Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

1 MB·English

by  Jean-Benoit Bost

HERMITIAN VECTOR BUNDLES AND EXTENSION GROUPS ON
ARITHMETIC SCHEMES. I. GEOMETRY OF NUMBERS
JEAN-BENOˆIT BOSTAND KLAUSKU¨NNEMANN
7
0
0
Abstract. WedefineandinvestigateextensiongroupsinthecontextofArakelovgeome-
2
i
try. The“arithmeticextensiongroups”Ext (F,G)weintroduceareextensionsbygroups
n dX
ofanalytictypesoftheusualextensiongroupsExti (F,G)attachedtoO -modulesF and
a X X
J G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension
1
2 groupEdxtX(F,G)—theelementsofwhichmaybedescribedintermsofadmissibleshort
1 exactsequencesofhermitianvectorbundlesoverX —andweespeciallyconsiderthecase
when X is an “arithmeticcurve”,namely thespectrumSpecO of theringof integersin
K
] some number field K. Then the study of arithmetic extensions over X is related to old
T
and new problems concerning lattices and thegeometry of numbers.
N
Namely,for any two hermitian vector bundlesF and G overX :=SpecO , we attach
K
h. a logarithmic size sF,G(α) to any element α of Edxt1X(F,G), and we give an upper bound
t on s (α) in terms of slope invariants of F and G. We further illustrate this notion by
a F,G
m relating the sizes of restrictions to points in P1(Z) of the universal extension over P1Z to
the geometry of PSL (Z) acting on Poincar´e’s upper half-plane, and by deducing some
2
[
quantitative results in reduction theory from our previous upperbound on sizes. Finally,

1 we investigate the behaviour of size by base change (i.e., under extension of the ground
v field K to a larger number field K′): when the base field K is Q, we establish that the
3 size, which cannot increase under base change, is actually invariant when the field K′ is
4 an abelian extension of K, or when F∨⊗G isa direct sum of root lattices and of lattices
3
of Voronoi’s first kind.
1
Theappendicescontainresultsconcerningextensionsincategoriesofsheavesonringed
0
spaces, and lattices of Voronoi’s first kindwhich might also beof independentinterest.
7
0
MSC: Primary 14G40; Secondary 11H31, 11H55, 14F05, 18G15.
/
h
t
a
m
:
v Contents
i
X
0. Introduction 3
r
a
1. Preliminaries 9
1.1. Arithmetic schemes 9
1.2. Hermitian coherent sheaves 12
1.3. Extensions 13
1
2. The arithmetic extension group Ext (F,G) 13
X
2.1. Basic definitions 14
d
2.2. The first exact sequence 15
2.3. The second exact sequence 17
Date: January 11,2007.
1
2 JEAN-BENOˆITBOSTANDKLAUSKU¨NNEMANN
2.4. Pushout, pullback, and inverse image 18
2.5. Arithmetic extensions as homomorphisms in the derived category 21
2.6. Admissible extensions 29
2.7. Arithmetic torsors 30
3. Slopes of hermitian vector bundles and splittings of extensions over arithmetic
curves 31
3.1. Arithmetic degree and slopes 31
3.2. Euclidean lattices and direct images 34
3.3. First minimum, upper degree, and maximum slope 37
3.4. The upper degree of a tensor product 39
3.5. The size of an arithmetic extension 41
3.6. Size and operations on extensions 45
3.7. Covering radius and size of admissible extensions 46
3.8. The geometric case I 48
3.9. The geometric case II 49
4. Sizes of admissible extensions: explicit computations and an application to
reduction theory 51
4.1. Some explicit computations of size 52
4.2. Universal extensions and heights over P1 54
Z
4.3. An application to reduction theory 58
5. Invariance of the size under base change and Voronoi cells of euclidean lattices 62
5.1. A geometric consideration 62
5.2. Size and base change 63
5.3. The condition P(K′/K, E) and Voronoi cells 65
5.4. Base change from euclidean lattices 66
5.5. Cyclotomic base change 69
5.6. Base change from root lattices 72
5.7. Base change from lattices of Voronoi’s first kind 77
Appendix A. Extension groups 77
A.1. Notation and sign conventions 78
A.2. Extension groups of sheaves of modules 79
A.3. Extension groups of quasi-coherent sheaves of modules over schemes 80
A.4. Groups of 1-extensions 80
A.5. Extension groups of holomorphic vector bundles 85
Appendix B. Lattices of Voronoi’s first kind 87
B.1. Selling parameters 87
B.2. Examples 89
B.3. The Voronoi cell of an euclidean lattice with strictly obtuse superbase 94
References 98
EXTENSIONS ON ARITHMETIC SCHEMES I 3
0. Introduction
Theaimofthispaperistointroduceandtostudyarithmetic extensions andtheextension
groups they define in the framework of Arakelov geometry.
