Comptes Rendus Mathématique

Saminathan Ponnusamy

Ramakrishnan Vijayakumar

Comptes Rendus
Mathématique
Saminathan Ponnusamy and Ramakrishnan Vijayakumar
Generalized versions of Lipschitz conditions on the modulus of
holomorphic functions
Volume 359, issue 5 (2021), p. 609-615
<https://doi.org/10.5802/crmath.200>
© Académie des sciences, Paris and the authors, 2021.
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Comptes Rendus
Mathématique
2021, 359, n 5, p. 609-615
https://doi.org/10.5802/crmath.200
Complex analysis and geometry, Functional analysis / Analyse et géométrie complexes,
Analyse fonctionnelle
Generalized versions of Lipschitz conditions
on the modulus of holomorphic functions
Saminathan Ponnusamya and Ramakrishnan Vijayakumara
a Department of Mathematics, Indian Institute of Technology Madras, Chennai-600
036, India
E-mails: samy@iitm.ac.in, mathesvijay8@gmail.com
Abstract. In this paper, we establish Lipschitz conditions for the norm of holomorphic mappings between
the unit ball Bn in Cn and X , a complex normed space. This extends the work of Djordjević and Pavlović.
Mathematical subject classification (2010). 30C80, 30H05, 32A10, 30G30, 46B20, 46E15, 46E40.
Manuscript received 15th September 2020, revised 9th February 2021, accepted 22nd March 2021.
1. Introduction and Preliminaries
Denote by Cn , the n-dimensional complex Hilbert space with the inner product and the norm
given by 〈z, w〉 :=∑nj=1 z j w j and ‖z‖ :=p〈z, z〉, where z, w ∈Cn , respectively. Write Bn := {z ∈Cn :
‖z‖ < 1} for the open unit ball in Cn so that B1 =: D denotes the open unit disk in C. If V and W
are two normed spaces and U ⊂V is open, then the Fréchet derivative of a holomorphic mapping
f : U →W is defined to be the unique linear map A = f ′(z) : V →W such that
f (z +h) = f (z)+ f ′(z) ·h +o(‖h‖2)
for h near the origin of V . The norm of such a map is defined by ‖A‖ = sup‖z‖=1 ‖Az‖.
In 1975, Globevnik [6] introduced the notion of uniform c-convexity and proved that L1-space
possesses this property. Namely, a complex normed space X is said to be uniformly c-convex if
there exists a positive increasing functionΩ(δ) (δ> 0) withΩ(0+) = 0 such that for all x, y ∈ X and
δ> 0 there holds the implication
max
|λ|≤1
‖x‖=1
‖x +λy‖ ≤ 1+δ =⇒ ‖y‖ ≤Ω(δ).
The smallest of the functionsΩ is denoted byΩX , i.e.,
ΩX (δ) := sup
‖y‖ : max|λ|≤1
‖x‖=1
‖x +λy‖ ≤ 1+δ
 .
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
610 Saminathan Ponnusamy and Ramakrishnan Vijayakumar
As mentioned in [3], it can be easily seen that
ΩC(δ) = δ and ΩH (δ) =
√
δ(2+δ),
where H is a Hilbert space of dimension at least two.
As in [4], we call a function ω : [0,∞) → R a majorant if ω is continuous, increasing, ω(0) = 0,
and t−1ω(t ) is nonincreasing on (0,∞). If, in addition, there is a constant C (ω) > 0 such that∫ δ
0
ω(t )
t
dt +δ
∫ ∞
δ
ω(t )
t 2
dt ≤C (ω) ·ω(δ)
whenever 0 < δ< 1, then we say that ω is a regular majorant.
Then the space Lip(ω,G , X ), where G is bounded subset of Cn , is defined to be the set of those
functions g : G → X for which
‖g (z)− g (w)‖ ≤ c ·ω(‖z −w‖),
where c is a constant. If ω(t ) = tα for some α ∈ (0,1], then we write Lip(ω,G , X ) =Λα(G , X ). If X is
uniformly c-convex, then ΩX is a majorant (cf. [2]). A majorant ω is said to be a Dini majorant if∫ 1
0
ω(t )
t dt <∞. For a Dini majorant, we define the majorant ω̃ by
ω̃(t ) =
∫ t
0
ω(x)
x
dx =
∫ 1
0
ω(t x)
x
dx.
A majorant ω is said to be fast [5] if∫ δ
0
ω(t )
t
dt ≤ const ·ω(δ), 0 < δ< δ0,
for some δ0 > 0. (Of course, if ω is fast, then it is a Dini majorant).
Dyakonov [4] gave some characterizations of the holomorphic functions of class Λω(D,C) in
terms of their moduli.
Theorem A (cf. [4]). Let ω be a regular majorant. A function f holomorphic in D is in Λω(D,C) if
and only if so is its modulus | f |.
The main ingredient in Dyakonov’s proof is a very complicated. However, Pavlovic [8] gave a
simple proof of Theorem A. The proof uses only the basic lemmas of [4] and the Schwarz lemma,
and is therefore considerably shorter than that of [4]. However, Theorem A does not extend to
Ck - valued functions (k ≥ 2). So we have to consider functions with additional properties (see
Theorems 5 and 6).
In [3], Djordjević and Pavlović extended to vector-valued functions of a theorem of
Dyakonov [4] on Lipschitz conditions for the modulus of holomorphic functions. Therefore,
it is natural for us to extend this result for holomorphic functions on Bn . Very recently, Kalaj [7]
established a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn
and Bm in the corresponding complex spaces.
Theorem B (cf. [7, Theorem 2.1]). If f is a holomorphic mapping of the unit ball Bn ⊂ Cn into
Bm ⊂Cm , then for z ∈Bn we have
‖ f ′(z)‖ ≤

