Log Abelian Varieties over a Log Point

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 de- generation in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In partic- ular, we prove the dual short exact sequence for isogenies, Poincar´e complete reducibility theorem for log abelian varieties, and the semi- simplicity 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 degenera- tion, endomorphism algebras, Poincar´e 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 geom- etry. 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 ´etale topol- ogy in [KKN08a], however the log flat topology is needed for studying some problems, for example finite group subobjects of log abelian varieties,l-adic re- alisations of log abelian varieties, logarithmic Dieudonn´e theory of log abelian varieties and so on. In section 2, we prove that various classical ´etale 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 ´etale locally finite rank free constant sheaves, for changing to the log flat site from the classical ´etale 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 de- generation. 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, Poincar´e 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 B¨ockle’s Arbeitsgruppe, and I thank him for his hospitality and kindness.

I would like to thank the anonymous referee, whose feedback has greatly im- proved 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 ´etale 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 ´etale topology, in the context of log flat topology.

Throughout this section, letSbe any fs log scheme with its underlying scheme locally noetherian, and (fs/S) be the category of fs log schemes overS. The log schemes in this section will always be fs log schemes unless otherwise stated.

Let S^cl_´

Et (resp. S^cl_fl, resp. S^log_´

Et, resp. S_fl^log) be the classical ´etale site (resp.

classical flat site, resp. log ´etale site, resp. log flat site)¹ associated to the category (fs/S), and let δ = m◦εfl : S_fl^log −→^εfl S_fl^cl −→^m S^cl_´

Et be the canonical map of sites. For any inclusion F ⊂Gof sheaves onS_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 ´etale 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_Et^cl_´ which are also sheaves on S_fl^log. Then we have Hom_S^cl

Et´ (F, G) =Hom_S^log

fl (F, G), in particular Hom_S^cl

Et´ (F, G) is a sheaf on S_fl^log.

Proof. This is clear.

Now we recall some definitions from [KKN08a]. LetGbe a commutative group scheme over the underlying scheme of S which is an extension of an abelian scheme B by a torusT. LetX be the character group ofT which is a locally constant sheaf of finite generated freeZ-modules for the classical ´etale topology.

The sheafG_m,log onS^cl_´

Etis defined by

G_m,log(U) = Γ(U, M_U^gp), the sheafTlog onS_Et^cl_´ is defined by

Tlog:=Hom_S^cl

Et´ (X,G_m,log),

and the sheafGlog is defined as the push-out ofTlog←T →Gin the category of sheaves onS_Et^cl_´ , see [KKN08a, 2.1].

Proposition 2.1. The sheaves G_m,log, X, Tlog and Glog on S_Et^cl_´ are also sheaves for the log flat topology. Moreover, Tlog can be alternatively defined as

Hom_S^log

fl (X,G_m,log),

and Glog can be alternatively defined as the push-out of Tlog ←T →Gin the category of sheaves on S_fl^log.

Proof. The statement for Gm,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

1Here we are following the terminology from [Kat91]. Note that in [KKN15]S^cl_´

Etis called the strict ´etale site, whileS^log_´

Et andS_fl^logare called the Kummer log ´etale site and the Kummer log flat site respectively.

by [Kat91, Thm. 3.1] and [KKN15, Thm. 5.2]. It follows then Tlog = Hom_S^cl

Et´ (X,G_m,log) =Hom_S^log

fl (X,G_m,log) is also a sheaf onS_fl^log. By its defini- tionGlog fits into a short exact sequence 0→Tlog→Glog→B→0 of sheaves onS_Et^cl_´ . Consider the following commutative diagram

0 //Tlog //

=

Glog //

B //

=

0

0 //Tlog //δ∗δ^∗Glog //B //R¹δ∗Tlog

with exact rows in the category of sheaves onS_Et^cl_´ , where the vertical maps come from the adjunction (δ^∗, δ∗). The sheafR¹δ∗Tlog is zero by Kato’s logarithmic Hilbert 90, see [Kat91, Cor. 5.2] or [Niz08, Thm. 3.20]. It follows that the canonical map Glog →δ∗δ^∗Glog is an isomorphism, whence Glog is a sheaf on S_fl^log. SinceGlog, as a push-out ofTlog←T →Gin the category of sheaves on S_Et^cl_´ , is already a sheaf onS_fl^log, it coincides with the push-out ofTlog ←T →G in the category of sheaves onS_fl^log.

