Seminar Algebra and Topology

The seminar usually takes place Wednesdays in F.13.11, 16:30 - 17:30.
Involved professors: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.

 

Marco D'Addezio: Injectivity failure in crystalline comparisons

I will report on a recent work with Daniel Caro, where we study the de Rham-to-crystalline comparison map for affine smooth schemes over the Witt vectors. We answer a question of Esnault-Kisin-Petrov by showing that this comparison map can fail to be injective, even in the presence of good compactifications. We identify an obstruction by looking at the slopes of the Frobenius action on crystalline cohomology. On the positive side, we prove that injectivity holds in cohomological degree 1 and for certain subspaces defined by slope conditions. In this talk, I will explain how the F-gauge structure of crystalline cohomology yields the relevant slope bounds. I will further explain how these techniques can be used to determine algebraic de Rham cohomology modulo torsion in terms of the slopes of rigid cohomology of the special fibre.

Krishna Kumar Madhavan Vijayalakshmi: Relative A^1-Contractibility of Smooth Schemes and Exotic Motivic Spaces

One of the emerging problems in motivic homotopy theory is to uniquely characterize the affine n-space A^n among smooth A^1-contractible (affine) schemes. In this talk, we shall learn the background of this problem and show that A^1-contractibility is strong enough to capture the aforementioned uniqueness in relative dimensions up to 2 over a base scheme. This is a joint work in progress with Paul Arne Østvær and Adrien Dubouloz. Following this, we illustrate the theory in higher dimensions by the family of Koras--Russell prototypes, smooth affine varieties that are A^1-contractible over a field of characteristic zero. We shall further extend their A^1-contractibility over a Noetherian scheme. A foundational consequence of this is the existence of "exotic spheres" in motivic homotopy theory in all dimensions >2. Preprint available at https://krishmv.github.io/research.html (arxiv: 2510:21594)

Christopher Deninger: Rational Witt vectors

We discuss various aspects of rational Witt vectors, in particular their connections with K-theory and algebraic cycles and their sheaf theoretical properties. We also explain how they can be used to construct dynamical systems attached to arithmetic schemes.

Christian Lehn: Tannakian groups of perverse sheaves and E6 geometry

In a joint work with Krämer and Maculan, we prove that the Fano surfaces of lines on smooth cubic threefolds are the only smooth (less than half-dimensional) subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. The proof uses an upgrade of the Krämer-Weissauer formalism from perverse sheaves to Hodge modules, the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi, an intensive computer search, and the geometry of lines on cubic surfaces.

Sebastian Bartling: Close local fields and fundamental lemmas

The idea of close local field goes back to work of Krasner, Deligne and Kazhdan and says that as ramification of p-adic fields grows to infinity, their theory approximates the theory of equal characteristic local fields. I want to explain how one can use this philosophy to put many moduli problems appearing in (local) arithmetic geometry (Rapoport-Zink spaces, Wittvector affine Grassmannians, Affine Deligne Lustzig varieties, Affine Springer fibers...) into profinite families. As applications one can deduce the missing function field case of the following statements: Wei Zhang's Arithmetic fundamental lemma, the base change fundamental lemma and remaining small p cases of the standard endoscopic fundamenta lemma as proven by Ngo. This is joint work with Andreas Mihatsch and also with Kazuhiro Ito.

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