sequence 0 → T → T_{log}^{(Y)} → G_{log}^{(Y)}/G → 0 gives rise to a long exact sequence

0 → Hom_{S_{fl}^{log}}(G_{log}^{(Y)}/G, H) → Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, H) → Hom_{S_{fl}^{log}}(T, H)
→ Ext_{S_{fl}^{log}}(G_{log}^{(Y)}/G, H) → Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, H).

Since we have Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, H) = 0 by part (1) of Lemma 2.5, it follows that Hom_{S_{fl}^{log}}(G_{log}^{(Y)}/G, H) = 0. Assuming R^{1}δ_{∗}H = 0, we get Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, H) = 0 by part (2) of Lemma 2.5. Hence we have

Ext_{S_{fl}^{log}}(G_{log}^{(Y)}, H) ≅ Ext_{S_{fl}^{log}}(B, H), Ext_{S_{fl}^{log}}(G_{log}^{(Y)}/G, H) ≅ Hom_{S_{fl}^{log}}(T, H).

Next we prove part (2). The exact sequence 0 → T → T_{log}^{(Y)} → T_{log}^{(Y)}/T → 0 gives rise to a long exact sequence

0 → Hom_{S_{fl}^{log}}(T_{log}^{(Y)}/T, 𝔾_{m,log}) → Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log})

⟶^{α} Hom_{S_{fl}^{log}}(T, 𝔾_{m,log}) → Ext_{S_{fl}^{log}}(T_{log}^{(Y)}/T, 𝔾_{m,log}).

Since the map Hom_{S_{Ét}^{cl}}(T_{log}^{(Y)}, 𝔾_{m,log}) → Hom_{S_{Ét}^{cl}}(T, 𝔾_{m,log}) is canonically identical to the identity map 1_{X} : X → X by [KKN08a, 7.20], so is the map α. Hence we have Hom_{S_{fl}^{log}}(T_{log}^{(Y)}/T, 𝔾_{m,log}) = 0. Since R^{1}δ_{∗}𝔾_{m,log} = 0 and Ext_{S_{Ét}^{cl}}((T_{log}^{(Y)}/T)_{S_{Ét}^{cl}}, 𝔾_{m,log}) = 0 by [KKN08a, Thm. 7.3 (2)], we get Ext_{S_{fl}^{log}}(T_{log}^{(Y)}/T, 𝔾_{m,log}) = 0 by the exact sequences (2.7) and (2.8). The inclusion δ_{∗}Ext_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}) ⊂ R^{2}δ_{∗}X is just part (3) of Lemma 2.5. The isomorphism Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, 𝔾_{m,log}) ≅ X has been proved in the proof of Lemma 2.5. At last, we show part (3). The isomorphism

Hom_{S_{fl}^{log}}(G, G′) ⟶^{≅} Hom_{S_{fl}^{log}}(G_{log}^{(Y)}, G′_{log})

comes from the corresponding isomorphism of [KKN08a, Thm. 7.3 (3)] with the help of Lemma 2.1. The short exact sequence

0 → T → T_{log}^{(Y)} → G_{log}^{(Y)}/G → 0

Documenta Mathematica 22 (2017) 505–550

gives an exact sequence

0 →Hom_{S_{fl}^{log}}(G_{log}^{(Y)}/G, G′_{log}/G′) → Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, G′_{log}/G′)
→Hom_{S_{fl}^{log}}(T, G′_{log}/G′).

The sheaf Hom_{S_{fl}^{log}}(T, G′_{log}/G′) is zero by part (2) of Lemma 2.2, hence we are

reduced to compute Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, G′_{log}/G′). We have

δ_{∗}Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, G′_{log}/G′) = Hom_{S_{Ét}^{cl}}(T_{log}^{(Y)}, δ_{∗}(G′_{log}/G′))
= Hom_{S_{Ét}^{cl}}(T_{log}^{(Y)}, (G′_{log}/G′)_{S_{Ét}^{cl}} ⊗_{ℤ} ℚ)

where the second equality comes from part (1) of Lemma 2.2. By [KKN08a, Thm. 7.3 (3)],

Hom_{S_{Ét}^{cl}}(X′, X) ⟶^{≅}Hom_{S_{Ét}^{cl}}((G_{log}^{(Y)}/G)_{S_{Ét}^{cl}}, (G′_{log}/G′)_{S_{Ét}^{cl}})
⟶^{≅}Hom_{S_{Ét}^{cl}}(T_{log}^{(Y)}, (G′_{log}/G′)_{S_{Ét}^{cl}}).

It follows that

Hom_{S_{fl}^{log}}(T_{log}^{(Y)}, G′_{log}/G′) ≅ Hom_{S_{Ét}^{cl}}(X′, X) ⊗_{ℤ} ℚ = Hom_{S_{fl}^{log}}(X′, X) ⊗_{ℤ} ℚ.

Since weak log abelian varieties with constant degeneration are defined in terms of log 1-motives, it is natural to try to relate every aspect of weak log abelian varieties with constant degeneration to the corresponding aspect of log 1-motives. In particular, we are keen on the relation on the homomorphisms. The following theorem is a combination of [KKN08a, Thm. 3.4] and [KKN15, 1.7].

Theorem 2.4. The functor [Y → G_{log}] ↦ (G_{log}^{(Y)}/Y)_{Scl Ét} = G_{log}^{(Y)}/Y induces an equivalence from the category of non-degenerate log 1-motives (resp. pointwise polarisable log 1-motives) over S to that of weak log abelian varieties with constant degeneration (resp. log abelian varieties with constant degeneration) over S.

Proof. See [KKN08a, §8].

Let f : A → A′ be a homomorphism between two weak log abelian varieties with constant degeneration over S, and let M = [Y → G_{log}], M′ = [Y′ → G′_{log}] be the log 1-motives defining A and A′ respectively. By Theorem 2.4, f comes from a homomorphism from M to M′, and we denote it by (f_{−1}, f_{0}). The proof of [KKN08a, Thm. 8.1] actually shows that f_{0} comes from a unique homomorphism from G to G′, and we denote it by f_{c} by convention ^{2}. The homomorphism f_{c} can also be obtained from the following diagram

\xymatrix{0 \ar[r] & G \ar[r] \ar@{-->}[d]^{f_{c}} & A \ar[r] \ar[d]^{f} & Q/Y \ar[r] \ar@{-->}[d]^{f_{d}} & 0 \\ 0 \ar[r] & G′ \ar[r] & A′ \ar[r] & Q′/Y′ \ar[r] & 0}

with exact rows, together with the vanishing of Hom_{S_{fl}^{log}}(G, Q′/Y′) (see Lemma 2.6 below). Here the exact rows come from part (3) of Theorem 2.1, and Q (resp. Q′) denotes the sheaf Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} (resp.

Hom_{S_{fl}^{log}}(X′, 𝔾_{m,log}/𝔾_{m})^{(Y′)}). Furthermore the diagram gives a homomorphism Q/Y → Q′/Y′ which we denote by f_{d} ^{3}. The procedure of getting f from (f_{−1}, f_{0}) also gives a homomorphism Ã := G_{log}^{(Y)} → G_{log}^{′(Y′)} =: Ã′, which we denote by f̃. The homomorphism f̃ induces a homomorphism Q → Q′ which we denote by f̃_{d} ^{4}.

Lemma 2.6. We have Hom_{S_{fl}^{log}}(G, Q′/Y′) = 0.

Proof. As before, let δ : S_{fl}^{log} → S_{Ét}^{cl} be the canonical map between these two sites. By part (3) of Lemma 2.2, we have a canonical isomorphism δ_{∗}Q′ ≅ Q′_{cl} ⊗_{ℤ} ℚ with Q′_{cl} := Hom_{S_{Ét}^{cl}}(X′, (𝔾_{m,log}/𝔾_{m})_{S_{Ét}^{cl}})^{(Y′)}. We also have δ_{∗}(Q′/Y′) = (δ_{∗}Q′/Y′)_{S_{Ét}^{cl}} by Lemma 2.4. Hence

Hom_{S_{fl}^{log}}(G, Q′/Y′) = Hom_{S_{Ét}^{cl}}(G, δ_{∗}(Q′/Y′))
= Hom_{S_{Ét}^{cl}}(G, (δ_{∗}Q′/Y′)_{S_{Ét}^{cl}})
= Hom_{S_{Ét}^{cl}}(G, ((Q′_{cl} ⊗_{ℤ} ℚ)/Y′)_{S_{Ét}^{cl}}).

The group Hom_{S_{Ét}^{cl}}(G, (Q′_{cl} ⊗_{ℤ} ℚ/Y′)_{S_{Ét}^{cl}}) equals zero by the same reason as in [KKN08a, 9.2]. Hence Hom_{S_{fl}^{log}}(G, Q′/Y′) vanishes.

For practical reason, we state the following proposition, which is nothing else than a tedious summary of various maps constructed out of f : A → A′.

Proposition 2.6. Let f : A → A′ be a homomorphism of weak log abelian varieties with constant degeneration over S. Then f induces the following four commutative diagrams

2Here the subscript c stands for connected.

3Here the subscript d stands for discrete.

4Here the symbol˜ for f̃ (resp. f̃_{d}) stands for the lifting of f (resp. f_{d}) to the “universal coverings”, and Ã (resp. Q) could be thought of as the “universal covering” of A (resp.

Q/Y).

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\xymatrix{0 \ar[r] & Y \ar[r] \ar[d]^{f_{−1}} & G_{log}^{(Y)} \ar[r] \ar[d]^{f̃} & A \ar[r] \ar[d]^{f} & 0 \\ 0 \ar[r] & Y′ \ar[r] & G_{log}^{′(Y′)} \ar[r] & A′ \ar[r] & 0}  (2.9)

\xymatrix{0 \ar[r] & G \ar[r] \ar[d]^{f_{c}} & A \ar[r] \ar[d]^{f} & Q/Y \ar[r] \ar[d]^{f_{d}} & 0 \\ 0 \ar[r] & G′ \ar[r] & A′ \ar[r] & Q′/Y′ \ar[r] & 0}  (2.10)

\xymatrix{0 \ar[r] & G \ar[r] \ar[d]^{f_{c}} & G_{log}^{(Y)} \ar[r] \ar[d]^{f̃} & Q \ar[r] \ar[d]^{f̃_{d}} & 0 \\ 0 \ar[r] & G′ \ar[r] & G_{log}^{′(Y′)} \ar[r] & Q′ \ar[r] & 0}  (2.11)

\xymatrix{0 \ar[r] & Y \ar[r] \ar[d]^{f_{−1}} & Q \ar[r] \ar[d]^{f̃_{d}} & Q/Y \ar[r] \ar[d]^{f_{d}} & 0 \\ 0 \ar[r] & Y′ \ar[r] & Q′ \ar[r] & Q′/Y′ \ar[r] & 0}  (2.12)

with exact rows.