Proposition2.2. We have canonical isomorphisms Hom_S^log

fl (X,Gm,log/Gm)∼=Tlog/T ∼=Glog/G.

Proof. By Proposition 2.1, Glog is the push-out of Tlog ← T → G in the category of sheaves onS_fl^log, so we get a commutative diagram

0 //T //

G //

B //0

0 //Tlog //Glog //B //0

with exact rows. Then the isomorphism Tlog/T ∼=Glog/G follows. Applying the functorHom_S^log

fl (X,−) to the short exact sequence 0→G_m→G_m,log →G_m,log/Gm→0, we get a long exact sequence

0→T →Tlog → Hom_S^log

fl (X,G_m,log/Gm)→ Ext_S^log

fl (X,G_m)

of sheaves onS^log_fl . SinceX is classical ´etale locally represented by a finite rank free abelian group, the sheafExt_S^log

fl (X,G_m) is zero. It follows that the sheaf Hom_S^log

fl (X,G_m,log/Gm) is canonically isomorphic toTlog/T.

It is obvious that the association ofGlogtoGis functorial inG. Hence we have a natural map Hom_S^log

fl (G, G^′) → Hom_S^log

fl (Glog, 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.

Proposition2.3. (1) The association ofGlog toGis functorial in G.

(2) The canonical map Hom_S^log

fl (G, G^′) → Hom_S^log

fl (Glog, G^′_log) is an isomor- phism.

(3) For a group schemeH of multiplicative type with character groupXH over the underlying scheme of S, let Hlog denote Hom_S^log

fl (XH,G_m,log). Let 0 →H^′ →H →H^′′ → 0 be a short exact sequence of group schemes of multiplicative type over the underlying scheme ofS such that their charac- ter groups are ´etale locally finite rank constant sheaves, then the sequences

0→H_log^′ →Hlog→H_log^′′ →0 and

0→ Hom_S^log

fl (XH^′,G_m,log/G_m)→ Hom_S^log

fl (XH,G_m,log/G_m)

→ Hom_S^log

fl (XH^′′,G_m,log/G_m)→0 are both exact.

(4) IfG→G^′ is injective, so is Glog →G^′_log. (5) IfG→G^′ is surjective, so isGlog→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→Glog→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^′ →Hlog→H_log^′′ → Ext_Slog

fl (XH^′,G_m,log), it suffices to show Ext_S^log

fl (XH^′,G_m,log) = 0. Since Ext_S^log

fl (Z,G_m,log) = 0, we are further reduced to show Ext_S^log

fl (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^log

fl (Z/nZ,G_m,log)→G_m,log −→ⁿ G_m,log →

Ext_S^log

fl (Z/nZ,G_m,log) → 0. Since G_m,log −→ⁿ G_m,log is surjective, the sheaf Ext_S^log

fl (Z/nZ,Gm,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 mapX^′→X on the character groups is surjective. It follows that the induced map

Glog/G=Hom_S^log

fl (X,G_m,log/G_m)→ Hom_S^log

fl (X^′,G_m,log/G_m) =G^′_log/G^′ is injective. HenceGlog →G^′_log is injective.

Now we prove part (5). Letf denote the mapG→G^′. Consider the torus and abelian variety decomposition off

0 //T //

ft

G //

f

B //

fab

0

0 //T^′ //G^′ //B^′ //0.

We first show thatftis surjective. Assume that the underlying scheme ofSis a point. The snake lemma gives an exact sequence Ker(fab)→Coker(ft)→0.

