Annales de la Faculté des Sciences de Toulouse

Gérard Gagneux

Olivier Millet

Sur le système de Nernst-Planck-Poisson-Boltzmann résultant de l’homogénéisation par convergence à double échelle

H. Gillet

C. Soulé

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
H. GILLET, C. SOULÉ
On the arithmetic Chern character
Tome XXIII, no 3 (2014), p. 611-619.
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Annales de la Faculté des Sciences de Toulouse Vol. XXIII, n◦ 3, 2014
pp. 611–619
On the arithmetic Chern character
H. Gillet(1), C. Soulé(2)
ABSTRACT. — We consider a short sequence of hermitian vector bundles
on some arithmetic variety. Assuming that this sequence is exact on the
generic fiber we prove that the alternated sum of the arithmetic Chern
characters of these bundles is the sum of two terms, namely the secondary
Bott Chern class of the sequence and its Chern character with support
on the finite fibers.
Next, we compute these classes in the situation encountered by the sec-
ond author when proving a ”Kodaira vanishing theorem” for arithmetic
surfaces.
RÉSUMÉ. — Nous considérons une suite courte de fibrés vectoriels her-
mitiens sur une variété arithmétique. Quand cette suite est exacte sur la
fibre générique nous montrons que la somme alternée des caractères de
Chern arithmétiques de ces fibrés est la somme de deux termes : la classe
secondaire de Bott-Chern de la suite et son caractère de Chern à support
dans les fibres finies. Nous calculons ces deux termes dans la situation ren-
contrée par le second auteur dans sa preuve d’un ’théorème d’annulation
de Kodaira’ pour les surfaces arithmétiques.
Let X be a proper and flat scheme over Z, with smooth generic fiber
XQ. In [4] we attached to every hermitian vector bundle E = (E, ‖ ‖) on X
a Chern character class lying in the arithmetic Chow groups of X:
ĉh(E) ∈ ⊕
p0
ĈH
p
(X)⊗Q .
(1) Department of Mathematics, Statistics, and Computer Science, University of Illi-
nois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street,
Chicago, IL 60607-7045, USA
gillet@uic.edu
(2) IHÉS, 35 route de Chartres, 91440 Bures-sur-Yvette, France
soule@ihes.fr
This material is based upon work supported in part by the National Science Foundation
under Grant No. DMS-0901373
– 611 –
H. Gillet, C. Soulé
Unlike the usual Chern character with values in the ordinary Chow groups,
ĉh is not additive on exact sequences; indeed suppose that Ei, i = 0, 1, 2 is
a triple of hermitian vector bundles on X, and that we are given an exact
sequence
0→ E0 → E1 → E2 → 0
of the underlying vector bundles onX, (i.e. in which we ignore the hermitian
metrics). Then the difference ĉh(E0) + ĉh(E2) − ĉh(E1), is represented by
a secondary characteristic class c̃h first introduced by Bott and Chern [1]
and defined in general in [2]. These Bott-Chern forms measure the defect
in additivity of the Chern forms associated by Chern-Weil theory to the
hermitian bundles in the exact sequence.
Assume now that the sequence
0→ E0 → E1 → E2 → 0 (∗)
is exact on the generic fiber XQ but not on the whole of X. We shall prove
here (Theorem 1) that ĉh(E0)+ĉh(E2)− ĉh(E1) is the sum of the class of c̃h
and the localized Chern character of (∗) (see [3], 18.1). This result fits well
