Comptes Rendus Mathématique

Jean Renault

Comptes Rendus
Mathématique
Jean Renault
Continuity of the dual Haar measure
Volume 359, issue 4 (2021), p. 415-419
<https://doi.org/10.5802/crmath.183>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 4, p. 415-419
https://doi.org/10.5802/crmath.183
Functional analysis, Harmonic analysis / Analyse fonctionnelle, Analyse harmonique
Continuity of the dual Haar measure
Continuité de la mesure de Haar duale
Jean Renaulta
a Institut Denis Poisson (UMR 7013) Université d’Orléans et CNRS, 45067 Orléans
Cedex 2, France.
E-mail: jean.renault@univ-orleans.fr
Abstract. Given a continuous field of locally compact groups, we show that the field of the Plancherel weights
of their C*-algebras is lower semi-continuous. As a corollary, we obtain that the dual Haar system of a
continuous Haar system of a locally compact abelian group bundle is also continuous.
Résumé. Étant donné un champ continu de groupes localement compacts, on montre que le champ des
poids de Plancherel de leurs C*-algèbres est semi-continu inférieurement. On en déduit que, lorsque les
groupes sont abéliens, le système de Haar dual d’un système de Haar continu est aussi continu.
Manuscript received 25th January 2021, accepted 31st January 2021.
Version française abrégée
Soient G un groupe localement compact abélien et Ĝ son groupe dual. La mesure de Haar duale
d’une mesure de Haar de G est la mesure de Haar de Ĝ qui rend la transformation de Fourier
isométrique. Supposons maintenant que G → X soit un fibré localement compact de groupes
abéliens muni d’un système de mesures de Haar continu pour la topologie vague des mesures.
Alors le fibré des groupes duaux a une topologie localement compact et est muni du système des
mesures de Haar duales. Nous montrons que ce système dual est lui aussi continu.
La démonstration est basée sur l’observation que, dans le cas d’un groupe localement compact
abélien G , la mesure de Haar duale peut être vue comme le poids de Plancherel du groupe G ,
c’est-à-dire le poids canonique de l’algèbre hilbertienne de la représentation régulière gauche
de G . La théorie de Tomita–Takesaki fournit la formule (1), valide pour tout groupe localement
compact G .
Si G → X est un fibré localement compact de groupes non nécessairement abéliens, muni
d’un système de Haar continu, on déduit de cette formule et de l’existence d’unité approchée
que, pour tout élément positif a de la C*-algèbre C∗(G), la fonction x 7→ϕx (ax ), oùϕx est le poids
de Plancherel du groupe Gx au dessus de x et ax est l’image de a dans C∗(Gx ), est semi-continue
inférieurement sur X . On définit ainsi un C0(X )-poids sur la C0(X )-algèbre C∗(G). Comme dans
la théorie usuelle, donnée dans [1], des poids scalaires sur les C*-algèbres, son domaine de
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
416 Jean Renault
définition est un idéal à gauche dense. Dans le cas commutatif, il contient l’idéal de Pedersen.
Cette propriété est exactement la continuité du système de Haar dual. Outre le C0(X )-poids de
Plancherel d’un fibré de groupes, il existe des exemples naturels de C0(X )-poids ; en particulier,
les Z-traces que Dixmier a introduites dans [3] dans le cadre des algèbres de von Neumann ont
leur analogue pour une C*-algèbre A : c’est une Cb(X )-trace où X est l’espace des idéaux primitifs
de A.
1. Introduction
Let G be a locally compact abelian group and let Ĝ be its dual group. A Haar measure λ of G
determines a dual Haar measure λ̂ of Ĝ : namely, this the Haar measure of Ĝ which makes the
Fourier transformF an isometry from L2(G ,λ) to L2(Ĝ , λ̂). Suppose now that p : G → X is a locally
compact abelian group bundle in the sense of [7]. A Haar system for G is a family of measures
(λx )x∈X , where for all x ∈ X , λx is a Haar measure of Gx = p−1(x); it is said to be continuous if
for all f ∈ Cc (G), the function x 7→
∫
f dλx is continuous. The dual group bundle p̂ : Ĝ → X is
then endowed with the dual Haar system. We show that if the initial Haar system is continuous,
then so is the dual Haar system. This was stated as [7, Proposition 3.6]. However it was recently
pointed to the authors by Henrik Kreidler that their proof is defective. This note corrects this
and gives a more general result, based on the fact that, given a locally compact abelian group G ,
the Haar weight of Ĝ corresponds via the Gelfand transform to the Plancherel weight of G . Our
main result says that given a a locally compact group bundle p : G → X with continuous Haar
system, the Plancherel weight of Gx varies continuously in a suitable sense. This lead us to the
definition of a lower semi-continuous C0(X )-weight on a C0(X )-C*-algebra which we illustrate by
three examples.
2. The Plancherel weight of a locally compact group
We recall first some elements of the Tomita–Takesaki theory, using the standard notation of [11].
Given a left Hilbert algebra A where the product, the involution and the scalar product are
respectively denoted by ab, a] and (a |b), we denote by H the Hilbert space completion of A
and by π : A → L(H) the left representation. We denote by M = π(A)′′ the left von Neumann
algebra of A. We denote by S the closure of the involution a 7→ a] and by F its adjoint. The
domain of S [resp. F ] is denoted by D] [resp. D[] and one writes ξ] = Sξ [resp. η[ = Fη] for ξ ∈D]
[resp. for η ∈D[]. An element η ∈H is called right bounded if there exists a bounded operator
π′(η) on H such that π′(η)a = π(a)η for all a ∈A. One then writes ξη = π′(η)ξ for all ξ ∈H. The
set of right bounded elements is denoted by B′. One shows that A′ = B′∩D[ with involution ξ[
is a right Hilbert algebra in the same Hilbert space H. An element ξ ∈H is called left bounded if
there exists a bounded operator π(ξ) on H such that π(ξ)η=π′(η)ξ for all η ∈A′. One then writes
ξη = π(ξ)η for all η ∈H. If ξ is left bounded and η is right bounded, the notation is consistent:
π(ξ)η= ξη=π′(η)ξ. The set of left bounded elements is denoted by B. According to [2, Section 2],
it is a linear subspace of H containing A, it is stable under M and π(B) is contained in M. We
recall that the canonical weight τ of the left Hilbert algebra A is the map τ :M+ → [0,∞] defined
by τ(T ) = ‖ξ‖2 if T is of the form π(ξ)∗π(ξ) for some ξ ∈B and τ(T ) =∞ otherwise. It is a faithful,
semi-finite, σ-weakly lower semi-continuous weight.
Using the polar decomposition of an operator in a von Neumann algebra and the techniques
of [11, Lemma 3.4], one obtains:
Lemma 1. For all ξ ∈B, one has τ(π(ξ)π(ξ)∗) = ‖ξ]‖2 if ξ ∈D] and is equal to ∞ otherwise.
C. R. Mathématique — 2021, 359, n 4, 415-419
Jean Renault 417
We consider now the left Hilbert algebra associated with the left regular representation of a
locally compact group G endowed with a left Haar measure λ as defined in [9]: the left Hilbert
algebra isA=Cc (G), where the product is the usual convolution product, the involution, denoted
by f ∗ instead of f ], is f ∗(γ) = f (γ−1) and the scalar product is ( f |g ) = ∫ f g dλ−1. The left
representation is denoted by L instead of π. It acts on the Hilbert space H = L2(G ,λ−1) by
L( f )g = f ∗ g for f , g ∈ Cc (G). The left von Neumann algebra M is the group von Neumann
algebra V N (G). The canonical weight τ of this left Hilbert algebra is called the Plancherel weight
of the group G ( [10, 12]). One has for f ∈Cc (G):
τ
(
L
(
f ∗∗ f ))= ‖ f ‖2 = ∫ | f |2dλ−1 = ( f ∗∗ f ) (e).
Let us express Lemma 1 in the case of the left regular representation of a locally compact group
G with a left Haar measure λ.
Corollary 2. For every left bounded element ξ of L2(G ,λ−1), one has
τ
(
L(ξ)L(ξ)∗
)= ∫ |ξ|2dλ. (1)
3. The Plancherel weight of a group bundle
We consider now the case of a locally compact group bundle p : G → X . We assume that the
groups Gx have a left Haar measure λx such that x 7→ λx is continuous. As a locally compact
groupoid with Haar system, we can construct its C*-algebra C∗(G) as in [9]. According to [5,
Section 5], this is a C0(X )-C*-algebra. Its elements are continuous fields x 7→ ax where ax ∈
C∗(Gx ). For x ∈ X , we let Lx be the left regular representation of Gx and τx be the Plancherel
weight of Gx .
Proposition 3. Let G → X be a locally compact group bundle with a continuous Haar system
(λx )x∈X . Given a ∈C∗(G)+, the function T (a) : x 7→ τx (Lx (ax )) is lower semi-continuous.
One first show that the function
x 7→ τx
(
Lx
(
a1/2x ∗
(
f ∗ f ∗)x ∗a1/2x ))
is lower semi-continuous for f ∈Cc (G) by using the equation (1). One deduces that
x 7→ τx
(
Lx
(
a1/2x ∗ (ei )x ∗a1/2x
))
is lower semi-continuous where (ei ) is an approximate unit as in [6, Proposition 2.10], which we
can assume bounded by 1. Then, our function can be written as a lower upper bound of a family
of lower semi-continuous functions.
4. Examples of C0(X )-weights on C0(X )-C*-algebras
Definition 4. Let X be a locally compact Hausdorff space. A C0(X )-weight on a C0(X )-C*-algebra
A is a map Φ : A+ → LSC (X )+, where LSC (X )+ denotes the convex cone of lower semi-continuous
functions f : X → [0,∞], which is additive and such that Φ(ha) = hΦ(a) for all h ∈ C0(X )+ and
a ∈ A+. It is called lower semi-continuous if an → a implies Φ(a) ≤ liminfΦ(an) and densely
defined if its domain P = {a ∈ A+ :Φ(a) is finite and continuous} is dense in A+.
We note as in [4, Lemme 4.4.2.i] that the domain is hereditary. Here are examples.
(1) The map T : C∗(G)+ → LSC (X )+ of the previous section is a lower semi-continuous and
densely defined C0(X )-weight. It can be called the Plancherel C0(X )-weight of the group
bundle G → X .
C. R. Mathématique — 2021, 359, n 4, 415-419
418 Jean Renault
(2) Let π : Y → X be a continuous, open and surjective map, where Y and X are locally
compact Hausdorff spaces. For x ∈ X , let Yx = π−1(x) be the fibre over x. Endowed with
the fundamental family of continuous sections Cc (Y ), x 7→C0(Yx ) is a continuous field of
C*-algebras. Its C*-algebra of continuous sections is identified with C0(Y ). Thus C0(Y ) is
a C0(X )-C*-algebra. Let (αx )x∈X be a family of Radon measures on Y , with αx supported
on π−1(x), and such that for all f ∈Cc (Y ), the function x 7→
∫
f dαx is continuous. Then
Φ : C0(Y ) → LSC (X )+ such that Φ( f )(x) =
∫
f dαx for f ∈ C0(Y ) is a densely defined and
lower semi-continuous C0(X )-weights of C0(Y ). The converse is also true and uses the
fact that Cc (Y ) is the Pedersen ideal, i.e. the minimal dense ideal, of C0(Y ) ( [8, 5.6.3.]). In
particular, given a a locally compact group bundle G → X , a continuous Haar system
(λx )x∈X defines a densely defined and lower semi-continuous C0(X )-weight of C0(G)
called the Haar C0(X )-weight of the group bundle G → X .
We now have all the elements to prove the continuity of the dual Haar system.
Corollary 5. Let p : G → X a locally compact bundle of abelian groups, equipped with a contin-
uous Haar system (λx )x∈X . Then the family of dual Haar measures (λ̂x )x∈X is a continuous Haar
system for p̂ : Ĝ → X .
Indeed, the Gelfand transform identifies the Plancherel C0(X )-weight of C∗(G) and the Haar
C0(X )-weight of C∗(Ĝ).
(3) In order to give our last example, we replace C0(X ) by its multiplier algebra Cb(X ) whose
elements are the complex-valued bounded continuous functions on X . We say that a
C*-algebra A is a Cb(X )-algebra if it is endowed with a nondegenerate morphism of Cb(X )
into the centre of the multiplier algebra of A. Then we define a Cb(X )-weight by replacing
C0(X ) by Cb(X ) in the above definition. By the Dauns–Hofmann theorem, a C*-algebra
A is a Cb(X )-algebra, where X is the primitive ideal space Prim(A) endowed with the
Jacobson topology. Given a ∈ A+, according to [8, Proposition 4.4.9], one can define
Φ(a)(P ) = Trace(π(a)) where P is a primitive ideal and π is any irreducible representation
admitting P as kernel, and Φ(a) is lower semi-continuous on X . Then Φ : A+ → LSC (X )+
is a lower semi-continuous Cb(X )-weight. It is densely defined if and only if A is a
continuous-trace C*-algebra. This is the C*-algebraic version of the Z-trace introduced
by Dixmier in [3] for von Neumann algebras.
Acknowledgements
I thank Henrik Kreidler for drawing the erroneous proof of [7, Proposition 3.6] to our attention,
Michel Hilsum for discussions which led to the present proof of Corollary 5 and Dana Williams
for sharing another proof and providing inspiring feedback and valuable comments.
References
[1] F. Combes, “Poids sur une C∗-algèbre”, J. Math. Pures Appl. 47 (1968), p. 57-100.
[2] ——— , “Poids associé à une algèbre hilbertienne à gauche”, Compos. Math. 23 (1971), no. 1, p. 49-77.
[3] J. Dixmier, Les algèbres de von Neumann, 2nd ed., Cahiers scientifiques, vol. 25, Gauthier-Villars, 1969.
[4] ——— , Les C∗-algèbres et leurs représentations, Cahiers scientifiques, vol. 29, Gauthier-Villars, 1969.
[5] N. P. Landsman, B. Ramazan, “Quantization of Poisson algebras associated to Lie algebroids”, in Groupoids in
analysis, geometry, and physics. AMS–IMS–SIAM joint summer research conference, University of Colorado, Boulder,
CO, USA, June 20-24, 1999, Contemporary Mathematics, vol. 282, American Mathematical Society, 2001, p. 159-192.
[6] P. S. Muhly, J. N. Renault, D. P. Williams, “Equivalence and isomorphism for groupoid C∗-algebras”, J. Oper. Theory
17 (1987), p. 3-22.
[7] ——— , “Continuous trace groupoid C∗-algebras. III”, Trans. Am. Math. Soc. 348 (1996), no. 9, p. 3621-3641.
C. R. Mathématique — 2021, 359, n 4, 415-419
Jean Renault 419
[8] G. Pedersen, C*-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14,
Academic Press Inc., 1979.
[9] J. N. Renault, A groupoid approach to C∗-algebras, Lecture Notes in Mathematics, vol. 793, Springer, 1980.
[10] Ş. Strătilă, Modular theory in operator algebras, Abacus Press, 1981.
[11] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, vol. 128,
Springer, 1970.
[12] ——— , Theory of operator algebras. II., Encyclopaedia of Mathematical Sciences, vol. 125, Springer, 2003.
C. R. Mathématique — 2021, 359, n 4, 415-419


