Basel-Dijon-EPFL meeting (23-24 March 2023)

Location:  GA 3 21 at the Bernoulli Center (click on the number of the room for a map)


23 March

14:00-15:00 Zsolt Patakfalvi (EPFL)

Title: Varieties with nef anti-canonical have surjective Albanese

Abstract: I present a joint work in progress with Sho Ejiri, where we aim to prove that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism. The statement has been known for a while in characteristic zero, so our proof deals with the positive characteristic case. The previous positive characteristic results had to exclude certain wild positive characteristic behavior: wild action of Frobenius on cohomology, wild singularities of the general fibers over the Albanese image, etc.. Ours is the first result on varieties with nef anti-canonical of any dimension that avoids any such assumption. Producing such results is usually the biggest challenge in positive characteristic algebraic geometry. 

15:00-15:30 coffee

15:30-16:30 Immanuel Van Santen (Universität Basel)

Title: Complements of Hypersurfaces in Projective Spaces

Abstract: This is joint work with Jérémy Blanc and Pierre-Marie Poloni. In this talk we will focus on the complement problem for the projective space P^n: If H, H’ are irreducible hypersurfaces of degree d in P^n such that the complements P^n\H and P^n\H are isomorphic, are the hypersurfaces H, H’ isomorphic?

This problem has been studied intensively for n=2 and we will focus in this talk on the case n > 2. In fact, in our main result, we provide counterexamples for all n, d > 2 provided that (n, d) is not equal to (3, 3) and give partial affirmative answers in case (n, d) = (3, 3) and d < 3. As a byproduct, we show that rational normal projective surfaces admitting a desingularisation by trees of smooth rational curves are piecewise isomorphic if and only if they coincide in the Grothendieck ring. 

16:30-17:00 coffee

17:00-18:00 Clémentine Lemarié-Rieusset (Université de Bourgogne)

Title: The quadratic linking degree

Abstract: In this talk I will present a new application of motivic homotopy theory: motivic knot theory. More specifically, I will present the quadratic linking degree, which is a counterpart in algebraic geometry of the linking number of two oriented disjoint knots (the number of times one of the knots turns around the other knot). To do this, I will recall some notions from knot theory and from quadratic intersection theory, and I will give several examples during my talk.

18:30 Dinner

24 March

8:45-9:00 coffee

9:00-10:00 Giulio Codogni (Università di Roma Tor Vergata)

Title: Isogeny graphs with level structure

Abstract: We study graphs whose vertexes are supersingular elliptic curves with level structure over a field of fixed positive characteristic, and edges are isogeny of fixed degree. Looking at the convenient moduli space in mixed characteristic, we will relate the eigenvalues of the adjacency matrix of the graph to the eigenvalues of a Frobenius on a convenient cohomology group. Combining this with the Weil conjecture, we will obtain a bound on the norm of the eigenvalue of the adjacency graph. This bound shows that the graph is Ramanujian, which, among the other things, means that random walks on it have an optimal mixing time. More generally, we will show that the specturm of the adjacency matrix behaves as the spectrum of the adjacency matrix a random graph. We will also present some applications of these results to cryptography. This is a joint work with Guido Maria Lido.

10:15-10:30 coffee

10:30-11:30 Iván Rosas Soto (Université de Bourgogne)

Title: Étale motives and the Hodge conjecture

Abstract: The Hodge conjecture, with rational coefficients, states that for a smooth projective variety over \mathbb{C} the image of the cycle class map from Chow groups to Betti cohomology is the group of Hodge classes. Although the integral version of the conjecture is false, Rosenschon and Srinivas proved that the étale version of the integral Hodge conjecture, i.e. using Lichtenbaum cohomology groups instead of Chow groups, is equivalent to the Hodge conjecture with rational coefficients. The goal of this talk is to give an overview of étale motivic cohomology and the Hodge conjecture, followed by a revisit to some of the counter-examples to the integral Hodge conjecture, such as the ones of Atiyah-Hirzebruch, Kollar and Benoist-Ottem, from an étale motivic point of view.

11:30-12:00 coffee

12:00-13:00 Jefferson Baudin (EPFL)

Title: On the Albanese morphism of varieties of Kodaira dimension zero in positive characteristic

Abstract:  Generic vanishing techniques have proven to be very useful in understanding the geometry of complex irregular varieties (i.e. those with a non-zero holomorphic global 1-form). One example of this is the proof of Ueno’s conjecture K: let X be a smooth complex variety with Kodaira dimension zero, then its Albanese morphism is an algebraic fiber space, and the Kodaira dimension of the general fiber is zero (in particular, Iitaka’s subadditivity conjecture holds for this fibration).

The goal of this talk is to explain this theory in positive characteristic, and give some hints on how these techniques can be used to prove Ueno’s conjecture in this setup.