Annales de l’institut Fourier (AIF)

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Transcendental versions in ${{\protect \mathbb{C}}}^n$ of the Nagata conjecture

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Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups

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Cohomology of the moduli space of non-hyperelliptic genus four curves

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Martín-Méndez, Pedro L.

Non-unimodular transversely homogeneous foliations

Guirardel, Vincent

Levitt, Gilbert

Random subgroups, automorphisms, splittings

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Classification of foliations of degree three on $\protect \mathbb{P}^{2}_{\protect \mathbb{C}}$ with a flat Legendre transform

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Ketterer, Christian

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Stratified spaces and synthetic Ricci curvature bounds

Bonnafé, Cédric

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ANNALES DE
L’INSTITUT FOURIER
Université Grenoble Alpes
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Bruno Kahn
On the universal regular homomorphism in codimension 2
Tome 71, no 2 (2021), p. 843-848.
<http://aif.centre-mersenne.org/item/AIF_2021__71_2_843_0>
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Ann. Inst. Fourier, Grenoble
71, 2 (2021) 843-848
ON THE UNIVERSAL REGULAR HOMOMORPHISM
IN CODIMENSION 2
by Bruno KAHN
Abstract. — We point out a gap in Murre’s proof of the existence of a universal
regular homomorphism for codimension 2 cycles on a smooth projective variety, and
offer two arguments to fill this gap.
Résumé. — On signale une lacune dans la preuve de l’existence d’un homomor-
phisme régulier universel pour les cycles de codimension 2 sur une variété projective
lisse par Murre, et on donne deux arguments différents pour combler cette lacune.
In [11], Jacob Murre shows the existence of a universal regular homo-
morphism for algebraically trivial cycles of codimension 2 on a smooth
projective variety over an algebraically closed field. This theorem has been
largely used in the literature, most lately in [1, 7] and [3]; for example, it
is essential in [3] for descending the method of Clemens and Griffiths [6]
to non-algebraically closed fields, thus allowing Benoist and Wittenberg to
obtain new examples of geometrically rational nonrational 3-folds.
Unfortunately its proof contains a gap, but fortunately this gap can be
filled, actually by two different methods. This is the purpose of this note,
which is a slight modification of a letter to Murre on December 5, 2018.
Recall the set-up, with the notation of [11]: V is a smooth projective
variety over an algebraically closed field k and An(V ) denotes the group of
codimension n cycles algebraically equivalent to 0 on V , modulo rational
equivalence. Following Samuel, given an abelian k-variety A, a homomor-
phism
φ : An(V )→ A(k)
is said to be regular if, for any pointed smooth projective k-variety (T, t0)
and any correspondence Z ∈ CHn(T × V ), the composition
(0.1) T (k) wZ−−→ An(V ) φ−→ A(k)
Keywords: Algebraic cycles, abelian varieties.
2020 Mathematics Subject Classification: 14C25, 14K30.
844 Bruno KAHN
is induced by a morphism f : T → A; here wZ is the composition
(0.2) T (k)→ A0(T )
Z∗−−→ An(V )