√
1−‖ f (z)‖2
1−‖z‖2 for m ≥ 2,
1−‖ f (z)‖2
1−‖z‖2 for m = 1.
In [1], Dai and Pan proved the following theorem which establishes a Schwarz–Pick type
estimates for gradient of the modulus of holomorphic mappings.
C. R. Mathématique — 2021, 359, n 5, 609-615
Saminathan Ponnusamy and Ramakrishnan Vijayakumar 611
Theorem C (cf. [1, Theorem 1]). Let f : Bn →Bm be a holomorphic mapping. Then∣∣∇‖ f ‖(z)∣∣≤ 1−‖ f (z)‖2
1−‖z‖2 for z ∈B
n .
For a holomorphic mapping f : Bn →Bm , we have∣∣∇‖ f ‖(z)∣∣= 1‖ f (z)‖
∥∥∥∥(〈∂ f (z)∂z1 , f (z)
〉
, . . . ,
〈
∂ f (z)
∂zn
, f (z)
〉)∥∥∥∥ if f (z) 6= 0. (1)
2. The main results
Theorem 1. Let X be uniformly c-convex and f : Bn → X be a holomorphic function satisfying∣∣‖ f (z)‖−‖ f (w)‖∣∣≤ c‖z −w‖α for z, w ∈Bn , (2)
where c ≥ 0 and α ∈ [0,1] are constants. Then
‖ f ′(z)‖ ≤ 2K ΩX (cK
−1(1−‖z‖)α)
1−‖z‖ for z ∈B
n , (3)
where K = ‖ f (0)‖+ c. Especially, if ‖ f (0)‖ = 1, then
‖ f ′(z)‖ ≤ 2(1+ c)ΩX (c(1−‖z‖)
α)
1−‖z‖ for z ∈B
n . (4)
Theorem 2. Let X be uniformly c-convex such that ΩX is a Dini majorant and f : Bn → X be a
holomorphic function such that the function ‖ f (z)‖ belongs to Λα(Bn ,R) for some α ∈ (0,1]. Then
f ∈ Lip(ωα,Bn , X ), where ωα(t ) = Ω̃X (tα).
In particular, the function f is uniformly continuous on Bn that has a continuous extension to
the closed disk.
Corollary 3. If ΩX is fast and f : Bn → X is a holomorphic function such that the function ‖ f (z)‖
belongs toΛα(Bn ,R) for some α ∈ (0,1]. Then f ∈ Lip(ωα,Bn , X ), where ωα(t ) =ΩX (tα).
Taking n = 1 and X =C, we get the following result of Dyakonov [4].
Corollary 4. If f : D→ C is a holomorphic function such that | f | belongs to Λα(D,R) for some
α ∈ (0,1]. Then f belongs toΛα(D,C).
Theorem 5. Let 0 <α≤ 1 and f : Bn →Cm be a holomorphic function such that
‖ f ′(z)‖‖ f (z)‖ ≤ K
∥∥∥∥(〈∂ f (z)∂z1 , f (z)
〉
, . . . ,
〈
∂ f (z)
∂zn
, f (z)
〉)∥∥∥∥ for z ∈Bn , (5)
where K is a constant independent of z. Then f ∈Λα(Bn ,Cm) if and only if ‖ f ‖ ∈Λα(Bn ,R).
Theorem 6. If f : Bn →Cm ,m ≥ 2, is holomorphic and if ‖ f ‖ ∈Λα(Bn ,R) for some α ∈ (0,1], then
we have f ∈Λα/2(Bn ,Cm).
The case n = 1 of Theorems 5 and 6 gives results of Pavlović [9].
3. Proofs of the Theorems
Theorem 1 is a direct consequence of the following lemma.
Lemma 7. If f : Bn → X is a holomorphic function satisfying the condition∣∣‖ f (z)‖−‖ f (w)‖∣∣≤ c(1−‖z‖)α whenever ‖w − z‖ ≤ 1−‖z‖, (6)
then there holds (3).
C. R. Mathématique — 2021, 359, n 5, 609-615
612 Saminathan Ponnusamy and Ramakrishnan Vijayakumar
Proof. Fix z ∈ Bn with f (z) 6= 0, and fix β ∈ Cn with ‖β‖ = 1. Let L ∈ X ∗, ‖L‖ = 1, where X ∗ is the
dual of X . Consider the scalar valued function
φ(z) = L ◦ f (z),
and introduce the following set for the given z ∈Bn ,
Dz := {w ∈Cn : ‖w − z‖ < 1−‖z‖} and Mz := sup{‖ f (w)‖ : w ∈ Dz }.
If z = 0 and M0 = 1, then the Schwarz–Pick lemma (see Theorem B) gives
|φ′(0)| ≤ 1−|φ(0)|2 ≤ 2(1−|φ(0)|), (7)
which is our inequality in this special case. The general case follows by applying the special case
to the functionΦ defined by
Φ(ζ) = φ(z + (1−‖z‖)βζ)
Mz
for ζ ∈Bn .
As
Φ(0) = L( f (z))
Mz
and Φ′(0) = (1−‖z‖)
Mz
L( f ′(z)β),
we deduce from (7) that
(1−‖z‖)|L( f ′(z)β/2)|+ |L( f (z))| ≤ Mz .
Hence, for every λ ∈D, we obtain
|λ(1−‖z‖)L( f ′(z)β/2)+L( f (z))| ≤ Mz .
Since this holds for every L of norm 1, by taking the supremum over all L with ‖L‖ = 1 and by
applying the Hahn–Banach theorem, we get∥∥∥∥λ (1−‖z‖) f ′(z)β2 + f (z)
∥∥∥∥≤ Mz , i.e., ∥∥∥∥ f (z)‖ f (z)‖ +λ (1−‖z‖) f ′(z)β2‖ f (z)‖
∥∥∥∥≤ Mz‖ f (z)‖ .
Now denoting
x = f (z)‖ f (z)‖ , y =
(1−‖z‖) f ′(z)β
2‖ f (z)‖ and δ=
Mz −‖ f (z)‖
‖ f (z)‖ ,
we see from the definition ofΩX that
(1−‖z‖)‖ f ′(z)β‖ ≤ 2‖ f (z)‖ΩX
(
Mz −‖ f (z)‖
‖ f (z)‖
)
.
Hence, the last inequality holds for every β ∈Cn with ‖β‖ = 1, we get
(1−‖z‖)‖ f ′(z)‖ ≤ 2‖ f (z)‖ΩX
(
Mz −‖ f (z)‖
‖ f (z)‖
)
. (8)
Therefore by (6) and (8), we obtain that
(1−‖z‖)‖ f ′(z)‖ ≤ 2‖ f (z)‖ΩX
(
c(1−‖z‖)α
‖ f (z)‖
)
.
Now (3) follows from the fact thatΩX (t )/t is a decreasing function and the inequality ‖ f (z)‖ ≤ K .
The proof is complete. 
Lemma 8. If a C 1-function u : Bn →R satisfies
‖∇u(z)‖ ≤ ω(1−‖z‖)
1−‖z‖ for z ∈B
n ,
where ω is a Dini majorant, then
|u(a)−u(b)| ≤ 3 ω̃(‖a −b‖) for a,b ∈Bn .
C. R. Mathématique — 2021, 359, n 5, 609-615
Saminathan Ponnusamy and Ramakrishnan Vijayakumar 613
Proof. We begin the proof with the following observation: ω≤ ω̃. In fact, we let t0 ∈ (0,∞). Since
ω(t )
t is decreasing on (0,∞), we have
ω(t0)
t0
≤ ω(t0x)
t0x
for x ∈ (0,1].
Integrating on both sides of the last inequality from 0 to 1, we obtain by definition of ω̃ that
ω(t0) ≤ ω̃(t0).
Let ‖a‖ ≤ ‖b‖ ≤ 1. By Lagrange’s mean-value theorem,
|u(a)−u(b)| ≤ ‖∇u(c)‖‖a −b‖,
where c = (1−λ)a +λb for some λ ∈ (0,1). Since ‖c‖ ≤ ‖b‖ and ω(t )/t decreases, we see that
ω(1−‖c‖)
1−‖c‖ ≤
ω(1−‖b‖)
1−‖b‖
and hence,
|u(a)−u(b)| ≤ω(‖a −b‖) ≤ ω̃(‖a −b‖),
under the condition ‖a −b‖ ≤ 1−‖b‖.
If 1−‖b‖ ≤ ‖a −b‖ ≤ 1−‖a‖, then
|u(a)−u(b)| ≤ |u(a)−u(b′)|+ |u(b′)−u(b)|,
where b′ = (1−δ)b‖b‖ and δ= ‖a −b‖. Using the Lagrange’s mean-value theorem as above we get