0.1. Arithmetic extensions are objects which arise naturally at various places in arithmetic
geometry. LetX bean arithmetic scheme –namely aseparated schemeof finitetypeover Z
– such that X is smooth, and let X(C) be the complex manifold of its complex points. By
C
definition, for any two locally free coherent -modules F and G, an arithmetic extension
X
O
( ,s) of F by G is given by an extension of -modules
X
E O
: 0 G E F 0
E −→ −→ −→ −→
together with a ∞-splitting over X(C)
C
s :F E ,
C C
−→
invariant undercomplex conjugation, of the extension of complex vector bundlesover X(C)
: 0 G E F 0
C C C C
E −→ −→ −→ −→
deduced from by extending the scalars from Z to C.
E
Recall that an hermitian vector bundle V := (V, . ) over X is the data of a locally free
kk
coherent sheaf V over X, together with a ∞-hermitian metric . on the attached vector
C kk
bundle V on X(C) that is invariant under complex conjugation. Arithmetic extensions
C
arise for instance from admissible extensions
(0.1) : 0 G E F 0,
E −→ −→ −→ −→
of hermitian vector bundles over X, namely from the data of an extension
: 0 G E F 0
E −→ −→ −→ −→
of the underlying -modules such that the hermitian metrics . and . on F and
OX kkF kkG C
G are induced (by restriction and quotients) by the metric . on E . In this case,
C kkE C
orthogonal projection determines a ∞-splitting s⊥ : F E of , and ( ,s⊥) is an
C C C
C → E E
arithmetic extensions of F by G.
It turns out that, by means of the Baer sum construction, one may define an addition
1
law on the set Ext (F,G) of isomorphism classes of arithmetic extensions of F by G,
X
which in this way is endowed with a natural structure of an abelian group. Moreover, in
1
analogy to the adrithmetic Chow groups, the arithmetic extension group Ext (F,G) is an
X
extension of the “classical” extension group Ext1 (F,G), defined in the context of sheaves
OX
of -modules, by a group of analytic type. More precisely, it fits into andexact sequence
X
O
(0.2) HomOX(F,G)−→HomCX∞(FC,GC)F∞ −→b Ext1X(F,G) −ν→ Ext1OX(F,G) −→ 0,
where F acts on X(C), F , and G by complex conjugation. We may also define an
∞ C C
d
homomorphism
Ψ : Ext1 (F,G) Z0,1(X ,F∨ G)
X −→ ∂ R ⊗
to the group Z0,1(X ,F∨ G) of F -invariant ∂-closed forms of type (0,1) on X(C) with
∂ R ⊗ d ∞
coefficients in F∨ G , by sending the class of an arithmetic extension ( ,s) to its “second
C ⊗ C E
fundamental form” ∂s.