where the first map sends t to [t] − [t0]. (Note that f is then unique, by
Zariski density of the rational points in T .)
Using fancy language, regular homomorphisms from An(V ) form a cat-
egory and a universal regular homomorphism is an initial object of this
category, if it exists. This initial object is well-known to exist when n = 0,
n = 1 (the Picard variety) and n = dimX (the Albanese variety). Murre’s
theorem is:
Theorem 0.1 ([11, Theorem 1.9]). — A universal regular homomor-
phism φ0 exists when n = 2 for any V (of dimension > 2).
Recall the main steps of his proof. First, given a regular homomorphism
φ, its image in A(k) is given by the points of some sub-abelian variety
A′ ⊆ A [11, Lemma 1.6.2(i)]. From this, one deduces [11, Proposition 2.1]
that φ0 exists if and only if dimA is bounded when φ runs through the sur-
jective regular homomorphisms. Now, Murre’s key idea is to bound dimA
by the torsion of A2(V ), which is controlled by the Merkurjev–Suslin the-
orem (Bloch’s observation).
Let us elaborate a little on this point, to avoid the l-adic argument of
loc. cit.: it suffices to prove that φ induces a surjection
(0.3) A2(V ){l} −→ A(k){l}
for some prime l 6= char k, where M{l} denotes the l-primary torsion of an
abelian groupM : indeed, corankA(k){l} = 2 dimA. Mainly by Merkurjev–
Suslin (Diagram in [11, Proposition 6.1])(1) ,
corankCH2(V ){l} 6 corankH3ét (V,Ql/Zl(2)) (= b3(V ))
so the same holds a fortiori for corankA2(V ){l}.
Now, in [11, Lemma 1.6.2(ii)], Murre constructs an abelian variety B
(pointed at 0) and a correspondence Z ∈ CH2(B × V ) such that (0.1) is
surjective for T = B. Since this map is induced by a morphism of abelian
varieties sending 0 to 0 (hence a homomorphism), it restricts to a surjection
(0.4) B{l} → A{l}.
This allows me to explain
(1)One could replace this diagram by the injection CH2(V ) ↪→ H4ét(V, Γ(2)) of [9, The-
orem 2.13(c)], together with the surjection H3ét(V,Ql/Zl(2)) → H
4
ét(V, Γ(2)){l}, cf. loc.
cit., proof of Theorem 2.15; here, Γ(2) is Lichtenbaum’s complex.
ANNALES DE L’INSTITUT FOURIER
ON THE UNIVERSAL REGULAR HOMOMORPHISM IN CODIMENSION 2 845
The gap
A priori (0.4) does not imply (0.3), because wZ is in general only a set-
theoretic map, not a group homomorphism (see e.g. [4, Theorem 3.1(a)]).
We now fix a surjective regular homomorphism φ as above. We shall give
two ways to fill this gap:
(A) construct (B,Z) such that wZ is a homomorphism;
(B) prove that wZ always sends torsion to torsion.
(A) was my initial idea, and (B) was inspired by a discussion with Murre.
1. Explanation of (A)
We have
Lemma 1.1. — Take (T, t0, z) with T of dimension 1 and z ∈ CH2(T ×
V ). Let J = J(T ) be the jacobian of T . Then the homomorphism z∗ :
A0(T ) = J(k) → A2(V ) is of the form wα for some correspondence α ∈
CH2(J × V ) (using 0 ∈ J(k) as base point).
Proof. — Let g be the genus of T . Recall from [10, Example 3.12] the
universal relative Cartier divisor Dcan on T ×T (g)/T (g), parametrising the
effective divisors of degree g on T . It defines a correspondenceDcan : T (g) →
T . Composing with the graph of the birational map J 99K T (g) inverse to
(t1, . . . , tg) 7→
∑
ti − gt0, we find a (Chow) correspondence D : J → T . I
claim that α = z ◦D answers the question. Indeed, one checks immediately
that the homomorphism
D∗ : A0(J)→ A0(T )
is the Albanese morphism for J ; hence the composition
J(k)→ A0(J)