3 Isogeny

In this section we study log abelian varieties over a log point. Note that in this case, log abelian varieties are necessarily log abelian varieties with constant degeneration by [KKN08a, Thm. 4.6 (2)]. Let k be a field, and S = (Spec k, M_{S}) an fs log point with log structure induced by a chart P → k, where P is a sharp fs monoid such that P → (M_{S}/O_{S}^{×})x̄ is an isomorphism. Here x denotes the underlying point of S and x̄ denotes a geometric point above x. Let (fs/S) be the category of fs log schemes over S, and log schemes in this section will always be fs log schemes unless otherwise stated. Let S_{fl}^{log} (resp. S_{fl}^{cl}) be the log flat (resp. classical flat) site on (fs/S), and let S_{Ét}^{log} (resp. S_{Ét}^{cl}) be the log étale (resp. classical étale) site on (fs/S).

3.1 Isogeny

Firstly we show a few properties of the category (fin/S)_{r}. See Definition A.1 in the Appendix for the categories (fin/S)_{f}, (fin/S)_{r}, (fin/S)_{d} and (fin/S)_{c}.

Lemma 3.1. Let F_{1} ∈ (fin/S)_{c} and let F_{2} be a subobject of F_{1} in (fin/S)_{r}. Then we have F_{2} ∈ (fin/S)_{c}. In other words, the category (fin/S)_{c} is closed under subobjects in (fin/S)_{r}.

Proof. If the field k is of characteristic zero, then F_{2} is a log flat locally constant sheaf of finite abelian groups. Hence it corresponds to a π_{1}^{log}(S)-module by Theorem A.2. Here π_{1}^{log}(S) denotes the logarithmic fundamental group of S with respect to some log geometric point of S, see [Ill02, §4] for its definition. We have a short exact sequence

1 → I^{log}(S) → π_{1}^{log}(S) ⟶^{forg} π_{1}(S) → 1

from [Ill02, (4.7.1)], where forg is the map to the classical étale fundamental group of S induced by the canonical map from the log étale site to the classical étale site, and the kernel I^{log}(S) of forg is called the log inertia group of S. Since F_{1} ∈ (fin/S)_{c}, the action of I^{log}(S) on F_{1} is trivial. It follows that the action of I^{log}(S) on the subobject F_{2} of F_{1} is also trivial. Hence we get F_{2} ∈ (fin/S)_{c}. Now we are left with the case that the field k is of positive characteristic. Let 0 → F_{i}^{∘} → F_{i} → F_{i}^{et} → 0 be the connected-étale short exact sequence of F_{i}, see Lemma A.1. It is obvious that we have F_{1}^{∘}, F_{1}^{et} ∈ (fin/S)_{c}. We have F_{2}^{et} ∈ (fin/S)_{r} and F_{2}^{∘} ∈ (fin/S)_{c} by Proposition A.2. The inclusion i : F_{2} ↪ F_{1} gives a commutative diagram

\xymatrix{0 \ar[r] & F_{2}^{∘} \ar[r] \ar@{^{(}->}[d]^{i^{∘}} & F_{2} \ar[r] \ar@{^{(}->}[d]^{i} & F_{2}^{et} \ar[r] \ar@{^{(}->}[d]^{i^{et}} & 0 \\ 0 \ar[r] & F_{1}^{∘} \ar[r] & F_{1} \ar[r] & F_{1}^{et} \ar[r] & 0}

with exact rows and injective vertical homomorphisms. Then we have F_{2}^{et} ∈ (fin/S)_{c} by applying the same argument as in the characteristic zero case to F_{2}^{et} ⊂ F_{1}^{et}. Let E be the pullback of the extension F_{1} along i^{et} : F_{2}^{et} ↪ F_{1}^{et}, then we have a commutative diagram

[figure: 0; F_{2}^{∘}; i^{∘}; F_{2}; F_{2}^{et}; 0; 0; F_{1}^{∘}; E; F_{2}^{et}; i^{et}; 0; 0; F_{1}^{∘}; F_{1}; F_{1}^{et}; 0]

with exact rows. Note that E lies in (fin/S)_{c}, and E is also the pushout of F_{2} along i^{∘} : F_{2}^{∘} ↪ F_{1}^{∘}. Now we make use of Kato’s classification theorem Theorem A.3. Note that the functor Φ of Theorem A.3 is compatible with pushout along the second argument, hence we have a commutative diagram

𝔈𝔵𝔱_{S_{fl}^{cl}}(F_{2}^{et}, F_{2}^{∘}) × ℌ𝔬𝔪_{S_{fl}^{cl}}(F_{2}^{et}(1), F_{2}^{∘}) ⊗ P^{gp} →^{≃} 𝔈𝔵𝔱_{S_{fl}^{log}}(F_{2}^{et}, F_{2}^{∘})
↓i_{∗}^{∘}  ↓i_{∗}^{∘}  ↓i_{∗}^{∘}
𝔈𝔵𝔱_{S_{fl}^{cl}}(F_{2}^{et}, F_{1}^{∘}) × ℌ𝔬𝔪_{S_{fl}^{cl}}(F_{2}^{et}(1), F_{1}^{∘}) ⊗ P^{gp} →^{≃} 𝔈𝔵𝔱_{S_{fl}^{log}}(F_{2}^{et}, F_{1}^{∘})

with rows equivalences of categories. Let [E] ∈ 𝔈𝔵𝔱_{S_{fl}^{log}}(F_{2}^{et}, F_{1}^{∘}) (resp. [F_{2}] ∈ 𝔈𝔵𝔱_{S_{fl}^{log}}(F_{2}^{et}, F_{2}^{∘})) denote the class represented by E (resp. F_{2}). With the help of the above commutative diagram, [E] ∈ 𝔈𝔵𝔱_{S_{fl}^{cl}}(F_{2}^{et}, F_{1}^{∘}) implies [F_{2}] ∈ 𝔈𝔵𝔱_{S_{fl}^{cl}}(F_{2}^{et}, F_{2}^{∘}).

Proposition 3.1. (1) The category (fin/S)_{f} is abelian.

(2) The category (fin/S)_{r} is a weak Serre subcategory of (fin/S)_{f}.

(3) The category (fin/S)_{c} is closed under subobjects and quotient objects in (fin/S)_{r}, but not closed under extensions in (fin/S)_{r}.

Proof. We first show part (1). Let f : F → F′ be in (fin/S)_{f}, and U → S a log flat cover such that both F_{U} and F′_{U} lie in (fin/U)_{c}. Since S is a log point, we may shrink U such that its underlying scheme is affine (in particular quasi-compact). For each positive integer n, let S_{n} := S ×_{Specℤ[P]} Specℤ[P^{1/n}] endowed with the log structure associated to P^{1/n} → O_{S_{n}}. By [Kat91, Prop. 2.7 (2)] or [Niz08, Cor. 2.16]^{5}, there exists a log flat cover V → U and some positive integer n_{0} such that W := V ×_{S} S_{n_{0}} → S_{n_{0}} is classically flat. Since both F_{W} and F′_{W} are represented by classical finite flat group schemes, so are F_{S_{n_{0}}} and F′_{S_{n_{0}}} by classical flat descent theory. Note that the underlying scheme of S_{n_{0}} is artinian. Since the category of commutative finite flat group schemes over an artinian base is abelian, part (1) follows. To show part (2), we need to check that (fin/S)_{r} is closed under kernels, cokernels and extensions. The closedness under kernels is trivial. The closedness under extensions is given by Proposition A.1. We are left to show the closedness under cokernels. It suffices to show that, for any short exact sequence 0 → F_{1} → F_{2} → F_{3} → 0 with F_{1}, F_{2} ∈ (fin/S)_{r}, we must have F_{3} ∈ (fin/S)_{r}. Let 0 → F_{i}^{∘} → F_{i} → F_{i}^{et} → 0 be the connected-étale short exact sequence of F_{i}, then we have the following commutative diagram

5The proof of [Kat91, Prop. 2.7] hasn’t been given in the very preprint. One may refer to [Niz08, Cor. 2.16] for the proof, however the statement might have missed the quasi-compact assumption. Nevertheless the proof works under the quasi-compact assumption.

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\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & F_{1}^{∘} \ar[r] \ar[d] & F_{1} \ar[r] \ar[d] & F_{1}^{et} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & F_{2}^{∘} \ar[r] \ar[d] & F_{2} \ar[r] \ar[d] & F_{2}^{et} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & F_{3}^{∘} \ar[r] \ar[d] & F_{3} \ar[r] \ar[d] & F_{3}^{et} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

with exact rows and columns. Both F_{1}^{∘} and F_{2}^{∘} lie in (fin/S)_{c}, so is F_{3}^{∘}. Hence F_{3} ∈ (fin/S)_{r} by Proposition A.2. Now we show part (3). The closedness under subobjects is just Lemma 3.1. The closedness under quotient objects can be proven by a similar argument as in the proof of Lemma 3.1. The non-closedness is clear by Kato’s classification theorem Theorem A.3.

Lemma 3.2. Let M = [Y → G_{log}], M′ = [Y′ → G′_{log}] be two non-degenerate log 1-motives over S, (f_{−1}, f_{0}) : M → M′ a homomorphism of log 1-motives, and f_{c} : G → G′ the map induced by f_{0}. Let X (resp. X′) be the character group of the torus part T (resp. T′) of G (resp. G′), Q (resp. Q′) the sheaf Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)} (resp. Hom_{S_{fl}^{log}}(X′, 𝔾_{m,log}/𝔾_{m})^{(Y′)}), f_{l} : X′ → X

the map induced by f_{c}, and f̃_{d} : Q → Q′ the map induced by f_{l}. If f_{c} is an isogeny, then the map f̃: G_{log}^{(Y)} → G_{log}^{′(Y′)} induced by (f_{−1}, f_{0}) is surjective with kernel Ker(f_{c}), and the map f̃_{d} is bijective.

Proof. Since f_{c} is an isogeny, the map f_{l} is injective and of finite cokernel. We consider the following commutative diagram

\xymatrix{0 \ar[r] & G \ar[r] \ar[d]^{f_{c}} & G_{log}^{(Y)} \ar[r] \ar[d]^{f̃} & Q \ar[r] \ar[d]^{f̃_{d}} & 0 \\ 0 \ar[r] & G′ \ar[r] & G_{log}^{′(Y′)} \ar[r] & Q′ \ar[r] & 0}

with exact rows. To show the surjectivity of f̃, it is enough to show the surjectivity of f̃_{d}.

