Seminar Algebra und Topologie
Das Seminar findet normalerweise am Mittwoch in F.13.11, 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.
Zock: The strong approximation property for ell-adic local systems
In 2001, Vladimir Drinfeld proved a set of conjectures about semisimple perverse sheaves on complex varieties posed by Masaki Kashiwara. Even though Drinfeld's proof involves a reduction modulo p, his methods are unsuited for proving the analogous conjectures for ell-adic perverse sheaves in positive characteristic. I will present joint work with Moritz Kerz in which we define a new class of ell-adic local systems amenable to methods inspired by Drinfeld's. Assuming a conjecture of Esnault-Kerz, this gives a way to attack the Kashiwara conjectures in positive characteristic.
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.
Vijaylaxmi Trivedi: Numerical characterization of integral dependence and density functions
We recall that two ideals \( I\subset J\) in a commutative Noetherian ring are integrally dependent if they have the same integral closure. Two integrally dependent ideals share many properties in common. This allows us in many situations to replace an ideal by a simpler ideal. A generalization is the notion of integral dependence of modules, which plays an important role in the study of equisingularity.
Integral dependence of ideals is characterized in terms of the well known numerical invariant, namely Hilbert-Samuel multiplicity, provided the ideals are of finite colength in an ambient local ring. Similarly for modules there is the notion of Buchsbaum-Rim multiplicity.
Here we work in a graded situation and give numerical characterizations, for integral dependence of ideals/modules which are not necessarily of finite colength in their ambient local ring (or free module). For this we introduce density Functions which are continuous real valued functions measuring the growth of graded components of ideals/modules on an \(\mathbb{R}\)-scale.
This talk is based on joint work with Suprajo Das and Sudeshna Roy.
Srinivas: Mordell-Weil groups of elliptic fibrations
This talk will give a report on a project with A. Grassi, R. Miranda, K. Paranjape and T. Weigand, studying the Mordell-Weil groups of rational sections of certain elliptic fibrations of complex projective varieties. Our work was motivated by redicted bounds for the ranks of such Mordell-Weil groups for elliptically fibred Calabi-Yau 3-folds, which were based on considerations from physics. We discuss two approaches to such questions, giving rise in particular to proofs of the expected bounds for Calabi-Yau 3-folds, and some related results, and also suggest natural directions for further research.
Zhang: Degrees of Fano quiver moduli
Computing the degree of a Fano variety is often time consuming and can only be done for individual cases. However, when a Fano variety happens to be a quiver moduli space, its degree is entirely encoded in the quiver and the dimension vector. In this talk, I will present the framework in which we can perform such computations, and give a method to compute it in the toric setting. In particular, we will see an explicit degree formula for the bipartite quiver (K(2,q))m with dimension vector \underline{1} and the subspace quiver Sq with dimension vector \underline{d} = (1q;2).
Picard: Construction of étale logarithmic Hodge numbers modulo an integer
The construction of Hodge numbers modulo an integer in positive characteristic was solved by Remy van Dobben de Bruyn and Matthias Paulsen in 2020. An interesting question is to ask whether something similar can also be achieved for the logarithmic Hodge numbers h^{r,q}_{log}, which are defined as the \F_p-vector space dimension of the étale cohomology groups of the sheaf of logarithmic differential forms. It turns out that it is necessary to replace the logarithmic Hodge numbers by a pair of integers (u^{r,q},d^{r,q}), the unipotent and étale logarithmic Hodge numbers, which come from the fact that every cohomology group of the sheaf of logarithmic differential forms is the group of k-rational points of a perfect algebraic group scheme. In this talk, we will see that the étale logarithmic Hodge numbers d^{r,q}(X), for 0<=r<=q<=dim(X) and r+q<=dim(X), can take arbitrary values modulo an integer.
Lorenz: Decomposability of central simple algebras
Let $A$ be a central division algebra over a field $F$ that has order 2 in the Brauer group of $F$. Then $A$ is Brauer equivalent to a tensor product of quaternion algebras, i.e. there are quaternion algebras $Q_1, \ldots, Q_k$ and positive integers $n, m$ such that $A \otimes M_m(F) \otimes \cong Q_1 \otimes_F \ldots \otimes_F Q_k \otimes M_n(F)$. While algebras of small dimension are even isomorphic to a tensor product of quaternion algebras, for higher dimensions indecomposable algebras that are not isomorphic to a product of proper subalgebras exist. In this talk, we will review the structure theory of central simple algebras and discuss their classification for small degrees. Afterwards, we will introduce an invariant for central simple algebras of degree 8 (i.e. dimension 64) that takes values in a certain quotient of a cohomology group (Galois or Kato-Milne cohomology, depending on the characteristic of the base field). We will study how this invariant interferes with the decomposability of $A$.
If time permits, we will further show how this invariant relates to descent problems for central simple algebras and quadratic forms along field extensions of odd degree.
Souly: On strong F-regularity of the zero locus of totally negative quivers moment maps and local Galois deformation rings
We show the strong F-regularity of the zero fiber of moment maps of totally negative quivers, assuming that the number of loops and arrows are large enough. Furthermore, we apply our results to establish the strong F-regularity of the special fibers of all irreducible components of local Galois deformation rings associated to any continuous residual representation, assuming that the base p-adic field is sufficiently large.
Kings: Algebraicity of Hecke L-values and p-adic interpolation
Deligne’s conjecture for special values of Hecke L-functions states roughly that the critical L-values divided by certain periods are algebraic numbers. If one knows moreover integrality of these values one can try to construct p-adic L-functions. For Hecke characters over CM fields and ordinary p these are known results by Shimura, Blasius and Katz.
In recent joint work with Sprang, we generalized these results to critical Hecke L-values over arbitrary fields. Using a new p-adic Fourier theory, which generalizes earlier results by Schneider and Teitelbaum, we can also construct for the first time a p-adic L-function for arbitrary prime numbers p. This was known so far only for Hecke characters over imaginary quadratic fields, by work of Katz and Boxall.
In this talk I will explain these results and some of the ideas going into its proof.