Since Coker(ft) is a torus and the reduced neutral component of Ker(fab) is an abelian variety by [Bri15, Lem. 3.3.7], we must have Coker(ft) = 0. Hence ft is surjective. In the general case,ft is fiberwise surjective, hence it is also set-theoretically surjective. The fibers of ft over S are all flat, hence ft is flat by the fiberwise criterion of flatness, see [Gro66, Cor. 11.3.11]. Then ft

is faithfully flat, hence it is surjective. Then we get a short exact sequence 0 → X^′ → X → X/X^′ → 0 of ´etale locally constant sheaves. Applying the functor Hom_S^log

fl (−,G_m,log/G_m) to this short exact sequence, we get a long exact sequence

→Glog/G→G^′_log/G^′→ Ext_S^log

fl (X/X^′,G_m,log/Gm).

LetZtor be the torsion part ofX/X^′, and letnbe a positive integer such that nZtor = 0. Since the multiplication-by-n map on G_m,log/Gm is an isomor- phism, we get that the sheafExt_S^log

fl (Ztor,G_m,log/Gm) is zero. The torsion-free nature of (X/X^′)/Ztor implies Ext_S^log

fl ((X/X^′)/Ztor,G_m,log/G_m) = 0, hence Ext_S^log

fl (X/X^′,Gm,log/Gm) = 0. It follows that Glog/G → G^′_log/G^′ is surjec- tive, henceGlog→G^′_log is surjective.

At last, we prove part (6). Consider the following commutative diagram 0

0

0

0 //G^′ //

G^′_log //

Hom_S^log

fl (X^′,G¯_m,log) //

0

0 //G //

Glog //

Hom_S^log

fl (X,G¯_m,log) //

0

0 //G^′′ //

G^′′_log //

Hom_S^log

fl (X^′′,G¯_m,log) //

0

0 0 0

with the first column and all rows exact, where ¯Gm,log denotes Gm,log/Gm. The maps G^′ →G→G^′′ induceT^′ →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, whereZ := Ker(X →X^′) is ´etale locally a finite rank free constant sheaf and Z/X^′′ is ´etale locally a finite torsion constant sheaf. By part (3), we get two short exact sequences

0→ Hom_S^log

fl (X^′,G¯_m,log)→ Hom_S^log

fl (X,G¯_m,log)→ Hom_S^log

fl (Z,G¯_m,log)→0 and

0→Hom_S^log

fl (Z/X^′′,G¯m,log)→ Hom_S^log

fl (Z,G¯m,log)

→Hom_S^log

fl (X^′′,G¯_m,log)→0.

But Hom_S^log

fl (Z/X^′′,G¯m,log) = 0, it follows that the third column of the dia- gram is exact. So is the middle column.

Recall that in [KKN08a, Def. 2.2], a log 1-motive M over S_Et^cl_´ is defined as a two-term complex [Y −→^u Glog] in the category of sheaves on S_Et^cl_´ , with the degree −1 term Y an ´etale locally constant sheaf of finitely generated free abelian groups and the degree 0 termGlog as above. Since both Y and Glog

are sheaves onS_fl^log,M can also be defined as a two-term complex [Y −→^u Glog] in the category of sheaves onS_fl^log. Parallel to [KKN08a, 2.3], we have a natural pairing

<, >:X×Y →X×(Glog/G) =X× Hom_S^log

fl (X,Gm,log/Gm)→Gm,log/Gm. (2.1)

It is clear that our pairing is induced from the one of [KKN08a, 2.3].

By our convention, Tlog/T denotes the quotient in the category of sheaves on S_fl^log. For the quotient of T ⊂Tlog in the category of sheaves on S_Et^cl_´ , we use the notation (Tlog/T)_S^cl

Et´ . 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^cl

Et´ (X,(Gm,log/Gm)_S^cl

´Et)^(Y) of the sheaf Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ ) on S_Et^cl_´ is defined by

Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )^(Y)(U) :=

{ϕ∈ Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )(U)|for everyu∈U andx∈Xu¯, there existyu,x, y_u,x^′ ∈Yu¯ such that< x, yu,x>|ϕu¯(x)|< x, y^′_u,x>}.