Continuity of the dual Haar measure

Mei Bai

Wenchang Chu

Comptes Rendus
Mathématique
Mei Bai and Wenchang Chu
Further Equivalent Binomial Sums
Volume 359, issue 4 (2021), p. 421-425
<https://doi.org/10.5802/crmath.184>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 4, p. 421-425
https://doi.org/10.5802/crmath.184
Combinatorics / Combinatoire
Further Equivalent Binomial Sums
Mei Bai∗, a andWenchang Chub
a School of Mathematics and Statistics, Zhoukou Normal University, Henan, China.
b Department of Mathematics and Physics, University of Salento, 73100 Lecce, Italy.
E-mails: baimei0418@163.com (M. Bai), chu.wenchang@unisalento.it (W. Chu)
Abstract. Five binomial sums are extended by a free parameter m, that are shown, through the generating
function method, to have the same value. These sums generalize the ones by Ruehr (1980), who discovered
them in the study of two unexpected equalities between definite integrals.
2020 Mathematics Subject Classification. 11B65, 05A10.
Manuscript received 22nd December 2020, revised and accepted 2nd February 2021.
In 1980, Kimura [14] proposed a monthly problem about two curious identities of definite
integrals. If f is continuous on [− 12 , 32 ], then for δ= 0, 1, prove that∫ 3
2
− 12
xδ f
(
3x2 −2x3)dx = 2∫ 1
0
xδ f
(
3x2 −2x3)dx.
In his (trigonometric) proof, Ruehr [14] observed by linearity that to prove these identities, it is
enough to verify them for monomials f (x) = xn . This led him to discover the following interesting
identities
An =Cn and Bn = Dn ,
where for a natural number n, the four binomial sums are defined by
An =
n∑
j=0
3 j
(
3n − j
2n
)
,
Bn =
n∑
j=0
2 j
(
3n +1
2n + j +1
)
,
Cn =
2n∑
j=0
(−3) j
(
3n − j
n
)
,
Dn =
2n∑
j=0
(−4) j
(
3n +1
n + j +1
)
.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
422 Mei Bai and Wenchang Chu
Allouche [1, 2] examined the related integrals and reviewed these identities in a more elegant
manner. These identities were also reconfirmed by Meehan et al [16] who found, through the
WZ-algorithm, that these four sequences satisfy also the common recurrence relation:
X0 = 1 and Xn+1 = 27
4
Xn −
3
(3n+1
n
)
4(n +1) .
By introducing a variable, Alzer and Prodinger [3] recently considered the polynomial analogues,
that were also examined by Kilic–Arikan [13] through bijections.
By applying the generating function approach to the binomial convolutions
Ωn =
n∑
k=0
(
3k
k
)(
3n −3k
n −k
)
the authors [4] not only confirmed the identities
Ωn = An = Bn =Cn = Dn ; (1)
but also found the two additional ones
Ωn = En = Fn , (2)
where
En =
n∑
k=0
3k
(
3n −k
2n
)
2k(k +1)
3n −k ,
Fn =
n∑
k=0
2k
(
3n +2
n −k
)
(k +1)(3k +2)
3n +2 .
Recall the following two binomial series due to Lambert [15] (see also [5–7, 11], [12, §5.4] and [17,
§5.4]):
∞∑
n=0
α
α+nβ
(
α+nβ
n
)
xn = (1+ y)α, (3)
∞∑
n=0
(
α+nβ
n
)
xn = (1+ y)
α+1
1+ y −βy ; (4)
where x and y are related by the equation x = y/(1+ y)β. According to the Lagrange inversion
theorem (cf. Comtet [8, §3.8]), both (3) and (4) can be considered as “formal power series”
equations in the variable x.
For two natural integers m and n, define the binomial sum
Ωm(n) =
n∑
k=0
(
mk
k
)(
mn −mk
n −k
)
.
Then we can compute, in view of (4), its generating function
Ωm[x] :=
∞∑
n=0
Ωm(n)x
n =
∞∑
n=0
xn
n∑
k=0
(
mk
k
)(
mn −mk
n −k
)
=
∞∑
k=0
(
mk
k
)
xk
∞∑
n=k
(
mn −mk
n −k
)
xn−k
= (1+ y)
2
(1+ y −my)2 , where x =
y
(1+ y)m .
By manipulating the above function, we shall prove that
Ωm(n) = Am(n) = Bm(n) = Em(n) = Fm(n), (5)
C. R. Mathématique — 2021, 359, n 4, 421-425
Mei Bai and Wenchang Chu 423
where the four binomial sums are defined by
Am(n) =
n∑
j=0
m j
(
nm − j
n − j
)
,
Bm(n) =
n∑
j=0
(m −1) j
(
nm +1
n − j
)
,
Em(n) =
n∑
k=0
mk
(
mn −k
mn −n
)
(m −1)k(k +1)
mn −k ,
Fm(n) =
n∑
k=0
(m −1)k
(
mn +2
n −k
)
(1+k)(2+mk)
2+mn .
When m = 3, we recover the previously known results
Ωn = An = Bn = En = Fn .
As done by Ekhad and Zeilberger [10] for the above identities, it should also be possible to give,
by the WZ-method, “automated” proofs for those displayed in (5).
§1. Firstly, we can reformulate the generating functionΩm[x] as
Ωm[x] = (1+ y)
2
(1+ y −my)2 =
1+ y
1+ y −my ×
1
1−my/(1+ y)
= 1+ y
1+ y −my ×
1
1−mx(1+ y)m−1 y = x(1+ y)
m
=
∞∑
j=0
(mx) j
(1+ y)(m−1) j+1
1+ y −my .
Letting α= (m −1) j and β= m in (4), we have further
Ωm[x] =
∞∑
j=0
(mx) j
∞∑
k=0
xk
(
(m −1) j +km
k
)
=
∞∑
n=0
xn
n∑
j=0
m j
(
nm − j
n − j
)
n = k + j
which implies thatΩm(n) = Am(n). 
§2. Secondly, the generating functionΩm[x] can also be restated as
Ωm[x] = (1+ y)
2
(1+ y −my)2 =
(1+ y)2
1+ y −my ×
1
1+ y −my
= (1+ y)
2
1+ y −my ×
1
1−x(1+ y)m(m −1) y = x(1+ y)
m
=
∞∑
j=0
(m −1) j x j (1+ y)
2+m j
1+ y −my .
By letting α= 1+m j and β= m in (4), we can expand further
Ωm[x] =
∞∑
j=0
(m −1) j x j
∞∑
k=0
xk
(
m j +km +1
k
)
=
∞∑
n=0
xn
n∑
j=0
(m −1) j
(
nm +1
n − j
)
. n = k + j .
This shows thatΩm(n) = Bm(n). 
C. R. Mathématique — 2021, 359, n 4, 421-425
424 Mei Bai and Wenchang Chu
§3. Thirdly, rewrite the generating functionΩm[x] as
Ωm[x] = (1+ y)
2
(1+ y −my)2 =
1(
1− my1+y
)2
= 1
(1−mx(1+ y)m−1)2 y = x(1+ y)
m
=
∞∑
k=0
(k +1)(mx)k (1+ y)(m−1)k .
Letting α= (m −1)k,β= m in (3), we have further
Ωm[x] =
∞∑
k=0
(mx)k
∞∑
i=0
xi
(
(m −1)k +mi
i
)
(m −1)k(k +1)
(m −1)k +mi
=
∞∑
n=0
xn
n∑
k=0
mk
(
mn −k
mn −n
)
(m −1)k(k +1)
mn −k . n = k + i
This gives rise to the equalityΩm(n) = Em(n). 
§4. Finally, the generating functionΩm[x] can be expressed alternatively as
Ωm[x] = (1+ y)
2
(1+ y −my)2 =
(1+ y)2
(1− (m −1)x(1+ y)m)2
=
∞∑
k=0
(k +1)(m −1)k xk (1+ y)mk+2. y = x(1+ y)m
By making use of (3) with α= 2+mk and β= m, we deduce that
Ωm[x] =
∞∑
k=0
(m −1)k xk
∞∑
i=0
xi
(
2+mk + i m
i
)
(k +1)(mk +2)
2+mk + i m
=
∞∑
n=0
xn
n∑
k=0
(m −1)k
(
2+mn
n −k
)
(k +1)(2+mk)
2+mn n = k + i
which leads us to the fourth identityΩm(n) = Fm(n). 
§5. Before concluding the paper, it is worthwhile making the following comments.
• As done by Duarte and Guedes de Oliveira [9], for an arbitrary real number λ, we have the
identity
Ωm(n) =
n∑
k=0
(
mk +λ
k
)(
mn −mk −λ
n −k
)
.
This can be justified easily by the functional relation
Ωm[x] = (1+ y)
2
(1+ y −my)2 =
(1+ y)1+λ
1+ y −my ×
(1+ y)1−λ
1+ y −my .
• Observe that the equalities in (5) hold for infinitely many integers m. According to the
fundamental theorem of polynomial algebra, these identities are valid also when m is
replaced by a variable and any complex number.
• For the remaining two sequences Cn and Dn , we failed to find their generalized expres-
sions Cm(n) and Dm(n). The interested reader is encouraged to make further attempts.

C. R. Mathématique — 2021, 359, n 4, 421-425
Mei Bai and Wenchang Chu 425
References
[1] J.-P. Allouche, “A generalization of an identity due to Kimura and Ruehr”, Integers 18A (2018), article no. a1.
[2] ——— , “Two binomial identities of Ruehr revisited”, Am. Math. Mon. 126 (2019), no. 3, p. 217-225.
[3] H. Alzer, H. Prodinger, “On Ruehr’s identities”, Ars Comb. 139 (2018), p. 247-254.
[4] M. Bai, W. Chu, “Seven equivalent binomial sums”, Discrete Math. 343 (2020), no. 2, article no. 111691.
[5] W. Chu, “Generating functions and combinatorial identities”, Glas. Mat., III. Ser. 33 (1998), no. 1, p. 1-12.
[6] ——— , “Some binomial convolution formulas”, Fibonacci Q. 40 (2002), no. 1, p. 19-32.
[7] ——— , “Logarithms of a binomial series: Extension of a series of Knuth”, Math. Commun. 24 (2019), no. 1, p. 83-90.
[8] L. Comtet, Advanced Combinatorics. The art of finite and infinite expansions, revised and enlarged ed., Reidel
Publishing Company, 1974, Translated from the French by J. W. Nienhuys.
[9] R. Duarte, A. G. de Oliveira, “Note on the convolution of binomial coefficients”, J. Integer Seq. 16 (2013), no. 7, article
no. 13.7.6.
[10] S. B. Ekhad, D. Zeilberger, “Some Remarks on a recent article by J.-P. Allouche”, https://arxiv.org/abs/1903.09511,
2019.
[11] H. W. Gould, “Some generalizations of Vandermonde’s convolution”, Am. Math. Mon. 63 (1956), no. 2, p. 84-91.
[12] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Group, 1989.
[13] E. Kiliç, T. Arikan, “Ruehr’s identities with two additional parameters”, Integers 16 (2016), article no. A30.
[14] N. Kimura, O. G. Ruehr, “Change of variable formula for definite integral”, Am. Math. Mon. 87 (1980), no. 4, p. 307-
308.
[15] J. H. Lambert, “Observationes variae in Mathesin puram”, Acta Helvetica 3 (1758), no. 1, p. 128-168, reprinted in his
Opera Mathematica, volume 1, p. 16–51.
[16] S. Meehan, A. Tefera, W. Michael, A. Zeleke, “Proofs of Ruehr’s identities”, Integers 14 (2014), article no. A10.
[17] J. Riordan, Combinatorial Identities, John Wiley & Sons, 1968.
C. R. Mathématique — 2021, 359, n 4, 421-425


