Séminaire Arithmétique et géométrie algébrique
organisé par l'équipe Arithmétique et géométrie algébrique
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Marco Robalo
HKR theorems and exponentials
3 avril 2025 - 14:00Salle de séminaires IRMA
In this talk we will explain a computation describing Hochschild-Kostant-Rosenberg isomorphism theorems as exponential maps. This computation uses the construction of a filtered circle obtained in collaboration with Moulinos and Toën. As two independent applications we will discuss a definition of motivic Donaldson-Thomas invariants in positive characteristic and the extension of Hochschild homology for elliptic curves. -
Andrea D'agnolo
An overview of Riemann-Hilbert
7 avril 2025 - 13:30Salle de conférences IRMA
The Riemann-Hilbert problem requires the existence of regular linear ordinary differential equations with prescribed singularity and monodromy. In higher dimensions, Deligne formulated it as a correspondence between regular meromorphic flat connections and local systems. Later, Kashiwara generalized it to a correspondence between regular holonomic D-modules and perverse sheaves on a complex manifold. Removing the regularity condition, Deligne and Malgrange proposed a correspondence between meromorphic connections on a curve and local systems endowed with a Stokes filtration. A few years ago, in joint work with Kashiwara, we dealt with general holonomic D-modules in any dimension. The target category of our correspondence is built on Kashiwara-Schapira's theory of ind-sheaves and Tamarkin's work on symplectic geometry. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya. In this lecture, I will give a brief overview of the above. -
Hélène Esnault
À déterminer
11 avril 2025 - 14:00Salle de séminaires IRMA
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Hélène Esnault
Vanishing at the generic point in cohomology
11 avril 2025 - 14:00Salle de séminaires IRMA
(1.5-hour talk) If X is a smooth projective variety defined over the field of complex numbers, its i-th Betti cohomology H^i(X, \mathbb C) is said to have coniveau one if there is a Zariski dense open U \subset X such that the restriction map H^i(X,\mathbb C) \to H^i(U,\mathbb C) dies. Equivalently, the restriction map to the generic point of X in i-th cohomology vanishes. Grothendieck’s generalized Hodge conjecture is in general difficult to express as one needs the notion of Hodge sub-structure, but one particular instance has a purely algebraic formulation. It predicts that if X has no non-trivial global differential forms of degree i, then H^i(X, \mathbb C) should have coniveau one. The converse is easily seen to be true. Aside of i=1,2, for which complex Hodge theory gives a positive answer, we know nothing. On the other hand, the philosophy behind is very useful to draw analogies, e.g. it helps to find rational points over finite fields of rationally connected varieties (Lang-Manin conjecture). So it is worth to try to understand whether more modern p-adic methods yield some non-trivial information. With Mark Kisin and Alexander Petrov, in work in progress, we formulate and prove a vanishing result in the separate quotient of p-completed de Rham cohomology, and a weaker version in the separate quotient of prismatic cohomology. I’ll present the ‘program’ and a few questions which at present are not understood. -
Matteo Penegini
Arithmetic Zariski multiplets of irreducible plane curves
24 avril 2025 - 14:00Salle de séminaires IRMA
Multicanonically embedded surfaces in projective space give rise to irreducible branch curves via projection from generic axes. Building on our previous work, we transfer results from the moduli space of surfaces to equisingular strata of plane curves. For instance, the faithful action of the Galois group on the connected components of the moduli spaces of surfaces isogenous to a product, as established by Bauer, Catanese, and Grunewald, gives rise to many arithmetic Zariski multiplets. This is a joint work with M. Loenne.
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Pol Van Hoften
À déterminer
28 mai 2025 - 14:00Salle de séminaires IRMA
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Philip Severin
TBA
12 juin 2025 - 14:00Salle de séminaires IRMA