4 JEAN-BENOˆITBOSTANDKLAUSKU¨NNEMANN
1
The arithmetic extension group Ext (F,G) actually admits an interpretation in terms of
homologicalalgebra,inthespiritofthewell-knownidentificationofthe“classical”extension
groupExt1 (F,G), originally defineddbyclasses of1-extensions equippedwiththeBaer sum,
X
with the “cohomological” extension group Hom (F,G[1]), defined as a group of
D(OX−mod)
morphismsinthederivedcategory of(sheaves of) -modulesover X . Indeed,if(X , ∞)
OX R CR
denotes the ringed space quotient of (X(C), ∞ ) by the action of complex conjugation
CX(C)
(acting both on X(C) and on values of C∞-functions), and if
ρ: (X , ∞) (X, )
R R X
C −→ O
is the natural map of ringed spaces, then, for any -module G on X, we may consider
X
O
the adjunction map
(0.3) ad :G ρ ρ∗G
G ∗
−→
— it maps any local section g of G to the section g , seen as a ∞-section of G , invariant
C C
C
under the complex conjugation F — and its cone C(ad ), namely (0.3) seen as complex
∞ G
of length 2, with G (resp. ρ ρ∗G) sitting in degree 1 (resp. 0). Then, for any two locally
∗
−
free coherent sheaves F and G on X, we have a natural isomorphism of abelian groups :
1 ∼
Ext (F,G) Hom F,C(ad )
X −→ D(OX−mod) G
between our arithmetic extension group and the group (cid:0)of morphism(cid:1)s from F to C(adG) in
d
the derived category D( mod) of the abelian category of sheaves of -modules over
X X
O − O
X (see 2.5 infra).
1
In a forthcoming part of this work, the above cohomological interpretation of Ext (F,G)
i
will motivate us to consider higher arithmetic extension groups Ext (F,G), defined for any
integer i 1 by means of the Dolbeault complex (A0,. ,∂) on X(C) and its sudbcomplex
≥ X(C)
0,. d
(A ,∂) of conjugation invariant forms, which defines a complex of sheaves of modules on
XR
the ringed space (X , ∞).
R CR
Forany ∞-moduleF onX ,wegetthe“Dolbeaultresolution” olb(F)ofF byapplying
CR R D
the functor F C∞ . to this complex. In particular, for any sheaf G of X-modules, we
⊗ R O
may consider the associated sheaf ρ∗G of ∞-modules over X , and the naive truncation
CR R
olb(ρ∗G) ofitsDolbeaultresolution. Theadjunctionmap(0.3)extendstoamorphism
≤i−1
D
of complexes
adi−1 :G ρ ( olb(ρ∗G) ),
G → ∗ D ≤i−1
and its cone C(adi−1) is a complex of (sheaves of) -modules.
G OX
For any two -modules F and G, we shall define
X
O
i
Ext (F,G) := Hom F,C(adi−1)[i 1] .
X D(OX−mod) G −
This group may be interpreted as an “hyper-extens(cid:0)ion group”: (cid:1)
d
Hom F,C(adi−1)[i 1] Exti F,C(adi−1)[ 1] ,
D(OX−mod) G − ≃ X G −
where, by the very definitions(cid:0)of the Dolbeault re(cid:1)solution a(cid:0)nd of a cone, the(cid:1)“shifted cone”
C(adi−1)[ 1] is the following complex of length i+1 of sheaves of -modules, with G
G − OX
sitting in degree 0:
0 −→ G −−a→dG ρ∗ρ∗G −−∂→G ρ∗(ρ∗G⊗CR∞ AX0,1R) −−∂→G ··· −−∂→G ρ∗(ρ∗G⊗CR∞ AX0,iR−1) −→ 0.