Here, ¯u denotes a classical ´etale geometric point above u, and for a, b ∈ (M_U^gp/O^×_U)u¯,a|b meansa⁻¹b∈(MU/O^×_U)u¯.

It is natural to define the analogue of Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

´Et)^(Y) in the log flat topology. We need the following lemma first.

Lemma 2.2. Let δ:S_fl^log→S_Et^cl_´ be the canonical map between these two sites.

(1) δ∗(Gm,log/Gm) = (Gm,log/Gm)_S^cl

Et´ ⊗_ZQ.

(2) Let H be a commutative group scheme over the underlying scheme of S with connected fibres. ThenHom_S^log

fl (H,G_m,log/G_m) = 0.

Proof. We denote the sheaf G_m,log/G_m on S_fl^log by ¯G_m,log. For any positive integern, we have the following commutative diagram

0

0

0 //Z/n(1) //G_m ⁿ //

G_m //

0

0 //Z/n(1) //G_m,log ⁿ //

G_m,log //

0

G¯_{m,log ∼}ⁿ

=

//

G¯_m,log

0 0

with exact rows and columns, whereZ/n(1) denotes the group scheme ofn-th roots of unity. Applying the functor εfl∗ to the above diagram, we get a new

commutative diagram

0 //Z/n(1) //Gm n //

 _

Gm

 _

0 //Z/n(1) //Gm,log n //

α

Gm,log

γ

εfl∗G¯_{m,log ∼}ⁿ

=

//

β

εfl∗G¯_m,log

δ

G_m ⁿ //

 _

G_m //

 _

R¹εfl∗Z/n(1) ^η //R¹εfl∗G_m ⁿ //

R¹εfl∗G_m

Gm,log n //Gm,log θ //R¹εfl∗Z/n(1) //R¹εfl∗Gm,log

with exact rows and columns. Since the map G_m −→ⁿ G_m is surjective and R¹εfl∗G_m,log= 0, we get a new commutative diagram

G ⊗_ZZ/n

θ¯

∼=

0 //G ⁿ //G _

1 nγ

ξ

//

ω

77

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦ R¹εfl∗Z/n(1) //

 _

η

0

0 //G ^α¯ //εfl∗G¯_m,log ^β //R¹εfl∗G_m //0 with exact rows, whereG denotes (G_m,log/G_m)_S^cl

fl, ¯α(resp. ¯θ) is the canonical map induced by α (resp. θ), ω is the canonical projection map andξ is the unique map guaranteed bynβ◦(¹_nγ) =δ◦(n(_n¹γ)) =δ◦γ= 0. Taking colimit of the above diagram with respect ton, we get a commutative diagram

0 //G //G ⊗_ZQ

//G ⊗_ZQ/Z //

0

0 //G ^α¯ //εfl∗G¯m,log β

//R¹εfl∗Gm //0

with exact rows. Since the map G ⊗_ZQ/Z → R¹εfl∗G_m is an isomorphism by Kato’s theorem [Kat91, Thm. 4.1] (see also [Niz08, Thm. 3.12]), we get G ⊗_ZQ∼=εfl∗(Gm,log/Gm). Then

δ∗(Gm,log/Gm) =m∗εfl∗(Gm,log/Gm) =m∗(G ⊗_ZQ)

= (Gm,log/Gm)_S^cl

Et´ ⊗_ZQ,

where the last equality follows from the following fact: for anyU ∈(fs/S), the sheaf M_U^gp/O_U^× on the small ´etale 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^log

fl (H,G¯_m,log) = Hom_S^cl

Et´ (H, δ∗G¯_m,log)

= Hom_S^cl

Et´ (H,(Gm,log/Gm)_S^cl

´Et⊗_ZQ).

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

´Et(H,(G_m,log/G_m)_S^cl

Et´ ⊗_ZQ) = 0.

Hence part (2) is proved.