Further Equivalent Binomial Sums

Jun-Yu Zhou

Quan-Hui Yang

Comptes Rendus
Mathématique
Jun-Yu Zhou and Quan-Hui Yang
On the structure of the h-fold sumsets
Volume 359, issue 4 (2021), p. 493-500
<https://doi.org/10.5802/crmath.191>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 4, p. 493-500
https://doi.org/10.5802/crmath.191
Number theory / Théorie des nombres
On the structure of the h-fold sumsets
Jun-Yu Zhoua and Quan-Hui Yang∗, a
a School of Mathematics and Statistics, Nanjing University of Information Science
and Technology, Nanjing 210044, China
E-mails: 1045908839@qq.com, yangquanhui01@163.com
Abstract. Let A be a set of nonnegative integers. Let (h A)(t ) be the set of all integers in the sumset h A
that have at least t representations as a sum of h elements of A. In this paper, we prove that, if k ≥ 2,
and A = {a0, a1, . . . , ak } is a finite set of integers such that 0 = a0 < a1 < ·· · < ak and gcd(a1, a2, . . . , ak ) = 1,
then there exist integers ct ,dt and sets Ct ⊆ [0,ct −2], Dt ⊆ [0,dt −2] such that
(h A)(t ) =Ct ∪
[
ct ,hak −dt
]∪ (hak−1 −Dt )
for all h≥∑ki=2(t ai −1)−1. This improves a recent result of Nathanson with the bound h≥(k−1)(t ak −1)ak +1.
2020 Mathematics Subject Classification. 11B13.
Funding. This author was supported by the National Natural Science Foundation for Youth of China, Grant
No. 11501299, the Natural Science Foundation of Jiangsu Province, Grant Nos. BK20150889, 15KJB110014.
Manuscript received 4th January 2021, revised 12th February 2021, accepted 15th February 2021.
1. Introduction
Let A and B be sets of integers. The sumsets and difference sets are defined by
A+B = {a +b : a ∈ A,b ∈ B}, A−B = {a −b : a ∈ A,b ∈ B}
respectively. For any integer t , we define the sets
t + A = {t }+ A, t − A = {t }− A.
For h ≥ 2, we denote by h A the h-fold sumset of A, which is the set of all integers n of the
form n = a1 +a2 +·· ·+ah , where a1, a2, . . . , ah are elements of A and not necessarily distinct.
In [3, 4], Nathanson proved the following fundamental beautiful theorem on the structure of
h-fold sumsets.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
494 Jun-Yu Zhou and Quan-Hui Yang
Nathanson’s Theorem A. Let A = {a0, a1, . . . , ak } be a finite set of integers such that
0 = a0 < a1 < ·· · < ak and gcd(A) = 1.
Let h1 = (k −1)(ak −1)ak +1. There are nonnegative integers c1 and d1 and finite sets C1 and D1
with C1 ⊆ [0,c1 −2] and D1 ⊆ [0,d1 −2] such that
h A =C1 ∪ [c1,hak −d1]∪ (hak −D1)
for all h ≥ h1.
Later, Wu, Chen and Chen [6] improved the lower bound of h1 to
∑k
i=2 ai−k. Recently, Granville
and Shakan [1], and Granville and Walker [2] gave some further results on this topic.
Let A be a set of integers. For every positive integer h, the h-fold representation function
r A,h(n) counts the number of representations of n as the sum of h elements of A. Thus,
r A,h(n) = ]
{(
a j1 , . . . , a jh
) ∈ Ah : n = h∑
i=1
a ji and a j1 ≤ ·· · ≤ a jh
}
.
For every positive integer t , let (h A)(t ) be the set of all integers n that have at least t representa-
tions as the sum of h elements of A, that is,
(h A)(t ) = {n ∈ Z : r A,h(n) ≥ t}
Recently, Nathanson [5] found that the sumsets (h A)(t ) have the same structure as the sumset
h A and proved the following theorem.
Nathanson’s Theorem B. Let k ≥ 2, and let A = {a0, a1, . . . , ak } be a finite set of integers such that
0 = a0 < a1 < ·· · < ak and gcd(A) = 1. For every positive integer t , let ht = (k−1)(t ak−1)ak+1. There
are nonnegative integers ct and dt , and finite sets Ct and D t with Ct ⊆ [0,ct −2] and D t ⊆ [0,dt −2]
such that
(h A)(t ) =Ct ∪ [ct ,hak −dt ]∪ (hak −D t )
for all h ≥ ht .
In this paper, motivated by the idea of Wu, Chen and Chen [6], we improved the lower bound
of h in Nathanson’s Theorem B.
Theorem 1. Let k ≥ 2, and let A = {a0, a1, . . . , ak } be a finite set of integers such that
0 = a0 < a1 < ·· · < ak and gcd(A) = 1
For every positive integer t , let
ht =
k∑
i=2
(t ai −1)−1
There are nonnegative integers ct and dt and finite sets Ct and D t with
Ct ⊆ [0,ct −2] and D t ⊆ [0,dt −2]
such that
(h A)(t ) =Ct ∪ [ct ,hak −dt ]∪ (hak −D t ) (1)
for all h ≥ ht .
Remark 2. Theorem 1 is optimal.
We shall prove Theorem 1 and Remark 2 in Section 3. In Section 2, we give some lemmas.
C. R. Mathématique — 2021, 359, n 4, 493-500
Jun-Yu Zhou and Quan-Hui Yang 495
2. Some Lemmas
Lemma 3 (see [5, Lemma 1]). Let A be a set of integers. For any positive integer h and t, we have
(h A)(t ) + A ⊆ ((h +1)A)(t ).
Lemma 4. Let k ≥ 2, and let A = {a0, a1, . . . , ak } be a set of integers satisfying 0 = a0 < a1 < ·· · < ak
and gcd(A) = 1. For every positive integer t , let ht =∑ki=2(t ai −1)−1 and c ′t =∑k−1i=1 ai (t ai+1−1). If
c ′t −ak < n < c ′t , then there exist at least t distinct nonnegative k-tuples (x1,s , x2,s , . . . , xk,s ) (1 ≤ s ≤ t )
satisfying
n = x1,s a1 +x2,s a2 +·· ·+xk,s ak
and x1,s +x2,s +·· ·+xk,s ≤ ht for s = 1,2, . . . , t .
Proof. Since gcd(a1, . . . , ak ) = 1, there exist integers x1, . . . , xk such that
n = x1a1 +·· ·+xk ak .
For any positive integer s, [(s−1)a2, sa2−1] is a complete residue system modulo a2. Hence there
exists an integer q such that x1 = a2q +x1,s with (s −1)a2 ≤ x1,s ≤ sa2 −1. This gives
n = x1,s a1 +
(
a1q +x2
)
a2 +·· ·+xk ak .
Let x ′2 = a1q + x2. Similarly, there exists an integer q ′ such that x ′2 = a3q ′+ x2,s with (s − 1)a3 ≤
x2,s ≤ sa3 −1. Now we have
n = x1,s a1 +x2,s x2 + (a2q ′+x3)a3 +·· ·+xk ak .
By continuing this process, we obtain
n = x1,s a1 +x2,s a2 +·· ·+xk,s ak
with (s−1)ai+1 ≤ xi ,s ≤ sai+1−1 for i = 1, . . . ,k −1 and xk,s is some integer. Hence, for any integer
s ∈ [1, t ], we have
0 ≤ xi ,s ≤ t ai+1 −1.
Since n > c ′t −ak , it follows that
xk,s ak = n −
(
x1,s a1 +x2,s a2 +·· ·+xk−1,s ak−1
)
≥ n − (t a2 −1) a1 −·· ·− (t ak −1) ak−1 = n − c ′t >−ak ,
and then xk,s > −1. Noting that xk,s is an integer, we have xk,s ≥ 0. By the bound of xi ,s , the
following nonnegative k-tuples
(x1,s , x2,s , . . . , xk−1,s , xk,s ) (1 ≤ s ≤ t )
are distinct.
Next, we shall prove that x1,s +x2,s +·· ·+xk,s ≤ ht for s = 1,2, . . . , t .
For any integer s ∈ [1, t ], let x1,s +x2,s +·· ·+xk,s = us . Since n < c ′t , it follows that
n = x1,s a1 +x2,s a2 +·· ·+xk,s ak
= x1,s a1 +·· ·+xk−1,s ak−1 +
(
us −x1,s −x2,s −·· ·−xk−1,s
)
ak
= us ak −x1,s (ak −a1)−·· ·−xk−1,s (ak −ak−1)
≥ us ak − (t a2 −1)(ak −a1)−·· ·− (t ak −1)(ak −ak−1)
= us ak −ak [(t a2 −1)+·· ·+ (t ak −1)]+a1 (t a2 −1)+·· ·+ak−1 (t ak −1)
= us ak − (ht +1) ak + c ′t
> us ak − (ht +1) ak +n.
Hence us ak − (ht +1) ak < 0, and then us < ht +1. Therefore, us ≤ ht .
This completes the proof of Lemma 4. 
C. R. Mathématique — 2021, 359, n 4, 493-500
496 Jun-Yu Zhou and Quan-Hui Yang
Lemma 5. Let c ′t and ht be defined in Lemma 4. Then
c ′t =
k−1∑
i=1
ai (t ai+1 −1) ∈ ((ht +1) A)(t ) .
Proof. For i = 1,2, . . . ,k −1, let pi = t ai+1 −1. Then
c ′t = (t a2 −1) a1 +·· ·+ (t ak −1) ak−1 = p1a1 +·· ·+pk−1ak−1.
Noting that
p1 +·· ·+pk−1 =
k∑
i=2
(t ai −1) = ht +1,
we have c ′t ∈ (ht +1) A.
Moreover, for any integer r ∈ [0, t −1], we have
c ′t =
k−1∑
i=1
(t ai+1 −1) ai =
k−1∑
i=1
((t − r )ai+1 −1) ai + r
k−1∑
i=1
ai ai+1
= ((t − r )a2 −1) a1 +
k−1∑
i=2
((t − r )ai+1 −1+ r ai−1)ai + r ak−1ak
:= p1,r a1 +p2,r a2 +·· ·+pk−1,r ak−1 +pk,r ak ,
where p1,r = (t − r )a2 − 1, pk,r = r ak−1 and pi ,r = (t − r )ai+1 − 1+ r ai−1 (2 ≤ i ≤ k − 1). Hence
pi ,r ≥ 0 for all i ∈ [1,k] and
k∑
i=1
pi ,r = (t − r )a2 −1+ (t − r )a3 −1+ r a1 +·· ·+ (t − r )ak −1+ r ak−2 + r ak−1
= ht +1− r (a2 +·· ·+ak )+ r (a1 +·· ·+ak−1)
= ht +1− r (ak −a1) ≤ ht +1.
Thus, r A,ht+1(c
′
t ) ≥ t , and so c ′t ∈ ((ht +1) A)(t ). 
Lemma 6. Let n and a1, a2 be positive integers with gcd(a1, a2) = 1. For any positive integer t , if
n > t a1a2 −a1 −a2, then the diophantine equation
a1x +a2 y = n (2)
has at least t nonnegative integer solutions. The lower bound of n is also best possible.
Proof. Suppose that n > t a1a2−a1−a2. Let (x0, y0) be a solution of the equation (2). Then all the
integer solutions of the equation (2) is{
x = x0 +ka2,
y = y0 −ka1,
k ∈Z. (3)
In order to have x ≥ 0 and y ≥ 0, we only need x >−1 and y >−1, that is,
−1−x0
a2
< k < y0 +1
a1
. (4)
Since
y0 +1
a1
− −1−x0
a2
= a1 +a2 +a1x0 +a2 y0
a1a2
= a1 +a2 +n
a1a2
> a1 +a2 + t a1a2 −a1 −a2
a1a2
= t ,
there exist at least t integers k such that (4) holds.
Therefore, the equation (2) has at least t nonnegative integer solutions.
C. R. Mathématique — 2021, 359, n 4, 493-500
Jun-Yu Zhou and Quan-Hui Yang 497
Now suppose that l = t a1a2 − a1 − a2. Then (t a2 −1,−1) is a solution of (2). Take x0 = t a2 −1
and y0 =−1. Then (3) becomes {
x = t a2 −1−ka2,
y =−1+ka1,
k ∈Z.
Since x ≥ 0 and y ≥ 0, it follows that 1 ≤ k ≤ t − 1. Hence there exist at most t − 1 nonnegative
integer solutions.
This completes the proof of Lemma 6. 
3. Proofs
Proof of Theorem 1. Let c ′t =
∑k−1
i=1 ai (t ai+1−1). By Lemma 4, there exist smallest integers ct and
dt satisfying [
c ′t −ak +1,c ′t −1
]⊆ [ct ,ht ak −dt ] ⊆ (ht A)(t ) .
It follows that ct −1 ∉ (ht A)(t ) and ht ak −dt +1 ∉ (ht A)(t ). Additionally
ct ≤ c ′t −ak +1, (5)
c ′t −1 ≤ ht ak −dt . (6)
Define the finite sets Ct and D t by
Ct = (ht A)(t ) ∩ [0,ct −2]
and
ht ak −D t = (ht A)(t ) ∩ [ht ak − (dt −2),ht ak ] .
Then
(ht A)
(t ) =Ct ∪ [ct ,ht ak −dt ]∪ (ht ak −D t ) . (7)
Therefore, (1) holds for h = ht .
Now we prove (1) by induction on h. Suppose that (1) holds for some h ≥ ht . Define
B (t ) =Ct ∪ [ct , (h +1)ak −dt ]∪ ((h +1)ak −D t ) .
Firstly we prove that B (t ) ⊆ ((h +1)A)(t ).
Take an arbitrary integer b ∈ B (t ).
Case 1: b ∈Ct ∪ [ct ,ht ak −dt ]. By (7), we have
b ∈ (ht A)(t ) ⊆ ((h +1)A)(t ).
Case 2: b ∈ [ct +ak , (h +1)ak −dt ]∪ ((h +1)ak −D t ). It follows that
b −ak ∈ [ct ,hak −dt ]∪ (hak −D t ) ⊆ (h A)(t ).
Thus, By Lemma 3, b ∈ (h A)(t ) +ak ⊆ ((h +1)A)(t ).
Case 3: ht ak −dt +1 ≤ b ≤ ct +ak −1. By (5) and (6), we have
ct +ak −1 ≤ c ′t ≤ ht ak −dt +1. (8)
Thus b = c ′t . By Lemma 5, we have
b = c ′t ∈ ((ht +1) A)(t ) ⊆ ((h +1)A)(t ).
Therefore, B (t ) ⊆ ((h +1)A)(t ).
Next we shall prove that ((h +1)A)(t ) ⊆ B (t ). Take an arbitrary integer a ∈ ((h +1)A)(t ).
C. R. Mathématique — 2021, 359, n 4, 493-500
498 Jun-Yu Zhou and Quan-Hui Yang
Case 1: a = c ′t . By (8) and h ≥ ht , we have
ct ≤ c ′t ≤ ht ak −dt +1 ≤ (h +1)ak −dt .
Hence a = c ′t ∈ B (t ).
Case 2: a 6= c ′t and a ∉ (h A)(t ). Since a ∈ ((h + 1)A)(t ), there exist t nonnegative integer k-tuples
(x1,s , x2,s , . . . xk,s ) (1 ≤ s ≤ t ) satisfying
a = x1,s a1 +x2,s a2 +·· ·+xk,s ak and x1,s +x2,s +·· ·+xk,s ≤ h +1.
Furthermore, we can get
0 ≤ xi ,s ≤ t ai+1 −1, 1 ≤ i ≤ k −1, 1 ≤ s ≤ t . (9)
Otherwise, without loss of generality, assume that x1,1 ≥ t a2, then for j = 1,2, . . . , t , we have
a = x1,1a1 +x2,1a2 +·· ·+xk,1ak
= (x1,1 − j a2)a1 + (x2,1 + j a1)a2 +·· ·+xk,1ak .
Noting that for j = 1,2, . . . , t ,
(x1,1 − j a2)+ (x2,1 + j a1)+x3,1 +·· ·+xk,1 = h +1− j (a2 −a1) < h +1,
we have a ∈ (h A)(t ), a contradiction. Hence the inequality (9) holds.
By a ∉ (h A)(t ), there exists s ∈ [1, t ] such that a = x1,s a1 +x2,s a2 +·· ·+xk,s ak and
x1,s +x2,s +·· ·+xk,s = h +1.
By (9), we have
a = x1,s a1 +x2,s a2 +·· ·+xk,s ak
= x1,s a1 +·· ·+xk−1,s ak−1 +
(
h +1−x1,s −x2,s −·· ·−xk−1,s
)
ak
= (h +1)ak −x1,s (ak −a1)−·· ·−xk−1,s (ak −ak−1)
≥ (h +1)ak − (t a2 −1)(ak −a1)−·· ·− (t ak −1)(ak −ak−1)
= (h +1)ak −ak [(t a2 −1)+·· ·+ (t ak −1)]+a1 (t a2 −1)+·· ·+ak−1 (t ak −1)
= (h +1)ak − (ht +1) ak + c ′t
≥ c ′t .
Since a 6= c ′t , it follows that a ≥ c ′t +1. By (5), we have a ≥ c ′t +1 ≥ ct +ak .
If xk,s = 0 for some s with 1 ≤ s ≤ t , by (9), then
a ≤ (t a2 −1) a1 +·· ·+ (t ak −1) ak−1 = c ′t ,
a contradiction.
Hence xk,s ≥ 1 for all integers s = 1,2, . . . , t .
Therefore, a −ak ∈ (h A)(t ) and a −ak ≥ ct . By the induction hypothesis,
a ∈ ak + [ct ,hak −dt ]∪ (hak −D t ) = [ct +ak , (h +1)ak −dt ]∪ ((h +1)ak −D t ) ⊆ B (t ).
Case 3: a 6= c ′t and a ∈ (h A)(t ). By the induction hypothesis, we have
(h A)(t ) =Ct ∪ [ct ,hak −dt ]∪ (hak −D t ) .
Since Ct ∪ [ct , (h +1)ak −dt ] ⊆ B (t ), we can suppose that a > (h +1)ak −dt . By a ∈ (h A)(t ), there
exist at least t distinct nonnegative k-tuples (x1,s , x2,s , . . . , xk,s ) (1 ≤ s ≤ t ) such that
a = x1,s a1 +x2,s a2 +·· ·+xk,s ak
and
x1,s +x2,s +·· ·+xk,s ≤ h.
C. R. Mathématique — 2021, 359, n 4, 493-500
Jun-Yu Zhou and Quan-Hui Yang 499
As in the proof of Lemma 4, assume that 0 ≤ xi ,s ≤ t ai+1−1 for i = 1,2, . . . ,k−1. If xk,s ≤ 0, then
by (6) we have
a ≤ x1,s a1 +x2,s a2 +·· ·+xk−1,s ak−1
≤ a1 (t a2 −1)+·· ·+ak−1 (t ak −1)
= c ′t ≤ ht ak −dt +1
≤ (ht +1) ak −dt ≤ (h +1)ak −dt ,
which contradicts with a > (h + 1)ak − dt . Therefore xk,s ≥ 1 and a − ak ∈ (h A)(t ). Since a >
(h +1)ak −dt , it follows that a −ak ∈ hak −D t . Hence
a ∈ (h +1)ak −D t ⊆ B (t ),
and so ((h +1)A)(t ) ⊆ B (t ).
This completes the proof of Theorem 1. 
Proof of Remark 2. Let n ≥ 3 be an integer and A = {0, n, n+1}. By Theorem 1, there exist integers
ct , dt and sets Ct ⊆ [0,ct −2], D t ⊆ [0,dt −2] such that
(h A)(t ) =Ct ∪ [ct ,hak −dt ]∪ (hak −D t )
for all h ≥ ht = t (n +1)−2.
For any integer m ≥ ct , choose an integer h′ ≥ t (n+1)−2 such that h′ak −dt ≥ m, then we have
m ∈ (h′A)(t ).
Hence, there exist t nonnegative integer tuples (ui , vi ) (1 ≤ i ≤ t ) such that m = ui n+vi (n+1).
On the other hand, there does not exist t nonnegative integer tuples (ui , vi ) (1 ≤ i ≤ t ) such
that ct − 1 = ui n + vi (n + 1). Otherwise, if exist, choose h > max1≤i≤t {ui + vi }, then we have
ct − 1 ∈ (h A)(t ), a contradiction. Hence, by Lemma 6, it follows that ct − 1 = t a1a2 − a1 − a2 =
tn(n +1)−n − (n +1), and then ct = tn(n +1)−2n.
Let p ∈ ((ht −1)A)(t ). Then there exist t nonnegative integer tuples (ui , vi ) (1 ≤ i ≤ t ) such that
p = ui n + vi (n +1) and u1 > u2 > ·· · > ut are the maximal t numbers in all the representations.
Hence
p = u1n + v1(n +1) = [u1 − (n +1)]n + (v1 +n)(n +1)
= [u1 −2(n +1)]n + (v1 +2n)(n +1)
= . . .
= [u1 − (t −1)(n +1)]n + [v1 + (t −1)n](n +1).
It follows that ut = u1 − (t −1)(n +1), vt = v1 + (t −1)n. Noting that
ut + vt < ut−1 + vt−1 < ·· · < u1 + v1 ≤ ht −1,
we have
ut + vt = u1 − (t −1)(n +1)+ v1 + (t −1)n
= u1 + v1 − (t −1) ≤ ht −1− (t −1) = tn −2.
Hence, for every p ∈ ((ht −1)A)(t ),
p = ut n + vt (n +1) ≤ (ut + vt )(n +1) ≤ (tn −2)(n +1)
= tn(n +1)−2(n +1) < tn(n +1)−2n = ct .
By (1), it follows that
((ht −1)A)(t ) ⊆ [0, tn(n +1)−2(n +1)].
Therefore, (1) cannot hold for h = ht −1, and so Theorem 1 is optimal. 
Acknowledgement
We would like to thank the referee for their useful comments.
C. R. Mathématique — 2021, 359, n 4, 493-500
500 Jun-Yu Zhou and Quan-Hui Yang
References
[1] A. Granville, G. Shakan, “The Frobenius postage stamp problem, and beyond”, https://arxiv.org/abs/2003.04076,
2020.
[2] A. Granville, A. Walker, “A tight structure theorem for sumsets”, https://arxiv.org/abs/2006.01041, 2020.
[3] M. B. Nathanson, “Sums of finite sets of integers”, Am. Math. Mon. 79 (1972), p. 1010-1012.
[4] ——— , Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, vol.
165, Springer, 1996.
[5] ——— , “Sums of finite sets of integers, II”, https://arxiv.org/abs/2005.10809v3, 2020.
[6] J. Wu, F. Chen, Y. Chen, “On the structure of the sumsets”, Discrete Math. 311 (2011), no. 6, p. 408-412.
C. R. Mathématique — 2021, 359, n 4, 493-500