EXTENSIONS ON ARITHMETIC SCHEMES I 5
0.2. Classical constructions inalgebraic anddifferential geometry providenaturalinstances
of admissible and arithmetic extensions. In the second part of this paper [BK], we shall
discuss three of these constructions, which give rise to the arithmetic Atiyah extension,
the arithmetic Hodge extension, and the arithmetic Schwarz extension. To advocate the
investigation of arithmetic extensions, we want to indicate briefly their constructions:
(i) LetE bean hermitian vector bundleon an arithmetic scheme X. Thebundleof 1-jets
of E induces an extension of -modules
X
O
0 Ω1 E 1 (E) E 0,
−→ X/Z⊗ −→ JX/Z −→ −→
the Atiyah extension of E. The holomorphic vector bundle E carries a unique ∞-
C
C
connection which is compatible with the metric and the complex structure, its so-called
Chern connection, which induces a ∞-splitting s of the Atiyah extension and yields a
C
canonical arithmetic extension class
1
at(E) Ext (E,Ω1 E).
∈ X X/Z ⊗
It is a refinement both of the algebraic Atiyah class at(E) = ν(at(E)) in Ext1(E,Ω1 E)
b d X/Z⊗
and of the curvature form of the Chern connection of E , which coincides with Ψ(at(E))
C
(up to some normalization factor). Applying a trace map to abt(E), we get an arithmetic
first Chern class in “arithmetic Hodge cohomology”: b
1 b
cˆH(E) Ext ( ,Ω1 ).
1 ∈ X OX X/Z
(ii) Let f : X Y be a smooth proper morphism of arithmetic schemes such that the
d
Hodge to de Rham→spectral sequence Ep,q = Rpf Ωq Rp+qf Ω· degenerates at E .
1 ∗ X/Y ⇒ ∗ X/Y 1
The spectral sequence defines the so-called Hodge extension
(0.4) Hdg(X/Y) : 0 f Ω1 R1f Ω· R1f 0
−→ ∗ X/Y −→ ∗ X/Y −→ ∗OX −→
whose interest was already advocated by Grothendieck in [Gro66]. Complex Hodge theory
equips Hdg(X/Y) with a canonical structure of an arithmetic extension1. We thus obtain
the class of the arithmetic Hodge extension
1
Hdg(X/Y) Ext (R1f ,f Ω1 ).
∈ Y ∗OX ∗ X/Y
(iii) Let f : C X be a smooth, projective curve of genus g 2 over an arithmetic
d d
→ ≥
scheme X. Using Deligne’s definition of the torsor of projective connections on relative
curves in [Del70], I.52, one obtains a canonical extension of -modules
X
O
: 0 f Ω⊗2 S 0,
SC/X −→ ∗ C/X −→ C/X −→ OX −→
the splittings of which correspond to projective connections on C/X. Complex uniformiza-
tion by the upper half-plane H induces a ∞ projective connection on C(C)/X(C) that is
C
1Namely,thevectorbundleoverY(C)definedbytherelativealgebraic deRhamcohomology R1f Ω•
∗ X/Y
may be identified with the relative first Betti cohomology with complex coefficients of X(C)/Y(C); the
complex conjugation on coefficients acts on Betti cohomology and maps the C-analytic sub-vector bundle
(f∗Ω1X/Y)C of(R1f∗Ω•X/Y)C ontoaC∞ directsummandof(f∗Ω1X/Y)C,whichprovidesaC∞-splittingofthe
extension of C-analytic vector bundlesover Y(C) defined byHdg(X/Y).
2Strictly speaking, the definition in loc. cit. is stated in the framework of complex analytic spaces.
However, it is formulated in a general geometric language, which makes it meaningful in the context of
smooth relative curvesoveran arbitrary scheme.
6 JEAN-BENOˆITBOSTANDKLAUSKU¨NNEMANN
holomorphic along the fibers — hence a ∞-splitting of over Y(C) — and allows one
C/X
C S
to define from the arithmetic Schwarz extension and its class
C/X
S
1
Ext ( ,f Ω⊗2 ).
