Séminaire Arithmétique et géométrie algébrique
organisé par l'équipe Arithmétique et géométrie algébrique
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Tess Bouis
Motivic cohomology of mixed characteristic schemes
11 janvier 2024 - 14:00Salle de séminaires IRMA
I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-A^1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A^1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field. -
Enrica Mazzon
A non-archimedean approach to the SYZ conjecture
18 janvier 2024 - 14:00Salle de séminaires IRMA
The SYZ conjecture concerns degenerations of complex Calabi-Yau manifolds and was proposed as a geometric explanation of mirror symmetry. Kontsevich and Soibelman introduced a non-archimedean approach to this conjecture, and more recently, Yang Li's work has connected the non-archimedean approach with the original SYZ conjecture.
In this talk, I will explain the key concepts of the non-archimedean approach and present recent developments in the context of hypersurfaces. This is based on a project in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey. -
Mirko Mauri
Hodge-to-singular correspondence
25 janvier 2024 - 14:00Salle de séminaires IRMA
We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria. -
Mingmin Shen
A degeneration argument and the integral Hodge conjecture
1 février 2024 - 14:00Salle de séminaires IRMA
An algebraic variety has the special property that its Zariski closure has no boundary in topological sense. I will explain how this can leads to non-algebraicity results via a degeneration argument. Then I will explain how this was recently extended by Kees Kok using refined unramified cohomology. -
Lucas Mann
Duality in p-Adic Cohomology of Rigid Varieties
8 février 2024 - 14:00None
In joint work with Anschütz and Le Bras we study pro-étale Q_p-cohomology on rigid varieties (i.e. "analytic spaces over C_p") and in particular investigate duality results. Examples show that even on proper smooth varieties, Poincaré duality for Q_p-line bundles does not hold in the naive sense and instead requires one to replace the coefficient category by so-called Banach-Colmez spaces. In joint work in progress we construct a general sheaf theory and associated six functors for a "relative" version of BC spaces. This will formally recover many existing duality results for p-adic geometry in the literature. -
Quentin Gazda
Quelques réflexions autour de la conjecture de Zagier — des Fonctions aux Nombres
15 février 2024 - 14:00Salle de séminaires IRMA
La conjecture de Zagier est une certaine formulation du slogan suivant : les relations linéaires entre les polylogarithmes évalués en des nombres algébriques proviennent de relations entre symboles de K-théorie. En identifiant les différents acteurs de cette conjecture, on peut énoncer et démontrer une version analogue en arithmétique des corps de fonctions. Les polylogarithmes classiques sont alors remplacés par ceux de Carlitz. La preuve, très différente des techniques développées jusqu’ici, utilise des ingrédients de la théorie des équations aux différences. Elle fait intervenir des déformations des polylogarithmes de Carlitz où figure une variable t. Dans cet exposé, je présenterai ces résultats, essentiellement fruits d’un travail en commun avec A. Maurischat. S’il on ne dispose pas aujourd’hui de la technologie nécessaires pour reproduire cet argument en théorie des nombres, il est amusant de spéculer sur une hypothétique transcription. Des q-déformations des polylogarithmes remplacent ces «t-déformations». Avec T. Bouis, on a récemment observé un q-Li_1 dans la classe de Chern syntomique introduite par Bhatt—Lurie. Encourageant…! Je mentionnerai ces travaux dans une deuxième partie d’exposé. -
Fangzhou Jin
The limit and boundary characteristic classes in Borel-Moore motivic homology
22 février 2024 - 11:00Salle de séminaires IRMA
Abstract: We define limit and boundary characteristic classes in Borel-Moore motivic homology, and compare them with Aluffi's pro-Chern-Schwartz-MacPherson class and Kato-Saito's Swan class respectively. This is a joint work with P. Sun and E. Yang. -
Marco D'addezio
Cohomologie cristalline bordée
22 février 2024 - 14:00Salle de séminaires IRMA
Je parlerai d'une nouvelle théorie cohomologique pour les variétés algébriques en caractéristique positive, appelée cohomologie cristalline bordée. Il s'agit d'une généralisation de la cohomologie cristalline qui dépend du choix d'une "fonction de décroissance à l'infini". Les fonctions de décroissance linéaires correspondent à des versions entières de la cohomologie rigide, tandis que les fonctions de décroissance logarithmiques produisent la famille conjecturée des théories de cohomologie cristalline à décroissance logarithmique. Au cours de l'exposé, j'expliquerai la construction de cette théorie après un bref rappel des théories classiques de cohomologie cristalline et rigide. -
Annalisa Grossi
Automorphisms of OG10 towards Enriques manifolds
29 février 2024 - 14:00Salle de séminaires IRMA
Automorphisms of HK manifolds have been studied for many different reasons: construct symplectic quotients or study fixed loci in order to find examples of irreducible symplectic varieties, define maps among different deformation families of HK manifolds, find new examples of Enriques manifolds, that are higher dimensional analogue of Enriques surfaces, and for which Pacienza and Sarti recently proved the Morrison-Kawamata cone conjecture. In the first part of the talk, I will show a recent result about symplectic rigidity of HK manifolds of OG10 type. Then I will show how to construct examples of Enriques manifolds considering nonysmplectic automorphisms of a Laza—Saccà—Voisin manifold that are induced by a nonysmplectic automorphism of the underlying cubic fourfold. The talk is based on a joint work with L. Giovenzana, Onorati, and Veniani and on a joint work in progress with Billi, F and L. Giovenzana. -