On the structure of the $h$-fold sumsets

Henry Wilton

Comptes Rendus
Mathématique
Henry Wilton
On the profinite rigidity of surface groups and surface words
Volume 359, issue 2 (2021), p. 119-122.
<https://doi.org/10.5802/crmath.121>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 2, p. 119-122
https://doi.org/10.5802/crmath.121
Group theory / Théorie des groupes
On the profinite rigidity of surface groups and
surface words
HenryWiltona
a DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3
0WB, UK.
E-mail: h.wilton@maths.cam.ac.uk
Abstract. Surface groups are determined among limit groups by their profinite completions. As a corollary,
the set of surface words in a free group is closed in the profinite topology.
Manuscript received 3rd September 2020, revised 23rd September 2020, accepted 15th October 2020.
Profinite rigidity is the study of the extent to which a group G is determined by its set of
finite quotients, or, in more sophisticated terminology, by its profinite completion Ĝ . Since Ĝ
only depends on the image of the natural map G → Ĝ , we should assume that G is residually
finite, which means that the map is injective. Remeslennikov asked one of the most provocative
questions in this area [10, Question 15].
Question 1 (Remeslennikov). Let F be a free (non-abelian) group, and let G be finitely generated
and residually finite. If Ĝ ∼= F̂ , does it follow that G ∼= F ?
In terminology that has recently become popular, Remeslennikov’s question asks if free groups
are absolutely profinitely rigid. Equally good questions can be formulated by replacing the free
group F by other groups of interest, such as surface groups.
Many finitely generated groups are known not to be absolutely profinitely rigid; the first
examples were a pair of meta-cyclic groups exhibited by Baumslag [3]. There are some easy
examples of absolutely profinitely rigid groups – for instance finite groups and finitely generated
abelian groups – but beyond these and their generalisations no finitely generated examples of
absolutely profinitely rigid groups were known until the recent work of Bridson, McReynolds,
Reid and Spitler [4,5], who showed that certain Kleinian groups, and even certain triangle groups,
are absolutely profinitely rigid. However, their techniques rely on delicate arithmetic information,
and an answer to Question 1 remains a distant prospect, as does its analogue for surface groups.
One way to make progress is to place more restrictive hypotheses on G . A group G is said to be
fully residually free if every finite subset S ⊆G can be mapped injectively by a homomorphism to a
free group F . The finitely generated fully residually free groups, called limit groups by Sela, form
a rich class of groups that are often difficult to distinguish from free groups, and that include
all surface groups except for the three non-orientable surfaces with χ ≥ −1 [1]. They came to
prominence because of the central role that they play in Sela’s solution to Tarski’s problems
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
120 Henry Wilton
on the elementary theory of free groups (see [12] et seq.; see also the work of Kharlampovich–
Miasnikov [8] et seq.).
As a consequence of the existence of surface subgroups in non-free limit groups, free groups
are profinitely rigid among limit groups [15, Corollary D]. The main theorem of this note extends
that result to surface groups.
Theorem 2. Let G be a limit group, and suppose that Ĝ ∼= π̂1Σ, where Σ is a closed surface. Then
G ∼=π1Σ.
Proof. Suppose that G is a limit group with Ĝ ∼= π̂1Σ. By [15, Corollary D], G is not free, so G
contains a non-trivial surface subgroup π1S by [15, Corollary C].
By [14], there is a retraction
r : K →π1S ,
where K is a subgroup of finite index in G . Unless K is a surface group, the kernel of r contains an
infinite cyclic subgroup 〈k〉, since limit groups are torsion-free. Passing to profinite completions,
r induces a continuous retraction
r̂ : K̂ → π̂1S
which contains the infinite pro-cyclic subgroup 〈k〉 ∼= Ẑ in its kernel. In particular, the supernat-
ural number 2∞ divides the index of π̂1S in K̂ .
This implies that K̂ is not a profinite surface group. Indeed, if it were, then K̂ would be a
profinite PD2 group, so by an observation of Serre [13, Exercise 5(b) on p. 44], and also [17,
Lemma 1.6], π̂1S would have trivial second continuous cohomology. But surface groups are
good [6, Proposition 3.7], so
H 2
(
π̂1S,Z/2
)∼= H 2 (π1S,Z/2) ∼=Z/2,
which is a contradiction.
Hence K is a surface group, so G is also a surface group by Nielsen realisation, since it is
torsion-free. But surface groups are distinguished from each other by their profinite completions,
so G ∼=π1Σ. 
Similar questions can also be asked fruitfully about subsets of a finitely generated free group
F . The profinite topology on a group is generated by the cosets of the subgroups of finite index,
and a subset is called separable if it is closed in the profinite topology. The following question is
then the analogue of absolute profinite rigidity for subsets of free groups (cf. [11, pp. 93–94]).
Question 3. Let w ∈ F . Is the automorphism orbit Aut(F ).w separable in F ?
This question appears to be very difficult for any w 6= 1. The case where w is a primitive
element of F – i.e. an element of a free basis – was answered affirmatively by Parzanchevski and
Puder [11], who proved that primitive elements are distinguished by the push-forward measures
they induce under random maps to finite symmetric groups. A different proof was later given
by the author [15, Corollary E], as a corollary of the profinite rigidity of free groups among limit
groups.
Surface words are often regarded as the “next simplest” class of words after primitive words.
For n even, the orientable surface words are those in the automorphism orbit of the product of
commutators [a1, a2] . . . [an−1, an]. For any n, the non-orientable surface words are those in the
orbit of the product of squares a21 . . . a
2
n . The terminology stems from the fact that, if w is a surface
word, then the presentation
F = 〈a1, . . . , an ,b |b = w〉
arises when one considers F as the fundamental group of a compact surface with a single
boundary component b.
C. R. Mathématique, 2021, 359, n 2, 119-122
Henry Wilton 121
Magee and Puder studied surface words in [9], and showed that they are characterised by
the measure that they induce under random homomorphisms to compact Lie groups. However,
since the Lie groups they consider are sometimes infinite, this does not imply that the set of
surface words is separable. Hanany, Meiri and Puder proved that the automorphism orbit of the
commutator [a1, a2] is separable [7], again by studying the push-forward measure associated to
a random map to a symmetric group.
It is an easy corollary of Theorem 2 that any automorphism orbit of surface words is separable.
Corollary 4. The sets of orientable and non-orientable surface words in a free group F are
separable.
Proof. A word w ∈ F is a surface word if and only if the double
D(w) = F ∗〈w〉 F
is isomorphic toπ1Σ, whereΣ is a closed hyperbolic surface. On the other hand, w is in the closure
of the set of surface words if and only if D(w) is a profinite hyperbolic surface group. Therefore,
it suffices to prove that, if D(w) is a profinite hyperbolic surface group, then D(w) is a hyperbolic
surface group.
If w were a proper power, say w = un for n > 1, then D(w) would contain a copy of the
torus-link group Z∗nZZ, and thence a copy of Z2. Every finite-index subgroup of this Z2 would
be separable in D(w) [18, Theorem 5.1], so D(w) would contain a copy of Ẑ2. But profinite
hyperbolic surface groups do not contain Ẑ2 [16, Theorem D], so w is not a proper power.1
Therefore, w is not a proper power, so D(w) is a limit group [2], and hence a surface group
by Theorem 2, so w is a surface word. Finally, (non)-orientability is determined by whether or
not the corresponding surface is orientable. Since this is detected by the continuous cohomology
group
H 2
(D(w),Z/3)∼= H 2 (D(w),Z/3)
it follows that the subsets of orientable and non-orientable surface words are separable. 
Remark 5. There is also a notion of a surface tuple of words, arising from surfaces with more than
one boundary component. The methods of this paper also show that surface tuples are separable,
but the details are left to the reader.
Remark 6. By [7, Theorem 1.7], Corollary 4 also extends to powers of surface words.
Acknowledgements
I am grateful to Doron Puder for pointing out [7, Theorem 1.7].
References
[1] B. Baumslag, “Residually free groups”, Proc. Lond. Math. Soc. 17 (1967), p. 402-418.
[2] G. Baumslag, “On generalised free product”, Math. Z. 78 (1962), p. 423-438.
[3] ——— , “Residually finite groups with the same finite images”, Compos. Math. 29 (1974), p. 249-252.
[4] M. R. Bridson, D. B. McReynolds, A. W. Reid, R. Spitler, “On the profinite rigidity of triangle groups”, https://arxiv.
org/abs/2004.07137, 2004.
[5] ——— , “Absolute profinite rigidity and hyperbolic geometry”, Ann. Math. 192 (2020), no. 3, p. 679-719.
[6] F. J. Grunewald, A. Jaikin-Zapirain, P. A. Zalesskii, “Cohomological goodness and the profinite completion of Bianchi
groups”, Duke Math. J. 144 (2008), no. 1, p. 53-72.
1The argument of this paragraph should also have been included in the proof of [15, Corollary E].
C. R. Mathématique, 2021, 359, n 2, 119-122
122 Henry Wilton
[7] L. Hanany, C. Meiri, D. Puder, “Some orbits of free words that are determined by measures on finite groups”, J. Algebra
555 (2020), p. 305-324.
[8] O. Kharlampovich, A. Myasnikov, “Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations
and Nullstellensatz”, J. Algebra 200 (1998), no. 2, p. 472-516.
[9] M. Magee, D. Puder, “Surface words are determined by word measures on groups”, https://arxiv.org/pdf/1902.04873.
pdf, to appear in Israel Journal of Mathematics, 2019.
[10] G. A. Noskov, V. N. Remeslennikov, V. A. Roman’kov, “Infinite groups”, Itogi Nauki Tekh., Ser. Algebra, Topologiya,
Geom. 17 (1979), p. 65-157.
[11] D. Puder, O. Parzanchevski, “Measure preserving words are primitive”, J. Am. Math. Soc. 28 (2015), no. 1, p. 63-97.
[12] Z. Sela, “Diophantine geometry over groups. I. Makanin–Razborov diagrams”, Publ. Math., Inst. Hautes Étud. Sci. 93
(2001), p. 31-105.
[13] J.-P. Serre, Galois cohomology, Springer Monographs in Mathematics, Springer, 1997, Translated from the French by
Patrick Ion and revised by the author.
[14] H. Wilton, “Hall’s Theorem for limit groups”, Geom. Funct. Anal. 18 (2008), no. 1, p. 271-303.
[15] ——— , “Essential surfaces in graph pairs”, J. Am. Math. Soc. 31 (2018), no. 4, p. 893-919.
[16] H. Wilton, P. Zalesskii, “Distinguishing geometries using finite quotients”, Geom. Topol. 21 (2017), no. 1, p. 345-384.
[17] H. Wilton, P. A. Zalesskii, “Profinite detection of 3-manifold decompositions”, Compos. Math. 155 (2019), no. 2,
p. 246-259.
[18] D. T. Wise, “Subgroup separability of graphs of free groups with cyclic edge groups”, Q. J. Math 51 (2000), no. 1,
p. 107-129.
C. R. Mathématique, 2021, 359, n 2, 119-122