with the idea that characteristic classes with support on the finite fibers of
X are the non-archimedean analogs of Bott-Chern classes (see [6]).
In Theorem 2 we compute more explicitly these secondary characteris-
tic classes in a situation encountered when proving a “Kodaira vanishing
theorem” on arithmetic surfaces ([7], 3.3).
Notation. — If A is an abelian group we let AQ = A⊗
Z
Q .
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 XQ is smooth and equidimensional of
dimension d. For every integer p  0 we denote by App(XR) the real vector
space of smooth real differential forms α of type (p, p) on the complex man-
ifold X(C) such that F ∗∞(α) = (−1)p α, where F∞ is the anti-holomorphic
involution of X(C) induced by complex conjugation. Let
A(X) = ⊕
p0
App(XR)
and
Ã(X) = ⊕
p1
Ap−1,p−1(XR)/(Im(∂) + Im(∂)) .
– 612 –
On the arithmetic Chern character
For every p  0 we let ĈHp(X) be the p-th arithmetic Chow homology group
of X ([5], §2.1, Definition 2). Elements of ĈHp(X) are represented by pairs
(Z, g) consisting of a p-dimensional cycle Z on X and a Green current g for
Z(C) on X(C). Recall that here a Green current for Z(C) is a current (i.e.
a form with distribution coefficients) of type (d− p− 1, d− p− 1) such that
ddc(g) + δZ is C
∞, δZ being the current of integration on Z(C). There are
canonical morphisms ([5], 2.2.1):
z : ĈHp(X) → CHp(X)
(Z, g) 	→ Z
and
ω : ĈHp(X) → Ad−p,d−p(XR)
(Z, g) 	→ ddc(g) + δZ .
Let CHfinp (X) be the Chow homology group of cycles on X the support
of which does not meet XQ. There is a canonical morphism
b : CHfinp (X)→ ĈHp(X)
mapping the class of Z to the class of (Z, 0). The composite morphism
z ◦ b : CHfinp (X)→ CHp(X)
is the obvious map. Let
a : Ad−p−1,d−p−1(XR)→ ĈHp(X)
be the map sending η to the class of (0, η). We have
ω ◦ a(η) = ddc(η) .
1.2. We assume given a sequence
0→ E0 → E1 → E2 → 0
of hermitian vector bundles on X, the restriction of which to XQ is exact.
Let
chfin(E•) ∩ [X] ∈ CHfin(X)Q = ⊕
p0
CHfinp (X)Q
be the localized Chern character of E• ([3] 18.1), and
c̃h(E•) ∈ Ã(X)Q
– 613 –
H. Gillet, C. Soulé
the Bott-Chern secondary characteristic class [2], such that
ddc c̃h(E•) =
2∑
i=0
(−1)i ch(Ei,C) ,
where ch(Ei,C) ∈ A(X) is the differential form representing the Chern char-
acter of the restriction Ei,C of Ei to X(C). Finally, if i = 0, 1, 2, we let
ĉh(Ei) ∩ [X] ∈ ĈH(X)Q = ⊕
p0
ĈHp(X)Q
be the arithmetic Chern character of Ei ([4] 4.1, [5] Theorem 4).
Theorem 1.1. — The following equality holds in ĈH(X)Q:
2∑
i=0
(−1)i ĉh(Ei) ∩ [X] = b (chfin(E•) ∩ [X]) + a (c̃h(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 1.1 we consider the Grassmannian graph con-
struction applied 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 φQ is
empty), the 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 X̃ of (X − φ)× {∞} with Y = π−1(φ×
{∞}). The sequence E• extends to a complex
0→ Ẽ0 → Ẽ1 → Ẽ2 → 0 ,
which is isomorphic to the pull-back of E• over X × A1. The restriction of
Ẽ• to X̃ is canonically split exact. On WQ = XQ × P1Q the sequence Ẽ• is
exact; it coincides with E• (resp. 0 → E0 → E0 ⊕ E2 → E2 → 0) when
restricted to XQ × {0} (resp. XQ × {∞}). We choose a metric on Ẽ• for