|u(a)−u(b′)| ≤ ω(1−‖b
′‖)
1−‖b′‖ ‖a −b
′‖ = ω(δ)
δ
‖a −b′‖ ≤ω(δ) ≤ ω̃(δ).
In the case of |u(b′)−u(b)|, we have
|u(b′)−u(b)| ≤
∫ ‖b‖
‖b′‖
ω(1− t )
1− t dt ≤
∫ 1
1−δ
ω(1− t )
1− t dt = ω̃(δ).
Finally, if δ> 1−‖a‖, we use the inequality
|u(a)−u(b)| ≤ |u(a)−u(a′)|+ |u(a′)−u(b′)|+ |u(b′)−u(b)|,
where a′ = (1−δ)a‖a‖ , and then proceed in a similar way as above, using the inequality ‖a′ −b′‖ ≤‖a −b‖. 
Lemma 9 can easily be proved by applying the previous lemma to the functions Re(L ◦ f (z))
and Im(L ◦ f (z)), where L ∈ X ∗ and ‖L‖ = 1.
Lemma 9. If f is an X -valued holomorphic function in Bn and satisfies the condition
‖ f ′(z)‖ ≤ ω(1−‖z‖)
1−‖z‖ for z ∈B
n ,
where ω is a Dini majorant, then f ∈ Lip(ω̃,Bn , X ).
Proof of Theorem 2. Let f satisfy the hypotheses of the theorem. Then
‖ f ′(z)/2K ‖ ≤ ω(1−‖z‖)
1−‖z‖ ,
by Theorem 1, where ω(t ) = ΩX (cK −1tα). But a simple calculation shows that ω̃(t ) =
α−1Ω̃X (cK −1tα) and so we can appeal to Lemma 9 to conclude the proof. 
Proof of Theorem 5. The “only if” part is trivial. Assume that ‖ f (z)‖ ∈Λα(Bn ,R) and we proceed
as in Theorem 1. Fix z ∈Bn with f (z) 6= 0, and consider the following sets for a given z ∈Bn ,
Dz := {w ∈Cn : ‖w − z‖ < 1−‖z‖} and Mz := sup{‖ f (w)‖ : w ∈ Dz }.
If z = 0 and M0 = 1, Theorem C gives
|∇‖ f ‖(0)| ≤ 1−‖ f (0)‖2 ≤ 2(1−‖ f (0)‖).
C. R. Mathématique — 2021, 359, n 5, 609-615
614 Saminathan Ponnusamy and Ramakrishnan Vijayakumar
Therefore, from (5) and the formula (1), we have that
‖ f ′(0)‖ ≤ 2K (1−‖ f (0)‖),
which is our inequality in this special case. The general case follows by applying the special case
to the function F defined by
F (ζ) = f (z +ζ(1−‖z‖))
Mz
for ζ ∈Bn , (9)
and obtain
1
2K
(1−‖z‖)‖ f ′(z)‖+‖ f (z)‖ ≤ Mz for z ∈Bn . (10)
Since ‖ f ‖ ∈Λα(Bn ,R), we have
‖ f (w)‖−‖ f (z)‖ ≤ c‖w − z‖α ≤ c(1−‖z‖)α,
for z ∈ Bn and w ∈ Dz . Taking the supremum over all w ∈ Dz and then using the inequality (10),
we get
‖ f ′(z)‖ ≤C ω(1−‖z‖)
1−‖z‖ ,
where C is a constant and ω(t ) = tα. The desired conclusion follows from Lemma 9. 
Proof of Theorem 6. Let z ∈ Bn and proceed the steps as in the above proof. If z = 0 and M0 = 1,
then the higher dimensional version of Schwarz–Pick lemma (Theorem C) gives
‖ f ′(0)‖ ≤
√
1−‖ f (0)‖2 ≤p2
√
1−‖ f (0)‖,
which is our inequality in this special case. The general case follows by applying the special case
to the function F defined by (9). Indeed, we obtain
(1−‖z‖)‖ f ′(z)‖ ≤ c
√
Mz −‖ f (z)‖, (11)
for some constant c. Since ‖ f ‖ ∈Λα(Bn ,R), we have
‖ f (w)‖−‖ f (z)‖ ≤ c‖w − z‖α ≤ c(1−‖z‖)α,
for z ∈ Bn and w ∈ Dz . Taking the supremum over w ∈ Dz and then using the inequality (11),
we get
‖ f ′(z)‖ ≤C ω(1−‖z‖)
1−‖z‖ ,
where C is a constant and ω(t ) = tα/2. Now the result follows from Lemma 9. 
Remark 10. The index α/2 in Theorem 6 is optimal as demonstrated by the following example
(see [9]). Consider the function f : D→C2 by f (z) = (1, (1− z)α/2), 0 <α≤ 1. We have∣∣∣‖ f (z)‖−‖ f (w)‖∣∣∣= ∣∣∣√‖1− z‖α+1−√‖1−w‖α+1∣∣∣
≤
∣∣∣‖1−w‖α−‖1− z‖α∣∣∣≤ ‖z −w‖α,
while ‖ f (1)− f (r )‖ = (1− r )α/2,0 < r < 1. This shows that the index α/2 is optimal.
C. R. Mathématique — 2021, 359, n 5, 609-615
Saminathan Ponnusamy and Ramakrishnan Vijayakumar 615
4. Concluding Remarks
As mentioned in [3], the inequality (4) is in a sense optimal for the case n = 1. To see this, let
ω(t ) > 0 be an arbitrary increasing function on (0,∞) such that ω(0+) = 0. We say that a Banach
space X has the property L (ω,α), if the following holds: For every c ∈ (0,1) and every analytic
function f : D→ X with ‖ f (0)‖ = 1, the inequality (2) implies that
‖ f ′(λ)‖ ≤ ω(c(1−|λ|)
α)
1−|λ| for λ ∈D.
It is well-known that, if the Banach space X has the property L (ω,α) (see [3, Proposition 10]),
then X is uniformly c-convex andΩX (δ) ≤ Bω(δ) for 0 < δ< 1, where B is a constant. This result is
to emphasize the fact that ‖ f (0)‖ = 1 provides condition for uniformly c-convexity of the Banach
space X .
References
[1] S. Dai, Y. Pan, “A Schwarz-Pick lemma for the modulus of holomorphic mappings”, Complex Var. Elliptic Equ. 60
(2015), no. 6, p. 864-874.
[2] W. J. Davis, D. J. H. Garling, N. Tomczak-Jaegermann, “The complex convexity of quasi-normed spaces”, J. Funct. Anal.
55 (1984), p. 110-150.
[3] O. Djordjević, M. Pavlović, “Lipschitz conditions for the norm of a vector valued analytic function”, Houston J. Math.
34 (2008), no. 3, p. 817-826.
[4] K. M. Dyakonov, “Equivalent norms on Lipschitz type spaces of holomorphic functions”, Acta Math. 178 (1997), no. 2,
p. 143-167.
[5] ——— , “Holomorphic functions and quasiconformal mappings with smooth moduli”, Adv. Math. 187 (2004), no. 1,
p. 146-172.
[6] J. Globevnik, “On complex strict and uniform convexity”, Proc. Am. Math. Soc. 47 (1975), p. 175-178.
[7] D. Kalaj, “Schwarz lemma for holomorphic mappings in the unit ball”, Glasg. Math. J. 60 (2018), no. 1, p. 219-224.
[8] M. Pavlović, “On K.M. Dyakonov’s paper: “Equivalent norms on Lipschitz type spaces of holomorphic functions””,
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C. R. Mathématique — 2021, 359, n 5, 609-615