SC/X ∈ X OX ∗ C/X
Thenon-vanishingofeachoftbheabovedclassescˆH(E),Hdg(X/Y),or isanintriguing
1 SC/X
issue, related to deep problems in Diophantine geometry and transcendence theory.
d b
1
0.3. In this paper, after introducing the arithmetic extension groups Ext (F,G) and
X
discussing their basic properties in Section 2, we concentrate on the case where X is an
“arithmetic curve”, namely the spectrum Spec of the ring of integers din some number
K
O
field K. It turns out that the study of arithmetic extensions over X is related to old and
new problems concerning lattices and the geometry of numbers.
Namely, ifF andGarevector bundlesover X := Spec (i.e.,projective -modules),
K K
O O
we obtain from the basic exact sequence (0.2) a canonical isomorphism
(0.5) Ext1 (F,G) HomOK(F,G)⊗ZR.
X ≃ Hom (F,G)
OK
d 1
Consequently the arithmetic extension group Ext (F,G) carries a canonical structure of a
X
real torus. Moreover, if F and G are equipped with hermitian metrics, which makes them
hermitian vector bundlesF and G, we get an indducedRiemannian metric on this real torus.
In Section 3, we define the size s ( ,s) of an arithmetic extension of F by G as the
F,G E
logarithm (in [ ,+ [) of the distance to zero of the corresponding point in the torus
−∞ ∞
(0.5). Let be an admissible extension (0.1) with associated arithmetic extension ( ,s⊥)
E E
as above, and let
∼
ϕ : E G F
−→ ⊕
be an isomorphism of -modules compatible with the extension (that is, such that
K
O E
ϕ−1 (Id ,0) : G E and pr ϕ : E F coincide with the morphisms defining ). Then
◦ G → 2◦ → E
1 1
ϕ 2 = ϕ−1 2 rk E,
[K :Q] k σkE∨⊗(G⊕F),σ [K :Q] k σ k(G⊕F)∨⊗E,σ ≥ OK
σ:K֒→C σ:K֒→C
X X
andtheminimumvalueachievedbytheleft-handsidewhenϕrunsoveralltheisomorphisms
of -modules as above is precisely
K
O
rk E+exp(2s ( ,s⊥))
OK F,G E
(see Proposition 3.5.3 and Corollary 3.5.5 infra).
Motivated by analogous results concerning vector bundles on projective curves over a
field, we show that the size of arithmetic extensions satisfies the following upper bound:
log ∆ rk F rk G
K K K K K
(0.6) s ( ,s) µ (F) µ (G)+ | | +log · ,
F,G E ≤ max − min [K : Q] 2
whereµmax(F)andµmin(bG)denotebthemaximalandminimalnormalizedslopesof F andG
(see 3.1, infra), and ∆ the discriminant of the number field K. To establish (0.6), we rely
K
on(i)someupperboundontheArakelov degreeofasub-linebundleinthetensorproductof
b b
two hermitian vector bundles over Spec , and (ii) some “transference theorem” from the
K
O
geometry of numbers, which relates the inhomogeneous minimum (also called the covering
EXTENSIONS ON ARITHMETIC SCHEMES I 7
radius) of a lattice in a euclidean vector space to the first of the successive minima of the
dual lattice.
Section 4 is devoted to further examples and applications of the notion of size. In partic-
ular, usingtheinequality (0.6), we derive an avatar, in theframework of Arakelov geometry
over arithmetic curves, of the main result of the classical reduction theory of positive qua-
dratic forms. It claims the existence of some “almost-splitting” for any hermitian vector
bundle E over Spec , namely the existence of n := rkE hermitian lines bundles L , ...,
K 1
O
L over Spec , and of an isomorphism of -modules
n K K
O O
n
∼
φ :E L
i
−→
i=1
M
suchthat thearchimedean normsofφandφ−1, computedby usingthehermitian structures
on E and on the orthogonal direct sum n L , are bounded in terms of K and n only
i=1 i
(Theorem 4.3.1 infra).