which these isomorphisms are isometries.
– 614 –
On the arithmetic Chern character
1.5. Let
x =
2∑
i=0
(−1)i ĉh(Ẽi) ,
and denote by t the standard parameter of A1. In the arithmetic Chow
homology of W we have
0 = x ∩ (W0 −W∞,− log |t|2) .
If x is the class of (Z, g), with Z meeting properly W0 and W∞, we get
x ∩ (W0 −W∞,− log |t|2) = (Z ∩ (W0 −W∞), g ∗ (− log |t|2)) ,
where the ∗-product is equal to
g ∗ (− log |t|2) = g(δW0 − δW∞)− ch(Ẽ•) log |t|2 .
Since W∞ = X̃ ∪ Y , with YQ = ∅, we get
0 = x ∩ (W0 −W∞,− log |t|2) (1.1)
= (Z ∩W0, g δW0)− (Z ∩ X̃, g δX̃)− (Z ∩ Y, 0)− (0, ch(Ẽ•) log |t|
2) .
The restriction of Ẽ• to X̃ is split exact, therefore
(Z ∩ X̃, g δ
X̃
) = 0 .
Applying π∗ to (1.1) we get
0 = ĉh(E•)− π∗((Z ∩ Y ), 0)− (0, π∗(ch(Ẽ•) log |t|2) . (1.2)
By definition of the localized Chern character ([3], 18.1, (14), which remains
valid over S by [3], 20.1, p. 395)
π∗(Z ∩ Y ) = chfin(E•) ∩ [X] (1.3)
in CHfin(X)Q. On the other hand we deduce from [4], (1.2.3.1), (1.2.3.2)
that
−π∗(ch(Ẽ1) log |t|2) = c̃h(E•) . (1.4)
and upon replacing t by 1/t, as in the proof of (1.3.2) in [4], we see that
π∗(ch(Ẽ•) log |t|2) = −π∗(ch(Ẽ1) log |t|2) . (1.5)
Theorem 1.1 follows from (1.2), (1.3), (1.4), (1.5).
– 615 –
H. Gillet, C. Soulé
2. A special case
2.1. We keep the hypotheses of the previous section, and we assume that
X is normal, d = 1, E0 and E2 have rank one and the metrics on E0 and
E2 are induced by the metric on E1. Finally, we assume that there exists a
closed subscheme φ in X which is 0-dimensional and such that there is an
exact sequence of sheaves on X
0→ E0 → E1 → E2 ⊗ Iφ → 0 , (2.6)
where Iφ is the ideal of definition of φ.
Let
c̃2 ∈ A1,1(XR)/(Im(∂) + Im(∂))
be the second Bott-Chern class of (2.6), Γ(φ,Oφ) the finite ring of functions
on φ and # Γ(φ,Oφ) its order. Let
f∗ : ĈH0(X)Q → ĈH0(S) = R
be the direct image morphism.
Theorem 2.1. — We have an equality of real numbers
f∗(ĉ2(E1) ∩ [X]) = f∗(ĉ1(E0) ĉ1(E2) ∩ [X])−
∫
X(C)
c̃2 + log # Γ(φ,Oφ) .
2.2. To prove Theorem 2.1 we remark first that
ĉ1(E1) = ĉ1(E0) + ĉ1(E2) ,
because the metrics on E0 and E2 are induced from E1. Therefore, since
ch2 = −c2 + c
2
1
2 , we get
ĉh2(E1) = −ĉ2(E1) +
(ĉ1(E0) + ĉ1(E2))
2
2
= −ĉ2(E1) + c1(E0) ĉ1(E2) + ĉh2(E0) + ĉh2(E2) .
By Theorem 1.1, this implies that
ĉ2(E1)∩ [X] = ĉ1(E0) ĉ1(E2)∩ [X]+ b (chfin(E•)∩ [X])+a (c̃h(E•)) . (2.7)
Since c̃h0(E•) and c̃h1(E•) vanish we have
c̃h(E•) = − c̃2 .
– 616 –
On the arithmetic Chern character
Therefore, if we apply f∗ to (2.7), we get
f∗(ĉ2(E1)∩[X]) = f∗(ĉ1(E0) ĉ1(E2)∩[X])−
∫
X(C)
c̃2+f∗(b(ch
fin(E•)∩[X])) ,
and we are left with showing that
f∗ ◦ b (chfin(E•) ∩ [X]) = log # Γ(φ,Oφ) . (2.8)
Let |φ| = {P1, · · · , Pn} ⊂ X be the support of φ and ψ = f(|φ|) ⊂ S. The
following diagram is commutative:
CH0(φ)
b−−−→ ĈH0(X)	 f∗
	 f∗
CH0(ψ)
b−−−→ ĈH0(S) = R ,
where
b : CH0(ψ) = Zψ → R
maps (np)p∈ψ to
∑
p
np log(p).
For any prime p ∈ ψ we let Z(p) be the local ring of S at p and we
let #p = #p(φ)  0 be the length of the finite Z(p)-module Γ(φ,Oφ) ⊗ Z(p).
Clearly
log # Γ(φ,Oφ) =
∑
p∈ψ
#p log(p) ,
hence it is enough to prove that
f∗(ch
fin(E•) ∩ [X]) = (#p) ∈ CH0(ψ)Q = Qψ . (2.9)
The complex E• defines an element