Generalized versions of Lipschitz conditions on the modulus of holomorphic functions

J. Fernando Barbero G.

Juan Margalef-Bentabol

Eduardo J.S. Villaseñor

Comptes Rendus
Mathématique
J. Fernando Barbero G., Juan Margalef-Bentabol and Eduardo J.S.
Villaseñor
A two-sided Faulhaber-like formula involving Bernoulli polynomials
Volume 358, issue 1 (2020), p. 41-44.
<https://doi.org/10.5802/crmath.10>
© Académie des sciences, Paris and the authors, 2020.
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
2020, 358, n 1, p. 41-44
https://doi.org/10.5802/crmath.10
Combinatorics, Mathematical Physics / Combinatoire, Physique mathématique
A two-sided Faulhaber-like formula involving
Bernoulli polynomials
Une formule bilatérale de type Faulhaber utilisant les
polynômes de Bernoulli
J. Fernando Barbero G.a, b, JuanMargalef-Bentabol b, c , d
and Eduardo J.S. Villaseñor b, e
a Instituto de Estructura de la Materia, CSIC. Serrano 123, 28006 Madrid, Spain
b Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M).
Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain
c Laboratory of Geometry and Dynamical Systems, Department of Mathematics,
EPSEB, Universitat Politècnica de Catalunya, BGSMath, Barcelona, Spain
d Institute for Gravitation and the Cosmos & Physics Department, Penn State,
University Park, PA 16802, USA
e Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la
Universidad 30, 28911 Leganés, Spain.
Abstract. We give a new identity involving Bernoulli polynomials and combinatorial numbers. This provides,
in particular, a Faulhaber-like formula for sums of the form 1m (n −1)m +2m (n −2)m + ·· ·+ (n −1)m 1m for
positive integers m and n.
Résumé. Nous donnons une nouvelle identité utilisant les polynômes de Bernoulli et les coefficient bino-
miaux. Ceci fournit, en particulier, une formule de type Faulhaber pour des sommes de la forme 1m (n −
1)m +2m (n −2)m +·· ·+ (n −1)m 1m où m et n sont des entiers positifs.
Manuscript received 8th May 2019, revised 26th September 2019, accepted 21st January 2020.
1. Introduction
Bernoulli numbers Bk are given by the exponential generating function z/(e
z −1),
Bk = k ![zk ]
z
ez −1 ,
where [zn] f (z) is the n-th coefficient of the Taylor expansion of f around z = 0.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
42 J. Fernando Barbero G., Juan Margalef-Bentabol and Eduardo J.S. Villaseñor
In the course of studying the distribution of the eigenvalues of the so-called area operator in
loop quantum gravity [1] we were led to believe that the following identity held
m∑
k=0
(
m
k
)
B2m−k+1
2m −k +1 2
k = (−1)
m
2
(
1−22m+1 Γ(1+m)
2
Γ(2m +2)
)
, (1)
for m ∈N∪ {0}. The purpose of this short note is to prove this formula by proving a generalization
of it. Particular cases of this general formula involve what we called a two-sided Faulhaber-like
formula. A Faulhaber formula (also called Bernoulli’s formula as Jacob Bernoulli was the first to
write it) is given by
n∑
k=1
kp = 1
p +1
p∑
j=0
(
p +1
j
)
B j n
p+1− j .
Notice that in
n−1∑
k=1
kp = 1p +2p +·· ·+ (n −2)p + (n −1)p
there is an increasing sequence of addends given by powers of the integers. A particular and
interesting case of the aforementioned generalized formula will involve instead a “two-sided”
version of it:
n−1∑
k=1
kp (n −k)p = 1p (n −1)p +2p (n −2)p +·· ·+ (n −2)p 2p + (n −1)p 1p .
Likewise, the Bernoulli numbers are generalized by considering the Bernoulli polynomials:
Bk (x) = k ![zk ]
zexz
ez −1 .
2. Main theorem
The main result of the paper is the following
Theorem 1. Given N ∈Z, m ∈N and w ∈C, we have
m∑
k=0
(
m
k
)
Bm+k+1
( N−w
2
)
m +k +1 w
m−k
= (−1)
m+1
22m+1
[
(2w)2m+1
2(2m +1)(2mm )− sign(N −1)
|N−1|∑
k=1
(
w2−(|N −1|−2k +1)2
)m]
. (2)
Before proceeding with the proof let us discuss some consequences of this formula
Remark 2. It is possible to get a number of Faulhaber-like formulas from (2). The simplest one
can be obtained by taking both w and N to be equal to a natural number n ≥ 2.
n−1∑
k=1
km(n −k)m = n
2m+1
(2m +1)(2mm ) +2(−1)m
m∑
k=0
(
m
k
)
Bm+k+1
m +k +1 n
m−k
= n
2m+1
(2m +1)(2mm ) −2(−1)m
m∑
k=0
(
m
k
)
ζ(−m −k)nm−k .
(3)
where we have used the well known relation between the zeta Riemann function and the
Bernoulli numbers
ζ(1−N ) =−BN
N
.
Equation (3) appears often in the literature obtained through different methods (see for in-
stance [2, p. 10]).
C. R. Mathématique, 2020, 358, n 1, 41-44
J. Fernando Barbero G., Juan Margalef-Bentabol and Eduardo J.S. Villaseñor 43
Remark 3. For N = 1, Equation (2) gives the beautiful expression (equivalent to equation (1.17)
of [4]) valid for any w ∈C,
m∑
k=0
(
m
k
)
Bm+k+1
( 1−w
2
)
m +k +1 w
m−k = (−1)
m+1
2
w2m+1
(2m +1)(2mm ) . (4)
Remark 4. Sums involving
Bβm+k+1
βm +k +1 =−ζ(−k −βm)
with integerβ≥ 2 can also be studied although a more complicated approach is needed involving
complex analysis and combinatorial identities. Nonetheless, the results are not as neat as (2) and
each case has to be studied separately.
Remark 5. It is also possible to generalize (2) for fractional values of N but, again, no systematic
approach has been found. One such expression is when w = N = 1/2
(−1)m+1
m∑
k=0
(
m
k
)
Bm+k+1
m +k +1 2
k = 1
2m+2(2m +1)(2mm ) +
1
23m+2
m∑
k=0
(−1)k
(
m
k
)
E2k
where the En are the Euler numbers [3, entry A122045].
Proof of Theorem 1. The result is a consequence, on one hand, of the following easy-to-prove
formula for the Bernoulli polynomials
Bn(x + r )−Bn(x) = n sign(r )
(|r |−1∑
k=1
(x +k sign(r )−1)n−1
+ 1+ sign(r )
2
(x + r −1)n−1 + 1− sign(r )
2
(x −1)n−1
)
, (5)
valid for r ∈Z and x ∈C, which is a direct consequence of
Bn(x +1)−Bn(x) = nxn−1 ,
and, on the other hand, of the remarkable identity obtained by Sun (equation (1.14) of [4])
(−1)k
k∑
j=0
(
k
j
)
xk− j
B`+ j+1(y)
`+ j +1 + (−1)
`
∑̀
j=0
(
`
j
)
x`− j
Bk+ j+1(z)
k + j +1 =
(−x)k+`+1
(k +`+1)(k+`k ) (6)
where k ,` ∈N and x + y + z = 1.
Taking now x = w , y = (N −w)/2, z = 1− (N +w)/2 and k = `= m ∈N in (6) we obtain
(−1)m
m∑
j=0
(
m
j
)
wm− j
Bm+k+1
( N−w
2
)
m + j +1 =
(−w)2m+1
(2m +1)(2mm ) + (−1)m+1
m∑
j=0
(
m
j
)
wm− j
Bm+ j+1
(
1− N+w2
)
m + j +1 .
Using now (5) to rewrite the last term in terms of Bm+ j+1
( N−w
2
)
, we finally obtain (2). 
Acknowledgments
This work has been supported by the Spanish Ministerio de Ciencia Innovación y Universidades-
Agencia Estatal de Investigación/FIS2017-84440-C2-2-P grant. Juan Margalef-Bentabol is
supported by 2017SGR932 AGAUR/Generalitat de Catalunya, MTM2015-69135-P/FEDER,
MTM2015-65715-P, and the ERC Starting Grant with number 335079. He is also supported
in part by the Eberly Research Funds of Penn State, by the NSF grant PHY-1806356, and by the
Urania Stott fund of Pittsburgh foundation UN2017-92945.
C. R. Mathématique, 2020, 358, n 1, 41-44
44 J. Fernando Barbero G., Juan Margalef-Bentabol and Eduardo J.S. Villaseñor
References
[1] J. F. Barbero G., J. Margalef-Bentabol, E. J. S. Villaseñor, “On the distribution of the eigenvalues of the area operator in
loop quantum gravity”, Class. Quant. Grav. 35 (2018), no. 6, article ID 065008 (17 pages).
[2] P. Kolosov, “On the relation between binomial theorem and discrete convolution of piecewise defined power function”,
https://arxiv.org/abs/1603.02468, 2016.
[3] N. J. A. Sloane, “The On-Line Encyclopedia of Integer Sequences”, 2010, http://oeis.org.
[4] Z.-W. Sun, “Combinatorial identities in dual sequences”, Eur. J. Comb. 24 (2003), no. 6, p. 709-718.
C. R. Mathématique, 2020, 358, n 1, 41-44


