Bielefeld Algebraic and Arithmetic Geometry

Wednesday 28 June, 2023
15:15	U2-200	Tobias Kreutz (Bonn) Motivic and arithmetic aspects of exceptional loci [Abstract] Given a family of smooth projective varieties defined over a number field, one often studies the Hodge locus (resp. the l-Galois exceptional locus), defined as the set of points in the base where exceptional Hodge (resp. l-adic Tate) classes occur in the cohomology of the fiber. The Hodge and Tate conjectures state that Hodge and Tate classes should be classes of algebraic cycles, and therefore make the following predictions about these loci: Both loci should in fact be equal, and moreover form a countable union of algebraic subvarieties of the base. In this talk, I want to discuss some recent progress on these questions: Firstly, we prove unconditionally that the l-Galois exceptional locus is indeed a countable union of algebraic subvarieties. The corresponding result for the Hodge locus, obtained by Cattani-Deligne-Kaplan in 1995, is often viewed as important evidence in favor of the Hodge conjecture. In addition, under a mild condition on the generic Mumford-Tate group, we prove that the expected equality between the Hodge locus and the l-Galois exceptional locus holds true if one restricts to their positive dimensional parts.
16:45	U2-200	Matthew Honnor (Imperial College) Formulas for Brumer—Stark units [Abstract] In the 1980's, Tate stated the Brumer--Stark conjecture, which conjectures the existence of an algebraic unit called the Brumer--Stark unit whose p-order relates to the value of a partial zeta function at 0. We begin by introducing the conjecture and recent progress by Dasgupta--Kakde. Before focusing on three formulas, of Dasgupta and Dasgupta--Spieß, which have been conjectured for the Brumer--Stark unit. We report on joint work with Samit Dasgupta which shows that these three formulas are equal. To finish, we give some consequences of our result.
Wednesday 14 June, 2023
15:15	U2-200	Alexander Efimov (Bonn) Localizing motives and their applications [Abstract] I will give an overview of my recent new results about the category of localizing motives -- the target of the universal localizing invariant (commuting with filtered colimits) of stable infinity-categories over some base ring. In particular, I will explain the most striking property of this category: it is rigid in the sense of Gaitsgory and Rozenblyum (considered as a "large" symmetric monoidal category). The proof of rigidity in fact also allows to effectively compute the morphisms between the motives of categories (a purely algebraic version of Kasparov's KK-theory and K-homology). As an application, we obtain corepresentability results for some classical invariants, such as TR and topological cyclic homology (previously it was only known that non-connective K-theory is corepresented by the unit object). These computations are closely related with the comparison theorem between the two approaches to K-theory of formal schemes: the classical continuous K-theory is equivalent to the K-theory of nuclear modules, defined by Clausen and Scholze. I will also explain how the rigidity of the category of motives gives refined versions of negative cyclic homology and periodic cyclic homology, and some potential applications of these invariants.
16:45	U2-200	Alexander Petrov (Bonn) Decomposability of the de Rham complex in positive characteristic [Abstract] Abstract: Deligne and Illusie proved that if a smooth variety over F_p admits a lift over Z/p^2 then the truncation of its de Rham complex in degrees <p is quasi-isomorphic to the direct sum of its cohomology sheaves. As a consequence, the Hodge-to-de Rham spectral sequence of a smooth proper liftable variety degenerates, provided that the dimension of the variety is ≤ p. However, further truncations of the de Rham complex of a liftable variety need not be decomposable. I will describe the obstruction to decomposing the truncation of the de Rham complex in degrees ≤p in terms of other invariants of the variety, and will use this to give an example of a smooth projective variety over F_p that lifts to Z_p but whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. The proof relies on the existence of prismatic cohomology, but the key argument is a computation in homotopical algebra, motivated by a construction of Steenrod operations on cohomology of topological spaces.
Wednesday 31 May, 2023
15:15	U2-200	Xianyu Hu (Bielefeld) Equivariant Kuznetsov component of special cubic hypersurfaces [Abstract] TBA
16:45	U2-200	Emanuel Reinecke (Bonn) Unipotent homotopy theory of schemes [Abstract] In this talk, I will present a notion of unipotent homotopy theory for schemes, which is based on Toen's work on affine stacks. I will discuss some general properties of the resulting unipotent homotopy group schemes and explain how over a field of characteristic p>0, they often recover the unipotent completion of the Nori fundamental group scheme, p-adic etale homotopy groups, and Artin-Mazur formal groups. As examples, we will see computations in the case of curves, abelian varieties, and Calabi-Yau varieties. Joint work with Shubhodip Mondal.
Wednesday 17 May, 2023
