Séminaire Arithmétique et géométrie algébrique
organisé par l'équipe Arithmétique et géométrie algébrique
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Livia Grammatica
Tate-linear formal varieties
15 janvier 2026 - 14:00Salle de séminaires IRMA
We work over a closed field of positive characteristic. A classic result of Serre-Tate says that the deformation space of an ordinary abelian variety has the structure of a formal torus, and one can consider the closed subvarieties which are given by formal subtori. Tate-linear formal varieties play the role of formal subtori in the deformation space of abelian varieties of arbitrary Newton polygon. Recent work of Chai-Oort established an important link between Tate-linear subvarieties and the Hecke orbit conjecture for \mathcal{A}_g, which then led to a full solution for Shimura varieties of Hodge type by D'Addezio and van Hoften. We will explain the role of Tate-linear varieties in the Hecke orbit conjecture, their conjectural link with special subvarieties of \mathcal{A}_g, and show how p-adic monodromy techniques can help shed light on their structure. -
Lin Zhou
On the infinite generation of morphic and motivic cohomology
22 janvier 2026 - 14:00Salle de séminaires IRMA
Since Mumford’s work in the 1960s, questions on the finite generation of Chow and Griffiths groups—such as the finiteness of dimension, the size of their torsion, or their divisibility—have been a central theme in the study of algebraic cycles. Motivic cohomology and morphic cohomology naturally generalize the Chow group (cycles modulo rational equivalence) and cycles modulo algebraic equivalence. In this talk, I will show how, over an algebraically closed field whose transcendence degree over its prime field is infinite (e.g., the complex number field), one can combine Schoen’s injectivity argument with Schreieder’s refined unramified cohomology to construct examples of motivic and morphic cohomology groups with infinitely many torsion elements. These appear to be the first known examples exhibiting infinite torsion in motivic or morphic cohomology. Joint work with Theodosis Alexandrou. -
Yuanyang Jiang
Locally analytic completed cohomology of Hilbert modular varieties
29 janvier 2026 - 14:00Salle de séminaires IRMA
The property of a Galois representation being de Rham is designed to capture the property of being of geometric origin. As an example, we have the following conjecture about classicality: a de Rham Galois representation that arises from a p-adic Hilbert modular form (i.e. inside the "completed cohomology" of Hilbert modular varieties) should actually arise from a classical Hilbert modular form. Following an idea of Lue Pan of realizing the Fontaine operator geometrically, we find a cohomological description of the property of being de Rham. We relate the relevant cohomology to classical Hilbert modular forms via a new type of locally analytic Jacquet-Langlands correspondence. As a corollary, we can deduce some cases of the Langlands-Clozel-Fontaine-Mazur conjecture. -
Jean Douçot
Transformée de Fourier des données de Stokes de connexions irrégulières
5 février 2026 - 14:00Salle de séminaires IRMA
Par la correspondance de Riemann-Hilbert, les connexions à singularités régulières sur les courbes sont caractérisées par leur monodromie. Cette description topologique des connexions admet une vaste généralisation au cas des singularités irrégulières, faisant intervenir des données de monodromie généralisées appelées données de Stokes. Par ailleurs, il existe une notion de transformée de Fourier pour les connexions irrégulières sur la droite projective complexe : celle-ci agit de manière non-triviale, modifiant le rang, le nombre de singularités, et l'ordre des pôles des connexions. Cela soulève la question de décrire directement l'action de la transformée de Fourier au niveau des données de Stokes. Dans cet exposé, je vais présenter un travail en commun avec Andreas Hohl, qui donne une méthode topologique pour déterminer explicitement les données de Stokes de la transformée de Fourier dans une nouvelle classe de cas, reposant sur des travaux de T. Mochizuki. En particulier, cela fournit des isomorphismes explicites entre les variétés de caractères sauvages correspondantes, qui conjecturalement sont compatibles avec leur structure symplectique. -
Matteo Verni
Galois covers between Calabi-Yau varieties
12 février 2026 - 14:00Salle de séminaires IRMA
The birational geometry of smooth projective complex varieties with trivial canonical bundle is a deep and intensively studied subject. While there has been a lot of work on birational maps between such varieties (for example, on birational automorphisms of Hyper-Kähler manifolds), less has been said about rational maps of degree at least two, which we will call rational covers. What restrictions do rational covers Y --> X (and especially their monodromy) impose on the geometry of X, when both X and Y have trivial canonical bundle? For a given X, when does there exists such a cover which is furthermore Galois?
In this talk, we will present and answer various questions around this theme, with a particular interest in the case where X is Hyper-Kähler. The main motivation is the following question of Laza: can it be that any Hyper-Kähler deforms to one birational to A/G, where A is an abelian variety and G a finite group acting on it? -
Abhishek Oswal
p-adic hyperbolicity of Shimura varieties
19 février 2026 - 14:00Salle de séminaires IRMA
A classical result of Borel states that a holomorphic map from a product of punctured disks into a Shimura variety (with torsion free level structure) extends across the punctures to a holomorphic map into the Baily-Borel compactification. As a consequence, all complex analytic maps from complex algebraic varieties into such Shimura varieties are algebraic. In this talk, I will report on joint work with Bakker, Shankar and Yao where we prove a p-adic analog of this algebraization and extension result. This builds on earlier joint work with Shankar, Zhu and Patel. -
Alessio Bottini
The period-index problem for hyper-Kähler varieties
5 mars 2026 - 14:00Salle de séminaires IRMA
The period-index conjecture is a fundamental problem concerning the Brauer group of algebraic varieties. For hyper-Kähler varieties, whose (birational) geometry is controlled by the second cohomology, it is expected that a stronger form of this conjecture holds. In this talk, I will present joint work with Daniel Huybrechts that provides new evidence for this expectation.
Following a brief introduction to the problem, I will discuss a proof of a variant of the conjecture where the classical index is replaced by a Hodge-theoretic one. Then, I will explain how to verify the conjecture for most Brauer classes on hyper-Kähler varieties of K3n-type. -
Enrico Fatighenti
Modular vector bundles on hyperkähler manifolds
12 mars 2026 - 14:00Salle de séminaires IRMA
We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3^[2]-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. Interestingly enough, these constructions are almost never infinitesimally rigid, and more precisely we show how to get (infinitely many) 20 and 40 dimensional families. This is a joint work with Claudio Onorati. Time permitting, I will also present a joint work with Alessandro D'Andrea and Claudio Onorati on a connection between discriminants of vector bundles on smooth and projective varieties and representation theory of GL(n). -
Cécile Gachet
Equivariant descent for the birational finiteness properties of certain Calabi—Yau pairs
19 mars 2026 - 14:00Salle de séminaires IRMA
In dimension 3 and higher, it is well-known that certain singular complex projective varieties do not admit a unique minimal resolution of singularities. Typically, there are small birational modifications which allow to toggle back and forth between different minimal models of the same variety. This framework is particularly well-understood for Calabi—Yau pairs, whose minimal models are connected by finite sequences of so-called flops. Some finite sequences of flops loop, and thereby define non-trivial birational automorphisms on one model; to that extent, it is not uncommon for a Calabi-Yau pair to have infinitely many marked minimal models. It is however conjectured that a klt Calabi—Yau pair has finitely many unmarked minimal models. As the class of klt Calabi—Yau pairs is naturally closed under quotients by finite group actions, it is reasonable to expect birational finiteness properties to descend under finite quotient. In that spirit, this talk presents a descent result for birational finiteness properties of a large class of varieties, both under the action of a finite group and under the action of the Galois group of a perfect field. We will provide examples and applications along the way.