The induced map T_{log} → T′_{log} is surjective by Proposition 2.3 (5). Furthermore we have the surjectivity of the map

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) → Hom_{S_{fl}^{log}}(X′, 𝔾_{m,log}/𝔾_{m}).

Thus for any φ′ ∈ Q′, there exists some φ ∈ Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) mapped

to φ′, i.e. φ′ = φ ∘ f_{l}. In order to show the surjectivity of f̃_{d}, it suffices to show that φ ∈ Q. Let n be a positive integer killing X/f_{l}(X′), and let <, >: X × Y → 𝔾_{m,log}/𝔾_{m} and <, >′: X′ × Y′ → 𝔾_{m,log}/𝔾_{m} be the pairings associated to M and M′ respectively. Given any U ∈ (fs/S), u ∈ U, x ∈ Xū, there exists x′ ∈ X′ ; ū such that nx = f_{l}(x′). By the definition of Q′, there exist

y′_{u,x′,1}, y′_{u,x′,2} ∈ Y ′ ; ū such that

< x′, y′_{u,x′,1} >′ | φ′(x′) | < x′, y′_{u,x′,2} >′ .  (3.1)

The map f_{l} being injective with finite cokernel, together with the nondegeneracy of M and M′, forces f_{−1} to be injective with finite cokernel. If necessary we enlarge n such that it also kills the cokernel of f_{−1}. Then there exist y_{u,x′,1}, y_{u,x′,2} ∈ Yū such that ny′_{u,x′,1} = f_{−1}(y_{u,x′,1}) and ny′_{u,x′,2} = f_{−1}(y_{u,x′,2}). Raising the relation (3.1) to n-th power, we get a new relation

< x′, f_{−1}(y_{u,x′,1}) >′ | φ′(x′)^{n} | < x′, f_{−1}(y_{u,x′,2}) >′ .  (3.2)

Since < f_{l}(−), − >=< −, f_{−1}(−) >′, the relation (3.2) can be rewritten as

< x, y_{u,x′,1} >^{n} | φ(x)^{n^{2}} | < x, y_{u,x′,2} >^{n} .  (3.3)

By [KKN15, 18.10], there exist y_{1}, y_{2} ∈ Yū such that

< x, y_{1}^{n} > | < x, y_{u,x′,1} > and < x, y_{u,x′,2} > | < x, y_{2}^{n} > .

Therefore relation (3.3) gives another relation

< x, y_{1} >^{n^{2}} | φ(x)^{n^{2}} | < x, y_{2} >^{n^{2}} .  (3.4)

Removing the exponents from (3.4), we get < x, y_{1} > | φ(x) | < x, y_{2} >, hence φ ∈ Q. The injectivity of f̃_{d} follows from the injectivity of

Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m}) → Hom_{S_{fl}^{log}}(X′, 𝔾_{m,log}/𝔾_{m}).

The identification Ker(f_{c}) = Ker(f̃) follows from the injectivity of f̃_{d}.

Proposition 3.2. Let A be a log abelian variety over S, and F ∈ (fin/S)_{r} be a subsheaf of A. Then:

(1) F is an extension of objects of (fin/S)_{c}.

(2) F ∈ (fin/S)_{d}.

(3) The quotient A/F is also a log abelian variety over S.

Proof. Let n be a positive integer such that nF = 0, then we have F ⊂ A[n] := Ker(A ⟶^{×n} A). Since the map G ⟶^{n_{G}} G is an isogeny, we get a short exact sequence 0 → G[n] → A[n] → (Q/Y)[n] → 0 by diagram (2.10), where Q denotes the sheaf Hom(X, 𝔾_{m,log}/𝔾_{m})^{(Y)}. Since the map Q ⟶^{×n} Q is an isomorphism, we get (Q/Y)[n] ≅ Y/nY by diagram (2.12). We also have A[n] ∈ (fin/S)_{r} by [KKN15, Prop. 18.1 (1) and (2)]. Now let F′ be the kernel of the composition F ↪ A[n] → (Q/Y)[n] ≅ Y/nY, and F′′ the image of F in Y/nY, then we have a commutative diagram

\xymatrix{0 \ar[r] & F′ \ar[r] \ar@{^{(}->}[d] & F \ar[r] \ar@{^{(}->}[d] & F′′ \ar[r] \ar@{^{(}->}[d] & 0 \\ 0 \ar[r] & G[n] \ar[r] & A[n] \ar[r] & Y/nY \ar[r] & 0}

with exact rows and injective vertical homomorphisms. As a kernel of a homomorphism between two representable objects, F′ ∈ (fin/S)_{r}; as a subobject of Y/nY which is a classical finite étale group scheme, F′′ ∈ (fin/S)_{c}. Applying Lemma 3.1 to the inclusion F′ ⊂ G[n], we conclude F′ ∈ (fin/S)_{c}, hence part (1) is proven. Part (2) follows from part (1) and Proposition A.1. Now we show part (3). It suffices to find a polarisable log 1-motive such that A/F is isomorphic to its associated quotient. Consider the pullback E of G_{log}^{(Y)} along F ⊂ A

[figure: 0; Y; E; F; 0; 0; Y; G_{log}^{(Y)}; A; 0,]

and let E_{tor} be the torsion subsheaf of E, Y′ := E/E_{tor}. Since the sheaf

G_{log}^{(Y)}/G = Hom_{S_{fl}^{log}}(X, 𝔾_{m,log}/𝔾_{m})^{(Y)}

is torsion-free, E_{tor} maps into G. So we get E_{tor} = F′ and Y′/Y = F′′ ⊂ Y/nY, and Y′ is étale locally constant. Let G′ = G/E_{tor} = G/F′, the inclusion E ↪ G_{log}^{(Y)} ↪ G_{log} gives a homomorphism Y′ → G_{log}/F′ = G′_{log} by taking the

quotient by F′. In this way, we get a log 1-motive M′ := [Y′ ⟶^{u′} G′_{log}] together with a homomorphism (f_{−1}, f_{0}) : M := [Y ⟶^{u} G_{log}] → M′. By the construction of the homomorphism (f_{−1}, f_{0}), it is clear that the multiplication by n map n_{M} on M factors through (f_{−1}, f_{0}). Let (g_{−1}, g_{0}) : M′ → M be the homomorphism such that n_{M} = (g_{−1}, g_{0}) ∘ (f_{−1}, f_{0}), let (h_{−1}, h_{0}) : M → M^{∗} = [X → G_{log}^{∗}] be a polarisation of M, then (g_{−1}^{∗} ∘ h_{−1} ∘ g_{−1}, g_{0}^{∗} ∘ h_{0} ∘ g_{0}) gives rise to a polarisation of M′, where (g_{−1}^{∗}, g_{0}^{∗}) : M^{∗} → M^{′∗} = [X′ → G_{log}^{′∗}] is the dual of (g_{−1}, g_{0}). By [KKN08a, Thm. 3.4], the homomorphism (f_{−1}, f_{0}) gives rise to a homomorphism f : A → A′ of log abelian varieties with constant degeneration, where A′ is the associated log abelian variety of M′. By the diagram (2.9) and Lemma 3.2, it is easy to see that A′ = A/F.

Definition 3.1. Let A, A′ be two log abelian varieties over S. An isogeny from A to A′ is a homomorphism f from A to A′ such that f is surjective for the log flat topology and Ker(f) ∈ (fin/S)_{r}.

Remark 3.1. Let f : A → A′ be an isogeny between two log abelian varieties over S. By Proposition 3.2 (2), we have Ker(f) ∈ (fin/S)_{d}. Hence we can replace the condition Ker(f) ∈ (fin/S)_{r} by the a priori stronger condition Ker(f) ∈ (fin/S)_{d} in the definition of isogeny. We can also replace the condition Ker(f) ∈ (fin/S)_{r} by the a priori weaker condition Ker(f) ∈ (fin/S)_{f}, since Ker(f) is automatically representable as the kernel of f|_{A[n]} : A[n] → A′[n] for n big enough. Here we have used the fact A[n], A′[n] ∈ (fin/S)_{r}, see [KKN15, Prop. 18.1].

Example 3.1. By Proposition 3.2, a subsheaf F ∈ (fin/S)_{r} of a log abelian variety A over S gives an isogeny A → A/F of log abelian varieties.

Isogenies between abelian varieties can be defined by several equivalent conditions, some of which concern the dimension. Here we show the same thing happens for log abelian varieties over S. Recall that the dimension of a log abelian variety is defined to be the dimension of its semi-abelian part, see [KKN08a, 4.4].

Proposition 3.3. Let f : A → A′ be a homomorphism of log abelian varieties over S. Let M = [Y ⟶^{u} G_{log}] (resp. M′ = [Y′ ⟶^{u′} G′_{log}]) be the log 1-motive defining A (resp. A′), and f_{−1} and f_{c} the homomorphisms induced by f as in Proposition 2.6. Consider the following conditions:

(1) f is an isogeny;

(2) f is surjective for the log flat topology and dimA = dimA′;

(3) Ker(f) ∈ (fin/S)_{r} and dimA = dimA′;

(4) f_{c} is an isogeny and f_{−1} is injective of finite cokernel.

Then we have (2) ⇐ (1) ⇔ (3) ⇔ (4).

Proof. If f is an isogeny, we have Ker(f) ∈ (fin/S)_{r} and A′ = A/Ker(f). By the construction of A/Ker(f) as a log abelian variety in the proof of part (3) of Proposition 3.2, we have that f_{c} is an isogeny of semi-abelian varieties, hence dimG = dimG′. This shows that (1) implies both (2) and (3). Now we show that (3) implies (1). The condition Ker(f) ∈ (fin/S)_{r} implies that Ker(f_{c}) is a finite group scheme. Since dimG = dimA = dimA′ = dimG′, f_{c} is an isogeny. Let f̃: G_{log}^{(Y)} → G_{log}^{′(Y′)} be the homomorphism induced by f as in Proposition 2.6. By Lemma 3.2, f̃ is surjective. Then the surjectivity of f follows from the surjectivity of f̃. Hence f is an isogeny. At last, we show the equivalence between (3) and (4). Assuming (3), we must have that f_{c} is an isogeny. Let f̃_{d} : Q → Q′ be the homomorphism induced by f as in Proposition 2.6. Then f̃_{d} is bijective by Lemma 3.2. Applying snake lemma to diagrams (2.10) and (2.12), we get Ker(f_{−1}) = 0 and a short exact sequence 0 → Ker(f_{c}) → Ker(f) → Coker(f_{−1}) → 0. Hence (3) implies (4). Conversely, assuming (4), we have that f̃_{d} is bijective by Lemma 3.2. Again applying snake lemma to diagrams (2.10) and (2.12), we have a short exact sequence 0 → Ker(f_{c}) → Ker(f) → Coker(f_{−1}) → 0. Hence Ker(f) ∈ (fin/S)_{r} by part (2) of Proposition 3.1, so we get (3).