On the profinite rigidity of surface groups and surface words

Samer Israwi

Henrik Kalisch

Comptes Rendus
Mathématique
Samer Israwi and Henrik Kalisch
A Mathematical Justification of the Momentum Density Function
Associated to the KdV Equation
Volume 359, issue 1 (2021), p. 39-45.
<https://doi.org/10.5802/crmath.143>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 1, p. 39-45
https://doi.org/10.5802/crmath.143
Partial differential equations / Équations aux dérivées partielles
A Mathematical Justification of the
Momentum Density Function Associated to
the KdV Equation
Samer Israwia and Henrik Kalisch∗, b
a Lebanese University, Faculty of Sciences 1, Department of Mathematics,
Hadath-Beirut, Lebanon
b Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen,
Norway
E-mails: s_israwi83@hotmail.com, henrik.kalisch@uib.no
Abstract. Consideration is given to the KdV equation as an approximate model for long waves of small
amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density
associated to the KdV equation, and the difference between this density and the physical momentum density
derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter.
Résumé. L’équation de KdV est considérée comme un modèle approximatif pour des ondes longues de
faible amplitude à la surface libre d’un fluide non visqueux. On montre qu’il y a une densité de moment
approximative associée à l’équation de KdV, et que la différence entre cette densité et la densité de de moment
physique dérivée dans le contexte du système d’Euler peut être estimée en fonction du paramètre d’onde
longue.
Funding. This research was supported by the Research Council of Norway under grant no. 239033/F20.
Manuscript received 16th August 2018, revised and accepted 2nd November 2020.
Version française abrégée
Dans le présent travail, nous prouverons qu’au moins pour la densité de moment, l’approxi-
mation entre l’équation de KdV et système d’Euler peut être rendue mathématiquement
rigoureuse. Le résultat principal montre que la densité I dans l’équation de KdV converge vers
la densité de moment physique définie en fonction de la solution du système d’Euler pour un
fluide parfait lorsque les paramètres physiques µ et ε tendent vers zéro. L’énoncé précis de notre
théorème est le suivant.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
40 Samer Israwi and Henrik Kalisch
Théorème. Soit (ζ,φ) une solution du système d’Euler, avec condition initiale (ζ0,φ0) suffisam-
ment régulière. Soit η une solution de l’équation de KdV (1) avec une condition initiale η0 = ζ0.
Alors, il existe une constante C tel que :∥∥∥∥∫ εζ(·, t )−1 ∂xφ(z, ·, t )d z −η(·, t )−ε34η2(·, t )−µ16ηxx (·, t )
∥∥∥∥
L∞
≤Cµ2(1+ t ).
La preuve de ce théorème est basée sur l’utilisation du système de Peregrine (9) comme
un modèle intermédiaire entre l’équation de KdV et le problème d’Euler. Le point clé est que
la densité de moment définie dans le contexte des équations d’Euler est reliée à la vitesse
moyenne V̄ . En effet, nous avons∫ εζ(x, t )
−1
∂xφ(z, x, t )d z = (1+εζ(x, t ))V̄ (x, t ),
et cette relation peut être utilisée d’une manière favorable dans la preuve car le système Pere-
grine (9) est également défini en fonction de la vitesse horizontale moyenne.
On définie la vitesse variable associée à une solution η de l’équation de KdV
vK dV = η−
ε
4
η2 − µ
6
ηxt
(voir [11]) par suite on obtient l’estimation suivante∥∥V̄ − vK dV ∥∥L∞ ≤Cµ2t ,
qui est l’ingrédient principal utilisé dans la preuve du résultat final.
1. Introduction
In the present contribution, we consider the question of momentum conservation in the context
of the Korteweg–de Vries equation. The KdV equation
ηt +ηx +ε3
2
ηηx +µ1
6
ηxxx = 0 (1)
is known to yield a valid description of surface waves for waves of small amplitude and large
wavelength at the free surface of an incompressible, inviscid fluid running in a narrow open
channel where transverse effects can be neglected.
Suppose h0 is the depth of the undisturbed fluid, and let λ denote a typical wavelength and by
a a typical amplitude of a wavefield to be described. The nondimensional number ε= a/h0 then
represents the relative amplitude. If we define the long-wave parameter by µ = h20/λ2, then the
KdV equation is known to be a good model for waves at the free surface of a fluid if the relations
µ << 1 and ε = O (µ). The approximation can be made rigorous using the techniques developed
in [7, 10, 11, 16, 17] and others.
It is well known that the KdV equation has an infinite number of formally conserved integrals
(indeed the conservation can be made rigorous by following the work of [8]). If the equation is
given in the non-dimensional form (1), the first three conserved integrals are∫ ∞
−∞
ηd x,
∫ ∞
−∞
η2 d x, and
∫ ∞
−∞
( µ
3εη
2
x −η3
)
d x. (2)
The first integral is found to be invariant with respect to time t as soon as it is recognized that the
KdV equation can be written in the form
∂
∂t
(
η
)+ ∂
∂x
(
η+ε3
4
η2 +µ1
6
ηxx
)
= 0, (3)
where the quantity appearing under the time derivative is interpreted as excess mass density,
and the term appearing under the spatial derivative is the mass flux through a cross section of
unit width due to the passage of a surface wave. The second and third integral are sometimes
C. R. Mathématique, 2021, 359, n 1, 39-45
Samer Israwi and Henrik Kalisch 41
called momentum and energy, but this terminology may be misleading since these integrals are
not readily interpreted as approximations of the physical momentum and energy appearing in
the context of the Euler equations. Indeed, the authors of [1] already state clearly that they do not
believe these integrals to be approximations of the physical momentum and density, and further
doubt was cast on this interpretation in more recent work [2, 12–14].
On the other hand, in physical flow problems, mass flux is often identical with momentum
density, so one might think that the term η + ε 34η2 + µ 16ηxx in (3) might be interpreted as
momentum flux. This is indeed correct as shown in the recent work [4], where based on ideas
developed in [3], it was shown how to find integral quantities that do represent approximations
to the physically relevant momentum and energy densities. In particular, following the procedure
laid out in [4] gives the expression for momentum density as
I = η+ε3
4
η2 +µ1
6
ηxx . (4)
Since the analysis in [4] was based on a formal asymptotic analysis, the question of whether
this identity can be made mathematically rigorous has so far remained open. In the present work
we will prove that a firm mathematical proof can indeed be given. The main result to be proved
thus states that the density I converges to the physical momentum density defined in terms of a
solutions of the governing Euler equation for a perfect fluid if µ and ε tend to zero. The precise
statement is as follows.
Theorem 1. Let (ζ,φ) be a solution of the water-wave problem defined below, with regular initial
data (ζ0,φ0). Let η be a solution of the KdV equation (1) with initial data η0 = ζ0. Then there exists
a constant C , so that we have the estimate∥∥∥∥∫ εζ(·,t )−1 ∂xφ(z, ·, t )d z −η(·, t )−ε34η2(·, t )−µ16ηxx (·, t )
∥∥∥∥
L∞
≤Cµ2(1+ t ). (5)
2. Auxiliary results
Denoting the original (dimensional) variables with a tilde, we introduce a scaling to make the
small amplitude and long wavelength relative to the undisturbed depth explicit. Thus we define
new variables (without a tilde) by x̃ =λx, z̃ = h0z, ζ̃= aζ, t̃ = λc0 t , φ̃=
aλg
c0
φ.
Then we obtain the system
µ∂2xφ+∂zφ2 = 0 in Ωt ,
∂zφ= 0, at z =−1,
∂tζ− 1µ
(−µ∂xζ∂xφ+∂zφ)= 0 at z = εζ,
∂tφ+ζ+ ε2
(
∂xφ
)2 + ε2µ (∂zφ)2 = 0 at z = εζ,
(6)
where Ωt = {(x, z),−1 < z < εζ(x, t )} is the fluid domain delimited by the free surface
{z = ζ(x, t )}, and the flat bottom {z =−1}, and whereφ(x, z, t ), defined onΩt is the velocity poten-
tial associated to the flow (that is, the two-dimensional velocity field v is given by v = (∂xφ,∂zφ)T ).
As is well known, the existence of the velocity potential is guaranteed by the assumption of irro-
tational flow. The equations above are formally equivalent to the Zakharov–Craig–Sulem (ZCS)
equations. They are written in terms of the trace Φ(x, t ) = φ(x,εζ(x, t ), t ) and the Dirichlet–
Neumann operator G(ζ) as  ζt −
1
µG(εζ)Φ= 0,
Φt +ζ+ ε2Φ2x − εµ [
G(εζ)Φ+εµζxΦx ]2
2(1+ε2µζ2x ) = 0.
(7)
C. R. Mathématique, 2021, 359, n 1, 39-45
42 Samer Israwi and Henrik Kalisch
Given a solution of this system, we reconstruct the potential φ by solving the Laplace equation in
the domain Ωt (cf. [15, 19]), and then define the average velocity in the context of the full water-
wave problem by
V̄ (x, t ) = 1
1+εζ
∫ εζ
−1
∂xφ(x, z, t )d z. (8)
From [16, Theorem 4.16], we have the following result.
Theorem 2.
(1) For large enough s, there exists a unique solution (ζ,Φ) ∈ C (0,T /ε, H s × Ḣ s+1) of the ZCS
water-wave system.
(2) For the average velocity, we have V̄ ∈C (0,T /ε, H s−3).
In the shallow-water small-amplitude regime specified above (µ¿ 1, ε=O (µ)), one can derive
the Peregrine system. For one-dimensional surfaces and flat bottoms, these equations couple the
free surface elevation ζ to the vertically averaged horizontal component of the velocity, and can
be written as {
ξt + [(1+εξ)u]x = 0,
ut +ξx +εuux = µ3 uxxt .
(9)
Based on results proved in [5, 18], the authors of [6] formulate the following result showing that
this system is globally well posed.
Theorem 3. Suppose s ≥ 1, and initial data (ξ0,u0) ∈ H s × H s+1 are given with the additional
assumption that infξ > −1. Then there is a unique solution (ξ,u) of the system (9) which for any
T > 0 lies in C (0,T , H s (R))×C (0,T , H s+1(R)), and such that ξ(x,0) = ζ0(x) and u(x,0) = u0(x).
Moreover the solution depends continuously on the initial data in the norm of
C
(
0,T , H s (R)
)×C (0,T , H s+1(R)) .
Proving convergence of the unknown quantities in the model system to the unknowns in the
full water-wave problem requires consistency and stability. Stability of the Peregrine system was
proved in [16, Proposition 6.5]. Restricting this result to a flat bottom and to one space dimension
yields the following Theorem 4.
Theorem 4. If a pair of functions (ξ̃, ũ) exists, such that{
ξ̃t +
[(
1+εξ̃) ũ]x = r,
ũt − µ3 ũxxt + ξ̃x +εũũ x̃ = R,
then ∥∥[(ξ̃, ũ)− (ξ,u)] (·, t )∥∥H s×H sµ ≤C (∥∥[(ξ̃, ũ)− (ξ,u)] (·,0)∥∥H s×H sµ + t ‖(r,R)‖L∞) , (10)
where the norm ‖ ·‖H sµ is defined by∥∥ f ∥∥2H sµ = ∥∥ f ∥∥2H s +µ∥∥ fx∥∥2H s .
Next following the procedure laid out in [7, 9, 16], we define consistency based on system (9).
Definition 5. A family of function pairs (ζε,µ, vε,µ) is consistent with (9) if for all ε> 0, we have{
ζ
ε,µ
t + [(1+εζε,µ) vε,µ]x = ε2r ε,µ,
vε,µt − µ3 v
ε,µ
xxt +ζε,µx +εvε,µvε,µx = ε2Rε,µ,
with (r ε,µ,Rε,µ) bounded in L∞(0,T /ε, H s (R)×H s (R)).
C. R. Mathématique, 2021, 359, n 1, 39-45
Samer Israwi and Henrik Kalisch 43
Clearly, if we are able to find a family of function pairs consistent with the Peregrine system,
then by the stability result, these functions will converge towards the unknowns of the Peregrine
system. It turns out that both the ZCS equations and the KdV equation can be shown to be
consistent with the Peregrine system. From [16, Corollary 5.20] (with flat bottom), we have the
following result.
Theorem 6. The water-wave equations are consistent with the Peregrine system. Indeed for a
solution ζ,Φ of the water wave problem, we can define φ and V̄ as explained above, and we have{
ζt +
[
(1+εζ)V̄ ]x = 0,
V̄t − µ3 V̄xxt +ζx +εV̄ V̄x̃ =µ2R,
with ‖R(·, t )‖H s bounded for t ∈ [0,T /ε].
Now following the proof of [16, Corollary 6.23], one may put all these theorems together to
obtain the following result.
Theorem 7. Suppose initial data (ζ0,Φ0) ∈ H N (R)× Ḣ N+1(R) are given for a large enough Sobolev
index N . Defining initial data for (9) by ξ0 = ζ0 and u0 = 11+εζ0
∫ εζ0
−1 ∂xφ(x, z,0)d z, there exists a
constant C depending only on N , such that the estimate∥∥(ζ,V̄ )− (ξ,u)∥∥L∞ ≤Cµ2t (11)
holds for the solutions of (7) and (9) and with V̄ defined by (8).
Next we turn to the uni-directional KdV model. Existence, uniqueness and continuous depen-
dence on the initial data follow from the results proved in [8], and are by now classical.
Theorem 8. For the KdV equation with initial data in H s , where s ≥ 2, there is a unique solution
η ∈C (0,T , H s ) for any T > 0, and the solution depends continuously on the initial data.
Note also that it was proved in [11] that η ∈ C 1(0,T /ε, H s−3). In order to prove consistency in
the sense of Definition 5, we need to define an appropriate velocity. Following [11], we define
vK dV = η−
ε
4
η2 − µ
6
ηxt . (12)
Now given initial data (ξ0,u0) ∈ H s × H s+1 and the solution (ξ,u) ∈ C (0,T , H s × H s+1) of
the system (9) we define initial data for (1) by η0 = ξ0, and using the solution η ∈ C (0,T , H s )
guaranteed by Theorem 8 we define vK dV by (12). Then following the proof laid out in [11], we
can obtain the following estimate.
Theorem 9. With the above provisos, we have the estimate∥∥(η, vK dV )− (ξ,u)∥∥L∞ ≤Cµ2t . (13)
Thus it becomes clear that since the Peregrine system approximates the full water wave
problem, so do solutions of the KdV equation.
3. Convergence of momentum density
We now give a proof of Theorem 1.
Proof. First of all, from (10) and (13) and the triangle inequality we have∥∥V̄ − vK dV ∥∥L∞ ≤Cµ2t and ∥∥ζ−η∥∥L∞ ≤Cµ2t . (14)
C. R. Mathématique, 2021, 359, n 1, 39-45
44 Samer Israwi and Henrik Kalisch
Next we have∫ εζ
−1
φx d z − I =
∫ εζ
−1
φx d z − 1
1+εζ
∫ εζ
−1
φx d z + 1
1+εζ
∫ εζ
−1
∂xφd z −
(
η− ε
4
η2 −ε1
6
ηxt
)
+
(
η− ε
4
η2 −ε1
6
ηxt
)
−
(
η+ 3ε
4
η2 +ε1
6
ηxt
)
=
(
1− 1
1+εζ
)∫ εζ
−1
φx d z + V̄ − vK dV −ε
(
1
4
+ 3
4
)
η2 − µ
6
(
ηxt +ηxx
)
.
Now observe that equation (1) together with the fact that solutions of (1) are bounded in H s (R) for
all time (cf. [8]) can be used to show that ‖ηxx (·, t )+ηxt (·, t )‖L∞ ≤Cµ. Using this estimate together
with the triangle inequality, and the fact that ‖V̄ (·, t )− vK dV (·, t )‖L∞ ≤Cµ2t from (14) leads to∥∥∥∥∫ εζ(·, t )−1 ∂xφ(·, z, t )d z − I (·, t )
∥∥∥∥
L∞
≤ ε∥∥ζ(·, t )V̄ (·, t )−η2(·, t )∥∥L∞ +Cµ2(1+ t ).
Finally, notice that
ζV̄ −η2 = ζ2 −ζ2 +ζV̄ −η2
= (ζ+η)(ζ−η)+ζ(−ζ+ V̄ )
= (ζ+η)(ζ−η)+ζ(V̄ − vK dV + vK dV −η+η−ζ) .
The function ζ is bounded in C (0,T /ε, H s ), V̄ is bounded in C (0,T /ε, H s−3) η is bounded in
C (0,T , H s ) and vK dV is bounded in C (0,T /ε, H
s−2), so that we have∥∥∥∥∫ εζ(·, t )−1 ∂xφ(·, z, t )d z − I (·, t )
∥∥∥∥
L∞
≤ εC {∥∥ζ−η∥∥L∞ +∥∥V̄ − vK dV ∥∥L∞ +∥∥vK dV −η∥∥L∞ +∥∥η−ζ∥∥L∞} .
Finally using (14), and the definition of vK dV (12) along with the fact that η ∈ C 1(0,T /ε, H s−3)
(cf. [11]) proves the required result. 
References
[1] M. J. Ablowitz, H. Segur, “On the evolution of packets of water waves”, J. Fluid Mech. 92 (1979), p. 691-715.
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(1975), no. 1287, p. 555-601.
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[10] W. Craig, “An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits”, Commun.
Partial Differ. Equations 10 (1985), no. 8, p. 787-1003.
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Numer. Anal. 44 (2010), no. 2, p. 347-370.
[12] S. Israwi, H. Kalisch, “Approximate conservation laws in the KdV equation”, Physics Letters A 383 (2019), no. 9, p. 854-
858.
[13] A. Karczewska, P. Rozmej, E. Infeld, “Energy invariant for shallow-water waves and the Korteweg–de Vries equation:
Doubts about the invariance of energy”, Phys. Rev. E 92 (2015), no. 5, article no. 053202 (15 pages).
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Samer Israwi and Henrik Kalisch 45
[14] A. Karczewska, P. Rozmej, E. Infeld, G. Rowlands, “Adiabatic invariants of the extended KdV equation”, Phys. Lett. 381
(2017), no. 4, p. 270-275.
[15] D. Lannes, “Well-posedness of the water-waves equations”, J. Am. Math. Soc. 18 (2005), no. 3, p. 605-654.
[16] ——— , The Water Waves Problem. Mathematical analysis and asymptotics, Mathematical Surveys and Monographs,
vol. 188, American Mathematical Society, 2013.
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p. 39-72.
C. R. Mathématique, 2021, 359, n 1, 39-45


