Log Abelian Varieties over a Log Point

46 Pages·2017·0.44 MB·English

Documenta Math. 505
Log Abelian Varieties over a Log Point
Heer Zhao
Received: October 25,2015
Revised: August26,2016
Communicated byTakeshiSaito
Abstract. We study (weak) log abelian varieties with constant de-
generationinthelogflattopology. Ifthebaseisalogpoint,wefurther
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.
2010Mathematics Subject Classification: Primary14D06;Secondary
14K99, 11G99.
Keywords and Phrases: log abelian varieties with constant degenera-
tion,endomorphismalgebras,Poincar´ecompletereducibilitytheorem,
dual short exact sequence for isogenies.
1 Introduction
As stated in [KKN08a], degenerating abelian varieties can not preserve group
structure,properness,andsmoothnessatthesametime. Logabelianvarietyis
aconstructionaimedtomaketheimpossiblepossibleinthe worldofloggeom-
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,forexamplefinitegroupsubobjectsoflogabelianvarieties,l-adicre-
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]arealsosheavesforthelogflattopology,inparticularweprove
that(weak)logabelianvarietieswithconstantdegenerationaresheavesforthe
log flat topology, see Theorem 2.1. We compute the first direct image sheaves
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506 Heer Zhao
of´etalelocallyfiniterankfreeconstantsheaves,forchangingtothelogflatsite
fromthe classical´etalesite, in Lemma 2.4. This lemma canbe consideredasa
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
abelianscheme. Nowonemaywonderifvariousresultsforabelianvarietyalso
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
groupoflogabelianvarietiesinTheorem3.4,Corollary3.3, Corollary3.4, 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
helpfulcommunicationsconcerningthepaper[KKN15]. IthankProfessorQing
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
ProfessorGebhardB¨ockle’sArbeitsgruppe, andI thank him forhis 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.
Throughoutthissection,letS be anyfslogschemewithitsunderlyingscheme
locallynoetherian,and(fs/S)bethecategoryoffslogschemesoverS. Thelog
schemes in this section will always be fs log schemes unless otherwise stated.
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Log Abelian Varieties over a Log Point 507
Let Scl (resp. Scl, resp. Slog, resp. Slog) be the classical ´etale site (resp.
E´t fl E´t fl
classical flat site, resp. log ´etale site, resp. log flat site)1 associated to the
category (fs/S), and let δ = m◦ε : Slog −ε→fl Scl −m→ Scl be the canonical
fl fl fl E´t
map of sites. For any inclusion F ⊂ G of sheaves on Slog, we denote by G/F
fl
the quotient sheaf in the category of sheaves on Slog by convention, unless
fl
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 Scl which are also sheaves on Slog.
E´t fl
Then we have Hom (F,G)=Hom (F,G), in particular Hom (F,G) is
SEc´lt Sfllog SEc´lt
a sheaf on Slog.
fl
Proof. This is clear.
Nowwerecallsomedefinitionsfrom[KKN08a]. LetGbeacommutativegroup
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
constantsheafoffinitegeneratedfreeZ-modulesfortheclassical´etaletopology.
The sheaf G on Scl is defined by
m,log E´t
G (U)=Γ(U,Mgp),
m,log U
the sheaf T on Scl is defined by
log E´t
T :=Hom (X,G ),
log Scl m,log
E´t
and the sheaf G is defined as the push-out of T ←T →G in the category
log log
of sheaves on Scl, see [KKN08a, 2.1].
E´t
Proposition 2.1. The sheaves G , X, T and G on Scl are also
m,log log log E´t
sheaves for the log flat topology. Moreover, T can be alternatively defined
log
as
Hom (X,G ),
Slog m,log
fl
and G can be alternatively defined as the push-out of T ← T → G in the
log log
category of sheaves on Slog.
fl
Proof. The statement for G is just [Kat91, Thm. 3.2], see also [Niz08,
m,log
Cor. 2.22]. Being representable by a group scheme, X is a sheaf on Slog
fl
1Herewearefollowingtheterminologyfrom[Kat91]. Notethatin[KKN15]Scl iscalled
E´t
thestrict´etalesite,whileSlogandSlogarecalledtheKummerlog´etalesiteandtheKummer
E´t fl
logflatsiterespectively.
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508 Heer Zhao
by [Kat91, Thm. 3.1] and [KKN15, Thm. 5.2]. It follows then T =
log
Hom (X,G )=Hom (X,G )isalsoasheafonSlog. Byitsdefini-
SEc´lt m,log Sfllog m,log fl
tion G fits into a shortexactsequence 0→T →G →B →0 of sheaves
log log log
on Scl. Consider the following commutative diagram
E´t
0 //T //G // B //0
log log
= =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // Tlog //δ∗δ∗Glog // B // R1δ∗Tlog
withexactrowsinthecategoryofsheavesonScl,wheretheverticalmapscome
E´t
from the adjunction (δ∗,δ∗). The sheaf R1δ∗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
Slog. Since G , as a push-out of T ←T →G in the category of sheaves on
fl log log
Scl,isalreadyasheafonSlog,itcoincideswiththe push-outofT ←T →G
E´t fl log
in the category of sheaves on Slog.
fl
Proposition 2.2. We have canonical isomorphisms
Hom (X,G /G )∼=T /T ∼=G /G.