Now we define the analogue ofHom_S^log

Et´

(X,(G_m,log/G_m)_S^cl

Et´ )^(Y). It is the sub- group sheaf Hom_S^log

fl (X,G_m,log/Gm)^(Y) of the sheaf Hom_S^log

fl (X,G_m,log/Gm) onS_fl^log given by

Hom_S^log

fl (X,Gm,log/Gm)^(Y)(U) :=

{ϕ∈ Hom_S^log

fl (X,Gm,log/Gm)(U)|after pushing forward toU_Et^cl_´ , for everyu∈U andx∈Xu¯, there existyu,x, y_u,x^′ ∈Yu¯ such that

< x, yu,x >|ϕu¯(x)|< x, y_u,x^′ >}.

Here ¯u still denotes a classical ´etale geometric point above u. Let F :=

δ∗(Gm,log/Gm) = (Gm,log/Gm)_S^cl

Et´ ⊗_ZQwithδthe canonical mapU_fl^log →U_Et^cl_´ . Fora, b∈(M_U^gp/O_U^×)u¯⊗_ZQ,a|bmeansa⁻¹b=α⊗_Zrfor someα∈(MU/O_U^×)¯u

andr∈Q.

Remark 2.1. In [KKN08a, 7.1], admissibility and non-degeneracy are defined for pairings into (Gm,log/Gm)_S^cl

Et´ in the classical ´etale site on (fs/S). We can define admissibility and non-degeneracy for pairings intoGm,log/Gmon the log flat site in the same way. Since bothX andY are classical ´etale 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^cl

Et´

(X,(Gm,log/Gm)_S^cl

Et´

)^(Y) on S^cl_´

Et

with the sheafHom_Slog

fl (X,G_m,log/Gm)^(Y) onS_fl^log.

Lemma2.3. Let X, Y be two free abelian groups of finite rank,<, >:X×Y → (G_m,log/G_m)_S^cl

Et´ an admissible pairing onS^cl_Et´ . Let Qcl:=Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )^(Y), Q:=Hom_S^log

fl (X,G_m,log/Gm)^(Y), and δ:S_fl^log →S_Et^cl_´ the canonical map between these two sites. Then we have Q=δ^∗Qclandδ∗Q=Qcl⊗_ZQ.

Proof. DenoteG_m,log/Gm by ¯G_m,log. We have δ^∗(Gm,log/Gm)_S^cl

Et´ = ¯Gm,log, and

δ∗( ¯G_m,log) = (G_m,log/G_m)_S^cl

Et´ ⊗_ZQ by part (1) of Lemma 2.2, hence

δ^∗Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

´Et) =Hom_S^log

fl (X,G¯_m,log), and

δ∗Hom_S^log

fl (X,G¯_m,log) =Hom_S^cl

Et´ (X,(G_m,log/G_m)_S^cl

Et´ )⊗_ZQ. Then by the definition ofQandQcl, we getQ=δ^∗Qclandδ∗Q=Qcl⊗_ZQ.

Recall that in [KKN08a, 3.2, Thm. 7.3], G^(Y_log⁾ ⊂Glog (resp. T_log^(Y) ⊂Tlog) on S_Et^cl_´ is defined to be the inverse image ofHom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )^(Y)under the map

Glog→(Glog/G)_S^cl

Et´

∼=Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ ) (resp. Tlog→(Tlog/T)_S^cl

Et´

∼=Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )).

We could also consider the inverse image sheaf of Hom_S^log

fl (X,G_m,log/Gm)^(Y) under the map

Glog →Glog/G∼=Hom_S^log

fl (X,G_m,log/Gm) (resp. Tlog →Tlog/T ∼=Hom_S^log

fl (X,G_m,log/Gm)).

The following proposition states that these two constructions coincide.

Proposition2.4. (1) The sheafG^(Y_log⁾ onS_Et^cl_´ is also a sheaf on S_fl^log. (2) The sheaf G^(Y_log⁾fits into a canonical short exact sequence

0→G→G^(Y_log⁾→ Hom_S^log

fl (X,G_m,log/G_m)^(Y)→0 (2.2) of sheaves onS_fl^log.

(3) The association ofG^(Y_log⁾ to a log 1-motive M = [Y →Glog]is functorial.