Remark 3.2. One might wonder if (2) implies (1) in Proposition 3.3. Note that the corresponding statement for abelian varieties holds. It is easy to see that the implication follows from the surjectivity of f_{c}. Unfortunately, it is not clear to the author how to deduce the surjectivity of f_{c} from the surjectivity of f.

Example 3.2. Let A be a log abelian variety over S. Let M = [Y → G_{log}] be the log 1-motive defining A, M^{∗} = [X → G_{log}^{∗}] the dual of M and (λ_{−1}, λ_{0}) : M → M^{∗} a polarisation, see [KKN08a, Def. 2.8] for the definition of polarisation. Then the map λ : A → A^{∗} induced by (λ_{−1}, λ_{0}) is an isogeny. One calls λ a polarisation of the log abelian variety A.

Proposition 3.4. Let A be a log abelian variety over S, let g be the dimension of A, and let n be a positive integer.

(1) The multiplication-by-n map n_{A} : A → A is an isogeny.

(2) The rank of A[n] := Ker(n_{A}) is n^{2g}.

(3) A[n] ∈ (fin/S)_{d}.

(4) Let (n_{A})^{∗} be the dual of the map n_{A}, then (n_{A})^{∗} = n_{A^{∗}}.

(5) If n is coprime to the characteristic of k, then Kummer étale locally on S, A[n] is isomorphic to (ℤ/nℤ)^{2g}.

Proof. By [KKN15, Prop. 18.1], we have A[n] ∈ (fin/S)_{r}, hence part (1) is a corollary of Proposition 3.3. For part (2) and part (5), we refer to [KKN15, Prop. 18.1]. Part (3) follows from part (1) and Remark 3.1. We are left with part (4). Let M = [Y → G_{log}] be the log 1-motive defining A, then n_{A} is the map induced by the map n_{M}. Since the dual of n_{M} is the map n_{M^{∗}}, where M^{∗} denote the dual of M, the dual of n_{A} is nothing but n_{A^{∗}}.

3.2 The dual short exact sequence

Recall that for an isogeny f : A → A′ between two abelian varieties over a field, we have that the dual f^{∗} of f is an isogeny with kernel (Ker(f))^{∗}. In this subsection we show that the same thing holds for log abelian varieties over S. Let f : A → A′ be an isogeny between two log abelian varieties over S, and let F be the kernel of f, then we get a short exact sequence 0 → F → A ⟶^{f} A′ → 0. Applying the functor Hom_{S_{fl}^{log}}(−, 𝔾_{m,log}) to this short exact sequence, we get a long exact sequence

→Hom_{S_{fl}^{log}}(A, 𝔾_{m,log}) → Hom_{S_{fl}^{log}}(F, 𝔾_{m,log}) → Ext_{S_{fl}^{log}}(A′, 𝔾_{m,log})
→Ext_{S_{fl}^{log}}(A, 𝔾_{m,log}).

(3.5)

By Theorem 2.2 (4), Hom_{S_{fl}^{log}}(A, 𝔾_{m,log}) = 0. By Theorem 2.2 (3), the map Ext_{S_{fl}^{log}}(A′, 𝔾_{m,log}) → Ext_{S_{fl}^{log}}(A, 𝔾_{m,log}) is just the map G_{log}^{′∗}/X′ → G_{log}^{∗}/X. The torsion-free nature of 𝔾_{m,log}/𝔾_{m} implies Hom_{S_{fl}^{log}}(F, 𝔾_{m,log}/𝔾_{m}) = 0, hence we have Hom_{S_{fl}^{log}}(F, 𝔾_{m,log}) = F^{∗}, where F^{∗} = Hom_{S_{fl}^{log}}(F, 𝔾_{m}) is the

Cartier dual of F (see Definition A.1). We have A^{∗} = G_{log}^{∗(X)}/X and A^{′∗} =

G_{log}^{′∗(X′)}/X′. And the map X′ → G_{log}^{′∗} having its image in G_{log}^{′∗(X′)} gives a short exact sequence

0 → A^{′∗} → G_{log}^{′∗}/X′ → R → 0,

where R denotes the quotient sheaf

Hom_{S_{fl}^{log}}(Y′, 𝔾_{m,log}/𝔾_{m})/Hom_{S_{fl}^{log}}(Y′, 𝔾_{m,log}/𝔾_{m})^{(X′)}.

Putting all these ingredients together, we get a commutative diagram

\xymatrix{ &  & 0 \ar[d] \\  &  & A^{′∗} \ar[r]^{f^{∗}} \ar@{^{(}->}[d] & A^{∗} \ar@{^{(}->}[d] \\ 0 \ar[r] & F^{∗} \ar[r] & G_{log}^{′∗}/X′ \ar[r] \ar[d] & G_{log}^{∗}/X \\  &  & R \ar[d] \\  &  & 0}  (3.6)

with exact rows and columns.

Lemma 3.3. The sheaf R is torsion-free.

Proof. We consider the following short exact sequence

0 → Q^{′∗} → Hom_{S_{fl}^{log}}(Y′, 𝔾_{m,log}/𝔾_{m}) → R → 0,

where Q^{′∗} denotes Hom_{S_{fl}^{log}}(Y′, 𝔾_{m,log}/𝔾_{m})^{(X′)}. To show R is torsion-free, it is enough to show that any section φ ∈ Hom_{S_{fl}^{log}}(Y′, 𝔾_{m,log}/𝔾_{m}), satisfying φ^{n} ∈ Q^{′∗}, actually lies in Q^{′∗}. For any U ∈ (fs/S), u ∈ U, y′ ∈ Y ′ ; ū, there exist

x′_{u,y′,1}, x′_{u,y′,2} ∈ X′ ; ū such that < x′_{u,y′,1}, y′ >′ | φ^{n}(y′) | < x′_{u,y′,2}, y′ >′. By

[KKN15, 18.10], there exist x_{1}, x_{2} ∈ X′ ; ū such that

< x_{1}, y′ >^{′n} | < x′_{u,y′,1}, y′ >′, < x′_{u,y′,2}, y′ >′ | < x_{2}, y′ >^{′n} .

Then we get a relation < x_{1}, y′ >^{′n} | φ^{n}(y′) | < x_{2}, y′ >^{′n}. Removing the exponents, we further get < x_{1}, y′ >′ | φ(y′) | < x_{2}, y′ >′, which shows that φ ∈ Q^{′∗}.

Theorem 3.1. We have a canonical short exact sequence

0 → F^{∗} → A^{′∗} ⟶^{f^{∗}} A^{∗} → 0,

in other words, f^{∗} is an isogeny with kernel the Cartier dual of F.

Proof. Since F ∈ (fin/S)_{d} by Proposition 3.2 (2), F^{∗} ∈ (fin/S)_{d}. The sheaf R is torsion-free by Lemma 3.3, hence we have Hom_{S_{fl}^{log}}(F^{∗}, R) = 0. It follows that the map F^{∗} ↪ G_{log}^{′∗}/X′ in the diagram (3.6) factors through A^{′∗}. Furthermore, F^{∗} is actually the kernel of f^{∗}. Since dimA^{′∗} = dimA^{∗}, f^{∗} is an isogeny by Proposition 3.3.

3.3 The Poincaré complete reducibility theorem

The Poincaré complete reducibility theorem for abelian varieties plays a very important role in the theory of abelian varieties. In this subsection, we formulate a Poincaré complete reducibility theorem for log abelian varieties admitting a polarisation over S.

Lemma 3.4. Let M = [Y → G_{log}] be a log 1-motive over S with a polarisation (λ_{−1}, λ_{0}) : M = [Y → G_{log}] → [X → G_{log}^{∗}] = M^{∗}, and let A be the log abelian variety associated to M. Let M_{1} = [Y_{1} → G_{1log}] be another log 1-motive with rank_{ℤ}Y_{1} = rank_{ℤ}X_{1}, where X_{1} is the character group of the torus part of G_{1}. Let (i_{−1}, i_{0}) : M_{1} → M be a homomorphism of log 1-motives and i_{c} : G_{1} → G the homomorphism corresponding to i_{0}, and let γ_{−1} := i_{−1}^{∗} ∘ λ_{−1} ∘ i_{−1}, γ_{0} := i_{0}^{∗} ∘ λ_{0} ∘ i_{0}. Suppose that i_{−1} is injective and i_{c} has finite kernel, then we have the following.

(1) The map (γ_{−1}, γ_{0}) : M_{1} = [Y_{1} → G_{1log}] → [X_{1} → G_{1log}^{∗}] = M_{1}^{∗} is a polarisation.

(2) Let A_{1} be the log abelian variety associated to M_{1} and i : A_{1} → A the map induced by (i_{−1}, i_{0}), then we have Ker(i) ∈ (fin/S)_{r}.