D∗−−→ A0(T )
is the identity. 
Remark 1.2. — On the other hand, the morphism T → A given by the
regularity of φ factors through a homomorphism
(1.1) J(T )→ A.
This homomorphism coincides with the one underlying φ ◦ z∗ in view of
Lemma 1.1. Indeed, by uniqueness, it suffices to see that (1.1) induces φ◦z∗
on k-points; this is clear since T (k) generates J(T )(k) as an abelian group.
TOME 71 (2021), FASCICULE 2
846 Bruno KAHN
Consider all triples (T, t0, z) with dimT = 1. The homomorphism
⊕
A0
(T ) (z∗)−−→ A2(V ) is surjective, hence so is
⊕
A0(T )
(z∗)−−→ A2(V )→A(k).
As in Remark 1.2, each summand of this homomorphism is induced by a
homomorphism ρT, t0, z : J(T )→ A, so
B :=
∏
(T, t0, z)∈S
J(T )
(ρT, t0, z)−−−−−−→ A
is surjective (faithfully flat) for a suitable finite set S. For each (T, t0, z), let
α = αz be a correspondence given by Lemma 1.1. Write πT, t0,z : B → J(T )
for the canonical projection, viewed as an algebraic correspondence. The
pair given by B and Z =
∑
(T, t0,z) αz ◦ πT, t0,z yields (A).
2. Explanation of (B)
It suffices to show that the map
f : B(k)→ A0(B)
sends l-primary torsion to l-primary torsion. Let d = dimB. By Bloch’s
theorem [4, Theorem (0.1)], we have A0(B)∗(d+1) = 0, where ∗ denotes
Pontrjagin product. In other words, f has “degree 6 d” in the sense that
its (d+ 1)st deviation [8, Section 8] is identically 0. It remains to show:
Lemma 2.1. — Let f : M → N be a map of degree 6 d between two
abelian groups, such that f(0) = 0. Let m0 ∈ M be an element such that
am0 = 0 for some integer a > 0. Then
a(
d+1
2 )f(m0) = 0.
Proof. — Induction on d. The case d = 1 is trivial. Assume d > 1. By
hypothesis, the dth deviation of f is multilinear, which implies that the
map
ga(m) = f(am)− adf(m)
is of degree6 d−1. By induction, a(
d
2)ga(m0) = 0, hence the conclusion. 
Remark 2.2. — Of course, either argument proves more generally the
following: the map φ : An(V ){l} → A(k){l} is surjective for any integer n,
any surjective regular homomorphism φ : An(V ) → A(k) and any prime
l 6= char k.
ANNALES DE L’INSTITUT FOURIER
ON THE UNIVERSAL REGULAR HOMOMORPHISM IN CODIMENSION 2 847
Remark 2.3. — In [2, Section 6, Lemma and Proposition 11], Beauville
gives a different proof that f sends torsion to torsion. Moreover, he observes
that Rǒıtman’s theorem [13] then implies that the restriction of f to torsion
is actually an isomorphism, hence a homomorphism.
If we apply Rǒıtman’s theorem together with Lemma 2.1, we obtain
the following stronger result: if m,m0 ∈ B(k) and m0 is torsion, then
f(m+m0) = f(m) + f(m0). (Fixing m, the map fm : m′ 7→ f(m+m′)−
f(m) − f(m′) is of degree < d, hence a(
d
2)fm(m0) = 0 if am0 = 0 by
Lemma 2.1, and therefore fm(m0) = 0 by Rǒıtman’s theorem.)
3. Some expectation
The landmark work of Bloch and Esnault [5] yields the existence of 4-
folds V over fields k of characteristic 0 such that the l-torsion of A3(V )
is infinite (hence its l-primary torsion has infinite corank). One example,
used by Rosenschon–Srinivas [14] and Totaro [16] and relying on Nori’s the-
orem [12] and Schoen’s results [15], is the following: start from the generic
abelian 3-fold A, whose field of constants k0 is finitely generated over Q;
choose an elliptic curve E/k0(t), not isotrivial with respect to k0, and take
V = Ak0(t) × E, k = algebraic closure of k0(t).
Conjecture 3.1. — For this V , a universal regular homomorphism on