14:15	U2-200	Amnon Neeman (Australian National University) Analytic ideas applied to triangulated categories, III [Abstract] TBA
Wednesday 10 May, 2023
15:15	U2-200	Patrick Bieker (Bielefeld) Integral models of moduli spaces of shtukas with deeper Bruhat-Tits level structures [Abstract] Integral models of Shimura varieties have been constructed by various people, culminating in the construction of integral models for Shimura varieties of abelian type with parahoric level by Kisin-Pappas. The purpose of this talk is to explain how to construct integral models for moduli spaces of shtukas - analogues of Shimura varieties over function fields - also for deeper Bruhat-Tits level structures. These integral models admit proper, surjective and generically étale level maps as well as a natural Newton stratification. Moreover, in the Drinfeld case they have a moduli interpretation in terms of Drinfeld level structures.
16:45	U2-200	Javier Carvajal-Rojas (EPFL and KU Leuven) Centers of F-purity and their behavior under finite covers [Abstract] Let X be an F-split variety (roughly, a strong form of ordinary log Calabi—Yau variety) and Y/X be a finite cover such that the F-splitting of X lifts to Y. How do the spectra of centers of F-purity (roughly, arithmetically tame log-canonical centers) of Y and X relate to one another? This turns out to be a very tricky question due to wild ramification. An optimal answer would look like Cohen—Seidenberg-type theorems for spectra of centers of F-purity. In this talk, I will report on work in progress with A. Fayolle (U. of Utah) where we address this question.
Wednesday 3 May, 2023
14:15	U2-200	Amnon Neeman (Australian National University) Analytic ideas applied to triangulated categories, II [Abstract] TBA
Wednesday 26 April, 2023
14:15	U2-200	Amnon Neeman (Australian National University) Analytic ideas applied to triangulated categories, I [Abstract] TBA
Wednesday 19 April, 2023
14:15	U2-200	Amnon Neeman (Australian National University) A one-hour introduction to the basics of derived and triangulated categories [Abstract] TBA
Wednesday 18 January, 2023
15:00	T2-220	Sabrina Pauli (Essen) A quadratically enriched Bézout theorem for tropical curves [Abstract] In enumerative geometry one counts the number of solutions to geometric questions. Classically, this is done over an algebraically closed field. Results from motivic homotopy theory allow to study these questions over an arbitrary field k and get a meaningful answer. One powerful tool to solve questions in enumerative geometry is the use of tropical methods: ``Tropicalization" turns algebraic varieties into polytopes and allows to solve these questions using merely combinatorics. In my talk I will present of one of the first examples of how to use tropical geometry for questions in enumerative geometry over k, namely a proof of the quadratically enriched Bézout's theorem for tropical curves.
16:45	T2-213	Mauro Varesco (Bonn) The Hodge conjecture for powers of K3 surfaces. [Abstract] In this talk, I will present some techniques that have proven to be effective in the study of the Hodge conjecture for powers of K3 surfaces. I will recall the construction of Kuga-Satake varieties and how it can be used to prove the Hodge conjecture for powers of K3 surfaces in some four-dimensional families. This talk is based on the paper [V]. [V] Varesco, M.: The Hodge conjecture for powers of K3 surfaces of Picard number 16. arXiv:2203.09778
Wednesday 14 December, 2022
16:15	T2-213	Sebastian Bartling (Essen) On the étale cohomology of the Fargues-Fontaine curve [Abstract] The Fargues-Fontaine curve associated to a p-adic field and an algebraically closed complete field in characteristic p behaves in many regards like a smooth projective curve over an algebraically closed field; this predicts a certain behavior of its étale cohomology (e.g. the étale cohomological dimension should be two), which was formulated in conjectures of Fargues. I will explain how to handle these conjectures in the prime to p torsion case and state what I know how to prove in the p-torsion case.
Wednesday 19 October, 2022
15:30	V5-148	Salvatore Floccari (Hanover) Sixfolds of generalized Kummer type and K3 surfaces [Abstract] I will explain a construction which associates to any hyper-Kähler sixfold K of generalized Kummer type a hyper-Kähler manifold of K3^[3] type. When K is projective, the associated variety is birational to a moduli space of sheaves on a uniquely determined K3 surface. As application of this construction I will show that the Kuga-Satake correspondence is algebraic for many K3 surfaces of Picard rank 16.
17:00	T2-213	Daiki Kawabe (Tohoku) Chow motives of genus one fibrations [Abstract] Genus 1 fibrations play a central role in the classification of surfaces. Let f : X -> C be a genus 1 fibration from a surface to a curve. Then there is a Jacobian genus 1 fibration j : J -> C that satisfies some good properties. In this talk, we prove that the motive of $X$ is isomorphic to the motive of J. As an application, we prove motivic finite-dimensionality for surfaces not of general type with geometric genus 0. This can be regarded as a generalization of Bloch-Kas-Lieberman's result to arbitrary characteristic.