Proof. To prove part (1), we need to verify the conditions (a), (b), (c) and (d) of [KKN08a, Def. 2.8]. We have the following commutative diagram

\xymatrix{Y_{1} \ar[r]^{i_{−1}} \ar[d] & Y \ar[r]^{λ_{−1}} \ar[d] & X \ar[r]^{i_{−1}^{∗}} \ar[d] & X_{1} \ar[d] \\ G_{1log} \ar[r]^{i_{0}} & G_{log} \ar[r]^{λ_{0}} & G_{log}^{∗} \ar[r]^{i_{0}^{∗}} & G_{1log}^{∗}.}

We also have the following commutative diagram

\xymatrix{0 \ar[r] & T_{1} \ar[r] \ar[d]^{i_{t}} & G_{1} \ar[r] \ar[d]^{i_{c}} & B_{1} \ar[r] \ar[d]^{i_{ab}} & 0 \\ 0 \ar[r] & T \ar[r] \ar[d]^{λ_{t}} & G \ar[r] \ar[d]^{λ_{c}} & B \ar[r] \ar[d]^{λ_{ab}} & 0 \\ 0 \ar[r] & T^{∗} \ar[r] \ar[d]^{i_{t}^{∗}} & G^{∗} \ar[r] \ar[d]^{i_{c}^{∗}} & B^{∗} \ar[r] \ar[d]^{i_{ab}^{∗}} & 0 \\ 0 \ar[r] & T_{1}^{∗} \ar[r] & G_{1}^{∗} \ar[r] & B_{1}^{∗} \ar[r] & 0}

with exact rows, where the rows are the torus and abelian variety decomposition exact sequences of semi-abelian varieties. By the construction of the duality theory of log 1-motives, we have that i_{−1}^{∗} (resp. i_{t}^{∗}) is induced by i_{t} (resp. i_{−1}). Then condition (d) follows. Let <, >: X × Y → 𝔾_{m,log}/𝔾_{m} (resp. <, >_{1}: X_{1} × Y_{1} → 𝔾_{m,log}/𝔾_{m}) be the pairing associated to M (resp. M_{1}). For y ∈ Y_{1s̄}\{0}, where s denotes the only point of S, we have i_{−1}(y) ≠ 0 by the injectivity of i_{−1}. Hence we have

< γ_{−1}(y), y >_{1s̄}=< λ_{−1} ∘ i_{−1}(y), i_{−1}(y) >s̄∈ (M_{S,s̄}/O_{S,s̄}^{×})\{1}

which gives condition (c). For condition (b), it suffices to show the injectivity of γ_{−1} because of rank_{ℤ}Y_{1} = rank_{ℤ}X_{1}. But this already follows from condition (c). At last we show condition (a). Since i_{c} has finite kernel, i_{ab} must have finite kernel by diagram chasing. Hence i_{ab} is a finite morphism. We want to show γ_{ab} = i_{ab}^{∗} ∘ λ_{ab} ∘ i_{ab} is a polarisation of B_{1}. Without loss of generality, we may assume λ_{ab} = φ_{L} for an ample line bundle L on B, where φ_{L} is defined by φ_{L}(b) := t_{b}^{∗}L⊗L^{−1} for b ∈ B. Then we have γ_{ab} = i_{ab}^{∗}∘φ_{L}∘i_{ab} = φ_{i_{ab}^{∗}L}. Clearly i_{ab}^{∗}L is ample, so γ_{ab} is a polarisation on B_{1}. This finishes the verification of condition (a). Hence (γ_{−1}, γ_{0}) is a polarisation of M_{1}. Now we prove part (2). By Proposition 2.6, the homomorphism i induces the following two commutative diagrams

\xymatrix{0 \ar[r] & G_{1} \ar[r] \ar[d]^{i_{c}} & A_{1} \ar[r] \ar[d]^{i} & Q_{1}/Y_{1} \ar[r] \ar[d]^{i_{d}} & 0 \\ 0 \ar[r] & G \ar[r] & A \ar[r] & Q/Y \ar[r] & 0}

and

\xymatrix{0 \ar[r] & Y_{1} \ar[r] \ar[d]^{i_{−1}} & Q_{1} \ar[r] \ar[d]^{ĩ_{d}} & Q_{1}/Y_{1} \ar[r] \ar[d]^{i_{d}} & 0 \\ 0 \ar[r] & Y \ar[r] & Q \ar[r] & Q/Y \ar[r] & 0}

with exact rows. Since i_{c} has finite kernel, i_{t} has finite kernel too. Hence ĩ_{d} is injective by part (3) Proposition 2.3. Applying snake lemma to the above two diagrams, we get exact sequences 0 → Ker(i_{c}) → Ker(i) → Ker(i_{d}) → Coker(i_{c}) and 0 → Ker(i_{d}) ⟶^{α} Y/Y_{1}.

Claim. The map α maps Ker(i_{d}) onto the torsion part (Y/Y_{1})_{tor} of Y/Y_{1}.

Proof of CLAIM. First we show (Y/Y_{1})_{tor} ⊂ Im(α). Let y ∈ Y be such that the quotient class ȳ represented by y lies in (Y/Y_{1})_{tor}. We have ny = i_{−1}(y_{1}) for some n ∈ ℕ and y_{1} ∈ Y_{1}. The map Q_{1} ⟶^{×n} Q_{1} is bijective by Lemma 3.2, hence there exists φ_{1} ∈ Q_{1} such that φ_{1}^{n} =< −, y_{1} >_{1}. It follows then ĩ_{d}(φ_{1})^{n} =< −, y >^{n}. Since the map Q ⟶^{×n} Q is bijective by Lemma 3.2, we get ĩ_{d}(φ_{1}) =< −, y >. Whence ȳ = α(φ̄_{1}), where φ̄_{1} denotes the quotient class of φ_{1} in Q_{1}/Y_{1}. Now we show the converse inclusion. It suffices to show, for φ_{1} ∈ Q_{1} with ĩ_{d}(φ_{1}) =< −, y > for some y ∈ Y, ȳ ∈ (Y/Y_{1})_{tor}. Since γ_{−1} = i_{−1}^{∗} ∘ λ_{−1} ∘ i_{−1} is of finite cokernel by part (1), there exists some positive integer m such that mi_{−1}^{∗} ∘ λ_{−1}(y) = i_{−1}^{∗} ∘ λ_{−1} ∘ i_{−1}(y_{1}) for some y_{1} ∈ Y_{1}. Let w := my − i_{−1}(y_{1}), and we have i_{−1}^{∗} ∘ λ_{−1}(w) = 0. Hence

< λ_{−1}(w), w > =< λ_{−1}(w), my > − < λ_{−1}(w), i_{−1}(y_{1}) >
= ĩ_{d}(φ_{1})(λ_{−1}(w))^{m}− < i_{−1}^{∗} ∘ λ_{−1}(w), y_{1} >_{1}
= φ_{1}(i_{−1}^{∗} ∘ λ_{−1}(w))^{m}− < 0, y_{1} >_{1}
= 0.

Since (λ_{−1}, λ_{0}) is a polarisation, we must have w = 0. This shows ȳ ∈ (Y/Y_{1})_{tor}.

By the above claim, we get an exact sequence

0 → Ker(i_{c}) → Ker(i) → (Y/Y_{1})_{tor} → Coker(i_{c}).

Let F be the kernel of (Y/Y_{1})_{tor} → Coker(i_{c}). We have F ∈ (fin/S)_{c}. Then the short exact sequence 0 → Ker(i_{c}) → Ker(i) → F → 0 forces Ker(i) ∈ (fin/S)_{r} by Proposition A.1.

Proposition 3.5. Let f : A → A′ be a homomorphism of log abelian varieties over S. Then there exists a log abelian subvariety j : A_{1} ↪ A such that f|_{A_{1}} = 0, and A_{1} possesses the following universal property: for any homomorphism g : A_{2} → A of log abelian varieties over S such that f ∘g = 0, g factors through A_{1} uniquely. In other words, A_{1} is the kernel of f in the category of log abelian varieties over S.

Proof. Let M = [Y → G_{log}] (resp. M′ = [Y′ → G′_{log}]) be the log 1-motive defining A (resp. A′), and let (f_{−1}, f_{0}) : M → M′ be the homomorphism defining f. We first construct the log 1-motive defining A_{1}. The homomorphism f induces a homomorphism f_{c} : G → G′. Let G_{1} be the reduced neutral component of Ker(f_{c}), then G_{1} is a semi-abelian variety by [Bri15, Rem. 5.4.7. (iii)]. Let j_{c} be the inclusion G_{1} ⊂ G, and let j_{0} : G_{1log} → G_{log} be the map induced by j_{c}. We consider the following commutative diagram

\xymatrix{0 \ar[r] & G_{1} \ar[r] \ar[d]^{j_{c}} & G_{1log} \ar[r] \ar[d]^{j_{0}} & Hom_{S_{fl}^{log}}(X_{1}, 𝔾̄_{m,log}) \ar[r] \ar[d]^{j̄_{0}} & 0 \\ 0 \ar[r] & G \ar[r] \ar[d]^{f_{c}} & G_{log} \ar[r] \ar[d]^{f_{0}} & Hom_{S_{fl}^{log}}(X, 𝔾̄_{m,log}) \ar[r] \ar[d]^{f̄_{0}} & 0 \\ 0 \ar[r] & G′ \ar[r] & G′_{log} \ar[r] & Hom_{S_{fl}^{log}}(X′, 𝔾̄_{m,log}) \ar[r] & 0}

with exact rows, where X_{1} (resp. X, resp. X′) is the character group of the torus part T_{1} (resp. T, resp. T′) of G_{1} (resp. G, resp. G′). By part (3) of Proposition 2.3, we have

Ker(f̄_{0}) = Hom_{S_{fl}^{log}}(Coker(f_{l}), 𝔾̄_{m,log}),

where f_{l} : X′ → X is the map induced by the torus part f_{t} : T → T′ of f_{c}. The above diagram gives rise to another commutative diagram

\xymatrix{0 \ar[r] & G_{1} \ar[r] \ar@{^{(}->}[d]^{j_{c}} & G_{1log} \ar[r] \ar[d]^{j_{0}} & Hom_{S_{fl}^{log}}(X_{1}, 𝔾̄_{m,log}) \ar[r] \ar[d]^{j̄_{0}} & 0 \\ 0 \ar[r] & Ker(f_{c}) \ar[r] & Ker(f_{0}) \ar[r]^{α} & Hom_{S_{fl}^{log}}(Coker(f_{l}), 𝔾̄_{m,log})}

with exact rows. Since the map Coker(f_{l}) → X_{1} induced by G_{1} → G → G′ is an isomorphism up to torsion, the map j̄_{0} in the above diagram is an isomorphism.

