On the arithmetic Chern character

0.12 MB·English

by  H. Gillet

ON THE ARITHMETIC CHERN CHARACTER
H.GILLETAND C.SOULE´
2
1
0
2 Let X be a proper and flat scheme over Z, with smooth generic fiber X . In [4]
Q

n weattachedtoeveryhermitianvectorbundle E =(E,kk)onX aCherncharacter
a class lying in the arithmetic Chow groups of X:
J
p
4 ch(E)∈ ⊕ CH (X)⊗Q.
2 p≥0
b d

Unlike the usual Chern character with values in the ordinary Chow groups, ch is
]
G not additive on exact sequences; indeed suppose that E , i = 0,1,2 is a triple of
i b
A hermitian vector bundles on X, and that we are give an exact sequence
h. 0→E0 →E1 →E2 →0
t
a ofthe underlyingvectorbundles onX,(i.e. inwhichweignorethe hermitianmet-
m
rics). Then the difference ch(E )+ch(E )−ch(E ), is representedby a secondary
0 2 1
[ characteristicclasschfirstintroducedbyBottandChern[1]anddefinedingeneral
b b b
1 in[2]. TheseBott-Cehernformsmeasurethe defectinadditivityofthe Chernforms
v associated by Chern-Weil theory to the hermitian bundles in the exact sequence.
8
Assume now that the sequence
6
9 0→E →E →E →0 (∗)
0 1 2
4
. is exact on the generic fiber X but not on the whole of X. We shall prove here
1 Q
0 (Theorem 1) that ch(E0)+ch(E2)−ch(E1) is the sum of the class of ch and the
2 localizedCherncharacterof(∗)(see[3],18.1). Thisresultfitswellwiththeideathat
b b b e
1 characteristicclasseswithsupportonthefinitefibersofX arethenon-archimedean
:
v analogs of Bott-Chern classes (see [6]).
i
X In Theorem 2 we compute more explicitly these secondary characteristic classes
in a situation encountered when proving a “Kodaira vanishing theorem” on arith-
r
a metic surfaces ([7], 3.3).
Notation. If A is an abelian group we let A =A⊗Q.
Q
Z
1. A general formula
1.1. Let S = Spec(Z) and f : X → S a flat scheme of finite type over S. We
assume that the generic fiber X is smooth and equidimensional of dimension d.
Q
For every integer p ≥ 0 we denote by App(X ) the real vector space of smooth
R
real differential forms α of type (p,p) on the complex manifold X(C) such that
Date:January25,2012.
ThismaterialisbaseduponworksupportedinpartbytheNationalScienceFoundationunder
GrantNo. DMS-0901373.
1
2 H.GILLETANDC.SOULE´
F∗(α) = (−1)pα, where F is the anti-holomorphic involution of X(C) induced
∞ ∞
by complex conjugation. Let
A(X)= ⊕ App(X )
R
p≥0
and
A(X)= ⊕ Ap−1,p−1(X )/(Im(∂)+Im(∂)).
R
p≥1
e
For every p ≥ 0 we let CH (X) be the p-th arithmetic Chow homology group of
p
X ([5], §2.1, Definition 2). Elements of CH (X) are represented by pairs (Z,g)
d p
consisting of a p-dimensional cycle Z on X and a Green current g for Z(C) on
d
X(C). Recall that here a Green current for Z(C) is a current (i.e. a form with
distributioncoefficients)oftype(p−1,p−1)suchthatddc(g)+δ isC∞,δ being
Z Z
the current of integration on Z(C) ). There are canonical morphisms ([5], 2.2.1):
z :CH (X) → CH (X)
p p
d(Z,g) 7→ Z
and
ω :CH (X) → App(X )
p R
d(Z,g) 7→ ddc(g)+δZ.