Slog m,log m log log
fl
Proof. By Proposition 2.1, G is the push-out of T ← T → G in the
log log
category of sheaves on Slog, so we get a commutative diagram
fl
0 //T // G //B // 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 // T // G //B // 0
log log
with exact rows. Then the isomorphism T /T ∼= G /G follows. Applying
log log
the functor Hom (X,−) to the short exact sequence
Slog
fl
0→G →G →G /G →0,
m m,log m,log m
we get a long exact sequence
0→T →T →Hom (X,G /G )→Ext (X,G )
log Slog m,log m Slog m
fl fl
ofsheavesonSlog. SinceX isclassical´etalelocallyrepresentedbyafiniterank
fl
free abelian group, the sheaf Ext (X,G ) is zero. It follows that the sheaf
Slog m
fl
Hom (X,G /G ) is canonically isomorphic to T /T.
Slog m,log m log
fl
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Log Abelian Varieties over a Log Point 509
ItisobviousthattheassociationofG toGisfunctorialinG. Hencewehave
log
a natural map Hom (G,G′) → Hom (G ,G′ ), where G′ is another
Slog Slog log log
fl fl
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 to G is functorial in G.
log
′ ′
(2) The canonical map Hom (G,G) → Hom (G ,G ) is an isomor-
Slog Slog log log
fl fl
phism.
(3) For a group scheme H of multiplicative type with character group X over
H
the underlying scheme of S, let H denote Hom (X ,G ). Let
log Slog H m,log
0 → H′ → H → H′′ → 0 be a short exact sequenceflof group schemes of
multiplicative type over the underlying scheme of S such that their charac-
ter groups are ´etale locally finite rank constant sheaves, then the sequences
′ ′′
0→H →H →H →0
log log log
and
0→HomSlog(XH′,Gm,log/Gm)→HomSlog(XH,Gm,log/Gm)
fl fl
→HomSlog(XH′′,Gm,log/Gm)→0
fl
are both exact.
′ ′
(4) If G→G is injective, so is G →G .
log log
(5) If G→G′ is surjective, so is G →G′ .
log 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′ →G →G′′ →0.
log log log
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→Hl′og →Hlog →Hl′o′g →ExtSlog(XH′,Gm,log),
fl
it suffices to show ExtSlog(XH′,Gm,log) = 0. Since ExtSlog(Z,Gm,log) = 0,
fl fl
we are further reduced to show Ext (Z/nZ,G ) = 0 for any positive
Slog m,log
fl
integer n. The short exact sequence 0 → Z −→n Z → Z/nZ → 0 gives rise
to a long exact sequence 0 → Hom (Z/nZ,G ) → G −→n G →
Slog m,log m,log m,log
fl
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Ext (Z/nZ,G ) → 0. Since G −→n G is surjective, the sheaf
Slog m,log m,log m,log
fl
Ext (Z/nZ,G ) must be zero. The other short exact sequence is proved
Slog m,log
fl
similarly.
We prove part (4). Since G → G′ is injective, then the corresponding map
T →T′ onthetoruspartsisalsoinjectiveandthecorrespondingmapX′ →X
on the character groups is surjective. It follows that the induced map
G /G=Hom (X,G /G )→Hom (X′,G /G )=G′ /G′
log Slog m,log m Slog m,log m log
fl fl
is injective. Hence G →G′ is injective.
log log
Nowweprovepart(5). Letf denotethemapG→G′. Considerthetorusand
abelian variety decomposition of f
0 // T // G //B //0
(cid:15)(cid:15) ft (cid:15)(cid:15)f (cid:15)(cid:15)fab
0 // T′ //G′ // B′ //0.