Then we have that α is surjective. By the snake lemma, we get that G_{1log} is canonically embedded into Ker(f_{0}) with finite cokernel Ker(f_{c})/G_{1}. Now let Y_{1} be the pullback of G_{1log} along Ker(f_{−1}) ↪ Ker(f_{0}), then rank_{ℤ}Y_{1} = rank_{ℤ}Ker(f_{−1}). Let j_{−1} be the canonical inclusion Y_{1} ⊂ Y, we get a log 1motive M_{1} := [Y_{1} → G_{1log}] together with a canonical map (j_{−1}, j_{0}) : M_{1} → M. In order to apply Lemma 3.4, we need to show rank_{ℤ}Y_{1} = rank_{ℤ}X_{1}. Without loss of generality, we may assume that both M and M′ admit a polarisation, in particular we have homomorphisms h : Y → X and h′ : Y′ → X′ which are both injective with finite cokernel. Consider the following diagram

Coker(f_{−1}^{∗}) × Ker(f_{−1})
↑d  ↓a
X  ×  Y  ⟶^{<,>} 𝔾_{m,log}/𝔾_{m}
↑f_{−1}^{∗}  ↓f_{−1}  =
X′  ×  Y′  ⟶^{<,>′} 𝔾_{m,log}/𝔾_{m}
↑c  ↓b
Ker(f_{−1}^{∗}) × Coker(f_{−1})

in which the parings <, > and <, >′ are compatible with the maps f_{−1}^{∗} and f_{−1}, we have the following relations

(1) the composition Ker(f_{−1}) ⟶^{a} Y ⟶^{h} X ⟶^{d} Coker(f_{−1}^{∗}) is injective, whence rank_{ℤ}Ker(f_{−1}) ≤ rank_{ℤ}Coker(f_{−1}^{∗});

(2) the composition Ker(f_{−1}^{∗})⊗ℚ ⟶^{c_{ℚ}} X′ ⊗ℚ ⟶^{h_{ℚ}^{′−1}} Y′ ⊗ℚ ⟶^{b_{ℚ}} Coker(f_{−1})⊗ℚ

is injective, whence rank_{ℤ}Ker(f_{−1}^{∗}) ≤ rank_{ℤ}Coker(f_{−1});

(3) rank_{ℤ}X = rank_{ℤ}Y, rank_{ℤ}X′ = rank_{ℤ}Y′;

(4) rank_{ℤ}Ker(f_{−1}) − rank_{ℤ}Coker(f_{−1}) = rank_{ℤ}Y − rank_{ℤ}Y′;

(5) rank_{ℤ}Coker(f_{−1}^{∗}) − rank_{ℤ}Ker(f_{−1}^{∗}) = rank_{ℤ}X − rank_{ℤ}X′.

The relations (3), (4) and (5) are trivial. For y ∈ Ker(f_{−1}) such that d∘h(y) = 0, we have h(y) = f_{−1}^{∗}(x′) for some x′ ∈ X′, hence

0 =< x′, f_{−1}(y) >′=< f_{−1}^{∗}(x′), y >=< h(y), y > .

This implies y = 0, hence relation (1). Relation (2) can be shown by a similar argument. Now these five relations together force rank_{ℤ}Ker(f_{−1}) = rank_{ℤ}Coker(f_{−1}^{∗}), so we get rank_{ℤ}Y_{1} = rank_{ℤ}X_{1}. Applying Lemma 3.4 to M_{1} (if necessary we take base change to k̄ in order to get a polarisation on M), we have that M_{1} defines a log abelian variety A_{1} and (j_{−1}, j_{0}) gives a homomorphism j : A_{1} → A. We leave the proof of the injectivity of j to Lemma 3.5.

Now we are left with checking the universal property. Let

(g_{−1}, g_{0}) : M_{2} = [Y_{2} → G_{2log}] → M

be the homomorphism defining g. By the equivalence of categories from Theorem 2.4, we have that f∘g = 0 implies f_{−1}∘g_{−1} = f_{0}∘g_{0} = f_{c}∘g_{c} = 0. Hence the map g_{c} factors through the map j_{c} uniquely, further the map g_{0} factors through j_{0} uniquely. The equality f_{−1} ∘g_{−1} = 0 implies that g_{−1}(Y_{2}) ⊂ Ker(f_{−1}). Since Y_{1} is defined as the pullback of G_{1log} along Ker(f_{−1}) ↪ Ker(f_{0}), the homomorphism Y_{2} → Ker(f_{−1}) factors through Y_{1} uniquely. It follows that (g_{−1}, g_{0}) factors through (j_{−1}, j_{0}) uniquely, and g factors through j uniquely.

Lemma 3.5. The homomorphism j : A_{1} → A in the proof of Proposition 3.5 is injective.

Proof. Let the notation be as in the proof of Proposition 3.5. We have the following commutative diagram

\xymatrix{Y_{1} \ar@{^{(}->}[r] \ar@{^{(}->}[d] & Ker(f_{−1}) \ar@{^{(}->}[r] \ar@{^{(}->}[d] & Y \ar[r]^{f_{−1}} \ar@{^{(}->}[d] & Y′ \ar@{^{(}->}[d] \\ G_{1log} \ar@{^{(}->}[r] & Ker(f_{0}) \ar@{^{(}->}[r] & G_{log} \ar[r]^{f_{0}} & G′_{1log}.}

Since the left square is a pullback diagram, we have that the canonical map G_{1log}/Y_{1} → Ker(f_{0})/Ker(f_{−1}) is injective. By diagram chasing, the canonical map Ker(f_{0})/Ker(f_{−1}) → G_{log}/Y is also injective. Hence the canonical map G_{1log}/Y_{1} → G_{log}/Y is injective. Then the injectivity of j follows.

Lemma 3.6. Let i : A_{1} ↪ A be an inclusion of log abelian varieties over S. Let (i_{−1}, i_{0}) : M_{1} = [Y_{1} → G_{1log}] → [Y → G_{log}] = M be the homomorphism of log 1-motives defining i, and let i_{c} : G_{1} → G be the homomorphism of semi-abelian varieties induced by i_{0}. Then i_{−1}, i_{0} and i_{c} are all injective.

Proof. The injectivity of i_{c} follows from the injectivity of i by diagram (2.10). The injectivity of i_{0} follows from the injectivity of i_{c} by part (4) of Proposition 2.3. The injectivity of i_{−1} follows from that of i_{0} by diagram (2.9).

Theorem 3.2 (Poincaré complete reducibility theorem). Let A be a log abelian variety over S with a polarisation λ : A → A^{∗}, and A_{1} a log abelian subvariety of A. Then there is another log abelian subvariety A_{2} such that A_{1} × A_{2} is isogenous to A.

Proof. Let M = [Y → G_{log}] and M_{1} = [Y_{1} → G_{1log}] be the log 1-motives defining A and A_{1} respectively. Let i be the inclusion A_{1} ⊂ A, let i^{∗} be the dual of i, and let f = i^{∗} ∘ λ. Let (f_{−1}, f_{0}) : M → M_{1}^{∗} be the homomorphism defining f. By Proposition 2.6, we have a commutative diagram

\xymatrix{0 \ar[r] & G \ar[r] \ar[d]_{λ_{c}} \ar@{.>}[dd]^{f_{c}} & A \ar[r] \ar[d]_{λ} \ar@{.>}[dd]^{f} & Q/Y \ar[r] \ar[d]_{λ_{d}} \ar@{.>}[dd]^{f_{d}} & 0 \\ 0 \ar[r] & G^{∗} \ar[r] \ar[d]_{i_{c}^{∗}} & A^{∗} \ar[r] \ar[d]_{i^{∗}} & Q^{∗}/X \ar[r] \ar[d]_{i_{d}^{∗}} & 0 \\ 0 \ar[r] & G_{1}^{∗} \ar[r] & A_{1}^{∗} \ar[r] & Q_{1}^{∗}/X_{1} \ar[r] & 0}

with exact rows. Firstly we study the homomorphism f_{c} via λ_{c} and i_{c}^{∗}. By Example 3.2, λ is an isogeny, hence Ker(λ) ∈ (fin/S)_{r} and dimG = dimG^{∗} by Proposition 3.3. So λ_{c} is an isogeny. The construction of i_{c}^{∗} gives a commutative diagram

\xymatrix{0 \ar[r] & T^{∗} \ar[r] \ar[d]^{i_{t}^{∗}} & G^{∗} \ar[r] \ar[d]^{i_{c}^{∗}} & B^{∗} \ar[r] \ar[d]^{i_{ab}^{∗}} & 0 \\ 0 \ar[r] & T_{1}^{∗} \ar[r] & G_{1}^{∗} \ar[r] & B_{1}^{∗} \ar[r] & 0}

with exact rows, where the map T^{∗} → T_{1}^{∗} is induced by i_{−1} : Y_{1} → Y and the map B^{∗} → B_{1}^{∗} is induced by the abelian variety part i_{ab} of i_{c}. The injectivity of i implies that i_{c} and i_{−1} are both injective by Lemma 3.6. It follows that i_{t}^{∗} is surjective. The injectivity of i_{c} implies that i_{ab} : B_{1} → B has finite kernel. Hence i_{ab}^{∗} : B^{∗} → B_{1}^{∗} is surjective, and so is i_{c}^{∗}. Then the surjectivity of f_{c} follows. Let j : A_{2} ↪ A be the kernel of f in the category of log abelian varieties guaranteed by Proposition 3.5. The proof of Proposition 3.5 tells us that the log abelian subvariety A_{2} could come from a log 1-motive M_{2} = [Y_{2} → G_{2log}] and a homomorphism (j_{−1}, j_{0}) : M_{2} → M, such that both j_{−1} and j_{0} are injective, and the semi-abelian variety G_{2} underlying G_{2log} is the reduced neutral component of Ker(f_{c}). Let Y′ := Y_{1} × Y_{2}, G′ := G_{1} × G_{2}, then we have a natural log 1-motive M′ = [Y′ → G′_{log}] and a homomorphism (α_{−1}, α_{0}) : M′ → M, where α_{−1} is the map Y_{1} × Y_{2} → Y, (y_{1}, y_{2}) ↦ y_{1} + y_{2}, and α_{0} is the map induced by α_{c} : G_{1} × G_{2} → G, (g_{1}, g_{2}) ↦ g_{1} + g_{2}. We claim that α_{−1} is injective and of finite cokernel. Note that (γ_{−1}, γ_{0}) := (i_{−1}^{∗} ∘ λ_{−1} ∘ i_{−1}, i_{0}^{∗} ∘ λ_{0} ∘ i_{0}) is a polarisation on M_{1} by Lemma 3.4. For y ∈ Y_{1s̄} ∩ Y_{2s̄}, γ_{−1}(y) = f_{−1}(y) = 0, hence 0 =< 0, y >_{1s̄}=< γ_{−1}(y), y >_{1s̄} implies y = 0, and Ker(α_{−1}) = Y_{1} ∩ Y_{2} = 0. Hence α_{−1} has finite cokernel due to rank reason. We also claim that the map α_{c} is an isogeny. Let γ_{c} : G_{1} → G_{1}^{∗} be the homomorphism corresponding to γ_{0}, γ_{ab} the abelian part of γ_{c}, and γ_{t} the torus part of γ_{c}. Since (γ_{−1}, γ_{0}) is a polarisation, γ_{ab} is a polarisation and γ_{−1} is injective and of finite cokernel. It follows that both γ_{ab} and γ_{t} are isogenies. Whence γ_{c} = i_{c}^{∗} ∘ λ_{c} ∘ i_{c} is also an isogeny. Hence for any g ∈ G, there exists g_{1} ∈ G_{1} such that i_{c}^{∗} ∘ λ_{c} ∘ i_{c}(g_{1}) = i_{c}^{∗} ∘ λ_{c}(g), it follows then g − i_{c}(g_{1}) ∈ Ker(f_{c}). Since Ker(f_{c})/G_{2} is a finite group scheme, the map α_{c} is an isogeny. Note that (α_{−1}, α_{0}) induces a homomorphism α : A_{1} × A_{2} → A of log abelian varieties. By the equivalence (1) ⇔ (4) of Proposition 3.3, we deduce that α is an isogeny.