L
Besides, for any rational point P P1(Q) = P1(Z), we calculate the size of the inverse
∈
image P∗ of the universal extension
E
:0 S ⊕2 (1) 0
E −→ −→ OX −→ OX −→
over the projective line X = P1 equipped with its natural structure of an admissible exten-
Z
sion. The extension class of P∗ is trivial iff P 0, . For P P1(Q) 0, , we show
E ∈ { ∞} ∈ \{ ∞}
that the size of P∗ is related to the usual height h(P) of P by the inequalities
E
1
log 2+h(P) s(P∗ ) log 2+2h(P).
−2 ≤ E ≤ −
We also give a geometric description of the size s(P∗ ) by means of so-called Ford circles
E
(namely the images under elements in SL (Z) of the horocycles Imz = 1 in Poincar´e’s
2
{ }
upper half-plane).
The final section of this paper is devoted to the intriguing question of the invariance of
size underbase change. Recall thatan extension of numberfieldsK′/K definesamorphism
g : Spec Spec of “arithmetic curves”. For hermitian vector bundles F and G
K′ K
O → O
over S, there is an induced morphism
1 1
g∗ : Ext (F,G) Ext (g∗F,g∗G).
S −→ S′
It is easy to see that the inequality
d d
(0.7) s (g∗e) s (e)
g∗F,g∗G ≤ F,G
1
holds for every extension class e Ext (F,G). Motivated again by geometric consider-
∈ S
ations, we ask – at least if K is the field Q – wether the size of extensions of F by G is
invariant underthebasechangeg, namdelywhethertheinequality (0.7)isindeedanequality
1
for any extension class e Ext (F,G).
∈ S
∨
LetE denotethehermitianvector bundleF Gover Spec . Theextension ofscalars
K
֒ defines a naturadl R-linear map ⊗ O
K K′
O → O
∆ : E = E R (g∗E) = (E ) R.
R ⊗Z −→ R ⊗OK OK′ ⊗Z
8 JEAN-BENOˆITBOSTANDKLAUSKU¨NNEMANN
Let (E) E denote theVoronoi cell of theeuclidean lattice E E underlyingE. Then
R R
V ⊆ ⊆
the size of extensions of F by G is invariant under the base change g if and only if
(0.8) ∆ (E) (g∗E).
V ∈V
Clearly (0.8) holds iff ∆ maps the set(cid:0)of ver(cid:1)tices of the polytope (E) to (g∗E).
V V
Here are some results which point towards a positive answer to our question in the case
where the base field is Q. Hence assume K =Q, put L = K′, and define E as above. Then
1
we show that (0.7) is an equality for any extension class e Ext (F,G) if either
∈ S
(i) L/Q is an abelian extension, or
d
(ii) E is an orthogonal direct sum of hermitian line bundles, or
(iii) E is a root lattice, or
(iv) E is a lattice of Voronoi’s first kind (hence in particular if rk E 3).
Z
≤
We use condition (0.8) to prove these results. For abelian extensions, we reduce to
the cyclotomic case and use some auxiliary results of Kitaoka, which he established when
investigating minimal vectors in tensor productsof euclidean lattices. Using theelementary
inequality
σ(α)2 Reσ(α) σ(α)2 σ(α) 0
| | − ≥ | | − | | ≥
σ:L֒→C σ:L֒→C σ:L֒→C σ:L֒→C
X X X X
satisfied by any integral element α , we show that (0.8) holds when E has rank one,
L
∈ O
and consequently when it splits as a direct sum of hermitian line bundles. Our proof for
root lattices relies on the computation of the vertices of the Voronoi cells of the irreducible
root lattices A , D , E , E , and E by Conway and Sloane ([CS99], Chapter 21). Our
n n 6 7 8
treatment of lattices of Voronoi’s first kind uses the description of the Voronoi cell of an
euclidean lattice with strictly obtuse superbase which is given in Appendix B.