Let CHfin(X)be the ChowhomologygroupofcyclesonX the supportof which
p
does not meet X . There is a canonical morphism
Q
b:CHfin(X)→CH (X)
p p
mapping the class of Z to the class of (Z,0).dThe composite morphism
z◦b:CHfin(X)→CH (X)
p p
is the obvious map. Let
a:Ad−p−1,d−p−1(X )→CH (X)
R p
be the map sending η to the class of (0,η). We hadve
ω◦a(η)=ddc(η).
1.2. We assume given a sequence
0→E →E →E →0
0 1 2
of hermitian vector bundles on X, the restriction of which to X is exact. Let
Q
chfin(E )∩[X]∈CH (X) = ⊕ CHfin(X)
• fin Q p Q
p≥0
be the localized Chern character of E ([3] 18.1), and
•
ch(E )∈A(X)
• Q
the Bott-Chern secondary characteeristic claess [2], such that
2
ddcch(E )= (−1)ich(E ),
• i,C
X
i=0
e
ON THE ARITHMETIC CHERN CHARACTER 3
where ch(E )∈A(X) is the differential form representing the Chern character of
i,C
the restriction E of E to X(C). Finally, if i=0,1,2, we let
i,C i
ch(E )∩[X]∈CH(X) = ⊕ CH (X)
i Q p Q
p≥0
b d d
be the arithmetic Chern character of E ([4] 4.1, [5] Theorem 4).
i
Theorem 1. The following equality holds in CH(X) :
Q
2 d
(−1)ich(E )∩[X]=b(chfin(E )∩[X])+a(ch(E )).
i • •
X
i=0
b e
1.3. This theorem is a special case of Lemma 21 in [5], though this may not be
immediately apparent. Therefore, for the sake of completeness, we give a proof
here.
1.4. To prove Theorem 1 we consider the Grassmannian graph construction ap-
plied to E ([3] 18.1, [5] 1.1). It consists of a proper surjective map
•
π :W →X ×P1
such that, if φ⊂X is the support of the homology of E (hence φ is empty), the
• Q
restriction of π onto (X −φ)×P1 and X ×A1 is an isomorphism. The effective
Cartier divisor
W =π−1(X ×{∞})
∞
is the union of the Zariski closure X of (X −φ)×{∞} with Y = π−1(φ×{∞}).
The sequence E extends to a complex
• e
0→E →E →E →0,
0 1 2
whichisisomorphictothe pull-baeckofEe overeX×A1. The restrictionofE toX
• •
is canonicallysplit exact. On W =X ×P1 the sequence E is exact;it coincides
Q Q Q • e e
with E (resp. 0→E →E ⊕E →E →0) when restricted to X ×{0} (resp.
• 0 0 2 2 e Q
X ×{∞}). WechooseametriconE forwhichtheseisomorphismsareisometries.
Q •
1.5. Let e
2
x= (−1)ich(E ),
i
X
i=0
b e
and denote by t the standard parameter of A1. In the arithmetic Chow homology
of W we have
0=x∩(W −W ,−log|t|2).
0 ∞
If x is the class of (Z,g), with Z meeting properly W and W , we get
0 ∞
x∩(W −W ,−log|t|2)=(Z ∩(W −W ),g∗(−log|t|2)),
0 ∞ 0 ∞
where the ∗-product is equal to
g∗(−log|t|2)=g(δ −δ )−ch(E )log|t|2.
W0 W∞ •
Since W =X ∪Y, with Y =∅, we get e
∞ Q
(1) 0 = xe∩(W −W ,−log|t|2)
0 ∞
= (Z∩W0,gδW0)−(Z ∩X,gδXe)−(Z∩Y,0)−(0,ch(E•)log|t|2).
e e
4 H.GILLETANDC.SOULE´
The restriction of E to X is split exact, therefore
•
e e (Z ∩X,gδe)=0.
X
Applying π to (1) we get e
∗
(2) 0=ch(E )−π (Z∩Y),0)−(0,π (ch(E )log|t|2).