[E•] =
n∑
i=1
[Oφ,Pi ] ∈ Kφ0 (X) =
n⊕
i=1
KPi0 (X) .
To prove (2.9), by replacing X by an affine neighbourhood of P , one can
assume that |φ| = {P}, and it is enough to show that, if p = f(P ) ,
f∗(ch
fin(Oφ,P ) ∩ [X]) = #p(Oφ,P )[p] .
– 617 –
H. Gillet, C. Soulé
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 ] =
∑
P∈|F|
#p(FP )[P ] ∈ Z0(X)
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
chP (Oφ,P ) = #p(Oφ,P )[P ] ∈ CH0(P )Q  Q . (2.10)
Replacing X by an affine neighbourhood of P , we may assume that we have
an exact sequence
0 −→ OX α−→O2X
β−→OX −→ Oφ −→ 0 . (2.11)
Hence the ideal Iφ ⊂ OX(X) is generated by two elements β1 and β2.
Since X is normal, its local rings satisfy Serre property S2 and, as dim(X) =
2, X is Cohen-Macaulay. Since β1 and β2 span an ideal of height two,
(β1, β2) is a regular sequence and the sequence (2.11) is isomorphic to the
Koszul resolution of Oφ = OX/(β1, β2). Now (2.10) can be deduced from
the following general fact:
Lemma 2.2. — Let X = Spec (A) be an affine scheme and Z ⊂ X a
closed subset such that the ideal IZ = (x1, . . . , xn) is generated by a regular
sequence (x1, . . . , xn). Let K•(x1, . . . , xn) be the Koszul complex associated
to (x1, . . . , xn). Then
chZn (K•(x1, . . . , xn)) = [OZ ] ∈ CH0(Z)Q .
Proof. — The Grassmannian-graph construction on K•(x1, . . . , xn) coin-
cides with the deformation to the normal bundle of Z in X. If W is defined
as in 1.4,
W∞ = X̃ ∪ P̂(NZ/X) ,
where X̃ is the blow up of X along Z, and P̂ (NZ/X) is the projective
completion of the normal bundle of Z in X. The pull back of the Koszul
complex K•(x1, . . . , xn) to W\W∞ extends to a complex K̃•(x1, . . . , xn) on
W . The restriction of K̃•(x1, . . . , xn) to X̃ is acyclic while the restriction of
K̃•(x1, . . . , xn) to P̂(NZ/X) is a resolution of the structure sheaf of the zero
section Z ⊂ NZ/X ⊂ P̂(NZ/X).
– 618 –
On the arithmetic Chern character
Now observe that Z ⊂ P̂(NZ/X) is an intersection of Cartier divisors
D1, . . . , Dn, hence
ch(K̃•(x1, . . . , xn) |P̂(NZ/X))
=
n∏
i=1
ch(O(−Di)→ OP̂(NZ/X))
=
n∏
i=1
ch(O(Di)) .
Since
ch(O(Di)) = ch1(O(Di)) + yi = [Di] + yi
where yi has degree  2, we get
ch(K̃•(x1, . . . , xn) |P̂(NZ/X)) = [D1] . . . [Dn] = [Z] .
This ends the proof of Lemma 1 and Theorem 2.
Bibliography
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of the zeroes of their holomorphic sections. Acta Math. 114, p. 71-112 (1965).
[2] Bismut (J.-M.), Gillet (H.), Soulé (C.). — Analytic torsion and holomorphic
determinant bundles I, II, III. Comm. Math. Physics 115, p. 49-78, p. 79-126,
p. 301-351 (1988).
[3] Fulton W.. — Intersection Theory. Springer 1984.
[4] Gillet (H.), Soulé (C.). — Characteristic classes for algebraic vector bundles
with hermitian metrics I, II. Annals of Math. 131, p. 163-203, p. 205-238 (1990).
[5] Gillet (H.), Soulé (C.). — An arithmetic Riemann-Roch theorem. Invent. Math.
110, p. 473-543 (1992).
[6] Gillet (H.), Soulé (C.). — Direct images in non-archimedean Arakelov theory.
Annales de l’institut Fourier, 50, p. 363-399 (2000).
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no. 1-3, p. 577-599 (1994).
– 619 –


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