Remark 3.3. Since abelian varieties over a field are always projective, they carry an ample line bundle, hence they are always polarisable. For log abelian varieties over a log point, they admit a polarisation after base change to the algebraic closure of the base field by definition. However it is not clear to the author if they actually carry a polarisation over the base log point. But he does think, over a log point, log abelian variety admitting a polarisation serves as the right counterpart of abelian variety for at least two reasons. Firstly, the canonical 1-parameter log abelian variety degeneration ([Zha14]) of an abelian variety transports a polarisation of the generic fibre to the special fibre. In other words the special fibre (which is a log abelian variety over a log point) as the degeneration of the generic fibre (which is an abelian variety over a trivial log point) is necessarily polarisable. Secondly, a polarisation is needed in the proof of Poincaré complete reducibility theorem (see Theorem 3.2), and we know that Poincaré complete reducibility theorem for abelian varieties plays a very important role in the theory of abelian varieties.

Definition 3.2. Let A be a log abelian variety over S, a log abelian subvariety of A is a subsheaf of A which is also a log abelian variety. The log abelian variety A is simple if it has no non-zero proper log abelian subvariety. In other words, if A_{1} is a log abelian variety properly contained in A, then A_{1} is zero.

Lemma 3.7. Let f : A → A′ be a non-zero homomorphism between log abelian varieties.

(1) If both A and A′ are simple, then f is an isogeny.

(2) If f is an isogeny, then there exists an isogeny g : A′ → A and a positive integer n such that g ∘ f = n_{A}.

Proof. We prove part (1) first. Let A_{1} be the kernel of f in the category of log abelian varieties, see Proposition 3.5 for the construction of A_{1}. Since A is simple, A_{1} has to be zero. Let F be the kernel of f in the category of sheaves of abelian groups, then F lies in (fin/S)_{r} by part (2) of Lemma 3.4. Then A/F gives rise to a log abelian subvariety of A′ by part (3) of Proposition 3.2. We must have A′ = A/F by the simplicity of A′. It follows that f is an isogeny. This shows part (1). Now we show part (2). Let n be a positive integer which kills F. Then there exists an epimorphism g : A′ → A such that g ∘ f = n_{A}. The kernel-cokernel exact sequence gives a short exact sequence 0 → F → A[n] → Ker(g) → 0. We get Ker(g) ∈ (fin/S)_{r} by part (2) of Proposition 3.1. Hence g is an isogeny.

Corollary 3.1. Let A be a log abelian variety over S admitting a polarisation. Then A is isogenous to a product A_{1}^{n_{1}} × ⋯ × A_{r}^{n_{r}}, where the A_{i}’s are simple log abelian varieties and not isogenous to each other. The isogeny type of the A_{i} and the integers n_{i}’s are uniquely determined.

Proof. If A is simple, there is nothing to prove. Otherwise, by Theorem 3.2, there exists nonzero log abelian subvarieties A′ and A′′ of A such that A is isogeneous to A′ × A′′. By Lemma 3.4 and Lemma 3.6, both A′ and A′′ admit polarisations. Hence applying Theorem 3.2 repeatedly, we have that A is isogenous to a product A_{1}^{n_{1}} × ⋯ × A_{r}^{n_{r}} for some simple log abelian varieties A_{i} and some positive integers n_{i}. By part (1) of Lemma 3.7, such a decomposition is unique up to isogeny.

Definition 3.3. Let A, A′ be two log abelian varieties over S, we abbreviate Hom_{S_{fl}^{log}}(A, A′) = Hom_{S_{Ét}^{cl}}(A, A′) as Hom(A, A′). We define Hom^{0}(A, A′) as

Hom(A, A′)⊗_{ℤ} ℚ, and End^{0}(A) as End(A)⊗_{ℤ} ℚ = Hom(A, A)⊗_{ℤ} ℚ. We define the category LAV_{S}^{0} of log abelian varieties up to isogeny over S, by localising the category LAV_{S} of log abelian varieties over S at the class of isogenies.

Corollary 3.2. Let A be a log abelian variety over S admitting a polarisation. If A is simple, the ring End^{0}(A) is a division ring. In general, if A is isogenous to A_{1}^{n_{1}} × ⋯ × A_{r}^{n_{r}} with A_{i} simple and not isogenous to each other, and D_{i} = End^{0}(A_{i}), then End^{0}(A) = M_{n_{1}}(D_{1}) × ⋯ × M_{n_{r}}(D_{r}).

Proof. For A simple, let f be a nonzero endomorphism of A. Then f is an isogeny by part (1) of Lemma 3.7. By part (2) of Lemma 3.7, f is invertible in the ring End^{0}(A). Hence the ring End^{0}(A) is a division ring. By part (1) of Lemma 3.7, we have Hom^{0}(A, A′) = 0 for two non-isogenous simple log abelian varieties. Hence the second part follows.

Lemma 3.8. The abelian group Hom(A, A′) is torsion-free.

Proof. Let f ∈ Hom(A, A′) such that nf = 0 for some positive integer n. Since 0 = nf = f ∘ n_{A} and n_{A} is surjective, f must be zero. Hence Hom(A, A′) is torsion-free.

Definition 3.4. (1) Let f : A → A′ be an isogeny between two log abelian varieties over S. The degree deg(f) of f is defined to be the rank of the finite log group object Ker(f). By convention, if f is not an isogeny, we let deg(f) = 0.

(2) Let f_{−1} : Y → Y′ be a monomorphism with finite cokernel between two étale locally finite rank free constant sheaf, the degree deg(f_{−1}) of f_{−1} is defined to be the determinant of f_{−1}. By convention, if f is not injective of finite cokernel, we let deg(f) = 0.

(3) Let f_{c} : G → G′ be an isogeny between semi-abelian varieties, the degree deg(f_{c}) of f_{c} is defined to be the rank of the finite group scheme Ker(f_{c}). By convention, if f_{c} is not an isogeny, we let deg(f_{c}) = 0.

Lemma 3.9. Let f : A → A be a homomorphism between two log abelian varieties over S. Let f_{−1} : Y → Y and f_{c} : G → G be the homomorphisms induced by f as in Proposition 2.6, and let f_{t} : T → T and f_{ab} : B → B be the homomorphisms induced by f_{c} on torus parts and abelian variety parts respectively. Then:

(1) deg(f) = deg(f_{−1})deg(f_{t})deg(f_{ab});

(2) let g : A → A be another homomorphism, and let h = f + g, then h_{−1} = f_{−1} + g_{−1}, h_{t} = f_{t} + g_{t} and h_{ab} = f_{ab} + g_{ab}.

Proof. Part (2) is obvious. We only need to show part (1). If f is not an isogeny, then deg(f_{−1})deg(f_{t})deg(f_{ab}) = 0 = deg(f) by Proposition 3.3. Now we suppose that f is an isogeny, so are f_{c}, f_{t} and f_{ab}. Also we have f_{−1} is injective and of finite cokernel. By diagram (2.10), we get a short exact sequence

0 → Ker(f_{c}) → Ker(f) → Ker(f_{d}) → 0.

Similarly, we have another short exact sequence

0 → Ker(f_{t}) → Ker(f_{c}) → Ker(f_{ab}) → 0.

By diagram (2.12), we get Ker(f_{d}) ≅ Coker(f_{−1}). Then

deg(f) = deg(f_{−1})deg(f_{t})deg(f_{ab}).

Theorem 3.3. The function f ↦ deg(f) on End(A) extends to a homogeneous polynomial function of degree 2g on End^{0}(A), where g is the dimension of A.

Proof. Since for any f ∈ End(A) and n ∈ ℤ,

deg(nf) = deg(n_{A}) · deg(f) = n^{2g} · deg(f),

it suffices to show that for f, g ∈ End(A), the function P (n) = deg(nf + g) is a polynomial function. By Lemma 3.9, we are reduced to show the functions deg(nf_{t}+g_{t}), deg(nf_{ab}+g_{ab}) and deg(nf_{−1}+g_{−1}) are all polynomial functions. The case for deg(nf_{ab} + g_{ab}) is a standard result for abelian varieties, see [Mum70, §19, Thm. 2]. And deg(nf_{−1} + g_{−1}) as a determinant function is clearly a polynomial function. The case for deg(nf_{t} + g_{t}) is reduced to the case for deg(nf_{−1} + g_{−1}) by taking the character groups of the tori.

Definition 3.5. Let l be a prime number which is coprime to the characteristic of k. The l-adic Tate module of A is defined to be

T_{l}(A)_{s̄(két)} = \underleftarrow{lim}_{n} A[l^{n}]_{s̄(két)},

where s̄(két) denotes a log geometric point of S for the log étale topology. Here we use the notation s̄(két) for the sake of coherence with [KKN15, 18.9], and the log étale topology is called the Kummer étale topology there.

Let π_{1}^{log} be the log fundamental group of S. By Proposition 3.4, we have T_{l}(A)_{s̄(két)} is a free ℤ_{l}-module of rank 2g endowed with a continuous π_{1}^{log}- action. Any homomorphism f : A → A′ induces a homomorphism

T_{l}(f) : T_{l}(A)_{s̄(két)} → T_{l}(A′)_{s̄(két)}

which is π_{1}^{log}-equivariant. It follows that we have a functor

T_{l} : LAV_{S} ⟶ (π_{1}^{log}, ℤ_{l})−Mod

from the category of log abelian varieties over the log point S to the category of finite rank ℤ_{l}-modules with continuous π_{1}^{log}-action. In particular, the functor T_{l} gives rise to a homomorphism

Hom(A, A′) → Hom_{(π_{1}^{log},ℤ_{l})−Mod}(T_{l}(A)_{s̄(két)}, T_{l}(A′)_{s̄(két)}).

