Scientific Committee
Comité scientifique
Sébastien Boucksom (CNRS, Sorbonne Université)
Stéphane Druel (CNRS, Université Claude Bernard Lyon 1)
Andreas Höring (Université Nice Sophia Antipolis)
Susanna Zimmermann (University of Basel)
Organizing Committee
Comité d’organisation
Benoît Cadorel (Université de Lorraine)
Thibaut Delcroix (Université de Montpellier)
Enrica Mazzon (Université Paris Cité)
IMPORTANT WARNING: Scam / Phishing / SMiShing ! Note that ill-intentioned people may be trying to contact some of participants by email or phone to get money and personal details, by pretending to be part of the staff of our conference center (CIRM). CIRM and the organizers will NEVER contact you by phone on this issue and will NEVER ask you to pay for accommodation/ board / possible registration fee in advance. Any due payment will be taken onsite at CIRM during your stay.
The conference aims at bringing together algebraic and complex geometers from France and abroad, to exchange on recent major developments in the field, and to introduce young researchers to this community. The program will consist of four mini-courses, eight research talks and a lightning talks session.
Topics will cover the broad spectrum of algebraic and complex geometry, from the Hodge theoretic aspects of singularities, through birational geometry over non algebraically closed fields, to the geometric analysis techniques of pluripotential theory. The research talks will be selected by the Scientific Committee on the basis of proposals from participants to present the most recent research work, and the lightning talks will offer young researchers the opportunity to present their first works. The conference is organized by the French research group Géométrie Algébrique et Singularités with the aims of federating complex and algebraic geometers, promoting research in this field, and fostering international interactions.
La conférence a pour but de rassembler les géomètres algébristes et complexes de France et de l’étranger, d’échanger sur les développements récents dans le domaine, et d’aider les jeunes chercheurs à s’intégrer dans cette communauté. Le programme comprendra quatre mini-cours, huit exposés de recherche et une session d’exposés courts.
Les sujets abordés couvriront le large spectre de la géométrie algébrique et complexe, de la théorie de Hodge appliquée aux singularités jusqu’aux techniques d’analyse géométrique de la théorie du pluripotentiel, en passant par la géométrie birationnelle sur des corps non algébriquement clos. Les exposés de recherche seront sélectionnés par le comité scientifique sur proposition des participants afin de présenter les travaux de recherche les plus récents, et les exposés courts offriront aux jeunes chercheurs l’occasion de présenter leurs premiers travaux. La conférence est organisée par le réseau thématique français Géométrie Algébrique et Singularités dans le but de fédérer les géomètres complexes et algébriques, de promouvoir la recherche dans ce domaine et de favoriser les interactions internationales.
MINI-COURSES
Eleonora Di Nezza — Introduction to pluripotential theory
Biography
Eleonora Di Nezza is a leading researcher in pluripotential theory and complex geometry.
Principal Investigator of an ERC grant on Singular Monge-Amp`ere equations, she is a Professor at Sorbonne Université and Ecole Normale Supérieure, as well as a Junior member of Institut Universitaire de France. She has received numerous prizes and honors for her work, such as the Médaille de Bronze du CNRS, the Prix Reine-Elizabeth g´en´eral veuve Le Conte of Académie des Sciences, and the Premio Bartolozzi. A major focus of her work has been on developing the pluripotential theory to allow to study Monge-Ampère equations in big cohomology classes, as well as Monge-Ampère equations with prescribed singularity type [10, 9, 8, 11].
Abstract
Fifty years ago, Aubin and Yau revolutionized the field of complex geometry by solving Calabi’s conjecture [31]. The conjecture was on the existence of Kähler-Einstein metrics, that is, Kähler metrics whose Ricci curvature is constant, or more generally on prescribing the Ricci
curvature of Kähler metrics in a given cohomology class on a compact Kähler manifold. Yau used the translation of this problem into a geometric PDE, the complex Monge-Amp`ere equation, and developed new a priori estimates for solutions to such equations. This approach is suited to smooth manifolds, but to tackle singular complex varieties, complex geometers were led to introduce weak formulations of Monge-Ampère equations in this setting [14]. The resulting pluripotential theory, taking its roots in the earlier works of Bedford and Taylor in the local case [1], is now a major asset in complex geometry, and has been developed to deal with other degenerate settings such as big cohomology classes (instead of Kähler), metrics with prescribed singularities etc.
Mirko Mauri — P=W conjecture
Biography
Mirko Mauri is Chaire de Professeur Junior at IMJ in Paris. Large part of his work focuses on birational geometry and Hodge theory of Calabi–Yau and symplectic varieties. Among his results, he proved, with Felisetti, the first version of the P=W conjecture for singular Betti moduli spaces [15] and, with Mazzon and Stevenson, investigated its geometric and non-Archimedean
significance [22]. He is currently working on a program to better understand the decomposition theorem for the Hitchin fibration, the key geometric input for defining the P-side of the conjecture [24, 23, 12].