We first show that f is surjective. Assume that the underlying scheme of S is
t
a point. The snake lemma gives anexact sequence Ker(f )→Coker(f )→0.
ab t
Since Coker(f ) is a torus and the reduced neutral component of Ker(f ) is
t ab
an abelian variety by [Bri15, Lem. 3.3.7], we must have Coker(f )=0. Hence
t
f is surjective. In the general case, f is fiberwise surjective, hence it is also
t t
set-theoretically surjective. The fibers of f over S are all flat, hence f is
t t
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 ´etale locally constant sheaves. Applying the
functor Hom (−,G /G ) to this short exact sequence, we get a long
Slog m,log m
exact sequencefl
→G /G→G′ /G′ →Ext (X/X′,G /G ).
log log Slog m,log m
fl
′
Let Z be the torsionpart of X/X , and let n be a positive integer such that
tor
nZ = 0. Since the multiplication-by-n map on G /G is an isomor-
tor m,log m
phism, wegetthatthe sheafExt (Z ,G /G )iszero. The torsion-free
Slog tor m,log m
nature of (X/X′)/Z implies Efxlt ((X/X′)/Z ,G /G ) = 0, hence
tor Slog tor m,log m
Ext (X/X′,G /G ) = 0. It ffolllows that G /G → G′ /G′ is surjec-
Slog m,log m log log
tive,fhl ence G →G′ is surjective.
log log
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Log Abelian Varieties over a Log Point 511
At last, we prove part (6). Consider the following commutative diagram
0 0 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //G′ //G′ // Hom (X′,G¯ ) //0
log Slog m,log
fl
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //G //G // Hom (X,G¯ ) //0
log Slog m,log
fl
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // G′′ //G′′ //Hom (X′′,G¯ ) //0
log Slog m,log
fl
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 0 0
with the first column and all rows exact, where G¯ denotes G /G .
m,log m,log 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 ´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 (X′,G¯ )→Hom (X,G¯ )→Hom (Z,G¯ )→0
Slog m,log Slog m,log Slog m,log
fl fl fl
and
0→Hom (Z/X′′,G¯ )→Hom (Z,G¯ )
Slog m,log Slog m,log
fl fl
→Hom (X′′,G¯ )→0.
Slog m,log
fl
But Hom (Z/X′′,G¯ ) = 0, it follows that the third column of the dia-
Slog m,log
fl
gram is exact. So is the middle column.
Recall that in [KKN08a, Def. 2.2], a log 1-motive M over Scl is defined as
E´t
a two-term complex [Y −→u G ] in the category of sheaves on Scl, with the
log E´t
degree −1 term Y an ´etale locally constant sheaf of finitely generated free
abelian groups and the degree 0 term G as above. Since both Y and G
log log
are sheaves on Slog, M can also be defined as a two-term complex [Y −→u G ]
fl log
inthecategoryofsheavesonSlog. Parallelto[KKN08a,2.3],wehaveanatural
fl
pairing
<,>:X ×Y →X×(G /G)=X×Hom (X,G /G )→G /G .
log Slog m,log m m,log m
fl
(2.1)
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512 Heer Zhao
It is clear that our pairing is induced from the one of [KKN08a, 2.3].
By our convention, T /T denotes the quotient in the category of sheaves on
log
Slog. For the quotient of T ⊂ T in the category of sheaves on Scl, we use
fl log E´t
the notation (T /T) . Now we assume that the pairing (2.1) is admissible
log Scl
E´t
(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 (X,(G /G ) )(Y) of the sheaf Hom (X,(G /G ) ) on
Scl m,log m Scl Scl m,log m Scl
E´t E´t E´t E´t
Scl is defined by
E´t
Hom (X,(G /G ) )(Y)(U):=
Scl m,log m Scl
E´t E´t
{ϕ∈Hom (X,(G /G ) )(U)|for every u∈U and x∈X ,
Scl m,log m Scl u¯
E´t E´t
′ ′
there exist y ,y ∈Y such that <x,y >|ϕ (x)|<x,y >}.
u,x u,x u¯ u,x u¯ u,x
Here, u¯ denotes a classical ´etale geometric point above u, and for a,b ∈
(Mgp/O×) , a|b means a−1b∈(M /O×) .
U U u¯ U U u¯
It is natural to define the analogue of Hom (X,(G /G ) )(Y) in the
Scl m,log m Scl
E´t E´t
log flat topology. We need the following lemma first.
Lemma 2.2. Let δ :Slog →Scl be the canonical map between these two sites.
fl E´t
(1) δ∗(Gm,log/Gm)=(Gm,log/Gm)Scl ⊗ZQ.
E´t
(2) Let H be a commutative group scheme over the underlying scheme of S
with connected fibres. Then Hom (H,G /G )=0.