15:15

U2-200

Tobias Kreutz (Bonn)
Motivic and arithmetic aspects of exceptional loci

[Abstract]
Given a family of smooth projective varieties defined over a number field, one often studies the Hodge locus (resp. the l-Galois exceptional locus), defined as the set of points in the base where exceptional Hodge (resp. l-adic Tate) classes occur in the cohomology of the fiber. The Hodge and Tate conjectures state that Hodge and Tate classes should be classes of algebraic cycles, and therefore make the following predictions about these loci: Both loci should in fact be equal, and moreover form a countable union of algebraic subvarieties of the base. In this talk, I want to discuss some recent progress on these questions: Firstly, we prove unconditionally that the l-Galois exceptional locus is indeed a countable union of algebraic subvarieties. The corresponding result for the Hodge locus, obtained by Cattani-Deligne-Kaplan in 1995, is often viewed as important evidence in favor of the Hodge conjecture. In addition, under a mild condition on the generic Mumford-Tate group, we prove that the expected equality between the Hodge locus and the l-Galois exceptional locus holds true if one restricts to their positive dimensional parts.

16:45

U2-200

Matthew Honnor (Imperial College)
Formulas for Brumer—Stark units

[Abstract]
In the 1980's, Tate stated the Brumer--Stark conjecture, which conjectures the existence of an algebraic unit called the Brumer--Stark unit whose p-order relates to the value of a partial zeta function at 0. We begin by introducing the conjecture and recent progress by Dasgupta--Kakde. Before focusing on three formulas, of Dasgupta and Dasgupta--Spieß, which have been conjectured for the Brumer--Stark unit. We report on joint work with Samit Dasgupta which shows that these three formulas are equal. To finish, we give some consequences of our result.

15:15

U2-200

Alexander Efimov (Bonn)
Localizing motives and their applications

[Abstract]
I will give an overview of my recent new results about the category of localizing motives -- the target of the universal localizing invariant (commuting with filtered colimits) of stable infinity-categories over some base ring. In particular, I will explain the most striking property of this category: it is rigid in the sense of Gaitsgory and Rozenblyum (considered as a "large" symmetric monoidal category). The proof of rigidity in fact also allows to effectively compute the morphisms between the motives of categories (a purely algebraic version of Kasparov's KK-theory and K-homology). As an application, we obtain corepresentability results for some classical invariants, such as TR and topological cyclic homology (previously it was only known that non-connective K-theory is corepresented by the unit object). These computations are closely related with the comparison theorem between the two approaches to K-theory of formal schemes: the classical continuous K-theory is equivalent to the K-theory of nuclear modules, defined by Clausen and Scholze. I will also explain how the rigidity of the category of motives gives refined versions of negative cyclic homology and periodic cyclic homology, and some potential applications of these invariants.