The Bielefeld Algebraic and Arithmetic Geometry Seminar 2023

Welcome to the website of the Bielefeld Algebraic and Arithmetic Geometry Seminar! For more information, please contact one of the organizers: Claudia Alfes-Neumann, Ana Botero, Eike Lau, Michael Spiess, Charles Vial, and Thomas Zink.

Talks

Wednesday 8 November, 2023
14:00	U2-200	Riccardo Pengo (Leibniz Universität Hannover) Standard conjectures in Arakelov geometry: from the projective space to Grassmannians [Abstract] The seminal work of Gross and Zagier, generalized by Zhang and encompassed in a celebrated program of Kudla, has shown how special values of L-functions at the center of the critical strip can be related to the Néron-Tate height of certain algebraic cycles on a given arithmetic variety. As Zhang has shown, this allows one to prove the positivity of these central values under the assumption that the arith- metic Chow groups of the aforementioned arithmetic variety satisfy a strong analogue of the standard conjectures in algebraic geometry, proposed by Soulé as a generalization of a weaker conjecture put forward by Gillet and Soulé. In this talk, based on joint work with Paolo Dolce and Roberto Gualdi, I will explain how to find an almost complete characterisation of the hermitian line bundles on the projective space which satisfy the arithmetic standard conjectures. Moreover, I will outline our strategy to give a similar characterisation for hermitian line bundles on general Grassmannians, based on a new presentation of their Arakelov Chow rings, which stems from work of Maillot and Tamvakis.
16:00	U2-200	Manuel Hoff (U Bielefeld) (G, mu)-displays and Rapoport-Zink spaces [Abstract] We formulate a Rapoport-Zink moduli problem using parahoric Hodge-type (G, mu)-displays and study its representability. This is work in progress.
Wednesday 6 December, 2023
14:15-15:15	U2-200	Léonard Pille-Schneider (U Regensburg) Continuity of families of Monge-Ampère measures [Abstract] Let X-> Spec R be a flat degeneration of projective varieties over a discrete valuation ring. Given an ample line bundle L on X and a semi-positively curved metric on the Berkovich analytification of L, we prove that the associated family of Monge-Ampère measures varies in a continuous way. All the relevant notions of Berkovich geometry will be introduced during the talk, and I will try to discuss some applications.
16:15-17:15	U2-200	Enrica Mazzon (U Regensburg) A non-archimedean approach to the SYZ conjecture [Abstract] The SYZ conjecture concerns degenerations of complex Calabi-Yau manifolds and was proposed as a geometric explanation of mirror symmetry. Kontsevich and Soibelman introduced a non-archimedean approach to this conjecture, and more recently, Yang Li's work has connected the non-archimedean approach with the original SYZ conjecture. In this talk, I will explain the key concepts of the non-archimedean approach and present recent developments in the context of hypersurfaces. This is based on a project in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey.
Wednesday 20 December, 2023
14:00	U2-200	Paul Kiefer(U Bielefeld) Injectivity of the Kudla-Millson-Lift in genus 2 [Abstract] In the eighties, Kudla and Millson constructed a linear map between a certain space of vector-valued Siegel modular cusp forms and the space of closed differential forms on some orthogonal Shimura variety. The injectivity of this map in genus 1 has been of great interest and has many applications, including the surjectivity of Borcherds' lift. The aim of this talk is to introduce orthogonal Shimura varieties and indicate why they might be of interest. We then proceed to explain the Kudla-Millson lift and its injectivity in genus 2. We end the talk with a cohomological application. This is joint work with Riccardo Zuffetti.
16:00	U2-200	Jens Funke (Durham U) Indefinite theta series via incomplete theta integrals [Abstract] In this talk we will discuss recent developments in the theory of indefinite theta series focusing on examples and highlighting the underlying geometric aspects. This is joint work with Steve Kudla.
Wednesday 17 January, 2024
14:00	U2-200	Timo Richarz(TU Darmstadt) Non-normality of Schubert varieties [Abstract] Schubert varieties in finite dimensional flag varieties are normal and satisfy many nice geometric properties regardless of the characteristic of the ground field. More generally, similar results hold true in the Kac-Moody setting by the works of Kumar, Mathieu and Littelmann. Later, Faltings, Pappas--Rapoport and very recently Lourenço in full generality proved the normality of Schubert varieties in affine flag varieties whenever the characteristic of the ground field is either zero or big enough. In my talk, I will discuss that such normality fails in small positive characteristic and give the classification of normal Schubert varieties for affine Grassmannians. This is based on joint works with Patrick Bieker, João Lourenço and Thomas Haines.
16:00	U2-200	Thibaud van den Hove (TU Darmstadt) Motivic Satake equivalences and central motives [Abstract] The spherical and Iwahori-Hecke algebras of a reductive group are of great importance in the Langlands program. They are categorified by equivariant sheaves on the affine Grassmannian and affine flag variety respectively, as introduced in the previous talk. Similarly, the Satake and Bernstein isomorphisms are categorified by the Satake equivalence and Gaitsgory's central functor, for which one needs a choice of cohomology theory. In this talk, I will explain that the choice of cohomology theory is irrelevant, in the sense that the Satake equivalence and central functor are of motivic origin. I will also sketch applications to Hecke algebras, obtained by decategorifying the motivic functors.