A two-sided Faulhaber-like formula involving Bernoulli polynomials

Hafedh Khalfoun

Ismail Laraiedh

The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology

Christophe Le Potier

A second order in space combination of methods verifying a maximum principle for the discretization of diffusion operators

Jaqueline da Costa Ferreira

Marcone Corrêa Pereira

A nonlocal Dirichlet problem with impulsive action: estimates of the growth for the solutions

Chimere Stanley Anabanti

Comptes Rendus
Mathématique
Chimere Stanley Anabanti
A question of Malinowska on sizes of finite nonabelian simple groups in
relation to involution sizes
Volume 358, issue 11-12 (2020), p. 1135-1138.
<https://doi.org/10.5802/crmath.130>
© Académie des sciences, Paris and the authors, 2020.
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
2020, 358, n 11-12, p. 1135-1138
https://doi.org/10.5802/crmath.130
Group Theory / Théorie des groupes
A question of Malinowska on sizes of finite
nonabelian simple groups in relation to
involution sizes
Chimere Stanley Anabantia
a Institut für Analysis und Zahlentheorie, Technische Universität Graz, Austria.
E-mail: anabanti@math.tugraz.at, chimere.anabanti@unn.edu.ng
Abstract. Let In (G) denote the number of elements of order n in a finite group G . Malinowska recently asked
“what is the smallest positive integer k such that whenever there exist two nonabelian finite simple groups
S and G with prime divisors p1, · · · , pk of |G| and |S| satisfying 2 = p1 < ·· · < pk and Ipi (G) = Ipi (S) for all
i ∈ {1, · · · , k}, we have that |G| = |S|?”. This paper resolves Malinowska’s question.
2020 Mathematics Subject Classification. 20D60,20D06.
Funding. The author is supported by both TU Graz and partial funding from the Austrian Science Fund
(FWF): P30934-N35, F05503, F05510. He is also at the University of Nigeria, Nsukka.
Manuscript received 25th May 2020, revised 6th October 2020, accepted 7th October 2020.
1. Introduction
In 1979, Herzog [6] conjectured that two finite simple groups containing the same number
of involutions have the same order. Zarrin [9], in 2018, disproved Herzog’s conjecture with a
counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group
such that I2(G) = I2(S) and Ip (G) = Ip (S) for some odd prime divisor p, then |G| = |S|”. Zarrin’s
conjecture was disproved by the recent author in [1]. In an attempt to reformulate the mentioned
conjecture of Zarrin (considering the works in [1,6,9]), Malinowska [8] asked: “what is the smallest
positive integer k such that whenever there exist two nonabelian finite simple groups S and G
with prime divisors p1, · · · , pk of |G| and |S| satisfying 2 = p1 < ·· · < pk and Ipi (G) = Ipi (S) for all
i ∈ {1, · · · ,k}, we have that |G| = |S|?”.
A simple computational check in GAP [5] or Magma [3] (see [1] for example) tells us that
I2(A8) = 315 = I2(PSL(3,4)), I7(A8) = 5760 = I7(PSL(3,4)) and |A8| = 20160 = |PSL(3,4)|. Also,
the only prime divisors of 20160 are 2,3,5 and 7, and that I3(A8) = 1232 < 2240 = I3(PSL(3,4))
and I5(A8) = 1344 < 8064 = I5(PSL(3,4)). One might think that the smallest such integer k in
Malinowska’s question is 2 (since k > 1 by hypothesis). This is not true since the results in [1]
implies that k > 2. In particular, PSL(4,3) and PSL(3,9) are nonabelian simple groups. Assume
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1136 Chimere Stanley Anabanti
for contradiction that k = 2. Then there exist prime divisors p1 and p2 of the orders of the groups
PSL(4,3) and PSL(3,9) such that 2 = p1 < p2 = 13 and I2(PSL(4,3)) = 7371 = I2(PSL(3,9)),
I13(PSL(4,3)) = 1866240 = I13(PSL(3,9)) but |PSL(4,3)| = 6065280 < 42456960 = |PSL(3,9)|.
Thus, the smallest value of k in Malinowska’s question cannot be 2. The goal of this paper is to
resolve Malinowska’s question.
For clarity, we reinstate Malinowska’s question as follows: “What is the smallest positive integer
k such that whenever there exist two nonabelian finite simple groups S and G that satisfy
Hypothesis(k), we have that |G| = |S|?”. We note that Hypothesis(k) says that there exist prime
divisors p1, · · · , pk of |G| and |S| satisfying 2 = p1 < ·· · < pk such that G and S have the same
numbers of elements of order pi for each i ∈ [1,k].
For the rest of this section, we state the main result of this paper.
Theorem 1. If G and S are two non-isomorphic finite simple nonabelian groups such that
|G| = |S|, then Hypothesis(k) does not hold for any k > 2.
2. Proof of Theorem 1
Before we give a proof of Theorem 1, we first recall some important results.
Theorem 2 ([7, Theorem 5.1]). If G and S are non-isomorphic finite nonabelian simple groups
such that |S| = |G|, then either G = PSL3(4) and S = PSL4(2) or G =Ω2n+1(q) and S = PSp2n(q) for
some odd prime power q, and some n ≥ 3.
As at 1955, Artin [2] gave a prototype of Theorem 2 for all the finite nonabelian simple groups
known then. The groups (PSL3(4) and PSL4(2) or Ω2n+1(q) and PSp2n(q) for some odd prime
power q , and some n ≥ 3) mentioned in Theorem 2 above were also given in Artin’s paper since
they were known at that time. As more finite simple groups were discovered, Tits et al. in many
papers reaffirmed that no other pair of finite simple groups other than the ones found by Artin
satisfies the hypothesis of Theorem 2.
Remark 3.
(i) The groups Ω2n+1(q) and PSp2n(q) are simple for all odd prime power q and n > 2.
Moreover, ∣∣Ω2n+1(q)∣∣= qn2 ∏ni=1
(
q2i −1)
2
= ∣∣PSp2n(q)∣∣ .
(ii) The involutions in Ω2n+1(q) and PSp2n(q) arise from subspace configurations of the
Fq -space V that O(V ) or Sp(V ) naturally acts on. The group Ω2n+1(q) has n conjugacy
classes of involutions and the group PSp2n(q) has bn2 c+ 1 conjugacy classes of involu-
tions. One can then use [4, Table 4.5.1] or otherwise to obtain the values of I2(Ω2n+1(q))
and I2(PSp2n(q)) according as q ≡ 1 mod 4 or q ≡ 3 mod 4. We give a summary of such
results in Proposition 4 below.
Proposition 4. Let q be any odd prime power and n ≥ 3. Then I2(Ω2n+1(q)) 6= I2(PSp2n(q)). In
particular, the following holds:
(a) if q ≡ 3 mod 4, then I2(Ω2n+1(q)) > I2(PSp2n(q));
(b) if q ≡ 1 mod 4, then I2(Ω2n+1(q)) < I2(PSp2n(q)).
Proof. Follows from computations using [4, Table 4.5.1]. 
We now give a proof of Theorem 1.
C. R. Mathématique, 2020, 358, n 11-12, 1135-1138
Chimere Stanley Anabanti 1137
Proof of Theorem 1. Let G and S be two non-isomorphic finite simple nonabelian groups such
that |G| = |S|. Suppose there exists an integer k > 2 satisfying Hypothesis(k). In the light of
Theorem 2, (G ,S) ∈ {(PSL3(4),PSL4(2)), (Ω2n+1(q),PSp2n(q))} for some odd prime power q , and
n ≥ 3. The exposition above tells us that G and S cannot be PSL3(4) and PSL4(2). So the only
possibilty is that G and S areΩ2n+1(q) and PSp2n(q) for some odd prime power q , and n ≥ 3. By
Proposition 4, I2
(
Ω2n+1(q)
) 6= I2(PSp2n(q)) for all such q and n; a contradiction to Hypothesis(k).
Therefore k 6> 2. 
An immediate consequence of Theorems 1 and 2 is the following:
Corollary 5. k = 3 in Malinowska’s question.
3. An observation
We close this paper with Observation 6 below which was motivated by the following question:
Are there non-isomorphic simple groups S and G with In(G) = In(S) for all orders of elements n?
The answer to this question is clearly ‘yes’ if we remove the word “simple” from the question. For
instance, the Heisenberg group mod 3 (given by the presentation: 〈x, y, z | x3 = y3 = z3 = [x, z] =
[y, z] = 1,[x, y] = z〉) and the elementary abelian 3-group of rank 3 are the only two groups of
size 27 whose exponent is 3; so they give rise to a positive answer to the question when the word
“simple” is removed from the question. However, if we leave the question ‘as it is’, then an answer
is given in the following result:
Observation 6. There are no non-isomorphic simple groups S and G with In(G) = In(S) for all
orders of elements n.
Proof. Suppose there are two non-isomorphic simple groups S and G with In(G) = In(S) for
all orders of elements n. Clearly, |S| = |G|. In the light of Theorem 2, either G = PSL(3,4) and
S = PSL(4,2) or G = Ω2n+1(q) and S = PSp2n(q) for some odd prime power q, and some n > 2.
But
I3(PSL(4,2)) = 1232 < 2240 = I3(PSL(3,4)).
So G and S cannot be PSL(3,4) and PSL(4,2); whence we take G =Ω2n+1(q) and S = PSp2n(q) for
some odd prime power q , and some n > 2. This choice is not possible by Proposition 4. Thus, no
such non-isomorphic simple groups exist. 
Acknowledgements
The author is grateful to Professors Sarah Hart, Richard Lyons and Geoff Robinson for useful
discussion during the writing of this paper.
References
[1] C. S. Anabanti, “A counterexample to Zarrin’s Conjecture on sizes of finite nonabelian simple groups in relation to
involution sizes”, Arch. Math. 112 (2019), no. 3, p. 225-226.
[2] E. Artin, “The Orders of the Classical Simple Groups”, Commun. Pure Appl. Math. 8 (1955), p. 455-472.
[3] W. Bosma, J. Cannon, C. Playoust, “The Magma algebra system”, 2019, Version 2.24–5, http://magma.maths.usyd.edu.
au/calc/.
[4] D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups. Part I, Chapter A: Almost simple
K -groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, 1998.
[5] T. G. Group, “GAP – Groups, Algorithms, and Programming”, 2020, Version 4.11.0, https://www.gap-system.org.
[6] M. Herzog, “On the classification of finite simple groups by the number of involutions”, Proc. Am. Math. Soc. 77 (1979),
no. 3, p. 313-314.
C. R. Mathématique, 2020, 358, n 11-12, 1135-1138
1138 Chimere Stanley Anabanti
[7] W. Kimmerle, R. Lyons, S. Robert, D. N. Teague, “Composition factors from the group ring and Artin’s theorem on
orders of simple groups”, vol. 60, 3, no. 1, London Mathematical Society, 1990, p. 89-122.
[8] I. A. Malinowska, “Finite groups with few normalizers or involutions”, Arch. Math. 112 (2019), no. 5, p. 459-465.
[9] M. Zarrin, “A counterexample to Herzog’s Conjecture on the number of involutions”, Arch. Math. 111 (2018), no. 4,
p. 349-351.
C. R. Mathématique, 2020, 358, n 11-12, 1135-1138


