Seminar Algebra und Topologie

Das Seminar findet normalerweise am Mittwoch in G.15.25, 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.

Zhu: Multiplication on the Real Brown–Peterson Spectrum

The development of brave new algebra was much guided by the problem of finding the full multiplicative structure of the Brown–Peterson spectrum and its truncated variants. While this has seen a thorough study throughout history, the C2-equivariant analogue for the Real Brown–Peterson spectrum has essentially been left completely open. In joint work with Ryan Quinn, we remedy this situation by developing an obstruction theory to lifting structured orientations. Powered by the engine of Wilson space theory, we manage to give the first structured versions of the Real Brown–Peterson spectrum and its truncated cousins.

Martin: The Enriques surface of minimal entropy

Salem numbers appear naturally as dynamical degrees of isometries of hyperbolic lattices and hence in the study of entropy of surface automorphisms. The conjecturally smallest Salem number is Lehmer's number $\lambda_{10}$, which can be realized by automorphisms of K3 surfaces and rational surfaces by work of McMullen. In this talk, I will explain how to generalize a result of Oguiso asserting the non-realizability of $\lambda_{10}$ for automorphisms of Enriques surfaces over the complex numbers to odd characteristics. Then, I will describe the unique counterexample in characteristic 2. This is joint work with Giacomo Mezzedimi and Davide Veniani.

Baudin: On the Euler characteristic of ordinary irregular varieties

Informally, a variety is "irregular" if it is closely related to an abelian variety (that is, a smooth projective variety which also admits the structure of a group). This is for example the case of non-rational curves, which embed in their Jacobian.

Over the complex numbers, several methods gave rise to a remarkable results in this field: characterization of abelian varieties by only fixing a few invariants, deeper understanding of the Euler characteristic of irregular varieties, study of their pluricanonical systems, and so on (in any dimension).

These theorems all rely on analytic techniques, making this whole topic harder to reach in positive characteristic. Our goal in this talk will be to explain purely positive characteristic methods that allow us to "approximate well enough the complex theory", in order to deduce geometric consequences. We will achieve this through presenting the following theorem: if X is a smooth projective ordinary variety of maximal Albanese dimension (i.e. dim(alb(X)) = dim(X)), then the Euler characteristic of the sheaf of top forms is non-negative. If in addition this quantity is zero, then the Albanese image of X is fibered by abelian varieties.

Viehmann: Automorphisms of generic supersingular abelian varieties

In 2001, Oort conjectured that generically on the supersingular locus of the moduli space of principally polarized abelian varieties of some dimension g and in characteristic p, the automorphism group of the universal principally polarized abelian variety consists only of ±1, except for few exceptional pairs (g,p). I will explain this question and its history and how to prove this conjecture.

• Gästeseminare vergangener Semester