The latter is clearly a ℤ_{l}-submodule of Hom_{ℤ_{l}}(T_{l}(A)_{s̄(két)}, T_{l}(A′)_{s̄(két)}) which is of finite ℤ_{l}-rank. Moreover, we have the following canonical homomorphism

T_{l} : Hom(A, A′) ⊗_{ℤ} ℤ_{l} → Hom_{ℤ_{l}}(T_{l}(A)_{s̄(két)}, T_{l}(A′)_{s̄(két)}).

We are going to use this map to investigate the finiteness of Hom(A, A′).

Theorem 3.4. For A, A′ two log abelian varieties over S admitting a polarisation, Hom(A, A′) is a finitely generated free abelian group, and the canonical map

T_{l} : Hom(A, A′) ⊗_{ℤ} ℤ_{l} → Hom_{ℤ_{l}}(T_{l}(A)_{s̄(két)}, T_{l}(A′)_{s̄(két)})

is injective, where l is a prime number different from the characteristic of k.

Proof. We have already proven the degree function on End^{0}(A) is a homogeneous polynomial function of degree 2g in Theorem 3.3. Now the proof of [Mum70, §19, Thm. 3] works verbatim here.

Corollary 3.3. Let A, A′ be two log abelian varieties over S. Then the canonical map

T_{l} : Hom(A, A′) ⊗_{ℤ} ℤ_{l} → Hom_{ℤ_{l}}(T_{l}(A)_{s̄(két)}, T_{l}(A′)_{s̄(két)})

is injective.

Proof. This follows from Theorem 3.4.

Corollary 3.4. Let A, A′ be two log abelian varieties over S. Then we have Hom(A, A′) ≅ ℤ^{r} with r ≤ 4dimA · dimA′.

Corollary 3.5. Let A be a log abelian variety over S admitting a polarisation. Then End^{0}(A) is a finite-dimensional semisimple algebra over ℚ.

Proof. This follows from Corollary 3.2 and Corollary 3.4.

A Appendix: Log finite flat group schemes

Since the theory of log finite flat group schemes is not well-known, we collect some results about them in this appendix. These results are all due to Kato, and the main references are [Kat92] and [MS]. Let S be an fs log scheme. We recall several kinds of finite group objects on S_{fl}^{log} defined by Kato.

Definition A.1. The category (fin/S)_{c} is the full subcategory of the category of sheaves of finite abelian groups over S_{fl}^{log} consisting of objects which are representable by a classical finite flat group scheme over S. Here classical means the log structure of the representing log scheme is the one induced from S. The category (fin/S)_{f} is the full subcategory of the category of sheaves of finite abelian groups over S_{fl}^{log} consisting of objects which are representable by a classical finite flat group scheme over a log flat cover of S. For F ∈ (fin/S)_{f}, let U → S be a log flat cover of S such that F_{U} := F ×_{S} U ∈ (fin/S)_{c}, then the rank of F is defined to be the rank of F_{U} over U. The category (fin/S)_{r} is the full subcategory of (fin/S)_{f} consisting of objects which are representable by a log scheme over S. Let F ∈ (fin/S)_{f}, the Cartier dual of F is the sheaf F^{∗} := Hom_{S_{fl}^{log}}(F, 𝔾_{m}). By the definition of (fin/S)_{f}, it is clear that F^{∗} ∈ (fin/S)_{f}. The category (fin/S)_{d} is the full subcategory of (fin/S)_{r} consisting of objects whose Cartier duals also lie in (fin/S)_{r}.

A.1 Logarithmic fundamental group

We have the following well-known theorem.

Theorem A.1. Let S̊ be a locally noetherian connected scheme. Let finet ; S̊ denote the category of finite étale group schemes over S̊, LC(S̊_{Ét}) (resp. LC(S̊_{fl})) the category of locally constant sheaves of finite abelian groups for the étale (resp. flat) topology, and π_{1}(S̊)−fMod the category of finite abelian groups endowed with a continuous π_{1}(S̊)-action.

(1) By the theory of fundamental group, there are equivalences of categories:

finet ; S̊ ⟶^{≅} LC(S̊_{Ét}) ⟶^{≅} π_{1}(S̊)−fMod.

(2) By flat descent, we further have an equivalence

LC(S̊_{Ét}) ⟶^{≅} LC(S̊_{fl}).

For the theory of logarithmic fundamental group, we have the following analogue of Theorem A.1 which is due to Kato.

Theorem A.2. Let S be an fs log scheme with underlying scheme locally noetherian and connected. Let π_{1}^{log}(S) be the logarithmic fundamental group of S, see [Ill02, 4.6] for its definition. Let fin_{S}^{ket} be the subcategory of (fin/S)_{r} consisting of objects which are Kummer log étale over S, LC(S_{Ét}^{log}) (resp. LC(S_{fl}^{log})) the category of locally constant sheaves of finite abelian groups on S_{Ét}^{log} (resp. S_{fl}^{log}), and π_{1}^{log}(S)−fMod the category of finite abelian groups endowed with a continuous π_{1}^{log}(S)-action. Then we have the following equivalences of categories:

(1)

fin_{S}^{ket} ⟶^{≅} LC(S_{Ét}^{log}) ⟶^{≅} π_{1}^{log}(S)−fMod;

(2)

LC(S_{Ét}^{log}) ⟶^{≅} LC(S_{fl}^{log}).

Proof. For part (1), see [Ill02, §4]. For part (2), see [MS, Thm. 1.4.5. (2)].

A.2 Structure of log finite flat group schemes

Proposition A.1. Suppose that the underlying scheme of S is locally noetherian. Then the category (fin/S)_{f} (resp. (fin/S)_{r}, resp. (fin/S)_{d}) is closed under extensions in the category of sheaves of abelian groups on S_{fl}^{log}.

Proof. See [Kat92, Prop. 2.3].

Lemma A.1. Assume that the underlying scheme of S is the spectrum of a henselian local ring. Let F ∈ (fin/S)_{f}, then there is a unique short exact sequence

0 → F^{∘} → F → F^{et} → 0  (A.1)

in (fin/S)_{f}, such that over any log flat cover S′ → S with F_{S′} ∈ (fin/S′)_{c}, this sequence restricts to the classical connected-étale sequence.

Proof. See [Kat92, 2.6], or [MS, Lem. 2.1.6].

Lemma A.2. Let the assumption be as in Lemma A.1. Assume further that F lies in (fin/S)_{r} and its underlying scheme is connected. Then F actually lies in (fin/S)_{c}.

Proof. See [MS, Prop. 2.1.7], see also [Kat92, Lem. 2.8] for the noetherian strict henselian case.

Proposition A.2. Let the notation and the assumption be as in Lemma A.1. We further assume that the underlying scheme is noetherian.

(1) The sheaf F^{et} lies in fin_{S}^{ket}, in particular it lies in (fin/S)_{r}.

(2) The sheaf F lies in (fin/S)_{r} if and only if F^{∘} lies in (fin/S)_{c}.

Proof. Part (1) follows from Theorem A.2. Suppose that F^{∘} ∈ (fin/S)_{c}, then F as an F^{∘}-torsor over F^{et} is representable by [Kat91, Thm. 9.1]. Hence F ∈ (fin/S)_{r}. This proves one direction of part (2). Conversely suppose that F ∈ (fin/S)_{r}, we want to show that F^{∘} ∈ (fin/S)_{c}. As the kernel of F → F^{et}, F^{∘} is representable, hence lies in (fin/S)_{r}. We claim that the underlying scheme of F^{∘} is connected. Taking a covering U → S in S_{fl}^{log} such that the underlying scheme of U is finite over that of S, and F_{U}^{∘} ∈ (fin/U)_{c}. Since the underlying scheme of F_{U}^{∘} is connected, so is that of F^{∘}. Now Lemma A.2 applies, and we get F^{∘} ∈ (fin/S)_{c}.

Remark A.1. The proof of Proposition A.2 is the same as the proof of the strict henselian case in [Kat92, Prop. 2.7]. However there is a small condition missing in the proof in [Kat92, Prop. 2.7], so we give a complete proof here.

Let S be an fs log scheme with underlying scheme SpecR, where R is a noetherian henselian local ring with residue characteristic p > 0. Let x be the closed point of S, and suppose that S admits a global chart P → O_{S} which induces an isomorphism P → (M_{S}/O_{S}^{×})x̄. Here P is an fs monoid, and x̄ is a classical geometric point over x. By Proposition A.2, any F ∈ (fin/S)_{r} is an extension of F^{et} ∈ fin_{S}^{ket} by F^{∘} ∈ (fin/S)_{c}. By Theorem A.2, F^{et} can be understood by the theory of logarithmic fundamental group. Let S′ be a Galois cover of S with Galois group G := π_{1}^{log}(S)/π_{1}^{log}(S′), such that F^{et} ×_{S} S′ ∈ (fin/S′)_{c}. Then one can recover F from F ×_{S} S′ and the G-action on F ×_{S} S′ induced from the G-action on F^{et} ×_{S} S′. In order to understand F, we may assume F^{et} ∈ (fin/S)_{c} after replacing S by S′. Hence to understand the category (fin/S)_{r}, we are reduced to understand the extensions of a classical finite étale group scheme by a classical connected finite flat group scheme on S_{fl}^{log}. This is done by Kato’s classification theorem of logarithmic finite flat group schemes as follows.

Theorem A.3. Let S be as above. Let G be a classical finite étale group scheme over S, H a classical finite flat group scheme over S. We denote by 𝔈𝔵𝔱_{S_{fl}^{log}}(G, H) (resp. 𝔈𝔵𝔱_{S_{fl}^{cl}}(G, H)) the category of extensions of G by H on

S_{fl}^{log} (resp. S_{fl}^{cl}), and ℌ𝔬𝔪_{S_{fl}^{cl}}(G(1), H) ⊗ P^{gp} the discrete category associated to the group Hom_{S_{fl}^{cl}}(G(1), H) ⊗ P^{gp}. Then we have an equivalence of categories

Φ : 𝔈𝔵𝔱_{S_{fl}^{cl}}(G, H) × ℌ𝔬𝔪_{S_{fl}^{cl}}(G(1), H) ⊗ P^{gp} ⟶^{≅} 𝔈𝔵𝔱_{S_{fl}^{log}}(G, H).

Proof. See [MS, Thm. 2.3.1]. See also [Kat92, Thm. 3.3] for the strict henselian case.

References

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Heer Zhao Fakultät für Mathematik Universität Duisburg-Essen Essen 45117 Germany heer.zhao@gmail.com