Wednesday 31 January, 2024
16:00	U2-200	Claudia Fevola (INRIA Saclay) KP solutions from nodal curves, and the Schottky problem [Abstract] The study of solutions to the The Kadomtsev-Petviashvili (KP) equation yields interesting connections between integrable systems and algebraic curves. In this talk, I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, the Riemann’s theta function becomes a finite exponential sum supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes KP solutions arising from such a sum. I will then focus on a special case: the Hirota variety associated to a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint works with Daniele Agostini, Yelena Mandelshtam, and Bernd Sturmfels.
17:00	U2-200	Andreas Gross (Goethe Universität Frankfurt am Main) Tropicalizing Psi Classes [Abstract] Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts. In work with Renzo Cavalieri and Hannah Markwig we defined, in a purely tropical way, tropical psi classes in arbitrary genus. They are operational cocycles on a stack of tropical curves, which enjoy several properties that we know from their algebraic ancestors. We also computed two examples in genus one and gave a tropical explanation for the psi class on the moduli space of 1-marked stable genus-1 curves to be 1/24 times a point. In my talk, I will report on joint work in progress with Renzo Cavalieri, where we explore the missing piece in the story: the link to algebraic geometry. I will explain how to obtain, if we are lucky, a family of tropical curves from a family of algebraic curves. Naturally, there also is a correspondence-type theorem that equates algebraic and tropical intersection products with psi classes, thus showing that the tropical computations done with Cavalieri and Markwig faithfully reflect the algebraic world.
Wednesday 14 February, 2024
14:00	V2-200	Jan Vonk (U Leiden) Around the class number one problem [Abstract] As part of a systematic computational study of equivalence classes of binary quadratic forms, Gauss stated several conjectures on their class numbers. In this talk, I will discuss some recent results related to these conjectures. We will begin with the case of negative discriminants, and its intimate connection with the important but challenging problem of finding rational points on modular curves with non-split Cartan level structures. We then turn to the more mysterious case of positive discriminants, where much less is known. The key missing notion is that of singular moduli for real quadratic fields, which is the subject of recent joint work with Darmon.
16:00	V2-200	Domenico Valloni (EPFL Lausanne) Noether problem in mixed characteristic [Abstract] Let $k$ be a field and let $V$ be a linear and faithful representation of a finite group $G$. The Noether problem asks whether $V/G$ is a (stably) rational variety over k. It is known that if $p = char(k) > 0$ and $G$ is a $p$-group, then $V/G$ is always rational. On the other hand, Saltman and later Bogomolov constructed many examples of $p$-groups such that $V/G$ is not stably rational over the complex numbers. In this talk we study the Noether problem over DVR of mixed characteristic $(0,p)$. We show for instance that for all the examples found by Saltman and Bogomolov, there cannot exist a smooth projective scheme over $R$ whose special resp. generic fibre are stably birational to $V/G$. The proof combines integral $p$-adic Hodge theory and the study of differential forms in positive characteristic.
Wednesday 15 May, 2024
14:00	H4	Fumiaki Suzuki (Hanover) TBD [Abstract] TBD
16:00	H4	Fabio Bernasconi (Neuchatel) On the failure of the Frobenius--stable version of the Grauert--Riemenschneider theorem [Abstract] The Grauert--Riemenschneider (GR) theorem is a vanishing theorem for the higher direct images of the canonical bundle for birational morphisms in characteristic 0, which is revealed to be incredibly powerful when applied to the study of singularities. While GR vanishing is well-known to fail in characteristic p, a Frobenius stable version of it was expected to hold In a recent work with Jefferson Baudin and Tatsuro Kawakami, we show this is not the case: there exists terminal 3folds in characteristic two violating this Frobenius stable version of GR vanishing. To do so, we develop a theory of F_p-rationality, motivated by the RH correspondence for étale F_p-sheaves.
Wednesday 29 May, 2024
14:00	H4	Anneloes Viergever (U Hannover) TBD [Abstract] TBD
16:00	H4	Nils Scheithauer (TU Darmstadt) TBD [Abstract] TBD
Wednesday 12 June, 2024
14:00	H4	Clémentine Lemarié-Rieusset (U Duisburg-Essen) TBD [Abstract] TBD
16:00	H4	Sara Mehidi (Utrecht University) TBD [Abstract] TBD
Wednesday 19 June, 2024
14:00	H4	TBD TBD [Abstract] TBD
16:00	H4	TBD TBD [Abstract] TBD
Wednesday 3 July, 2024
14:00	H4	Andreas Mihatsch (University of Bonn) TBD [Abstract] TBD
16:00	H4	Joahnnes Sprang (U Duisburg-Essen) TBD [Abstract] TBD
Wednesday 17 July, 2024
14:00	H4	Xiaolei Zhao (UCSB) TBD [Abstract] TBD
16:00	H4	Steven Charlton (MPIM Bonn) TBD [Abstract] TBD

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The Bielefeld Algebraic and Arithmetic Geometry Seminar 2023

Talks

Wednesday

8 November, 2023

Wednesday

6 December, 2023

Wednesday

20 December, 2023

Wednesday

17 January, 2024

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31 January, 2024

Wednesday

14 February, 2024

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15 May, 2024

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29 May, 2024

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12 June, 2024

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19 June, 2024

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3 July, 2024

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17 July, 2024