Proof. LetT (resp. B) be the torus (resp. abelian scheme) part ofG, then we have a short exact sequence 0 → T_log^(Y) →G^(Y_log⁾ → B →0 of sheaves on S_Et^cl_´ . To show that G^(Y_log⁾ on S_Et^cl_´ is a sheaf onS^log_fl , it suffices to show thatT_log^(Y) is so. By [KKN08a, 7.7], locally onS^cl_´

Etthe sheafT_log^(Y)is a union of representable sheaves. Hence it is also a sheaf onS^log_fl . So part (1) is proven.

By the definition ofG^(Y_log⁾, we have a pullback diagram 0 //G //G^(Y_log⁾ //

 _

Hom_S^cl

Et´ (X,(G_m,log/G_m)_S^cl

´Et)^(Y) //

 _

0

0 //G //Glog //Hom_S^cl

´Et(X,(Gm,log/Gm)_S^cl

Et´ ) //0 in the category of sheaves on S_Et^cl_´ . Since G, G^(Y_log⁾ and Glog are all sheaves on S_fl^log, applying the functor δ^∗ to the above commutative diagram, we get the following commutative diagram

0 //G //G^(Y_log⁾ //

 _

δ^∗Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )^(Y) //

 _

0

0 //G //Glog //δ^∗Hom_S^cl

Et´ (X,(G_m,log/G_m)_S^cl

Et´ ) //0.

Since we have canonical isomorphisms δ^∗Hom_S^cl

´Et(X,(G_m,log/G_m)_S^cl

Et´ )∼=Hom_S^log

fl (X,G_m,log/G_m) and

δ^∗Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )^(Y)∼=Hom_S^log

fl (X,G_m,log/Gm)^(Y), part (2) follows.

Now we prove part (3). It is enough to prove that for a given homomorphism (f−1, f0) : M = [Y → Glog] →M^′ = [Y^′ → G^′_log], the composition G^(Y_log⁾ ֒→ Glog

f0

−→G^′_log factors throughG^′_log^(Y′) ֒→G^′_log. LetX and X^′ be the character groups of the torus parts ofGandG^′respectively, letfl:X^′ →X be the map induced fromf0, and let

f˜d:Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )→ Hom_S^cl

Et´ (X^′,(Gm,log/Gm)_S^cl

Et´ ) be the map induced from fl. By the definition of G^(Y_log⁾ and G^′(Y_log^′), we are reduced to show the composition

Hom_S^cl

Et´ (X,(G_m,log/G_m)_S^cl

Et´ )^(Y)֒→ Hom_S^cl

Et´ (X,(G_m,log/G_m)_S^cl

Et´ )−→^f˜d Hom_S^cl

Et´ (X^′,(Gm,log/Gm)_S^cl

Et´ ) factors through

Hom_S^cl

Et´ (X^′,(G_m,log/G_m)_S^cl

Et´ )^(Y′)֒→ Hom_S^cl

Et´ (X^′,(G_m,log/G_m)_S^cl

Et´ ).

Let<, >:X×Y →(Gm,log/Gm)_S^cl

Et´

(resp. <, >^′:X^′×Y^′ →(Gm,log/Gm)_S^cl

Et´

) be the pairing associated toM (resp. M^′), then we have

< fl(x^′), y >=< x^′, f−1(y)>^′ for anyx^′∈X^′, y∈Y. For anyU ∈(fs/S),

ϕ∈ Hom_S^cl

Et´ (X,(Gm,log/Gm)_S^cl

Et´ )^(Y)(U), we need to show

ψ:=ϕ◦fl∈ Hom_S^cl

Et´ (X^′,(G_m,log/G_m)_S^cl

Et´ )^(Y′)(U).