A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation

Guangdong Jing

Penghui Wang

Comptes Rendus
Mathématique
Guangdong Jing and Penghui Wang
A note on “Problem of eigenvalues of stochastic Hamiltonian systems
with boundary conditions”
Volume 359, issue 1 (2021), p. 99-104.
<https://doi.org/10.5802/crmath.103>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, 1, p. 99-104
https://doi.org/10.5802/crmath.103
Stochastic differential equations / Équations différentielles stochastiques
A note on “Problem of eigenvalues of
stochastic Hamiltonian systems with
boundary conditions”
Guangdong Jinga and Penghui Wang ∗, a
a School of Mathematics, Shandong University Jinan, Shandong 250100, The People’s
Republic of China
E-mails: 201511240@mail.sdu.edu.cn (G. Jing), phwang@sdu.edu.cn (P. Wang)
Abstract. The eigenvalue problem of stochastic Hamiltonian systems with boundary conditions was studied
by Peng [4] in 2000. For the one-dimensional case, denoting by {λn }∞n=1 all the eigenvalues of such an
eigenvalue problem, Peng proved that λn →+∞ as n →∞. In this short note, we prove that the growth order
of λn is the same as n2. Apart from the interest of this result in itself, the statistic periodicity of solutions of
FBSDEs can be estimated directly by corresponding coefficients and time duration.
2020 Mathematics Subject Classification. 34L15, 60H10.
Funding. Guangdong Jing is partially supported by NSFC (No.11871308). Penghui Wang is partially supported
by NSFC (No.11471189, No.11871308).
Manuscript received 23rd October 2019, revised 17th April 2020 and 28th July 2020, accepted 28th July 2020.
1. Introduction and main results
Let (Ω,F ,F,P) be a complete filtered probability space, on which a standard one-dimensional
Brownian motion B = {Bt }t ≥0 is defined, and F= {Ft }t ≥0 is the natural filtration of B augmented
by all the P-null sets in F . Let T > 0 be any fixed time horizon.
In [4], Peng considered the following eigenvalue problem of stochastic Hamiltonian system
with boundary conditions:
dxt =
[
Hλ21xt +Hλ22 yt +Hλ23zt
]
dt + [Hλ31xt +Hλ32 yt +Hλ33zt ]dBt , t ∈ [0, T ],
−dyt = [Hλ11xt +Hλ12 yt +Hλ13zt ]dt − zt dBt , t ∈ [0, T ],
x(0) = 0, y(T ) = 0,
(1)
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
100 Guangdong Jing and Penghui Wang
where Hλ = H −λH̄ ,
H =
H11 H12 H13H21 H22 H23
H31 H32 H33
 , H̄ =
H̄11 H̄12 H̄13H̄21 H̄22 H̄23
H̄31 H̄32 H̄33
 ,
Hλi j = Hi j −λH̄i j , Hi j = H Ti j , H̄i j = H̄ Ti j , i , j = 1,2,3, which are constant matrices.
Definition 1. A real number λ is called an eigenvalue of linear stochastic Hamiltonian system
with boundary conditions (1) if there exists a nontrivial solution (x, y, z) of (1). This solution is
called an eigenfunction corresponding to λ. All eigenfunctions associated with the eigenvalue λ
constitute a linear subspace of M 2(0,T ;Rn), called the eigenfunction subspace corresponding toλ.
The above eigenvalue problem is a stochastic analogue of classical eigenvalue problem of
mechanic systems, and it is closely related to the existence of solutions to Forward–Backward
Stochastic Differential Equations (FBSDEs in short). Please refer to [1–5] and references therein
for the well-posedness of FBSDEs, among which the monotonicity condition is an important
sufficient condition for the existence and uniqueness of solutions to FBSDEs. In particular, for
linear FBSDE (1) with H̄ = 0, the monotonicity condition has the following concrete form:−H11 −H12 −H13H21 H22 H23
H31 H32 H33
6−αI3n , (2)
where α> 0 is a constant.
For the case
H̄ =
0 0 00 H22 0
0 0 0
 , (3)
by using the blow-up time (as usual, the blow-up time denotes the endpoints of the maximum
existing interval of the solution to certain ODEs) of the solution for the related Riccati equation,
Peng proved the following
Theorem 2. [4, Theorem 3.2] For (1) of one dimension with perturbation (3), assume that (2) is
satisfied as well as H23 =−H33H13. Then there exist {λn}+∞n=1, all the eigenvalues of the problem (1),
such that λn →+∞ as n →+∞. Moreover, the eigenfunction space corresponding to each λn is of
one dimension.
Such a theorem was generalized to the eigenvalue problem of stochastic Hamiltonian system
driven Poison process in [6].
The existence of eigenvalues is given in the above Theorem 2. Then it is natural and meaningful
to ask that whether those eigenvalues have any relationship with the coefficients of systems and
how they tend to infinity. Towards solving these problems we have the following
Theorem 3. Under the same assumptions in Theorem 2,
λn =O
(
n2
)
, asn →+∞.
In detail,
π2
−2H11H22T 2
≤ lim
n→+∞
λn
n2
≤ lim
n→+∞
λn
n2
≤ 4π
2
−H11H22T 2
.
Remark 4. The result about order in Theorem 3 can be considered as an analogue of the well-
known result in deterministic case.
C. R. Mathématique, 2021, 359, 1, 99-104
Guangdong Jing and Penghui Wang 101
In literature, the study of FBSDEs is mainly focused on the existence and uniqueness of
solutions and hardly any on properties of solutions. From the point of view of eigenvalue problem
of stochastic Hamiltonian system with boundary conditions, what is different is that some
concrete characteristic such as statistic periodicity and stochastic oscillations of solutions of
FBSDEs can be given in [4]. By the proof of [4, Theorem 3.2], the eigenfunction associated to the
nth eigenvalue has the statistic periods of n. Combining with Theorem 3, we have the following
Corollary 5.
Corollary 5. Let λ be an eigenvalue of the stochastic Hamiltonian system in Theorem 3, for
sufficiently large n, if
λ< n
2π2
−2H11H22T 2
,
(
resp. λ> 4n
2π2
−H11H22T 2
,
)
the statistic periods of the associate eigenfunctions (i.e., the solutions of FBSDEs) is less (resp.
greater) than n.
The rest of the paper is organized as follows. In Section 2, we recall some preliminary results
and give several lemmata. The proof of Theorem 3 is given in Section 3.
2. Preliminaries and several lemmata
For one-dimensional case with perturbation (3), the eigenvalue problem of stochastic Hamilton-
ian system with boundary conditions (1) is rewritten as{
dxt =
[
H21xt + (1−λ)H22 yt +H23zt
]
dt + [H31xt +H32 yt +H33zt ]dBt , t ∈ [0,T ],
−dyt =
[
H11xt +H12 yt +H13zt
]
dt − zt dBt , t ∈ [0,T ], x(0) = 0, y(T ) = 0.
(4)
As given in [4, Subsection 4.2], through Legendre transformation, the dual Hamiltonian H̃ of the
original Hamiltonian H corresponding to (4) is
H̃ =
 H−133 H 232 −ρH22 H−133 H32H31 −H21 −H−133 H32H−133 H32H31 −H21 H−133 H 231 −H−133 H31
−H−133 H32 −H−133 H31 H−133
 ,
where ρ = 1−λ, and the relation between solution (x, y, z) of original Hamiltonian system and
solution (x̃, ỹ , z̃) of dual Hamiltonian system is:{
x(t ) = ỹ(t ), y(t ) = x̃(t ),
z(t ) =−H−133 H32x̃(t )−H−133 H31 ỹ(t )+H−133 z̃(t ).
In Peng [4], the idea to study the eigenvalue problem of stochastic Hamiltonian system is to
deal with the blow-up time of the following Riccati equations with terminal conditions [4, (6.2)]:{
dk
dt =−
(
2H21 +H 213
)
k −H11 −
(
ρH22 −H33H 213
)
k2, t ≤ T,
k(T ) = 0, (5)
and dual Riccati equations with terminal conditions [4, (6.4)]:{
dk̃
dt =
(
2H21 +H 213
)
k̃ +H11k̃2 +
(
ρH22 −H33H 213
)
, t ≤ T,
k̃(T ) = 0. (6)
The following two lemmata 6 and 7 from [4] are needed.
Lemma 6 (See [4, Lemma 6.1]). For the Riccati equation (5), the blow-up time tρ is continuous
and strictly decreasing with respect to ρ when ρ < ρ0, where ρ0 = H−122 H33H 213. Besides,
lim
ρ→−∞ tρ = T, limρ→ρ−0
tρ =−∞. (7)
C. R. Mathématique, 2021, 359, 1, 99-104
102 Guangdong Jing and Penghui Wang
Lemma 7 (See [4, Lemma 6.2]). For the dual Riccati equation (6), the blow-up time t̃ρ is
continuous and strictly decreasing with respect to ρ when ρ < ρ0. Besides,
lim
ρ→−∞ t̃ρ = T, limρ→ρ−0
t̃ρ =−∞. (8)
To simplify the notation, denote
q(ρ) =−(ρH22 −H33H 213) , r =−H11,
r̃ (ρ) = (ρH22 −H33H 213) , q̃ = H11,
p̃ =−p = 2H21 +H 213,
ρ∗ = (4H11H22)−1
(
2H21 +H 213
)2
.
(9)
By (2),
r =−H11 < 0, H22 < 0, H33 < 0.
Besides, by [4, Page 278], ρ0 is the critical point, which implies that all the eigenvalues of
problem (4) are located in (−∞,ρ0). Moreover, by Theorem 2, there are at most finite eigenvalues
{λi }mi=1 of problem (4), such that 1−λi ∈ [ρ0 +ρ∗,ρ0). Since we merely try to depict the growth
order of λn in this paper, it is reasonable for us to only check those ρ ∈ (−∞,ρ0 +ρ∗). Actually,
that ρ < ρ0 +ρ∗ guarantees
1− p
2
4r q(ρ)
= 1− p̃
2
4r̃ (ρ)q̃
> 0. (10)
In what follows, for simplicity, we sometimes omit the ρ in q(ρ) and r̃ (ρ).
The following two Lemmata 8 and 9 depict the solutions of (5) and (6), which are essential in
proving Theorem 3.
Lemma 8. The blow-up time tρ of solution k of (5) satisfies√
r q(ρ)− p
2
4
(T − tρ)+arctan −p√
4r q(ρ)−p2
= π
2
. (11)
Proof. Under (10), pk + r +qk2 < 0 and 1− p24r q > 0. Then
dk
p
r k +
q
r k
2 +1 =
dk(√
q
r k −
p
2
p
r q
)2
+
(
1− p24r q
) = r dt .
Combined with terminal condition k(T ) = 0,
k =−
√
4r q −p2
2q
tan
√r q − p2
4
(T − t )+arctan −p√
4r q −p2
− p
2q
.
Therefore, for any fixed ρ (ρ < ρ0 +ρ∗), the blow-up time tρ of k satisfies√
r q − p
2
4
(
T − tρ
)+arctan −p√
4r q −p2
= π
2
.

Lemma 9. The blow-up time t̃ρ of solution k̃ of (6) satisfies√
q̃ r̃ (ρ)− p̃
2
4
(
t̃ρ −T
)+arctan p̃√
4q̃ r̃ (ρ)− p̃2
=−π
2
. (12)
C. R. Mathématique, 2021, 359, 1, 99-104
Guangdong Jing and Penghui Wang 103
Proof. Under (10), r̃ + p̃k̃ + q̃ k̃2 > 0 and 1− p̃24r̃ q̃ > 0. Then
dk̃
1+ p̃r̃ k̃ +
q̃
r̃ k̃
2
= dk̃(√
q̃
r̃ k̃ +
p̃
2
p
q̃ r̃
)2
+
(
1− p̃24q̃ r̃
) = r̃ dt .
Combined with terminal condition k̃(T ) = 0,
k̃ =
√
4q̃ r̃ − p̃2
2q̃
tan
[√
4q̃ r̃ − p̃2
2
(t −T )+arctan p̃√
4q̃ r̃ − p̃2
]
− p̃
2q̃
.
For any fixed ρ (ρ < ρ0 +ρ∗), the blow-up time t̃ρ of solution k̃ satisfies√
q̃ r̃ − p̃
2
4
(t̃ρ −T )+arctan p̃√
4q̃ r̃ − p̃2
=−π
2
.

3. Proof of Theorem 3
In this section, we prove Theorem 3.
Proof of Theorem 3. By [4, Section 6, Proof of Theorem 3.2.], the nth eigenvalue λn of (4) is
uniquely determined by
t 2n−1ρn = 0, λn = 1−ρn , ∀ n ∈N+,
where
t 2 j−1ρn = T − j
(
T − tρn
)− ( j −1)(T − t̃ρn ) , j = 1,2, · · ·n,
t 2 j−2ρn = T − ( j −1)
(
T − tρn
)− ( j −1)(T − t̃ρn ) , t 1ρ1 = tρ .
By Lemma 8 and Lemma 9, we obtain√
r q − p
2
4
(
T − tρ
)+arctan −p√
4r q −p2
= π
2
and √
q̃ r̃ − p̃
2
4
(
t̃ρ −T
)+arctan p̃√
4q̃ r̃ − p̃2
=−π
2
.
Then
lim
n→+∞
T − tρn
T − t̃ρn
= 1
and √
r q − p
2
4
=
π
2 +arctan
pp
4r q−p2
T − tρ
=
π
2 −arctan
pp
4r q−p2
T − t̃ρ
. (13)
Besides, following the method in [4, Subsection 6.2],
T −n(T − tρn )− (n −1)
(
T − t̃ρn
)= 0. (14)
Then
max
{
T − tρn ,T − t̃ρn
}≥ T
2n −1
and for sufficiently large n,
max
{
T − tρn ,T − t̃ρn
}≤ T
2n −2 .
C. R. Mathématique, 2021, 359, 1, 99-104
104 Guangdong Jing and Penghui Wang
Then by (13), for sufficiently large n,
(n −1)π
T
≤
√
r q(ρn)− p
2
4
≤ (2n −1)π
T
.
Moreover, by (9),
r q(ρn)− p
2
4
= (−H11H22)λn +H11H22 −H11H33H 213 −
(
2H21 +H 213
)2
4
.
Then
π2
−2H11H22T 2
≤ lim
n→+∞
λn
n2
≤ lim
n→+∞
λn
n2
≤ 4π
2
−H11H22T 2
.