A question of Malinowska on sizes of finite nonabelian simple groups in relation to involution sizes

Christian Genest

Magid Sabbagh

Comportement extrémal des copules diagonales et de Bertino

Alexander Olevskii

Alexander Ulanovskii

Comptes Rendus
Mathématique
Alexander Olevskii and Alexander Ulanovskii
Fourier Quasicrystals with Unit Masses
Volume 358, issue 11-12 (2020), p. 1207-1211.
<https://doi.org/10.5802/crmath.142>
© Académie des sciences, Paris and the authors, 2020.
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
2020, 358, n 11-12, p. 1207-1211
https://doi.org/10.5802/crmath.142
Harmonic Analysis / Analyse harmonique
Fourier Quasicrystals with Unit Masses
Alexander Olevskiia and Alexander Ulanovskii∗, b
a School of Mathematics, Tel Aviv University, 69978 Ramat Aviv, Israel.
b Stavanger University, 4036 Stavanger, Norway.
E-mails: olevskii@post.tau.ac.il (A. Olevskii), Alexander.Ulanovskii@uis.no
(A. Ulanovskii)
Abstract. The sum of δ-measures sitting at the points of a discrete set Λ ⊂ R forms a Fourier quasicrystal if
and only ifΛ is the zero set of an exponential polynomial with imaginary frequencies.
Manuscript received 19th October 2020, accepted 30th October 2020.
1. Introduction
By a Fourier quasicrystal one usually means a complex measure with discrete support and
spectrum. This concept goes back to works of Yves Meyer in the 1970-ies and it reappeared later in
connection with an unexpected phenomenon in crystallography discovered by Dan Shechtman
in the 1980-ies, see [5].
More precisely, following [6] we call a measure µ on R a crystalline measure, if it is an atomic
measure which is a tempered distribution, its distributional Fourier transform µ̂ is an atomic
measure and both the support Λ and the spectrum S of µ are locally finite sets. If in addition the
measures |µ| and |µ̂| are also tempered, then µ is called a Fourier quasicrystal (FQ).
The classical example of an FQ is the Dirac comb (the crystal)
µ= ∑
k ∈Z
δk ,
where δx is the unit mass at point x. Then the Poisson summation formula reads µ̂=µ.
Examples of aperiodic quasicrystals were presented in [3] and then in [1, 6, 7]. Recently a new
progress was achieved by P. Kurasov and P. Sarnak [2] who discovered examples of FQs with unit
masses
µ= ∑
λ∈Λ
δλ, (1)
where Λ ⊂ R is a uniformly discrete aperiodic set. An alternative construction of such measures
was suggested by Y.Meyer [8].
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1208 Alexander Olevskii and Alexander Ulanovskii
Below we present one more construction and prove that it characterizes all FQs of form (1).
A preliminary publication of our results was given in arXiv [9, 10].
The Theorem 1 below reveals a fundamental connection between FQs with unit masses and
the zero sets Z (p) := {z ∈C : p(z) = 0} of exponential polynomials p with imaginary frequencies.
Theorem 1.
(i) Let p be an exponential polynomial
p(t ) = ∑
1≤ j ≤N
c j e
2π i γ j t , N ∈N,c j ∈C,γ j ∈R, (2)
which has only simple real zeros. Then the measure µ defined in (1) with Λ = Z (p) is an
FQ.
(ii) Conversely, letµ be an FQ of form (1). Then there is an exponential polynomial p of form (2)
with real simple zeros such thatΛ= Z (p).
We will sketch the proof of part (ii), see [10] for the proof of part (i).
Using Theorem 1(i) one may construct simple examples of aperiodic FQs.
Lemma 2. Fix a real number ε satisfying 0 < |ε| ≤ 1/2 and set
pε(t ) := sin(πt )+εsin t . (3)
Then pε has only simple real zeros and
Z (pε) = {k +εk : k ∈Z} , εk ∈ [−1/6,1/6].
For a proof see [10].
Theorem 1 and Lemma 2 show that the sum of δ-measures sitting at the points of Z (pε) is an
FQ.
Let pε be given in (3). One may check that the numbers εk in Lemma 2 satisfy maxk |εk |→ 0 as
ε→ 0. Therefore, the set Z (pε) “approaches” the set of integers Z:
Corollary 3. For every ε> 0 there is an aperiodic set
Λ= {k +εk : k ∈Z} , 0 < |εk | < ε,k ∈Z,
such that the corresponding measure in (1) is an FQ.
2. Proof of Part (ii) of Theorem 1
In what follows we consider the standard form of the Fourier transform
ĥ(u) :=
∫
R
e−2πi ut h(t )d t , h ∈ L1(R).
Let us start with a result which may have intrinsic interest:
Proposition 4. Let µ be a positive measure which is a tempered distribution, such that its
distributional Fourier transform µ̂ is a measure satisfying∣∣µ̂∣∣ (−R,R) =O (Rm) , R →∞, for some m > 0, (4)
which means that |µ̂| is a tempered distribution. Then there exists C such that
µ(a,b) ≤C (1+b −a), −∞< a < b <∞. (5)
C. R. Mathématique, 2020, 358, n 11-12, 1207-1211
Alexander Olevskii and Alexander Ulanovskii 1209
Proof. It suffices to prove (5) for every interval (a,b) satisfying b −a ≥ 2.
Fix any non-negative Schwartz function g (x) supported by [−1/2,1/2] and such that∫
R
g (x)d x = 1.
Set
f (x) := (g ∗1(a−1/2,b+1/2)) (x) ∈ S(R).
Clearly, ∣∣ f̂ (t )∣∣= ∣∣ĝ (t )1̂(a−1/2,b+1/2)(t )∣∣≤ (1+b −a) ∣∣ĝ (t )∣∣ .
Using this inequality and (4), we get∫
R
f (x)µ(d x) =
∫
R
f̂ (t )µ̂(d t ) ≤ (1+b −a)
∫
R
∣∣ĝ (t )∣∣ ∣∣µ̂∣∣ (d t ) =C (1+b −a).
On the other hand, clearly,
f (x) = g (x)∗1(a−1/2,b+1/2)(x) = 1, x ∈ (a,b).
Hence, ∫
R
f (t )µ(d t ) ≥µ(a,b),
which proves the Proposition 4.
Recall that a setΛ⊂R is called uniformly discrete, if
inf
λ′,λ∈Λ,λ 6=λ′
∣∣λ−λ′∣∣> 0.
A setΛ is called relatively uniformly discrete if it is a union of finite number of uniformly discrete
sets.
Proposition 4 implies
Corollary 5. Let µ be a measure of form (1) whose distributional Fourier transform is a measure
satisfying (4). Then its supportΛ is a relatively uniformly discrete set.
Assume µ is an FQ of form (1). This means that a Poisson-type formula∑
λ∈Λ
f (λ) = ∑
s∈S
as f̂ (s), f ∈ S(R), (6)
is true where S(R) denotes the Schwartz space, S is locally finite set and the coefficients as satisfy∑
s∈S, |s|<R
|as | ≤C Rm , R > 1, for some C ,m > 0. (7)
To prove part (ii) of Theorem 1 we have to show that Λ= Z (p) for some exponential polynomial
p of form (2). We will prove this under the additional restrictions that Λ is a symmetric set,
−Λ=Λ and 0 6∈Λ. For the general case see [9].
Set
ψ(z) := ∏
λ∈Λ
(
1− z
λ
)
= ∏
λ∈Λ,λ>0
(
1− z
2
λ2
)
, z ∈C. (8)
The product converges (uniformly on compacts) due to Corollary 5.
Lemma 6. ψ is an entire function of order one and finite type, i.e. there exist C ,σ> 0 such that∣∣ψ(z)∣∣≤Ceσ|z|, z ∈C.
This lemma follows from Corollary 5 and the symmetry ofΛ by standard estimates.
Lemma 7. The following representation is true:
ψ′(z)
ψ(z)
=−2πi
(
a0/2+
∑
s∈S∩(−∞,0)
as e
−2π i s z
)
, Im z > 0, (9)
where as are the coefficients in (6).
C. R. Mathématique, 2020, 358, n 11-12, 1207-1211
1210 Alexander Olevskii and Alexander Ulanovskii
By (7), the series in (9) converges absolutely for every z, Im z > 0.
Let us sketch a proof of Lemma 7. It follows from (8) that
ψ′(z)
ψ(z)
= ∑
λ∈Λ
1
z −λ , z ∈C. (10)
The next step is to check that∑
λ∈Λ
1
z −λ =−2πi
(
a0/2+
∑
s∈S∩(−∞,0)
as e
−2π i s z
)
, Im z > 0. (11)
This can be done as follows: For every fixed z, Im z > 0, set
ez (u) =
{
2πe−2πi zu u < 0
0 u ≥ 0
Then the inverse Fourier transform of ez is the function i /(z − t ). Fix any function h ∈ S(R) such
that h(0) = 1 and the Fourier transform H := ĥ is even, non-negative and vanishes outside (−1,1).
Then use (6) with f (t ) = h(εt )/(z − t ):∑
λ∈Λ
h(ελ)
z −λ =−i
∑
s∈S
as
(
ez (u)∗ 1
ε
H(u/ε)
)
(s).
Finally, to prove Lemma 7 one lets ε → 0 and checks that the right and left hand-sides above
converge to the corresponding sides of (11).
Now, it follows from (9) that there exists K ∈C such that
ψ(z) = K exp
(
−πi a0z +
∑
s∈S∩(−∞,0)
(as /s)e
−2π i s z
)
, Im z > 0.
Set
p(z) := eπi a0zψ(z)/K = exp
( ∑
s∈S∩(−∞,0)
(as /s)e
−2π i s z
)
, Im z > 0. (12)
Recall that S is a locally finite set. Therefore, by (7) the series above converges absolutely for every
z, Im z > 0.
Denote by Sk the sets
S1 := S ∩ (−∞,0), S2 := S1 +S1, S3 := S1 +S1 +S1, . . .
Denote by as,k the coefficients of the series
1
k !
( ∑
s∈S∩(−∞,0)
(as /s)e
−2πi sz
)k
= ∑
s∈Sk
as,k e
−2π i sz , k ∈N, Im z > 0.
Then by (12) we get a representation
p(z) = 1+
∞∑
k=1
∑
s∈Sk
as,k e
−2π i sz ,
where the double series converges absolutely for every z, Im z > 0. Set
U := {0} ∞∪
j=1
S j ⊂ (−∞,0].
One may check that U is a locally finite set and that p admits a representation
p(z) = ∑
u∈U
due
−2π i uz , Im z > 0, (13)
where the series converges absolutely.
To prove part (ii) of Theorem 1 it remains to check that the series in the right hand-side
of (13) contains only a finite number of terms. This can be done as follows: Since ψ is an entire
C. R. Mathématique, 2020, 358, n 11-12, 1207-1211
Alexander Olevskii and Alexander Ulanovskii 1211
function of order one and finite type, the same is true for p. By (13), p is bounded on every
line Im z = const > 0. It follows that (see [4, Lecture 6, Theorem 2]) p is an entire function of
exponential type, i.e. it satisfies ∣∣p(x + i y)∣∣≤Ceσ|y |, x, y ∈R,
with some C ,σ > 0. Now, to check that in (13) we have du = 0 for every u ∈ U , |u| > σ, one
simply integrates both sides against e2πi uz (sinεz/εz)2, where ε > 0 is so small that |u| − ε > σ
and U ∩ (u −ε,u +ε) = {u}.
We note that one can extend Theorem 1 to measures with integer masses,
µ= ∑
λ∈Λ
cλδλ, cλ ∈N,λ ∈Λ. (14)
Theorem 8.
(i) If a measure µ of form (14) is an FQ, then there is an exponential polynomial p of form (2)
with real zeros such thatΛ= Z (p) and c(λ) is the multiplicity of zero λ.
(ii) Conversely, let p be an exponential polynomial of form (2) with real zeros and let c(λ) be
the multiplicity of zero λ. Then the measure µ of form (14) whereΛ= Z (p) is an FQ.
References
[1] M. N. Kolountzakis, “Fourier pairs of discrete support with little structure”, J. Fourier Anal. Appl. 22 (2016), no. 1,
p. 1-5.
[2] P. Kurasov, P. C. Sarnak, “Stable polynomials and crystalline measures”, J. Math. Phys. 61 (2020), no. 8, article
no. 083501 (13 pages).
[3] N. Lev, A. Olevskii, “Quasicrystals with discrete support and spectrum”, Rev. Mat. Iberoam. 32 (2016), no. 4, p. 1341-
1352.
[4] B. Y. Levin, Lectures on Entire Fuctions, in collab. with yu. lyubarskii, m. sodin, v. tkachenko. transl. by v. tkachenko
from an original russian manuscript ed., Translations of Mathematical Monographs, vol. 150, American Mathemati-
cal Society, 1996.
[5] Y. F. Meyer, “Quasicrystals, diophantine approximation and algebraic numbers”, in Beyond quasicrystals. Papers of
the winter school, Les Houches, France, March 7-18, 1994, Springer, 1995, p. 3-16.
[6] ——— , “Measures with locally finite support and spectrum”, Proc. Natl. Acad. Sci. USA 113 (2016), no. 12, p. 3152-
3158.
[7] ——— , “Measures with locally finite support and spectrum”, Rev. Mat. Iberoam. 33 (2017), no. 3, p. 1025-1036.
[8] ——— , “Curved model sets and crystalline measures”, to be published in Applied and Numerical Harmonic Analysis,
Springer, 2020.
[9] A. Olevskii, A. Ulanovskii, “Fourier quasicrystals with unit masses”, https://arxiv.org/abs/2009.12810, 2020.
[10] ——— , “A Simple Crystalline Measure”, https://arxiv.org/abs/2006.12037, 2020.
C. R. Mathématique, 2020, 358, n 11-12, 1207-1211