For every u∈ U and every x^′ ∈X_u^′_¯, there exist yu,x^′,1, yu,x^′,2∈ Yu¯ such that

< fl(x^′), yu,x^′,1>|ϕu¯(fl(x^′))|< fl(x^′), yu,x^′,2>. The relation can be rewritten as

< x^′, f−1(yu,x^′,1)>^′ |ψu¯(x^′)|< x^′, f−1(yu,x^′,2)>^′, which implies thatψ∈ Hom_Scl

Et´

(X^′,(Gm,log/Gm)_Scl Et´

)^(Y′)(U). This finishes the proof of part (3).

Remark 2.2. Clearly, the image ofu:Y →Glog is contained in G^(Y_log⁾.

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_Slog

fl (Y,G_m,log/Gm) and Y → Hom_Slog

fl (X,G_m,log/Gm) 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 onS_Et^cl_´ which is isomorphic to the quotient sheaf (G^(Y_log⁾/Y)_S^cl

Et´ for a pointwise polarisable (resp. non-degenerate) log 1-motive M = [Y −→^u Glog]. 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 overS.

Theorem 2.1. Let A be a weak log abelian variety with constant degeneration over S. Suppose A = (G^(Y_log⁾/Y)_S^cl

Et´ for a non-degenerate log 1-motive M = [Y −→^u Glog]. Then

(1) Ais a sheaf on S_fl^log;

(2) A=G^(Y_log⁾/Y, in other words Afits into a canonical short exact sequence 0→Y →G^(Y_log⁾→A→0 (2.3) in the category of sheaves of abelian groups onS_fl^log;

(3) Afits into a canonical short exact sequence 0→G→A→ Hom_S^log

fl (X,G_m,log/Gm)^(Y)/Y →0 (2.4) in the category of sheaves of abelian groups onS_fl^log.

Proof. Part (2) follows from part (1). Since the log 1-motive M is non- degenerate, the map Y → Hom_S^log

fl (X,G_m,log/Gm)^(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 ´etale case in [KKN15,§5].

Consider the short exact sequence 0→Y →A˜→A→0 of [KKN15, 5.3]. Note that ˜Ais nothing but G^(Y_log⁾in our situation, however we stick to the notation A˜for the sake of coherence with [KKN15, 5.3]. The argument showing that ˜A is a log ´etale sheaf, also shows that ˜A is a log flat sheaf, since representable functors are sheaves for the log flat topology by [KKN15, Thm. 5.2]. We have the canonical mapδ:=m◦εfl:S_fl^log −→^εfl S_fl^cl−→^m S_Et^cl_´ of sites. Applyingδ^∗ and δ∗ to 0→Y →A˜→A→0, we get a commutative diagram

0 //Y //

=

A˜ //

=

A //

0

0 //Y //A˜ //δ∗δ^∗A //R¹δ∗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 mapA→δ∗δ^∗Ais an isomorphism. This follows from the above commutative diagram with the help of the lemma below.

Lemma 2.4. The sheaf R¹δ∗Y is zero.

Proof. Since Y is ´etale locally isomorphic to a finite rank free abelian group, we are reduced to the caseY =Z. Note thatY 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 thatY is ´etale overS.

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, ˆB its completion, and let α ∈ H¹((SpecB)^log_fl ,Z) such that it vanishes in H¹((Spec ˆB)^log_fl ,Z). By fpqc descent, αis a class of a representable Z-torsor over SpecB such that its structure morphism is ´etale. Since B is a strict local ring, the torsor admits a section by [Gro67, Prop. 18.8.1], so αis zero. It follows that [Kat91, Thm.

4.1] also holds for the caseG=Z, soR¹εfl∗Z= 0. The Leray spectral sequence

gives a short exact sequence 0 →R¹m∗Z→R¹δ∗Z→m∗R¹εfl∗Z. The sheaf R¹m∗Z= 0 by [Gro68, Thm. 11.7], it follows thatR¹δ∗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 ´etale 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 → Glog] 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^(Y_log⁾/Y. Then we have:

(1) Ext_S^log

fl (A,Z)∼=Hom_S^log

fl (Y,Z);

(2) the sheaf δ∗Ext_S^log

fl (A,G_m)fits into an exact sequence 0→G^∗→δ∗Ext_S^log

fl (A,G_m)→ Hom_S^cl

Et´ (A, R¹δ∗G_m);

(3) Ext_S^log

fl (A,G_m,log)∼= (G^∗_log/X)_S^cl

Et´

∼=G^∗_log/X;

(4) Hom_S^log

fl (A,Z) =Hom_S^log

fl (A,G_m) =Hom_S^log

fl (A,G_m,log) = 0.