References
[1] Y. Hu, S. Peng, “Solution of forward-backward stochastic differential equations”, Probab. Theory Relat. Fields 103
(1995), no. 2, p. 273-283.
[2] J. Ma, P. Protter, J. Yong, “Solving forward-backward stochastic differential equations explicitly — a four step scheme”,
Probab. Theory Relat. Fields 98 (1994), no. 3, p. 339-359.
[3] J. Ma, Z. Wu, D. Zhang, J. Zhang, “On well-posedness of forward-backward SDEs — a unified approach”, Ann. Appl.
Probab. 25 (2015), no. 4, p. 2168-2214.
[4] S. Peng, “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”, Stoch. Proc. Appl. 88
(2000), no. 2, p. 259-290.
[5] S. Peng, Z. Wu, “Fully coupled forward-backward stochastic differential equations and applications to optimal
control”, SIAM J. Control Optimization 37 (1999), no. 3, p. 825-843.
[6] H. Wang, Z. Wu, “Eigenvalues of stochastic Hamiltonian systems driven by Poisson process with boundary condi-
tions”, Bound. Value Probl. 2017 (2017), article no. 164.
C. R. Mathématique, 2021, 359, 1, 99-104


A note on “Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions”

Sheng Rao

Runze Zhang

$L^2$ extension theorem for jets with variable denominators

Federico Scavia

Comptes Rendus
Mathématique
Federico Scavia
Motivic classes and the integral Hodge Question
Volume 359, issue 3 (2021), p. 305-311.
<https://doi.org/10.5802/crmath.178>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 3, p. 305-311
https://doi.org/10.5802/crmath.178
Algebraic geometry / Géométrie algébrique
Motivic classes and the integral Hodge
Question
Federico Scaviaa
a Department of Mathematics, University of British Columbia, Vancouver, BC V6T
1Z2, Canada
E-mail: scavia@math.ubc.ca
Abstract. We prove that the obstruction to the integral Hodge Question factors through the completion of
the Grothendieck ring of varieties for the dimension filtration. As an application, combining work of Peyre,
Colliot-Thélène and Voisin, we give the first example of a finite group G such that the motivic class of
its classifying stack BG in Ekedahl’s Grothendieck ring of stacks over C is non-trivial and BG has trivial
unramified Brauer group.
Manuscript received 17th March 2020, revised 24th October 2020, accepted 8th January 2021.
1. Introduction
Let X be a smooth projective complex variety, and let d := dim(X ). For every integer i , we write
C H i (X ) for the group of algebraic cycles of codimension i on X modulo rational equivalence,
and we set C Hi (X ) :=C H d−i (X ). We have the cycle class maps
cliX : C H
i (X ) → H 2i (X (C),Z).
By convention, we set C H i (X ) = 0 and H 2i (X (C),Z) = 0 when i < 0 and i > d .
A cohomology class α ∈ H 2i (X (C),Z) is called an integral Hodge class if its image in
H 2i (X (C),C) is of type (i , i ) with respect to the Hodge decomposition of H 2i (X (C),C). We de-
note by Hdg2i (X ,Z) the subgroup of integral Hodge classes of H 2i (X (C),Z). We have an inclusion
Im(cliX ) ⊆ Hdg2i (X ,Z). We set
Z 2i (X ) := Hdg2i (X ,Z)/Im(cliX ), Z2i (X ) := Z 2d−2i (X ).
For every integer i , the abelian group Z 2i (X ) is finitely generated. The Hodge Conjecture for
cycles of codimension i on X predicts that Z 2i (X ) is finite. The integral Hodge Question for
cycles of codimension i on X asks whether Z 2i (X ) is zero. By the Lefschetz Theorem on (1,1)-
classes, the integral Hodge Question has an affirmative answer when i = 1. When i = 2, the
integral Hodge Question has a negative answer in general, as shown by examples of M. Atiyah
and F. Hirzebruch [1].
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
306 Federico Scavia
We denote by K0(VarC) the Grothendieck ring of complex varieties, introduced by
A. Grothendieck in a 1964 letter to J.-P. Serre. We also let K0(Ab) the abelian group gener-
ated by isomorphism classes [A] of finitely generated abelian groups A, modulo the relations
[A ⊕B ] = [A]+ [B ]. We show that the obstruction to the integral Hodge Question factors through
the completion K̂ 0(VarC) of K0(VarC)[L−1] with respect to the dimension filtration topology.
Theorem 1. Let i be an integer.
(a) There exists a unique group homomorphism
Z2i : K0(VarC)[L
−1] → K0(Ab)
which sends {X }/Lm 7→ [Z2i+2m(X )] for every smooth projective complex variety X and
every m ≥ 0.
(b) The homomorphism Z2i is continuous with respect to the dimension filtration topology
on K0(VarC)[L−1] and the discrete topology on K0(Ab). It thus extends uniquely to a group
homomorphism
Ẑ 2i : K̂ 0(VarC) → K0(Ab).
We now describe the application that motivated Theorem 1. The Grothendieck ring of complex
algebraic stacks K0(StacksC) was defined by T. Ekedahl in [6]. By definition, every algebraic
stack X of finite type over C and with affine stabilizers has a class {X } in K0(StacksC). The
multiplicative identity of K0(StacksC) is 1 = {SpecC}.
Let G be a finite group. It is an interesting problem to compute the class {BG} in K0(StacksC),
and in particular to understand whether the equality {BG} = 1 holds. Although no formal impli-
cation is known, the equality {BG} = 1 appears to be related with the stable rationality of the field
of invariants C(V )G , where V is a faithful complex representation of G . (By the no-name lemma,
the stable rationality of C(V )G does not depend on the faithful representation V .)
It turns out that {BG} = 1 in K0(StacksC) in many cases. For example, this is true when G
is a cyclic group, a symmetric group, or a finite subgroup of GL3(C); see [5, Proposition 3.2,
Theorem 4.1] and [7, Theorem 2.4]. For all these G , the fields of invariants C(V )G are known to
be stably rational.
There are also examples of finite groups G for which {BG} 6= 1 in K0(StacksC). It was shown by
Ekedahl that, if the unramified Brauer group Brnr(C(V )G /C) is not trivial, then {BG} 6= 1; see [5,
Theorem 5.1]. The unramified Brauer group had famously been used in [10] by D. J. Saltman to
give the first examples of finite groups G for which C(V )G is not stably rational. It follows from
Ekedahl’s result that Saltman’s examples also satisfy {BG} 6= 1.
In [9], E. Peyre gave the first examples of finite groups G for which C(V )G is not stably rational
and Brnr(C(V )G /C) = 0. It is natural to wonder whether similar counterexamples to {BG} = 1 can
be constructed.
Question 2. Does there exist a finite group G such that Brnr(C(V )G /C) is trivial, but {BG} 6= 1 in
K0(StacksC)?
To our knowledge, this question was first asked by Ekedahl. It was posed to us by A. Vistoli.
If K /C is a finitely generated field extension, we denote by H inr(K /C,Q/Z) the i -th unramified
cohomology group of K over C with Q/Z coefficients. We have H 2nr(K /C,Q/Z) = Brnr(K /C). If
H inr(C(V )
G /C,Q/Z) 6= 0 for some i , thenC(V )G is not stably rational; see [8, Proposition 3.4]. When
i ≥ 3, it is not known whether H inr(C(V )G /C,Q/Z) 6= 0 implies {BG} 6= 1 in K0(StacksC). We show
that this is the case if i = 3.
Theorem 3. Let V be a faithful complex representation of G. Assume that H 3nr(C(V )
G /C,Q/Z) is
not trivial. Then {BG} 6= 1 in K0(StacksC).
C. R. Mathématique — 2021, 359, n 3, 305-311
Federico Scavia 307
Our proof of Theorem 3 combines Theorem 1 with a result of J.-L. Colliot-Thélène and
C. Voisin [4]; see Theorem 5 below.
For every odd prime p, Peyre constructed a central extension
1 → (Z/pZ)6 →G → (Z/pZ)6 → 1
such that H 3nr(C(V )
G /C,Q/Z) 6= 0 for any faithful complex representation V of G ; see [9, Theo-
rem 6.1]. By Theorem 3, the examples of Peyre provide an affirmative answer to Question 2.
In [11] B. Totaro asked, among other things, whether the stable rationality of C(V )G is equiv-
alent to the condition {BG} = 1 in K0(StacksC). An affirmative answer to Totaro’s question is sup-
ported by all known examples, and also by Theorem 3. However, a proof of the equivalence seems
to be out of reach of current techniques.
2. The Grothendieck rings of varieties and stacks
By definition, the Grothendieck ring of varieties K0(VarC) is the abelian group generated by
isomorphism classes {X } of schemes X of finite type over C, subject to the relations {X } =
{Y }+ {X \ Y } for every closed immersion Y ,→ X . The multiplication in K0(VarC) is defined on
generators by {X } · {Y } := {X ×C Y }, and we have 1 = {SpecC}. We set L := {A1C}.
Following Ekedahl [6], we define the Grothendieck ring of stacks K0(StacksC) as the abelian
group generated by isomorphism classes {X } of algebraic stacks X with affine stabilizers and
of finite type over C, modulo the relations {X } = {Y } + {X \ Y } for every closed immersion
Y ,→ X , and the relations {E } = {Ar
C
×C X } for every vector bundle E → X of constant rank r .
The multiplication is defined on generators by {X } · {Y } := {X ×CY }, and we have 1 = {SpecC}.
By [6, Theorem 1.2], the canonical ring homomorphism K0(VarC) → K0(StacksC) induces an
isomorphism
K0(StacksC) ∼= K0(VarC)[{L−1, (Ln −1)−1 : n ≥ 1}].
The dimension filtration Fil• K0(VarC)[L−1] of K0(VarC)[L−1] is defined as follows: for every
n ∈Z, Filn K0(VarC)[L−1] is the subgroup generated by the elements {X }/Lm , where X is a complex
variety and dim(X )−m ≤ n. Using resolution of singularities, we see that Filn K0(VarC)[L−1] is
generated by elements of the form {X }/Lm , where X is a smooth projective complex variety and
dim(X )−m ≤ n; see [6, Lemma 3.1]. We denote by K̂ 0(VarC) the completion of K0(VarC) with
respect to the dimension filtration. For every n,n′ ∈Z, we have
Filn K0(VarC)[L
−1] ·Filn′ K0(VarC)[L−1] ⊆ Filn+n
′
K0(VarC)[L
−1].
It follows that the multiplication on K0(VarC) extends to K̂ 0(VarC), making the latter into a
commutative ring with identity.
For every n ≥ 1, we have (1−Ln)∑i≥0Lni = 1 in K̂ 0(VarC). Therefore, we have canonical ring
homomorphisms
K0(VarC) → K0(StacksC) → K̂ 0(VarC).
The following result was observed by Ekedahl in [6, p. 14]. It follows from Bittner’s presentation
of K0(VarC), given in [2, Theorem 3.1].
Lemma 4. As an abelian group, K0(VarC)[L−1] may be presented as the abelian group generated
by formal fractions of the form {X }/Lm , where X is a smooth projective complex variety and m ≥ 0,
modulo the following relations:
(i) {;} = 0,
(ii) {X̃ }/Lm − {X }/Lm = {E }/Lm − {Y }/Lm , for every smooth projective complex variety X , every
blow-up X̃ → X at a smooth closed subscheme Y ⊆ X , with exceptional divisor E → Y , and
every m ≥ 0,
C. R. Mathématique — 2021, 359, n 3, 305-311
308 Federico Scavia
(iii) {X ×CP1C}/Lm+1 − {X }/Lm+1 = {X }/Lm , for every smooth projective complex variety X and
every m ≥ 0.
3. Proof of Theorem 1
Proof of Theorem 1. (a). To show that Z2i : K0(VarC)[L−1] → K0(Ab) is well-defined, we verify
that the association {X }/Lm 7→ [Z2i+2m(X )] respects the relations of Lemma 4. It is clear that (i) is
satisfied.
Let m ≥ 0, let Y ,→ X be a closed immersion of smooth projective complex varieties, let X̃ → X
be the blow-up of X at Y , and let E be the exceptional divisor of the blow-up. Denote by d the
dimension of X , and by r the codimension of Y in X . We want to show that
[Z2i+2m(X )]− [Z2i+2m(Y )] = [Z2i+2m(X̃ )]− [Z2i+2m(E)]. (1)
in K0(Ab). Letting j = d − i −m, we see that (1) is equivalent to:
[Z 2 j (X )]− [Z 2 j−2r (Y )] = [Z 2 j (X̃ )]− [Z 2 j−2(E)]. (2)
By [13, Theorem 9.27], we have a group isomorphism
ϕ j :
⊕
0≤h≤r−2
C H j−1−h(Y )⊕C H j (X ) ∼−→C H j (X̃ ).
By [12, Theorem 7.31], we have an isomorphism of Hodge structures⊕
0≤h≤r−2
H 2 j−2−2h(Y (C),Z)⊕H 2 j (X (C),Z) ∼−→ H 2 j (X̃ (C),Z),
where the Hodge structure on H 2 j−2−2h(Y (C),Z) is shifted by (h +1,h +1), and so has weight 2 j .
In particular, we have an isomorphism of groups
ψ j :
⊕
0≤h≤r−2
Hdg2 j−2−2h(Y ,Z)⊕Hdg2 j (X ,Z) ∼−→ Hdg2 j (X̃ ,Z).
Comparing the explicit descriptions of these isomorphisms, as given in the references, we see
that ϕ j and ψ j are compatible with the cycle class maps. In other words, we have a commutative
square ⊕
0≤h≤r−2 C H j−1−h(Y )⊕C H j (X ) C H j (X̃ )
⊕
0≤h≤r−2 Hdg2 j−2−2h(Y ,Z)⊕Hdg2 j (X ,Z) Hdg2 j (X̃ ,Z).
ϕ j
(
⊕
h cl
h
Y )⊕cl
j
X
cl
j
X̃
ψ j
We deduce that
Z 2 j (X̃ ) ∼=
⊕
0≤h≤r−2
Z 2 j−2−2h(Y )⊕Z 2 j (X ). (3)
The morphism E → Y identifies E with the projectivization of the normal bundle of Y inside X .
By [13, Theorem 9.25]1 and [12, Lemma 7.32], the pullback along E → Y induces a commutative
diagram ⊕
0≤h≤r−1 C H j−1−h(Y ) C H j−1(E)
⊕
0≤h≤r−1 Hdg2 j−2−2h(Y ,Z) Hdg2 j−2(E ,Z),
∼
⊕
h cl
h
Y cl
j−1
E
∼
where the horizontal arrows are isomorphisms. We deduce that
Z 2 j−2(E) ∼=
⊕
0≤h≤r−1
Z 2 j−2−2h(Y ). (4)
1Note that the formula of [13, Theorem 9.25] contains a typographical error: C Hl−r−1+k (−) should be C Hl−r+1+k (−).
C. R. Mathématique — 2021, 359, n 3, 305-311
Federico Scavia 309
Now (2) follows from (3) and (4). Therefore, Z2i respects all relations of type (ii).
It remains to show that Z2i is compatible with relations of type (iii). Let X be a smooth
projective variety of dimension d , and let m ≥ 0 be an integer. We must show that
[Z2i+2m+2(X ×CP1C)]− [Z2i+2m+2(X )] = [Z2i+2m(X )].
Setting j = d − i −m, the claim becomes
[Z 2 j (X ×CP1C)]− [Z 2 j−2(X )] = [Z 2 j (X )]. (5)
Applying [13, Theorem 9.25] and [12, Lemma 7.32] to the trivial projective bundle X ×C P1C → X ,
we obtain a commutative square
C H j (X )⊕C H j−1(X ) C H j (X ×CP1C)
Hdg2 j (X ,Z)⊕Hdg2 j−2(X ,Z) Hdg2 j (X ×CP1C,Z).
∼
cl
j
X ⊕cl
j−1
X
cl
j
X×CP1C
∼
Thus
Z 2 j (X ×CP1C) ∼= Z 2 j (X )⊕Z 2 j−2(X ),
which implies (5). It follows that Z2i respects relations of type (iii) as well, hence Z2i is a well-
defined group homomorphism.
(b). Let X be a smooth projective variety of dimension d , and let m ≥ d − i . Then 2i +2m ≥ 2d ,
and so
Z2i ({X }/L
m) = [Z2i+2m(X )] = 0.
This means that Z2i sends Fil
i K0(VarC)[L−1] to zero. Therefore, if we endow K0(VarC)[L−1] with
the dimension filtration topology and K0(Ab) with the discrete topology, the homomorphism
Z2i is continuous. It follows that Z2i extends uniquely to a homomorphism Ẑ 2i : K̂ 0(VarC) →
K0(Ab). 
We also denote by Z2i the composition
K0(StacksC) → K̂ 0(VarC) Ẑ 2i−−→ K0(Ab).
4. Proof of Theorem 3
Theorem 5 (Colliot-Thélène, Voisin). Let X be a smooth projective complex variety, of dimension
d. Assume that there exists a smooth closed subvariety S ⊆ X of dimension ≤ 2, such that the
pushforward map C H0(S) →C H0(X ) is surjective. Then we have an isomorphism of finite groups
H 3nr(C(X )/C,Q/Z)
∼= Z 4(X ).
Proof. See [4, Théorème 1.1]. 
Remark 6. It is well known that the assumptions of Theorem 5 are satisfied when X is unira-
tional. We have learned the following argument from J.-L. Colliot-Thélène.
If X is a smooth projective unirational complex variety, then there exist a dense open sub-
scheme U ⊆ X and a surjective morphism ϕ : V → U , where V is an open subscheme of some
affine space. If p1, p2 ∈U (C), we may find q1, q2 ∈V (C) such that ϕ(qi ) = pi for i = 1,2. There is a
line connecting q1 and q2, hence, since X is complete, we find a morphism P1C→ X whose image
contains p1 and p2. It follows that any two zero-cyles of degree 1 in U are rationally equivalent.
Now, if p ∈ X (C), a moving lemma shows that p is rationally equivalent to a zero-cycle whose
support is contained in U ; see [3, Complément, p. 599]. We conclude that the degree map
deg : C H0(X ) → Z is an isomorphism. Thus, the hypotheses of Theorem 5 are satisfied, with S
a closed point of X .
C. R. Mathématique — 2021, 359, n 3, 305-311
310 Federico Scavia
Proposition 7. Let G be a finite group and let V be a faithful complex representation of G. Then
Z2i ({BG}) = 0 for every i ≥−1, and
Z−4({BG}) = [H 3nr(C(V )G /C,Q/Z)].
Proof. Let d ≥ 1 be the dimension of V . By [5, Proposition 3.1(ii)], we have
{BG} = lim
m→∞{V
m/G}L−md .
Fix an integer i . By Theorem 1(b), the homomorphism Ẑ 2i : K̂ 0(VarC) → K0(Ab) is continuous.
Therefore, if m is sufficiently large, we have
Z2i ({BG}) = Ẑ 2i ({BG}) = Ẑ 2i ({V m/G}L−md ) = Z2i ({V m/G}L−md ).
We fix one such m. Using resolution of singularities, we may write
{V m/G} = {X }+∑
q
nq {Xq } (6)
in K0(VarC), where X and the Xq are smooth projective varieties over C, X is birationally equiva-
lent to V m/G , dim(Xq ) ≤ md −1, and nq ∈Z for every q . Applying Z2i , we obtain:
Z2i ({BG}) = [Z2i+2md (X )]+
∑
q
nq [Z2i+2md (Xq )]. (7)
By the Lefschetz Theorem on (1,1)-classes, we have Z2md−2(X ) = Z 2(X ) = 0. Therefore, if
i ≥−1 every term on the right hand side of (7) is zero. This shows that Z2i ({BG}) = 0 for all i ≥−1.
If i = −2, another application of the Lefschetz Theorem on (1,1)-classes shows that the right
hand side of (7) reduces to [Z2md−4(X )] = [Z 4(X )]. Since X is birationally equivalent to V /G , by
Theorem 5 we have:
Z 4(X ) ∼= H 3nr(C(X )/C,Q/Z) ∼= H 3nr(C(V )G /C,Q/Z).
Using (7) with i =−2, we conclude that
Z−4({BG}) = [Z 4(X )] = [H 3nr(C(V )G /C,Q/Z)]. 
Proof of Theorem 3. As an abelian group, K0(Ab) is freely generated by [Z] and [Z/pnZ], where p
ranges among all prime numbers and n ≥ 1; see [6, Proposition 3.3(i)]. Since H 3nr(C(V )G /C,Q/Z)
is not trivial, [H 3nr(C(V )
G /C,Q/Z)] 6= 0 in K0(Ab). By Proposition 7, we deduce that Z−4({BG}) 6= 0
in K0(Ab). On the other hand, it is clear that Z−4({SpecC}) = 0. We conclude that {BG} 6= 1 in
K0(StacksC). 
Acknowledgements
I am grateful Jean-Louis Colliot-Thélène for making me aware of the results of [4], and for a
conversation which eventually led me to Theorem 1. I thank Angelo Vistoli for posing Question 2
to me, and my advisor Zinovy Reichstein and the anonymous referee for helpful suggestions on
the exposition.
References
[1] M. F. Atiyah, F. Hirzebruch, “Analytic cycles on complex manifolds”, Topology 1 (1962), p. 25-45.
[2] F. Bittner, “The universal Euler characteristic for varieties of characteristic zero”, Compos. Math. 140 (2004), no. 4,
p. 1011-1032.
[3] J.-L. Colliot-Thélène, “Un théorème de finitude pour le groupe de Chow des zéro-cycles d’un groupe algébrique
linéaire sur un corps p-adique”, Invent. Math. 159 (2005), no. 3, p. 589-606.
[4] J.-L. Colliot-Thélène, C. Voisin, “Cohomologie non ramifiée et conjecture de Hodge entière”, Duke Math. J. 161
(2012), no. 5, p. 735-801.
C. R. Mathématique — 2021, 359, n 3, 305-311
Federico Scavia 311
[5] T. Ekedahl, “A geometric invariant of a finite group”, https://arxiv.org/abs/0903.3148, 2009.
[6] ——— , “The Grothendieck group of algebraic stacks”, https://arxiv.org/abs/0903.3143, 2009.
[7] I. Martino, “The Ekedahl invariants for finite groups”, J. Pure Appl. Algebra 220 (2016), no. 4, p. 1294-1309.
[8] A. S. Merkurjev, “Invariants of algebraic groups and retract rationality of classifying spaces”, in Algebraic Groups:
Structure and Actions, Proceedings of Symposia in Pure Mathematics, vol. 94, American Mathematical Society, 2017,
p. 277-294.
[9] E. Peyre, “Unramified cohomology of degree 3 and Noether’s problem”, Invent. Math. 171 (2008), no. 1, p. 191-225.
[10] D. J. Saltman, “Noether’s problem over an algebraically closed field”, Invent. Math. 77 (1984), no. 1, p. 71-84.
[11] B. Totaro, “The motive of a classifying space”, Geom. Topol. 20 (2016), no. 4, p. 2079-2133.
[12] C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76,
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[13] ——— , Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, vol. 77,
Cambridge University Press, 2007.
C. R. Mathématique — 2021, 359, n 3, 305-311