Abstract
Nonabelian Hodge theory gives a real analytic isomorphism between two algebraic varieties of very different origins associated with a Riemann surface: the Betti moduli space, which parametrizes representations of the fundamental group of a curve into a complex reductive group G, and the Dolbeault moduli space, which parametrizes G-Higgs bundles over the curve.
This correspondence has remarkable applications, such as constraining which groups can appear as fundamental groups of compact Kähler manifolds [30] or proving the abundance conjecture for Calabi–Yau varieties: if the canonical bundle of a compact Kähler manifold has degree zero when restricted to any subcurve, then it must be torsion.
Despite its significance, the correspondence is transcendental and somewhat mysterious. The P=W conjecture seeks to clarify its behavior at least at the cohomological level. In 2006, Hausel and Rodriguez-Villegas discovered an unexpected symmetry in the cohomology of Betti moduli spaces [17]. In 2010, de Cataldo, Hausel and Migliorini proposed a conjectural relation between the cohomology of these spaces, the so-called P=W conjecture, explaining the curious property on the Betti side in term of a standard symmetry of the Dolbeault side [13]. In a short span of time between 2022 and 2023, three independent proofs of the conjecture emerged [16, 21, 20] – evident signs of the great interest around the conjecture. Today, the P=W paradigm continues to inspire the study of more general integrable systems; see for instance [18, 25].
Mircea Mustață — Singularities and Hodge theory
Bibliography
Mircea Mustață is a leading researcher in the theory of D-modules and the interplay between these objets and the theory of singularities in birational geometry. He was an invited speaker at the ECM in 2004, and gave lectures at the ICM twice in 2006 and 2014. In the recent few years, he has devoted a lot of effort to the study of the minimal exponent and the Bernstein-Sato polynomial [7, 5, 6]
Abstract
Since Saito’s pioneering work [27, 26], we know that a singular algebraic hypersurface gives rise on the ambient manifold to a Hodge module, which provides in turn a lot of information on the possible singularities of the given hypersurface. The natural setting for all this story is the theory of D-modules, that was used earlier by Kashiwara and Mebkhout to extend the famous Riemann-Hilbert correspondance between local systems and flat holomorphic vector bundles, to a more general (singular) setting. We now know that the invariants arising from Saito’s theory can be used to refine substantially a lot of important and classical constructions employed to study singularities in algebraic geometry. For example, the theory of Hodge ideal sheaves offers a generalization of the more classical multiplier ideal sheaves, which have proved to be of great usefulness to quantify the “complexity” of a given singularity. Similarly, the minimal exponent is a numerical invariant refining the log-canonical threshold. Studying the properties of this invariant has led many important recent developments (e.g. behaviour under restriction to a submanifold, semicontinuity in families, description of this invariant in terms of birational geometry…)
Julia Schneider — Cremona groups
Biography
Julia Schneider is a rising young researcher in birational geometry, who has just taken up a position of Charg´ee de Recherche CNRS at Universit´e de Bourgogne. She completed her PhD thesis under the supervision of Jeremy Blanc at University of Basel in 2020, followed by postdocs at the University of Toulouse, EPFL and the University of Zurich. Her research focuses on the birational geometry of surfaces over non-algebraically closed, even sometimes non-perfect fields [2]. Among her results, in addition to the major achievements mentionned in the previous paragraph, she classified maximal algebraic subgroups of the Cremona group of rank two over perfect fields with Susanna Zimmerman [29], and exhibited new normal subgroups of the Cremona group of rank two in [28].
Abstract
A birational transformation between algebraic varieties is a map which is locally described by rational functions (quotients of polynomials), and which is an isomorphism over a dense open set. Birational geometry is the study of algebraic varieties up to birational transformations, it has been at the center stage of algebraic geometry with the development of Mori’s program, the Minimal Model Program. The Cremona group of rank n is the group of birational transformations of the projective space of dimension n, and as such, it contains all the birational geometry of rational varieties of dimension n. It is also a group in the classical sense, which motivated several natural questions: is it a simple group? does it admit a nice set of generators?
These questions were extremely hard to answer, but in the recent years there has been decisive progress, by using deep techniques from the minimal model program. Blanc, Lamy and Zimmermann [3] proved that the Cremona group over a field of characteristic zero is not simple. Lamy and Schneider [19] proved that the Cremona group of rank two is generated by involutions over any perfect field. In rank four or higher, this is no longer the case as shown by Blanc, Schneider and Yasinsky who further showed that these Cremona groups admit surjective homomorphisms to any group of cardinality less than the cardinality of C [4].
RESEARCH TALKS
To be announced
SHORT TALKS
To be announced
SPONSORS