Finally, as aconsequenceofour“reductiontheorem”andofcase(ii), weshow,inthecase
where the base field K is Q, that equality holds in (0.7) “up to some constant”. Namely,
we derive the existence of a non-negative real constant c(rkF,rkG) — depending on the
ranks of F and G only — such that the inequality
s (e) s (g∗e)+c(rkF,rkG)
F,G ≤ g∗F,g∗G
1
holds for any class e Ext (F,G).
∈ S
Appendix A gathers “well known” facts concerning extension groups of sheaves of mod-
ules. In particular, it spdecifies sign conventions which enter in the construction of canonical
isomorphisms between variously defined extension groups.
Appendix B contains a self-contained presentation of lattices of Voronoi’s first kind, a
description of their Voronoi cells, and various facts concerning these lattices which might
be of independent interest.
0.4. The starting points of this paper have been, in 1998, (i) the observation that, for any
two hermitian vector bundles F and G over an arithmetic curve X, the set of isomorphism
classes of admissible extensions of F by G becomes an abelian group when the Baer sum of
twoadmissibleextensionsisequippedwiththehermitianstructuredefinedbyformula(2.37)
infra, and (ii) Grothendieck’s remark in [Gro66] on the non-trivial information encoded in
the extension class of the Hodge extension (0.4).
EXTENSIONS ON ARITHMETIC SCHEMES I 9
Related ideas have been investigated in Mochizuki’s preprints [Moc99]. Let us emphasize
amajor differencebetween hisapproach andours: Mochizuki thinksoftheHodgeextension
in the context of Arakelov geometry as some kind of non-linear geometric object, while
we see it as an element of some naturally defined abelian extension group. Moreover,
Mochizuki’s earlier work [Moc96] has been an inspiration for considering the arithmetic
Schwarz extension.
Let us finally indicate that, in [CLT01], Chambert-Loir and Tschinkel have defined and
investigated “arithmetic torsors” under some group scheme G on an arithmetic scheme X,
at least whenGis deduced by basechange froma groupscheme over an “arithmetic curve”.
Their definition easily extends to the case of general smooth affine group schemes over X,
and specialized to vector groups of the form Eˇ F, where E and F are vector bundles over
⊗
X, is equivalent to our definition of arithmetic extensions of E by F (see 2.7, infra).
ItisapleasuretothankE.Bayerforhelpfulremarksoneuclideanlattices, andR.Bostfor
his help in the preparation of the figure. We are grateful to the TMR network ‘Arithmetic
geometry’ and the DFG-Forschergruppe ‘Algebraische Zykel und L-Funktionen’ for their
support and to the universities of Regensburg and Paris (6,7,11,13) for their hospitality.
1. Preliminaries
1.1. Arithmetic schemes. We work over an arithmetic ring R = (R,Σ,F ) in the sense
∞
of Gillet and Soul´e, [GS90, 3.1.1]. Recall that this means that R is an excellent regular
noetherian integral domain, Σ is a finitenonempty set of monomorphismsfrom R to C, and
F is a conjugate-linear involution of C-algebras F : CΣ CΣ such that F δ = δ for
∞ ∞ ∞
→ ◦
the canonical map δ :R CΣ = C.
→ σ∈Σ
LetS bethespectrumofan arithmetic ringR, andK its fieldof fractions. An arithmetic
Q
scheme3 X over R is a separated S-scheme X of finite type such that each base change
X = X C, σ Σ, is smooth over SpecC (or equivalently such that X is smooth
σ R,σ K
× ∈
over K). For σ in Σ, we write X = X C. We obtain a scheme
σ R,σ
⊗
X = X CΣ = X
Σ R,δ σ
⊗
σ∈Σ
a
and a complex manifold
X (C)= X (C).
Σ σ
σ∈Σ
a
We write X(C) instead of X (C) if Σ = σ :R ֒ C .
Σ
{ → }
1.1.1. The most prominent example of an arithmetic ring is = ( ,Σ,F ) where
K K ∞ K
O O O
is the ring of integers in a number field K, Σ is the set of complex embeddings σ : K ֒ C,
→
and F is given by
∞
F : CΣ CΣ , (z ) (z ) .