Proof. By Proposition 2.1 and Theorem 2.1, the sheavesZ, G_m,log and A on S_Et^cl_´ are also sheaves onS_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.

LetF1(resp. F2) be a sheaf onS_Et^cl_´ (resp. S_fl^log), then we have δ∗Hom_S^log

fl (δ^∗F1, F2) =Hom_S^cl

Et´

(F1, δ∗F2).

Letθ be the functor sending F2 to δ∗Hom_S^log

fl (δ^∗F1, F2) =Hom_S^cl

Et´ (F1, δ∗F2), then we get two Grothendieck spectral sequences

E₂^p,q =R^pδ∗R^qHom_S^log

fl (δ^∗F1,−)⇒R^p+qθ (2.5) and

E₂^p,q=R^pHom_S^cl

Et´ (F1,−)R^qδ∗⇒R^p+qθ. (2.6) These two spectral sequences give two exact sequences

0→R¹δ∗Hom_S^log

fl (δ^∗F1, F2)→R¹θ(F2)→δ∗Ext_S^log

fl (δ^∗F1, F2)

→R²δ∗Hom_S^log

fl (δ^∗F1, F2) (2.7)

and

0→ Ext_S^cl

Et´ (F1, δ∗F2)→R¹θ(F2)→ Hom_S^cl

Et´ (F1, R¹δ∗F2). (2.8)

Let F1 =A, and let F2 be Z, G_m or G_m,log, part (4) together with Theorem 2.1 and exact sequence (2.7) imply

R¹θ(A)∼=δ∗Ext_S^log

fl (A, F2), so we get an exact sequence

0→ Ext_S^cl

Et´ (A, δ∗F2)→δ∗Ext_S^log

fl (A, F2)→ Hom_S^cl

Et´ (A, R¹δ∗F2).

SinceExt_S^cl

Et´

(A,G_m)∼=G^∗by [KKN08a, Thm. 7.4 (2)], the caseF2=G_mgives part (2). SinceR¹δZ= 0 by Lemma 2.4 andExt_S^cl

Et´ (A,Z)∼=Hom_S^log

fl (Y,Z) by [KKN08a, Thm. 7.4 (1)], the caseF2=Zgives part (1). The sheafR¹δ∗G_m,log equals zero by Kato’s logarithmic Hilbert 90 [Kat91, Cor. 5.2]. And we have Ext_S^cl

Et´ (A,G_m,log)∼= (G^∗_log/X)_S^cl

Et´ by [KKN08a, Thm. 7.4 (3)]. Then part (3) follows from the caseF2=Gm,log.

Let A be a weak log abelian variety with constant degeneration over S, and letM = [Y →Glog] be the log 1-motive of type (X, Y) definingA. 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 ofM, then the pairing associated to M^∗ is the same (up to switching the positions of X and Y) as the paring associated toM, hence it is automatically non-degenerate. IfAis further a log abelian variety with constant degeneration, i.e. the log 1-motiveMis pointwise polarisable, then M^∗ is also pointwise polarisable.

Definition 2.1. LetA be a weak log abelian variety with constant degener- ation (resp. log abelian variety with constant degeneration) overS. 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^∗(X)_log /X associated to the log 1-motive M^∗ = [X → G^∗_log]. We denote the dual ofA byA^∗.

Let WLAV^CD_S (resp. LAV^CD_S ) denote the category of weak log abelian varieties with constant degeneration (resp. log abelian varieties with constant degener- ation) overS. Then we have the following proposition.

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

(−)^∗: WLAV^CD_S →WLAV^CD_S which restricts to a contravariant functor

(−)^∗: LAV^CD_S →LAV^CD_S .

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