• ∗ ∗ •
By definition of theblocalized Chern character ([3], 18.1e, (14))
(3) π (Z∩Y)=chfin(E )∩[X]
∗ •
in CHfin(X) . On the other hand we deduce from [4], (1.2.3.1), (1.2.3.2) that
Q
(4) − π (ch(E )log|t|2)=ch(E ).
∗ 1 •
and upon replacing t by 1/t, as in tehe proof of (1.e3.2) in [4], we see that
(5) π (ch(E )log|t|2)=−π (ch(E )log|t|2).
∗ • ∗ 1
Theorem 1 follows from (2)e, (3), (4), (5). e
2. A special case
2.1. We keep the hypotheses of the previous section, and we assume that X is
normal,d=1, E andE haverankoneandthe metricsonE andE areinduced
0 2 0 2
by the metric on E . Finally, we assume that there exists a closed subscheme φ in
1
X whichis 0-dimensionalandsuchthatthere isanexactsequenceofsheavesonX
(6) 0→E →E →E ⊗I →0,
0 1 2 φ
where I is the ideal of definition of φ.
φ
Let
c ∈A1,1(X )/(Im(∂)+Im(∂))
2 R
be the second Bott-Chern class of (6), Γ(φ,O ) the finite ring of functions on φ
e φ
and #Γ(φ,O ) its order. Let
φ
f :CH (X) →CH (S)=R
∗ 0 Q 0
be the direct image morphism.d d
Theorem 2. We have an equality of real numbers
f (c (E )∩[X])=f (c (E )c (E )∩[X])− c +log#Γ(φ,O ).
∗ 2 1 ∗ 1 0 1 2 Z 2 φ
X(C)
b b b e
2.2. To prove Theorem 2 we remark first that
c (E )=c (E )+c (E ),
1 1 1 0 1 2
because the metrics on E0 band E2 arbe inducedbfrom E1. Therefore, since ch2 =
−c + c21, we get
2 2
(c (E )+c (E ))2
1 0 1 2
ch (E ) = −c (E )+
2 1 2 1
2
b b
b = −cb (E )+c (E )c (E )+ch (E )+ch (E ).
2 1 1 0 1 2 2 0 2 2
By Theorem 1, this impliesbthat b c c
(7) c (E )∩[X]=c (E )c (E )∩[X]+b(chfin(E )∩[X])+a(ch(E )).
2 1 1 0 1 2 • •
b b b e
ON THE ARITHMETIC CHERN CHARACTER 5
Since ch (E ) and ch (E ) vanish we have
0 • 1 •
e e ch(E )=−c .
• 2
Therefore, if we apply f∗ to (7)e, we get e
f (c (E )∩[X])=f (c (E )c (E )∩[X])− c +f (b(chfin(E )∩[X])),
∗ 2 1 ∗ 1 0 1 2 Z 2 ∗ •
X(C)
b b b e
and we are left with showing that
(8) f ◦b(chfin(E )∩[X])=log#Γ(φ,O ).
∗ • φ
Let|φ|={P ,··· ,P }⊂X bethesupportofφandψ =f(|φ|)⊂S. Thefollowing
1 n
diagram is commutative:
b //
CH (φ) CH (X)
0 0
d
f∗ f∗
(cid:15)(cid:15) (cid:15)(cid:15)
CH (ψ) b //CH (S)=R,
0 0
where d
b:CH (ψ)=Zψ →R
0
maps (n ) to n log(p).
p p∈ψ p
Pp
For any prime p ∈ ψ we let Z be the local ring of S at p and we let ℓ =
(p) p
ℓ (φ)≥0 be the length of the finite Z -module Γ(φ,O )⊗Z . Clearly
p (p) φ (p)
log#Γ(φ,O )= ℓ log(p),
φ p
X
p∈ψ
hence it is enough to prove that
(9) f (chfin(E )∩[X])=(ℓ )∈CH (ψ) =Qψ.