Johannes Sprang
Irrationality and linear independence of p-adic zeta values
14 mars 2024 - 14:00Salle de séminaires IRMA
The celebrated theorem of Ball-Rivoal shows that there are infinitely many irrational odd zeta values. More precisely, it provides asymptotically lower bounds on the dimension of the Q-vector space spanned by these numbers. Although we are still far from fully understanding the structure of odd zeta values, the corresponding question for p-adic zeta values is even more difficult. For example, the question of the non-vanishing of these numbers is still open today, although it has interesting consequences e.g. for the non-vanishing of p-adic regulators. In this talk we would like to report recent progress on the irrationality and linear independence of p-adic zeta values. -
Vonk Jan
Around the class number one problem
21 mars 2024 - 14:00Salle de séminaires IRMA
As part of a systematic computational study of equivalence classes of binary quadratic forms, Gauss stated several conjectures on their class numbers. In this talk, I will discuss some recent results related to these conjectures. We will begin with the case of negative discriminants, and its intimate connection with the important but challenging problem of finding rational points on modular curves with non-split Cartan level structures. We then turn to the more mysterious case of positive discriminants, where much less is known. The key missing notion is that of singular moduli for real quadratic fields, which is the subject of recent joint work with Darmon. -
Junliang Shen
Cohomology and motives for Hitchin systems
27 mars 2024 - 09:00Salle de séminaires IRMA
In the last a few decades, rich cohomological structures have been found for the topology of the Hitchin system from various perspectives. For example, the work of Ngô reduces the fundamental lemma of the Langlands program to a study of the topology of the Hitchin system; the P=W conjecture (now a theorem) connecting the topology of the Hitchin system to Hodge theory of the character variety via non-abelian Hodge correspondence. I will first discuss some new cohomological structures led by these progress. Then I will explain that tools in these developments further yield a proof of Corti-Hanamura’s motivic decomposition conjecture for the Hitchin system, which states that the decomposition theorem associated with the Hitchin system is induced by algebraic cycles. Based on joint work in progress with Davesh Maulik and Qizheng Yin. -
Yagna Dutta
Twists of intermediate Jacobian fibrations
28 mars 2024 - 14:00Salle de séminaires IRMA
Given an elliptic fibration of a K3 surface, one can reglue the fibres of the elliptic fibration differently to obtain different K3 surfaces. The data of such regluing are dictated either by degree twists or by Brauer twists. A similar story exists for other curves on K3 surfaces. Moving from curves to 3-folds, I will report on a joint work in progress with Mattei and Shinder where we look at the cubic 3-folds obtained as the hyperplane sections of a fixed smooth cubic 4-fold. The relative intermediate Jacobians of the universal hyperplanes induces a well behaved abelian group scheme over the projective space of dimension 5. The total space this time is a hyperKähler manifold of dimension 10. This group scheme gives rise to a Brauer type group associated to the K3-type Hodge structure of the cubic 4-fold. -
Matilde Maccan
Variétés homogènes en petite caractéristique
4 avril 2024 - 14:00Salle de séminaires IRMA
Toute variété homogène, projective et rationnelle peut s’écrire comme quotient d’un groupe algébrique semi-simple par un sous-groupe dit parabolique. Dans cet exposé, on généralisera les résultats de Wenzel, Haboush et Lauritzen en traitant le cas des sous-groupes paraboliques sur un corps algébriquement clos de caractéristique petite (deux et trois), achevant ainsi leur classification en toute caractéristique. Si le temps le permet, on mentionnera quelques consequences géométriques. -
Anna Bot
Real forms and automorphism groups: some open questions
18 avril 2024 - 14:00Salle de séminaires IRMA
A real form of a complex variety X is a real variety whose complexification is isomorphic to X. Many varieties, for example curves or abelian varieties, are known to only have finitely many isomorphism classes of real forms. Recently, there has been some development in the field: now, there are examples of projective varieties (in fact, even surfaces) with infinitely many nonisomorphic real forms, and of affine surfaces with uncountably many. There are still some interesting questions one can pose, especially in connection with the automorphism group of the variety. I will give an overview of the field and relevant notions, and survey some open problems. -
Nirvana Coppola
On perfect powers that are sums of cubes of a nine term arithmetic progression
25 avril 2024 - 14:00Salle de séminaires IRMA
Solving Diophantine equations has always been one of the most fascinating problems in number theory, since even if it can be easily formulated, it almost always requires advanced techniques. In this talk, I will focus on equations that relate sums of powers to perfect powers. After showing some examples that are in the literature, I will discuss the (non-)existence of perfect powers that are sums of cubes of a nine-term arithmetic progression. The proof involves a battery of techniques and both theoretical and computational tools. This is joint work with Mar Curcó-Iranzo, Maleeha Khawaja, Vandita Patel, Özge Ülkem.