Fourier Quasicrystals with Unit Masses

Song Yang

Xun Yu

Rational cubic fourfolds in Hassett divisors

Indranil Biswas

Sorin Dumitrescu

Laurent Meersseman

Comptes Rendus
Mathématique
Indranil Biswas, Sorin Dumitrescu and Laurent Meersseman
Stability and holomorphic connections on vector bundles over LVMB
manifolds
Volume 358, issue 2 (2020), p. 151-157.
<https://doi.org/10.5802/crmath.24>
© Académie des sciences, Paris and the authors, 2020.
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
2020, 358, n 2, p. 151-157
https://doi.org/10.5802/crmath.24
Analytic Geometry / Géométrie analytique
Stability and holomorphic connections on
vector bundles over LVMB manifolds
Fibrés vectoriels holomorphes sur les variétés LVMB
Indranil Biswasa, Sorin Dumitrescub and Laurent Meerssemanc
a School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha
Road, Mumbai 400005, India
b Université Côte d’Azur, CNRS, LJAD
c Laboratoire Angevin de Recherche en Mathématiques, Université d’Angers,
Université de Bretagne-Loire F-49045 Angers Cedex, France.
E-mails: indranil@math.tifr.res.in, dumitres@unice.fr,
laurent.meersseman@univ-angers.fr.
Abstract. We characterize all LVMB manifolds X such that the holomorphic tangent bundle T X is spanned at
the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus.
We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds
with this previous property are always flat.
Résumé. Nous caractérisons les variétés LVMB qui ont la propriété de positivité P suivante : le fibré tangent
holomorphe est engendré au point générique par une famille de champs de vecteurs holomorphes (globa-
lement définis) {vi }, tel que chaque vi s’annule en au moins un point de X . Nous en déduisons que, sur les
variétés LVMB avec la propriété P , les connexions holomorphes sur les fibrés vectoriels holomorphes semi-
stables sont nécessairement plates.
2020 Mathematics Subject Classification. 32Q26, 32M12, 53B05.
Manuscript received 30th April 2019, accepted 24th February 2020.
Version française abrégée
Les variétés LVMB [3, 4, 10, 11] (voir aussi [14]) forment une classe de variétés complexes com-
pactes (en général, non–Kähler) qui généralise les variétés toriques et aussi les variétés de Hopf
diagonales. Les variétés LVMB sont des compactifications équivariantes lisses de groupes de Lie
complexes abéliens.
Cet article étudie la géométrie complexe des fibrés holomorphes au-dessus d’une variété
LVMB.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
152 I. Biswas, S. Dumitrescu and L. Meersseman
Rappelons qu’une variété LVMB X est biholomorphe à un quotient d’un ouvert V de l’espace
projectif complexe Pn−1 par une action holomorphe, libre et propre du groupe de Lie complexe
Cm , engendré par m (avec n > 2m) champs de vecteurs linéaires diagonaux qui commutent.
L’ouvert V est le complémentaire dans Pn−1 d’une union E de k hyperplans de coordonnées.
Par conséquent, l’ouvert V contient le tore T := (C∗)n−1 et est invariant par l’action naturelle de
T sur Pn−1. Le quotient de T par l’action de Cm s’identifie à un groupe de Lie complexe abélien
G . Cela implique que X est une compactification équivariante lisse (de dimension ν= n −m −1)
de G [3, 4, 10, 11].
Le premier résultat principal de l’article (Théorème 4) est une caractérisation de variétés
LVMB X qui ont la propriété de positivité P suivante : le fibré tangent holomorphe T X est
engendré au point générique par une famille de champs de vecteurs holomorphes (globalement
définis) (v1, . . . , vn−m−1), tel que chaque vi s’annule en au moins un point de X . Le théorème 4
affirme qu’une variété LVMB a la propriété P si et seulement si k ≤ m +1.
Le cas k = m + 1 est particulièrement intéressant car les variétés LVMB correspondantes
possèdent des structures affines complexes (notez que la preuve du Lemme 12.1 faite dans [4]
pour les variétés LVM se généralise sans difficulté au cadre LVMB). Les variétés LVMB les plus
simples avec k = m +1 sont les variétés de Hopf diagonales, mais il y a également des exemples
dont la topologie peut-être compliquée (variétés difféomorphes à un produit entre (S1)m et une
somme connexe de produits de sphères, ainsi que des variétés avec torsion arbitraire dans le sens
du Théorème 14.1 et du Corollaire 14.3 dans [4]).
Dans la deuxième partie de l’article nous étudions les fibrés holomorphes E munis de connex-
ions holomorphes au-dessus des variétés LVMB qui ont la propriété P . Plus précisément, nous
faisons des liens avec la notion de stabilité au sens des pentes. Mentionnons que les notions de
degré, pente et (semi)-stabilité sont considérées ici par rapport à une métrique de Gauduchon.
En particulier, le degré n’est pas, en général, un invariant topologique (comme c’est le cas dans
le contexte Kähler).
Nous montrons que chaque sous-faisceau Ei dans la filtration de Harder–Narasimhan de E
0 = E0 ⊂ E1 ( E2 ( · · ·( Eb ( Eb+1 = E
est invariant par la connexion holomorphe et est, par conséquent, localement libre. Nous util-
isons ensuite la propriété de positivité P de T X pour en déduire que la connexion holomorphe
induite sur chaque quotient semi-simple Ei /Ei−1 est nécessairement plate. Par conséquent, nous
obtenons le Corollaire 10 qui affirme que toute connexion holomorphe sur un fibré semi-stable E
au-dessus d’une variété LVMB avec la propriété P est nécessairement plate.
1. Introduction
An important class of compact complex manifolds (in general, non–Kähler) generalizing the
toric varieties and diagonal Hopf manifolds are the so-called LVMB manifolds [3, 4, 10, 11] (see
also [14]). They are smooth equivariant compactifications of complex abelian Lie groups.
This article deals with the geometry of holomorphic vector bundles over LVMB manifolds.
Our first result (Theorem 4) is a geometric characterization of LVMB manifolds X admitting a
family of global holomorphic vector fields spanning the holomorphic tangent bundle T X at the
generic point and satisfying the condition that each vector field in the family has a non-empty
zero locus.
We use this positivity result and stability arguments to show that holomorphic connections on
semi-stable holomorphic vector bundles over the LVMB manifolds of the above type are always
flat (see Corollary 10).
C. R. Mathématique, 2020, 358, n 2, 151-157
I. Biswas, S. Dumitrescu and L. Meersseman 153
2. Geometry of the tangent bundle of LVMB manifolds
We refer to [11] and [3] for more details on the properties of LVMB manifolds used below. It
should be clarified that although the article [11] deals with LVM manifolds, most results there
hold verbatim, and with the same proof, for the more general LVMB case.
A LVMB manifold X is a a quotient of some open and dense subset V of Pn−1 by a free
and proper holomorphic Cm-action generated by m commuting linear diagonal vector fields
ξ1, . . . ,ξm (with n > 2m). More precisely, the action is given by
T · [z] := [zi e〈Λi ,T 〉]ni=1 (1)
where the m rows of the matrix (Λ1, . . . ,Λn) are the coefficients of the m vector fields ξ1, . . . ,ξm as
linear combinations of z1
∂
∂z1
, . . . , zn
∂
∂zn
.
The subset V is the complement in Pn−1 of a union E of coordinate subspaces:
V =Pn−1 \ E . (2)
It follows that it contains a torus T := (C∗)n−1 and is preserved under the natural action of T onto
Pn−1. The quotient ofT by theCm-action is a complex abelian Lie group, say G , and X is a smooth
equivariant compactification of G .
The following alternative description of G can be found in [10, p. 27] (and which itself is a
straightforward generalization of Theorem 1 in [11]).
Proposition 1. Assume that
rankC
(
Λ1 . . . Λm+1
1 . . . 1
)
= m +1. (3)
Then G is isomorphic to the quotient of Cn−m−1 by the Zn−1 abelian subgroup generated by
(Id,B A−1), where
A =t(Λ2 −Λ1, . . . ,Λm+1 −Λ1)
and
B =t(Λm+2 −Λ1, . . . ,Λn−1 −Λ1).
Remark 2. It is easy to prove that
rankC
(
Λ1 . . . Λn
1 . . . 1
)
= m +1.
(cf. [12, Lemma 1.1] in the LVM case). Hence, up to a permutation, condition (3) is always fulfilled.
Let G be the Lie algebra of G . Given a ∈G, let
v(x) := d
d t
(
x ·exp(t a)) |t=0, x ∈ X
be the corresponding fundamental vector field.
Definition 3. We say that X has the non-zero vanishing property if we may find a basis
(a1, . . . , an−m−1) of G such that the corresponding fundamental vector fields (v1, . . . , vn−m−1) all
have a non-empty vanishing locus.
Let k denote the number of coordinate hyperplanes in E ; see (2). We have:
Theorem 4. A LVMB manifold X has the non-zero vanishing property if and only if the inequality
k ≤ m +1 is fulfilled.
C. R. Mathématique, 2020, 358, n 2, 151-157
154 I. Biswas, S. Dumitrescu and L. Meersseman
Proof. Observe that the natural map V −→ X induces a Lie group morphism T −→ G . Hence,
using homogeneous coordinates [z1, . . . , zn] inPn−1, the image of the vector fields z1 ∂∂z1 , . . . , zn
∂
∂zn
form a generating family of fundamental vector fields of X .
Observe also that such a vector field zi
∂
∂zi
vanishes on some point of X if and only if (the
projectivization of) {zi = 0} is not included in E .
Assume firstly that k ≤ m+1. For simplicity, assume that the coordinate hyperplanes belonging
to E are {zn−k−1 = 0}, . . . , {zn = 0}. Observe that the Cm-action (1) preserves the zero and the non-
zero coordinates. Given [z] ∈ V , since k ≤ m + 1, we may thus find some T ∈ Cm such that the
image w of z under the action of T has the form
[w1, . . . , wn−k−2,1, . . . ,1].
Such a T is not unique but unique up to some additive subgroup H of Cm of dimension m−k−1,
cf. the statement of Proposition 1 whose proof follows the same line of arguments. In other words,
setting
V0 := {[w1, . . . , wn−k−2,1, . . . ,1] ∈V }
then X is isomorphic to V0 quotiented by the action of H . As a consequence, the smaller family
z1
∂
∂z1
, . . . , zn−k−2 ∂∂zn−k−2 still descends as a generating family of fundamental vector fields of X .
Hence we may find n−m−1 vector fields in this family which form a basis of fundamental vector
fields. Now they all vanish on X by the above observation.
To prove the converse, assume that k > m +1. Then in the generating family of fundamental
vector fields, we have strictly less than n −m − 1 vector fields with non-empty vanishing locus.
Moreover, a linear combination of vector fields of the family, say
n∑
i=1
αi zi
∂
∂zi
has vanishing locus
V ∩P({zi = 0 |αi 6= 0})
Hence to have non-empty vanishing locus, a fundamental vector field must be a linear combi-
nation of the n −k vector fields with non-empty vanishing locus of the generating family. Since
n −k < n −m −1, we cannot find a basis of such vector fields. 
Remark 5. The case k = m + 1 in Theorem 4 is of special interest, because the corresponding
LVMB manifolds admit an affine atlas, see Lemma 12.1 of [4] for a proof in the LVM case, which
trivially generalizes to LVMB manifolds. Examples of LVMB fulfilling k = m +1 include diagonal
Hopf manifolds, but also manifolds diffeomorphic to the product of (S1)m with connected sums
of sphere products and even manifolds with arbitrary torsion in the sense of Theorem 14.1 and
Corollary 14.3 of [4].
3. Stability and holomorphic connections on LVMB manifolds
As in Theorem 4, let X be a LVMB manifold, of complex dimension ν := n − m − 1, such that
k ≤ m + 1. A Gauduchon metric on X is a Hermitian structure h such that the corresponding
(1,1)-form ωh satisfies the equation ∂∂ω
ν−1
h = 0; Gauduchon metrics exist [7]. Fix a Gauduchon
form ωh on X . The degree of a torsionfree coherent analytic sheaf F on X is defined to be
degree(F ) :=
p−1
2π
∫
X
K (detF )∧ων−1h ∈R,
where detF is the determinant line bundle for F [8, Ch. V, §6] (or [5, Definition 1.34]) and K is the
curvature for a hermitian connection on detF compatible with the Dolbeault operator ∂detF . This
C. R. Mathématique, 2020, 358, n 2, 151-157
I. Biswas, S. Dumitrescu and L. Meersseman 155
degree is independent of the Hermitian metric on detF , because any two such curvature forms
differ by a ∂∂-exact 2–form on X :∫
X
(∂∂u)∧ων−1h =−
∫
X
u ∧∂∂ων−1h = 0.
A torsionfree coherent analytic sheaf F on X is called semistable if
degree(V )
(V )
≤ degree(F )
(F )
for all nonzero coherent analytic subsheaf V ⊂ F (see [9, p. 44, Definition 1.4.3], [8, Ch. V, §7]).
Let
T0 ( T1 ( · · ·( T`( T`+1 = T X (4)
be the Harder–Narasimhan filtration of the holomorphic tangent bundle T X . We note that T X is
semistable if and only if `= 0.
Lemma 6. The subsheaf T` in (4) satisfies the following condition:
degree(T X /T`) > 0.
Proof. Fix a basis (a1, . . . , aν) of G such that the corresponding fundamental vector fields
(v1, . . . , vν) have the property that each of them has a non-empty vanishing locus. Theorem 4 en-
sures that such a basis exists. Let r be the rank of T X /T`. Fix r elements from (v1, . . . , vν) satisfying
the following condition: their projections to T X /T` together generate T X /T` over a Zariski open
subset. Let (w1, . . . , wr ) denote this subset of (v1, . . . , vν). Therefore, w1∧·· ·∧wr is a holomorphic
section of
∧r T X which is not identically zero.
Consider the holomorphic line bundle det(T X /T`) over X . The quotient map T X −→ T X /T`
produces a natural projection q :
∧r T X −→ det(T X /T`). Let
σ := q(w1 ∧·· ·∧wr ) ∈ H 0(X ,det(T X /T`))
be the nonzero holomorphic section given by w1∧·· ·∧wr . If any of (w1, . . . , wr ) vanishes at a point
x ∈ X , then clearly we have σ(x) = 0. Since each of (w1, . . . , wr ) has the property that it vanishes at
some point of X , we conclude that the divisor div(σ) is nonzero. Consequently, we have
degree(T X /T`) = Volumeh(div(σ)) > 0;
see [9, p. 35, Proposition 1.3.5] and [6, p. 628, Corollary 1] for it. This completes the proof. 
Let E be a holomorphic vector bundle on X equipped with a holomorphic connection D
(see [1] for holomorphic connections). Let
0 = E0 ⊂ E1 ( E2 ( · · ·( Eb ( Eb+1 = E (5)
be the Harder–Narasimhan filtration of E , so E is semistable if and only if b = 0.
Proposition 7. For every 1 ≤ i ≤ b −1, the subsheaf Ei ⊂ E in (5) is preserved by the holomorphic
connection D.
Proof. Take any 1 ≤ i ≤ b −1. Let
Si : T X ⊗Ei −→ E/Ei
be the homomorphism given by the second fundamental form of Ei for the connection D . We
recall that this second fundamental form is given by the composition
Ei
D−→ E ⊗Ω1X
qi⊗Id−→ (E/Ei )⊗Ω1X ,
where qi : E −→ E/Ei is the quotient map. Tensoring both sides of it with T X and taking trace,
the above homomorphism Si is obtained from the second fundamental form.
C. R. Mathématique, 2020, 358, n 2, 151-157
156 I. Biswas, S. Dumitrescu and L. Meersseman
For any holomorphic vector field v on X , consider the composition
Ei
v⊗−−→ T X ⊗Ei Si−→ E/Ei . (6)
From the properties of the Harder–Narasimhan filtration we know that there is no nonzero homo-
morphism from Ei to E/Ei . Hence the composition homomorphism in (6) vanishes identically.
Since global holomorphic vector fields generate T X over a Zariski open subset, the proposition
follows. 
Corollary 8. Each subsheaf Ei in (5) is a holomorphic subbundle of E.
Proof. By Lemma 4.5 in [2], any coherent analytic subsheaf Ei invariant by the holomorphic
connection D is locally free. 
For each 1 ≤ i ≤ b, let Di be the holomorphic connection, induced by D , on the quotient
bundle Ei /Ei−1 in (5).
Proposition 9. The holomorphic connection Di on Ei /Ei−1 is flat for every i .
Proof. A holomorphic vector bundle V on X is semistable if and only if V admits an approxi-
mate Hermitian–Einstein structure [13, p. 629, Theorem 1.1] (set the Higgs field φ in [13, Theo-
rem 1.1] to be zero). If V1 and V2 are semistable holomorphic vector bundles on X , then approx-
imate Hermitian–Einstein structures on V1 and V2 together produce an approximate Hermitian–
Einstein structure on V1 ⊗V2. Hence V1 ⊗V2 is also semistable.
Since Ei /Ei−1 is semistable, from the above observation we conclude that End(Ei /Ei−1) is also
semistable. Note that we have
degree(End(Ei /Ei−1)) = 0,
because detEnd(Ei /Ei−1) =OX .
Let
K (Di ) ∈ H 0(X ,Ω2X ⊗End(Ei /Ei−1))
be the curvature of the holomorphic connection Di on Ei /Ei−1. For any holomorphic vector field
v on X , the contraction iv K (Di ) of this curvature form gives a holomorphic homomorphismãiv K (Di ) : T X −→ End(Ei /Ei−1).
Now, since End(Ei /Ei−1) is semistable of degree zero, from Lemma 6 it follows immediately that
there is no nonzero holomorphic homomorphism from T X to End(Ei /Ei−1). Hence the above
homomorphism ãiv K (Di ) vanishes identically. Since global holomorphic vector fields generate
T X over a Zariski open subset, we conclude that K (Di ) = 0. 
The following is an immediate consequence of Proposition 9.
Corollary 10. Let E be a semistable holomorphic vector bundle on X , and let D be a holomorphic
connection on E. Then D is flat.
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Stability and holomorphic connections on vector bundles over LVMB manifolds

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