∞ σ σ∈Σ σ σ∈Σ
−→ 7→
Then an arithmetic scheme over is precisely a separated Z-scheme X of finite type
K
O
such that X is smooth, equipped with a scheme morphism to Spec , and X (C) is the
Q K Σ
O
complex manifold X(C) of all complex points of X.
3Weusetheterminology arithmetic scheme forwhatiscalled anarithmetic variety in[GS90]andsubse-
quent papers by Gillet and Soul´e, in order to avoid confusion with quotients of symmetric domains by the
action of arithmetic groups.
10 JEAN-BENOˆITBOSTANDKLAUSKU¨NNEMANN
1.1.2. There are natural morphisms of locally ringed spaces
j :(X (C), hol) (X, )
Σ OXΣ −→ OX
where hol denotes thesheaf of holomorphic functionson thecomplex manifold X (C) and
OXΣ Σ
κ :(X (C), ∞ ) (X (C), hol)
Σ CXΣ −→ Σ OXΣ
where ∞ denotesthesheafofcomplexvaluedsmoothfunctions. Themorphismj isflatby
CXΣ
[SGA03, Exp. XII]. To any -module F on X is associated an hol-module Fhol = j∗F
OX OXΣ C
on X (C) and an ∞ -module F = κ∗j∗F. The so-defined functor F F is exact, as
Σ CXΣ C 7−→ C
a consequence of following Lemma:
Lemma 1.1.3. The morphism κ is flat, i.e. ∞ is a flat hol -module for each p in
CXΣ,p OXΣ,p
X (C).
Σ
Proof. We consider for n 0 the R-algebra (resp. ) of germs of real analytic
R2n,0 R2n,0
≥ O E
(resp. real valued ∞) functions around 0 in R2n, and the C-algebra hol of germs of
C OCn,0
holomorphic functions around 0 in Cn. The canonical map from to is flat by
R2n,0 R2n,0
O E
[Tou72, VI Cor. 1.3]. We have C = ∞ under the canonical identification of Cn
ER2n,0⊗R CCn,0
with R2n. Therefore κ is flat if we show that C is flat over hol . This can be
OR2n,0 ⊗R OCn,0
checked on completions (which are faithfully flat). We have
\ C= C[[z ,..,z ,z ,..,z ]]
R2n,0 R 1 n 1 n
O ⊗
and \hol = C[[z ,..,z ]]. Our claim follows. (cid:3)
OCn,0 1 n
1.1.4. Let F denote the anti-holomorphic involution of the complex manifold X (C)
∞ Σ
which maps s : Spec C X to the composition of complex conjugation in C with s. We
→
obtain an induced C-antilinear involution
F : Ak(X (C),C) Ak(X (C),C) , α F∗ (α)
∞ Σ −→ Σ 7→ ∞
on the space of smooth complex valued k-forms on X (C). One checks easily that this map
Σ
is C-anti-linear and Γ(X, )-linear. Furthermore it respects the (p,q)-type and commutes
X
O
with d, ∂, and ∂.
1.1.5. For any -module F on X, we consider the sheaf
X
O
Ak( ,F) := FChol ⊗OXhoΣl Ak( ,C) = FC ⊗CX∞Σ Ak( ,C)
on X (C). It may be decomposed according to types:
Σ
(1.1) Ak( ,F) = Ap,q( ,F)
p+q=k
M
where, for any two non-negative integers p and q:
Ap,q( ,F) := FChol⊗OXhoΣl Ap,q( ,C)= FC ⊗CX∞Σ Ap,q( ,C).
The space of sections Ak(X (C),F) is endowed with the C-antilinear involution F
Σ ∞
(which specializes to the one considered above when F = ), defined by complex conju-
X
O
gation, acting both on X (C) and on the coefficients (namely, k-forms and fibers of F ).
Σ C