Motivic classes and the integral Hodge Question

Patricia Reynaud-Bouret

Alexandre Muzy

Ingrid Bethus

Towards a mathematical definition of functional connectivity

Vamsi Pritham Pingali

Comptes Rendus
Mathématique
Vamsi Pritham Pingali
A note on Demailly’s approach towards a conjecture of Griffiths
Volume 359, issue 4 (2021), p. 501-503
<https://doi.org/10.5802/crmath.192>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 4, p. 501-503
https://doi.org/10.5802/crmath.192
Complex analysis and geometry / Analyse et géométrie complexes
A note on Demailly’s approach towards a
conjecture of Griffiths
Vamsi Pritham Pingalia
a Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
E-mail: vamsipingali@iisc.ac.in
Abstract. We prove that a “cushioned” Hermitian–Einstein-type equation proposed by Demailly in an ap-
proach towards a conjecture of Griffiths on the existence of a Griffiths positively curved metric on a
Hartshorne ample vector bundle, has an essentially unique solution when the bundle is stable. This result
indicates that the proposed approach must be modified in order to attack the aforementioned conjecture of
Griffiths.
Funding. This work is partially supported by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India),
and a MATRICS grant MTR/2020/000100 from SERB (Govt. of India).
Manuscript received 4th February 2021, accepted 16th February 2021.
1. Introduction
The notion of ampleness/positivity is paramount in algebraic geometry. For a holomorphic line
bundle, there is only one notion of differentio-geometric positivity, i.e., there is a smooth Hermit-
ian metric whose curvature form is a Kähler form. By the Kodaira embedding theorem, it coin-
cides with algebro-geometric ampleness. A holomorphic vector bundle E is said to be Hartshorne
ample if OE∗ (1) is an ample line bundle over P(E∗). There is no unique differentio-geometric no-
tion of positivity of curvature Θ of a smooth Hermitian metric h. There are several competing
inequivalent notions. The most natural of these notions are Griffiths positivity (〈v,p−1Θv〉 is a
Kähler form for all v 6= 0), Nakano positivity (the bilinear form defined by p−1Θ on T 1,0M ⊗E
is positive-definite), and dual-Nakano positivity (the Hermitian holomorphic bundle (E∗,h∗) is
Nakano negative). Nakano positivity and dual-Nakano positivity imply Griffiths positivity and all
three of them imply Hartshorne ampleness. A famous conjecture of Griffiths [5] asks whether
Hartshorne ample vector bundles admit Griffiths positively curved metrics. This conjecture is
still open. However, a considerable amount of work has been done to provide evidence in its
favour [1–3, 6, 8–10, 13].
Relatively recently, Demailly [3] proposed a programme to prove the aformentioned conjec-
ture of Griffiths for a holomorphic rank−r vector bundle E on a compact Kähler manifold (X ,ω0).
In fact, if Demailly’s method works, it will end up proving a stronger conjecture : Do Hartshorne-
ample bundles admit dual-Nakano positively curved metrics? Demailly’s approach involves solv-
ing a family (depending on a parameter 0 ≤ t ≤ 1) of vector bundle Monge–Ampère equations
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
502 Vamsi Pritham Pingali
(distinct from the one introduced in [11]) in conjunction with “cushioned” Hermitian–Einstein-
type equations (see [3, Theorem 2.17]):
det
T X⊗E∗
(
ΘTht + (1− t )αω0 ⊗ IE∗
)1/r = ft (deth0)λ
(detht )λ
ωn0 , (1)(p−1Fht − p−1r trFht
)
ωn−10 =−ε
(deth0)µ
(deth)µ
ln
(
hh−10
det(hh−10 )1/r
)
ωn0 , (2)
where h0 is a smooth background Hermitian metric, µ,λ ≥ 0 are fixed constants, α > 0 is a
large enough constant so that Θh0 +αω is dual-Nakano positively curved, and ft > 0 are smooth
positive functions. We focus on the cushioned Hermitian–Einstein-type equation in the following
theorem.
Theorem 1. Let E be an ω0-stable rank−r holomorphic bundle on X . Let H0 be a Hermitian–
Einstein metric on E with respect to ω0, that is,
p−1FH0ωn−10 = λωn0 . Let h be a smooth metric on
E solving the following cushioned Hermitian–Einstein equation for given parameters ε≥ 0,µ≥ 0.(p−1Fh − p−1r trFh
)
ωn−10 =−ε
(det H0)µ
(deth)µ
ln
(
hH−10
det(hH−10 )1/r
)
ωn0 , (3)
where h, H0 are matrices (any holomorphic trivialisation will do). Then h = H0e− f for some
smooth function f .
As a result, if we consider the system of the vector bundle Monge–Ampère equation and the
cushioned Hermitian–Einstein-type equation on an ω0-stable ample E , and if solutions exist all
the way till t = 1, the final t = 1 solution, by virtue of the fact that it satisfies the cushioned
Hermitian–Einstein-type equation, has to be of the form H0e f . This condition might be a strong
restriction (which is unlikely to be met owing to [7, 12] without a restriction on the second Chern
character). On the other hand, if we replaceω0 by say (1− t )ω0+ t
p−1tr(Fht ) (or the choice in [3,
Section 2.19] for instance), the above argument will not be applicable and there might be some
hope for the approach to yield an affirmative solution to the Griffiths Conjecture.
Acknowledgements
The author thanks Jean-Pierre Demailly for fruitful discussions and for encouraging me to write
up this note.
2. Proof of uniqueness
In a holomorphic trivialisation, our conventions are : 〈v, w〉H = vT H w̄ , if g is an endomorphism
then g .s = [g ]T~s, ∇s = d s + AT s, A = ∂H H−1, F = d A− A∧ A = ∂̄A, and ∇g = d g + [g , A].
The proof is motivated by a similar one by Donaldson for Riemann surfaces [4]. In general,
h = q H0 where q is some smooth H0-Hermitian positive-definite endomorphism of E . We
decompose q further as q = e− f g where det(g ) = 1 and f is a smooth function. Thus, Fh =
FH0 +∂∂̄ f + ∂̄(∂0g g−1). The trace-free part of the curvature is F ◦h = F ◦0 + ∂̄(∂0g g−1). Substituting
these expressions in 3 and using the fact that H0 is Hermitian–Einstein with respect to ω0, we getp−1∂̄(∂0g g−1)ωn−10 =−εerµ f ln gωn0 . (4)
Now we compute
1
2
p−1∂̄∂ tr(g 2)ωn−10 =
p−1∂̄ tr(g∂0g )ωn−10 =
p−1tr(∂̄g∂0g )ωn−10 +
p−1tr(g ∂̄∂0g )ωn−10
=p−1tr(∂̄g∂0g )ωn−10 −εerµ f tr(g 2 ln g )ωn0 −
p−1tr(g∂0g g−1∂̄g )ωn−10
≤−εerµ f tr(g 2 ln g )ωn0 . (5)
C. R. Mathématique — 2021, 359, n 4, 501-503
Vamsi Pritham Pingali 503
Note that tr(g 2 ln g ) =∑i λ2i ln(λi ) where λi > 0 are the eigenvalues of g such that λ1 ≤ λ2 . . .. The
product of the λi is 1. Thus, ∑
i | λi<1
|ln(λi )| =
∑
i | λi>1
ln(λi )
which implies that∑
1≤i≤p | λp≤1, λp+1>1
λ2i |ln(λi )| ≤
∑
1≤i≤p | λp≤1, λp+1>1
λ2p |ln(λi )| =λ2p
n∑
i=p+1
ln(λi )
≤
n∑
i=p+1
λ2i ln(λi ). (6)
Therefore, tr(g 2 ln g ) ≥ 0 and hence
1
2
p−1∂̄∂ tr(g 2)ωn−10 ≤ 0. (7)
The strong maximum principle then implies that actually
1
2
p−1∂̄∂ tr(g 2)ωn−10 = 0, (8)
and hence g = I . 
References
[1] B. Berndtsson, “Curvature of vector bundles associated to holomorphic fibrations”, Ann. Math. 169 (2009), no. 2,
p. 531-560.
[2] F. Campana, H. Flenner, “A characterization of ample vector bundles on a curve”, Math. Ann. 287 (1990), no. 4, p. 571-
575.
[3] J.-P. Demailly, “Hermitian–Yang–Mills approach to the conjecture of Griffiths on the positivity of ample vector
bundles”, https://arxiv.org/abs/2002.02677, 2020.
[4] S. K. Donaldson, “A new proof of a theorem of Narasimhan and Seshadri”, J. Differ. Geom. 2 (1983), p. 269-277.
[5] P. A. Griffiths, “Hermitian differential geometry, Chern classes and positive vector bundles”, in Global analysis,
Princeton University Press, 1969, p. 185-251.
[6] K. Liu, X. Sun, X. Yang, “Positivity and vanishing theorems for ample vector bundles”, J. Algebr. Geom. 22 (2013), no. 2,
p. 303-331.
[7] M. Lübke, “A note on positivity of Einstein bundles”, Indag. Math. 2 (1991), no. 3, p. 311-318.
[8] C. Mourougane, S. Takayama, “Hodge metrics and positivity of direct images”, J. Reine Angew. Math. 606 (2007),
p. 167-178.
[9] P. Naumann, “arXiv preprint”, https://arxiv.org/abs/1710.10034, 2017.
[10] V. P. Pingali, “Representability of Chern–Weil forms”, Math. Z. 288 (2018), no. 1-2, p. 629-641.
[11] ——— , “A vector bundle version of the Monge–Ampère equation”, Adv. Math. 360 (2020), article no. 106921
(40 pages).
[12] M. Schneider, A. Tancredi, “Positive vector bundles on complex surfaces”, Manuscr. Math. 50 (1985), no. 1, p. 133-144.
[13] H. Umemura, “Some results in the theory of vector bundles”, Nagoya Math. J. 52 (1973), p. 97-128.
C. R. Mathématique — 2021, 359, n 4, 501-503


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