∗ • p 0 Q
The complex E defines an element
•
n n
[E ]= [O ]∈Kφ(X)= KPi(X).
• φ,Pi 0 0
X M
i=1 i=1
Toprove(9),byreplacingX byanaffineneighbourhoodofP,onecanassumethat
|φ|={P}, and it is enough to show that, if p=f(P),
f (chfin(O )∩[X])=ℓ (O )[p].
∗ φ,P p φ,P
Now recall that, if F is a coherent sheaf on a scheme X of finite type over S,
supported on a finite set of closed points, the associated 0-cycle
[F]= ℓ (F )[P]∈Z (X)
p P 0
X
P∈|F|
is such that, if f :X →Y is a proper morphism of schemes of finite type over S,
f [F]=[f (F)]
∗ ∗
([3] , 15.1.5). Hence it is enough to show that
(10) chP(O )=ℓ (O )[P]∈CH (P) ≃Q.
φ,P p φ,P 0 Q
6 H.GILLETANDC.SOULE´
Replacing X by an affine neighbourhood of P, we may assume that we have an
exact sequence
(11) 0−→O −α→O2 −β→O −→O −→0.
X X X φ
HencetheidealI ⊂O (X)isgeneratedbytwoelementsβ andβ . SinceX is
φ X 1 2
normal, its local rings satisfy Serre property S and, as dim(X)= 2, X is Cohen-
2
Macaulay. Sinceβ andβ spananidealofheighttwo,(β ,β )isaregularsequence
1 2 1 2
andthe sequence (11) is isomorphic to the Koszulresolutionof O =O /(β ,β ).
φ X 1 2
Now (10) can be deduced from the following general fact:
Lemma 1. Let X =Spec(A) be an affine scheme and Z ⊂X a closed subset such
that the ideal I =(x ,...,x ) is generated by a regular sequence (x ,...,x ). Let
Z 1 n 1 n
K (x ,...,x ) be the Koszul complex associated to (x ,...,x ). Then
• 1 n 1 n
chZ(K (x ,...,x ))=[O ]∈CH (Z) .
n • 1 n Z 0 Q
Proof. The Grassmannian-graph construction on K (x ,...,x ) coincides with
• 1 n
the deformation to the normal bundle of Z in X. If W is defined as in 1.4,
W =X ∪P(N ),
∞ Z/X
e b
where X is the blow up of X along Z, and P(N ) is the projective comple-
Z/X
tion of the normal bundle of Z in X. The pull back of the Koszul complex
e b
K (x ,...,x ) to W\W extends to a complex K (x ,...,x ) on W. The re-
• 1 n ∞ • 1 n
striction of K (x ,...,x ) to X is acyclic while the restriction of K (x ,...,x )
• 1 n e • 1 n
to P(N ) is a resolution of the structure sheaf of the zero section Z ⊂ N ⊂
Z/X e e e Z/X
P(NbZ/X).
b Now observe that Z ⊂ P(N ) is an intersection of Cartier divisors D ,...,
Z/X 1
D , hence
n b
ch(K•(x1,...,xn)|bP(NZ/X))
n e
= Ych(O(−Di)→ObP(NZ/X))
i=1
n
= ch(O(D )).
i
Y
i=1
Since
ch(O )=ch (O )+x =[D ]+x
Di 1 Di i i i
where x has degree ≥2, we get
i
ch(K•(x1,...,xn)|bP(NZ/X))=[D1]...[Dn]=[Z].
e
This ends the proof of Lemma 1 and Theorem 2.
ON THE ARITHMETIC CHERN CHARACTER 7
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I,II,III.Comm. Math. Physics115(1988), 49–78, 79–126, 301–351.
[3] Fulton,W.: IntersectionTheory.Springer1984.
[4] Gillet,H.,Soul´e,C.: Characteristicclassesforalgebraicvectorbundleswithhermitianmetrics
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577-599.
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at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street,
Chicago,IL 60607-7045,USA
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