Slog m,log m
fl
Proof. We denote the sheaf G /G on Slog by G¯ . For any positive
m,log m fl m,log
integer n, we have the following commutative diagram
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 // Z/n(1) //G n // G //0
m m
(cid:15)(cid:15) (cid:15)(cid:15)
0 // Z/n(1) // G n //G // 0
m,log m,log
(cid:15)(cid:15) (cid:15)(cid:15)
G¯ n // G¯
m,log ∼ m,log
=
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
with exact rows and columns, where Z/n(1) denotes the groupscheme of n-th
roots of unity. Applying the functor εfl∗ to the above diagram, we get a new
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Log Abelian Varieties over a Log Point 513
commutative diagram
0 // Z/n(1) //Gm(cid:127)_ n //Gm(cid:127)_
(cid:15)(cid:15) (cid:15)(cid:15)
0 // Z/n(1) //G n // G
m,log m,log
α γ
(cid:15)(cid:15) (cid:15)(cid:15)
εfl∗G¯m,log ∼n //εfl∗G¯m,log
=
β δ
(cid:15)(cid:15) (cid:15)(cid:15)
Gm(cid:127)_ n // Gm(cid:127)_ //R1εfl∗Z/n(1) η // R1εfl∗Gm n // R1εfl∗Gm
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Gm,log n // Gm,log θ //R1εfl∗Z/n(1) // R1εfl∗Gm,log
with exact rows and columns. Since the map G −→n G is surjective and
m m
R1εfl∗Gm,log =0, we get a new commutative diagram
G77 ⊗ZZ/n
0 //G n //G(cid:127)_♦♦♦♦♦ξ♦♦ω♦♦♦//♦R♦1♦εfl∼=∗Z(cid:15)(cid:15)(cid:127)θ¯/_ n(1) // 0
n1γ η
(cid:15)(cid:15) (cid:15)(cid:15)
0 //G α¯ //εfl∗G¯m,log β // R1εfl∗Gm // 0
with exact rows,where G denotes (G /G ) , α¯ (resp. θ¯) is the canonical
m,log m Scl
fl
map induced by α (resp. θ), ω is the canonical projection map and ξ is the
unique mapguaranteedby nβ◦(1γ)=δ◦(n(1γ))=δ◦γ =0. Takingcolimit
n n
of the above diagram with respect to n, we get a commutative diagram
0 // G // G⊗ZQ //G⊗ZQ/Z //0
(cid:15)(cid:15) (cid:15)(cid:15)
0 //G α¯ // εfl∗G¯m,log β // R1εfl∗Gm //0
with exact rows. Since the map G ⊗Z Q/Z → R1εfl∗Gm 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)Scl ⊗ZQ,
E´t
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where the last equality follows fromthe following fact: for any U ∈(fs/S), the
sheaf Mgp/O× on the small ´etale site of U is constructible. This proves part
U U
(1).
Now we prove part (2) which corresponds to [KKN08a, Lem. 6.1.1]. We have
HomSfllog(H,G¯m,log)=HomSEc´lt(H,δ∗G¯m,log)
=HomScl(H,(Gm,log/Gm)Scl ⊗ZQ).
E´t E´t
By the same argument of the proof of [KKN08a, Lem. 6.1.1], we have
HomScl(H,(Gm,log/Gm)Scl ⊗ZQ)=0.
E´t E´t
Hence part (2) is proved.
Now we define the analogue of Hom (X,(G /G ) )(Y). It is the sub-
SEl´otg m,log m SEc´lt
group sheaf Hom (X,G /G )(Y) of the sheaf Hom (X,G /G )
Slog m,log m Slog m,log m
fl fl
on Slog given by
fl
Hom (X,G /G )(Y)(U):=
Slog m,log m
fl
{ϕ∈Hom (X,G /G )(U)|after pushing forward to Ucl,
Slog m,log m E´t
fl
′
for every u∈U and x∈X , there exist y ,y ∈Y such that
u¯ u,x u,x u¯
′
<x,y >|ϕ (x)|<x,y >}.
u,x u¯ u,x
Here u¯ still denotes a classical ´etale geometric point above u. Let F :=
δF∗o(rGam,b,lo∈g/(GMmUg)p/=O(U×G)mu¯,⊗loZg/QG,ma|)bSEc´mlte⊗aZnsQaw−1itbh=δαth⊗eZcranfoorniscoamlemαap∈U(flMlogU→/OUU×E´c)ltu¯.
and r∈Q.
Remark 2.1. In [KKN08a, 7.1], admissibility and non-degeneracy are defined
for pairings into (G /G ) in the classical ´etale site on (fs/S). We can
m,log m Scl
E´t
defineadmissibilityandnon-degeneracyforpairingsintoG /G onthelog
m,log m
flatsiteinthesameway. SincebothX andY areclassical´etalelocallyconstant
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 (X,(G /G ) )(Y) on Scl
SEc´lt m,log m SEc´lt E´t
with the sheaf Hom (X,G /G )(Y) on Slog.
Slog m,log m fl
fl
Lemma 2.3. Let X,Y be two free abelian groups of finite rank, <,>:X×Y →
(G /G ) an admissible pairing on Scl. Let
m,log m SEc´lt E´t
Q :=Hom (X,(G /G ) )(Y), Q:=Hom (X,G /G )(Y),
cl SEc´lt m,log m SEc´lt Sfllog m,log m
and δ : Slog → Scl the canonical map between these two sites. Then we have
Q=δ∗Qcfll and δ∗E´Qt =Qcl⊗ZQ.
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