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. -
Andrea Gallese
How to compute the connected monodromy field of a CM abelian variety
2 avril 2026 - 14:00Salle de séminaires IRMA
Let A be an abelian variety defined over a number field k. The connected monodromy field k(eA) is the minimal extension of k over which every \ell-adic Galois representation attached to A has connected image, or equivalently, the minimal extension over which all Tate classes on all self-products A^r are defined. When k(eA)/k(End A) has positive degree, "exotic" Tate classes arise on certain powers A^r — classes not explained by the endomorphism algebra alone. I will explain how to compute k(eA) when A is the Jacobian of a curve with complex multiplication. We will exploit CM theory to describe the algebra of Tate classes and make the Galois action on this algebra explicit in terms of periods — suitable integrals of algebraic differential forms. Though periods are generally transcendental, those attached to Tate classes are algebraic, so computing k(eA) reduces to identifying these periods as exact algebraic numbers. -
German Stefanich
Higher algebraic geometry
9 avril 2026 - 14:00Salle de séminaires IRMA
The goal of this talk is to describe joint work with Scholze where we study a version of algebraic geometry which is built, not out of spectra of commutative rings, but out of spectra of symmetric monoidal higher categories. The resulting higher geometry contains the usual category of qcqs schemes, but also provides a home to new and interesting objects which cannot be studied with more classical means. Time permitting, I will sketch the role that some of these objects play in ongoing work joint with Ben-Zvi and Nadler on various Langlands duality statements in the context of three dimensional topological field theory. -
Eric Chen
Langlands functoriality of Hitchin systems
16 avril 2026 - 14:00Salle de séminaires IRMA
Traditionally, Langlands functoriality refers to the identification of automorphic forms whose parameters take a special shape. In this talk, we explain how to ask analogous questions on the Hitchin moduli space using perspectives from the relative Langlands program. We gain, in this setting, the advantage of working with a version of Langlands duality which is readily computable, and if time permits we will discuss the ramifications of these calculations for automorphic periods and L-functions. -
Ivan Rosas-Soto
Etale motivic cohomology of (some) Fano fivefolds
30 avril 2026 - 14:00Salle de séminaires IRMA
In 2016, Rosenschon and Srinivas proved that, for a smooth, projective, complex variety X, the Hodge conjecture for X with rational coefficients is equivalent to an integral version if we replace the Chow groups with the étale motivic cohomology groups. The natural question that arises is whether an improvement can be obtained in the setting of classical motives with integral coefficients by replacing Chow groups with étale motivic cohomology. In this talk, I will begin by presenting results concerning the existence of projectors for integral étale motives over non-algebraically closed fields. I will then come back to the complex case and discuss the integral Hodge conjecture of a family of Fano fivefolds, as well as the comparison map between their Chow and étale motivic cohomology groups. This is based on joint work with Pedro Montero. -
Mattia Morbello
Compactification du feuilletage Painlevé V
7 mai 2026 - 14:00Salle de séminaires IRMA
Les connexions du type Painlevé V sont une classe particulier de connexions irrégulières de rang deux sur la sphère de Riemann, avec un précise diviseur des poles. Leur espace des modules est doté d'un feuilletage en courbes dont les feuilles contiennent les connections avec même monodromie. Ce feuilletage est induit par une équation différentielle classique, l'équation Painlevé V. Le but de cet exposé est de présenter une compactification de l'espace des modules qui nous permettra d'étudier le comportement asymptotique des feuilles isomonodromiques sur les composants du bord. -
Andrea Ricolfi
Moduli spaces of semiorthogonal decompositions
21 mai 2026 - 14:00Salle de séminaires IRMA
The bounded derived category of coherent sheaves on a smooth projective variety X is a sensible and somewhat subtle invariant of X. Its study is tightly related to rationality problems, MMP, Mirror Symmetry, Enumerative Geometry. Semiorthogonal decompositions (SODs) are a gadget allowing one to "decompose" this category into smaller pieces. Proving the very existence of SODs is often a delicate question. In this talk we shall explain how to construct a "moduli space of SODs" attached to a smooth proper morphism of schemes; we will also discuss its main properties, and how to use it to detect indecomposability of derived categories of some smooth projective varieties. Joint work with Pieter Belmans and Shinnosuke Okawa.