A3(V ) does not exist.
Ackowledgements
I am indebted to Jacob Murre for discussions around this problem, and
for his encouragement to publish this note. I am also indebted to the ref-
eree for a careful reading, pointing out an incorrect earlier formulation of
Lemma 1.1, as well as the reference to [2]. (The referee credits in turn
Charles Vial for this reference.)
BIBLIOGRAPHY
[1] J. D. Achter, S. Casalaina-Martin & C. Vial, “On descending cohomology
geometrically”, Compos. Math. 153 (2017), no. 7, p. 1446-1478.
[2] A. Beauville, “Quelques remarques sur la transformation de Fourier dans l’anneau
de Chow d’une variété abélienne”, in Algebraic geometry (Tokyo/Kyoto, 1982),
Lecture Notes in Mathematics, vol. 1016, Springer, 1982, p. 238-260.
TOME 71 (2021), FASCICULE 2
848 Bruno KAHN
[3] O. Benoist & O. Wittenberg, “The Clemens-Griffiths method over non-closed
fields”, Algebr. Geom. 7 (2020), no. 6, p. 696-721.
[4] S. Bloch, “Some elementary theorems about algebraic cycles on Abelian varieties”,
Invent. Math. 37 (1976), no. 3, p. 215-228.
[5] S. Bloch & H. Esnault, “The coniveau filtration and non-divisibility for algebraic
cycles”, Math. Ann. 304 (1996), no. 2, p. 303-314.
[6] C. H. Clemens & P. A. Griffiths, “The intermediate Jacobian of the cubic three-
fold”, Ann. Math. 95 (1972), p. 281-356.
[7] J.-L. Colliot-Thélène & A. Pirutka, “Troisième groupe de cohomologie non
ramifiée d’un solide cubique sur un corps de fonctions d’une variable”, Épijournal
de Géom. Algébr., EPIGA 2 (2018), article no. 13 (13 pages).
[8] S. Eilenberg & S. Mac Lane, “On the groups H(Π, n), II; methods of computa-
tion”, Ann. Math. 60 (1954), p. 49-139.
[9] S. Lichtenbaum, “New results on weight-two motivic cohomology”, in The
Grothendieck Festschrift, Vol. III, Progress in Mathematics, vol. 88, Birkhäuser,
1990, p. 35-55.
[10] J. S. Milne, “Jacobian varieties”, in Arithmetic Geometry (G. Cornell & S. J. H.,
eds.), Springer, revised and second ed., 1998, p. 167-212.
[11] J. P. Murre, “Applications of algebraic K-theory to the theory of algebraic cycles”,
in Algebraic geometry, Proc. Conf., Sitges (Barcelona)/Spain 1983, Lecture Notes
in Mathematics, vol. 1124, Springer, 1985, p. 216-261.
[12] M. V. Nori, “Cycles on the generic abelian threefold”, Proc. Indian Acad. Sci.,
Math. Sci. 99 (1989), no. 3, p. 191-196.
[13] A. A. Rǒıtman, “The torsion of the group of 0-cycles modulo rational equivalence”,
Ann. Math. 111 (1980), no. 3, p. 553-569.
[14] A. Rosenschon & V. Srinivas, “The Griffiths group of the generic abelian 3-
fold”, in Cycles, motives and Shimura varieties. Proceedings of the international
colloquium, Mumbai, India, January 3–12, 2008, Tata Institute of Fundamental
Research Studies in Mathematics, vol. 21, Narosa Publishing House / Tata Institute
of Fundamental Research, 2010, p. 449-467.
[15] C. Schoen, “On certain exterior product maps of Chow groups”, Math. Res. Lett.
7 (2000), no. 2-3, p. 177-194.
[16] B. Totaro, “Complex varieties with infinite Chow groups modulo 2”, Ann. Math.
183 (2016), no. 1, p. 363-375.
Manuscrit reçu le 21 juin 2019,
révisé le 7 janvier 2020,
accepté le 13 mai 2020.
Bruno KAHN
IMJ-PRG
Case 247
4 place Jussieu
75252 Paris Cedex 05
France
bruno.kahn@imj-prg.fr
ANNALES DE L’INSTITUT FOURIER


On the universal regular homomorphism in codimension $2$

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