Annales de la Faculté des Sciences de Toulouse

Shin-ichi Matsumura

Open problems on structure  of positively curved projective varieties

Long Li

Strict convexity of the Mabuchi functional for energy minimizers

Sébastien Boucksom

Mattias Jonsson

Global pluripotential theory over a trivially valued field

Dan Coman

George Marinescu

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

Youness Boutaib

Terry Lyons

A new definition of rough paths on manifolds

Eduard Curcă

The divergence equation with $L^{\infty }$ source

Julien Bichon

Maeva Paradis

Some isomorphism results for graded twistings of function algebras on finite groups

Sławomir Dinew

$m$-subharmonic and $m$-plurisubharmonic functions: on two problems of Sadullaev

Hoang Chinh Lu

Henri Guenancia

Vincent Guedj

Publication membre du centre
Mersenne pour l’édition scientifique ouverte
http://www.centre-mersenne.org/
ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
HOANG CHINH LU, HENRI GUENANCIA AND VINCENT GUEDJ
Analysis of Monge–Ampère, a tribute to Ahmed Zeriahi
Tome XXXI, no 3 (2022), p. i–v.
https://doi.org/10.5802/afst.1703
© les auteurs, 2022.
Les articles des Annales de la Faculté des Sciences de Toulouse sont mis
à disposition sous la license Creative Commons Attribution (CC-BY) 4.0
http://creativecommons.org/licenses/by/4.0/
– i –
Preface
The volume gathers some articles that were presented at the conference
AMAZER held in Toulouse in June 2021, in honor of Ahmed Zeriahi on the
occasion of his retirement. AMAZER is an acronym for Analysis of Monge–
Ampère, a tribute to Ahmed Zeriahi. It is also reminiscent of “Amazigh”, the
name of the Berber people, which means “free man” in tamazight.
Ahmed spent most of his career at the University Paul Sabatier in
Toulouse, where he arrived in 1973 after completing his baccalaureate in
Rabat (Morocco). He defended his “doctorat d’état” in 1986 under the su-
pervision of Nguyen Van Thanh. He took a position as Professor at the
University of Rabat at the end of the 80s, before returning to Toulouse as
full Professor from 1992 until his retirement in September 2020.
Most of Ahmed’s work is devoted to complex analysis in several variables
and its many applications to dynamical systems and geometry. He has be-
come a leading expert in pluripotential theory, establishing fine properties of
the complex Monge–Ampère operator which plays a central role in complex
analytic geometry.
Ahmed has supervised a dozen of PhD theses of French, Moroccan and
Vietnamese students. He has established friendships and rich collabora-
tions with researchers from many European countries (France, Italy, Iceland,
Poland, Sweden), and has worked hard to develop exchanges with coun-
tries in the process of scientific development (Morocco, Uzbekistan, Senegal,
Turkey, Vietnam).
His good mood, his generosity and his energy have greatly contributed to
the success of these scientific exchanges. Many authors (in particular some
of his many collaborators!) wished to pay tribute to him, leading to this
volume. Let us now briefly describe the contributions it contains.
Berman and Berndtsson establish Moser–Trudinger type inequalities for
Hodge classes on a compact Kähler manifold, positively answering a conjec-
ture of Aubin. They also study the local setting of bounded pseudoconvex
domains in Cn in connection with a Brezis–Merle inequality of Ahmed and
collaborators, obtaining a sharp Moser–Trudinger inequality for S1-invariant
plurisubharmonic functions
Boucksom and Jonsson establish the foundations of a global non-archim-
edean pluripotential theory, generalizing the work of Ahmed and his col-
laborators in the framework of Berkovich geometry over a trivially valued
– ii –
field. They define and study functions and measures of finite energy, the non-
Archimedean Monge–Ampère operator and investigate the topology of the
space of valuations of linear growth, and the behavior of plurisubharmonic
functions.
Boyer and Tonnesen-Friedman explain how iterated S3 Sasaki joins can
be fit into a categorical framework closely related to the category of Bott
orbifold towers. In this framework, they outline various results concerning the
existence and non-existence of Sasaki–Einstein, and more generally, constant
scalar curvature Sasaki metrics.
Generalizing a landmark result of Berndtsson, Cao, Guenancia and Păun
establish several formulas for the curvature of the L2 metric on the direct
image of the relative canonical bundle twisted by a holomorphic line bundle
endowed with a positively curved metric with analytic singularities. A very
explicit lower bound is moreover obtained when the twist is relatively big.
Collins and Tosatti define and explore the notion of numerical restricted
volume of a pseudoeffective class in a compact Kähler manifold, along a given
irreducible subvariety. They relate that notion to the well-known notion of
restricted volume of a line bundle on a projective manifold, and conjecture
that the non-Kähler locus of a class coincides with its null locus, which they
show to hold for classes admitting a Zariski decomposition.
Bergman kernels on smooth compact complex polarized manifolds have
been thoroughly studied in the last decades. The growing interest in mildly
singular varieties naturally motivates to investigate the asymptotic proper-
ties of Bergman currents on compact Kähler spaces. Coman and Marinescu
study these questions on non-normal algebraic curves.
In a recent breakthrough, Chen and Cheng have shown that coercivity of
the Mabuchi K-energy ensures the existence of a constant scalar curvature
Kähler metric. Deruelle and Di Nezza explain the main steps of the proof
of Chen–Cheng, and provide, using a generalized Monge–Ampère capacity,
a new and streamlined proof of the crucial uniform L∞-estimate.
The article by Dinew studies the connection between m-subharmonic
and k-plurisubharmonic functions, answering by the negative two questions
raised by Sadullaev. In particular the comparison principle does not hold
for the twisted Hessian operator, unless the latter equals either the complex
Monge–Ampère operator or the Laplacian.
By a famous result of Demailly–Peternell–Schneider the Albanese map
of a compact Kähler manifold with nef tangent bundle is a locally trivial
fibration, and up to taking a finite étale cover, the fibre is a Fano manifold.
Matsumura surveys some recent developments of the theory where one aims
– iii –
at replacing the nefness of the tangent bundle by other conditions (pseu-
doeffectivity, semi-positive holomorphic sectional curvature, nefness of the
relative anticanonical bundle etc).
Acknowledgements
We thank all the authors for their wonderful contributions, as well as the
referees and the scientific committee for their work. A big thank you to Mar-
tine Klughertz and Omar Alehyane for co-organizing an amazing conference
and for providing the nice photos!
Hoang Chinh Lu, Henri Guenancia, Vincent Guedj.
– iv –
– v –


Analysis of Monge–Ampère, a tribute to Ahmed Zeriahi

Guy David

A local description of 2-dimensional almost minimal sets bounded by a curve

Not a member yet

Open problems on structure of positively curved projective varieties