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, 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
14:30	H4	Fumiaki Suzuki (Hanover) Non-injectivity of the cycle class map in continuous l-adic cohomology [Abstract] Jannsen asked whether the rational cycle class map in continuous l-adic cohomology is injective over a finitely generated field. As recently pointed out by Schreieder, the integral version of Jannsen's question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. This is a joint work with Federico Scavia.
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) Motivic Donaldson-Thomas invariants in low degrees for the three-dimensional projective space. [Abstract] Suppose that we would like to count ideal sheaves with zero-dimensional support of a given length on a smooth projective toric complex threefold with an action of the 3-dimensional torus on it. One could try to do this by doing intersection theory on a suitable moduli space, i.e. the Hilbert scheme of points on our threefold. However, these Hilbert schemes might have all sorts of bad singularities. One way to remedy this issue, is to use virtual fundamental classes, introduced by Behrend-Fantechi. The degrees of those classes - which are integers - are called Donaldson-Thomas invariants. Maulik, Nekrasov, Pandharipande and Okounkov have computed the generating series of these Donaldson-Thomas invariants in terms of a power of the MacMahon function. Levine has introduced a motivic analogue of virtual fundamental classes, of which the degrees land in the Witt ring of the base field when the scheme has an orientation in some sense. By another result of Levine, there is such an orientation for the threedimensional projective space, so that one can define its motivic Donaldson-Thomas invariants. If we take the base field to be the real numbers, the degrees of these classes are again integers. We discuss a computation of the first three motivic Donaldson-Thomas invariants for the threedimensional projective space, and a conjecture for an analogue of the formula of Maulik, Nekrasov, Pandharipande and Okounkov in this setting.
16:00	H4	Nils Scheithauer (TU Darmstadt) Reflective modular varieties and their cusps [Abstract] Automorphic forms for orthogonal groups are natural generalisations of elliptic modular forms. An important class of these functions is given by automorphic products. We show that under some natural assumptions there are exactly 11 reflective automorphic products of singular weight. The corresponding modular varieties have a very rich geometry. Surprisingly their 1-dimensional type-0 cusps are naturally parametrised by Schellekens' list which classifies certain 2-dimensional quantum field theories. This is joint work with Janik Wilhelm and Thomas Driscoll-Spittler.
Wednesday 12 June, 2024
14:15	H4	Clémentine Lemarié-Rieusset (U Duisburg-Essen) Motivic Linking [Abstract] In classical knot theory and in real algebraic knot theory, one can associate to an oriented link with two components (i.e. two disjoint oriented knots) its linking number: an integer which counts how many times one of the components turns around the other component. In this talk, I will present motivic counterparts, over a perfect field F, to the linking number, which describe how two (nice) disjoint oriented closed F-subschemes of an ambient F-scheme can be intertwined, i.e. linked together. These counterparts, called quadratic linking degrees, take values in the Witt group of F, in the Grothendieck-Witt group of F or in the first Milnor-Witt K-theory group of F and can be computed in a similar way to the classical case, by replacing the singular complex with the Rost-Schmid complex. There will be several examples, some of which will feature affine planes minus the origin in the affine four-space minus the origin (the setting closest to classical knot theory) as well as other motivic spheres, and some of which will feature projective lines in the projective three-space (the setting closest to real algebraic knot theory).
15:45	H4	Sara Mehidi (Utrecht University) The Logarithmic Jacobian and extension of torsors over families of degenerating curves [Abstract] The problem of extending fppf torsors has been largely encountered in the literature, without giving a satisfactory solution in the classical setting. Given a torsor on the smooth locus of a family of nodal curves, the question consists in extending each of the structural group and the total space of the torsor above the family. It turns out that adding a log structure to the family of nodal curves allows to enlarge the category of classical (fppf) torsors to what is called log torsors. In particular, this provides a new framework for studying the aforementioned problem. On the other hand, Molcho and Wise constructed the log Picard group, a canonical compactification of the Picard group of families of nodal curves that is smooth, proper and possesses a group structure, but which can only be represented in the category of logarithmic spaces. Afterwards, it was shown that the restriction of this log Picard group to the degree zero log line bundles - which gives the so-called Logarithmic Jacobian- is in fact the log Néron model of the Jacobian of the smooth locus. In this talk, I will show that the Logarithmic Jacobian classifies finite log torsors over families of nodal curves, generalizing the classical situation for smooth curves. Then, we will see that the Néron property of the Log Jacobian allows to get a result on extending fppf torsors into log torsors over families of nodal curves. This is a joint work with Thibault Poiret.
Wednesday 3 July, 2024
14:15-15:15	H4	Andreas Mihatsch (University of Bonn) Generating series of complex multiplication cycles [Abstract] A classical result of Zagier states that the degrees of Heegner divisors on the modular curve form the positive Fourier coefficients of a modular form. In my talk, I will define complex multiplication cycles on higher-dimensional Shimura varieties and show that their degrees have a similar modularity property. I will then discuss a possible extension to integral models. This is based on joint work with Lucas Gerth, Tonghai Yang and Siddarth Sankaran.
15:45-16:45	H4	Joahnnes Sprang (U Duisburg-Essen) Irrationality and linear independence of p-adic zeta values [Abstract] The celebrated theorem of Ball-Rivoal shows that there are infinitely many irrational odd zeta values. More precisely, it provides asymptotically lower bounds on the dimension of the Q-vector space spanned by these numbers. Although we are still far from fully understanding the structure of odd zeta values, the corresponding question for p-adic zeta values is even more difficult. For example, the question of the non-vanishing of these numbers is still open today, although it has interesting consequences e.g. for the non-vanishing of p-adic regulators. In this talk we would like to report recent progress on the irrationality and linear independence of p-adic zeta values. Parts of this talk are based on joint work with Li Lai.
Wednesday 10 July, 2024
14:15-15:15	H4	Sebastián Olano (University of Toronto) Singularities of Secant Varieties from a Hodge theoretic perspective [Abstract] Given a smooth projective variety $X$ and a sufficiently positive embedding into $\mathbb{P}^N$, we can define the associated secant variety. These varieties have been extensively studied, yet many aspects of their singularities remain to be understood. The singular locus admits a simple description since it is the original variety $X$. However, its singularities are more intricate, as the variety is generally not a local complete intersection. In this talk, I will explore the singularities of secant varieties from a Hodge-theoretic perspective, focusing on generalizations of notions well-understood in the context of local complete intersections. Specifically, I will describe the du Bois complex and its relationship with the sheaves of differential forms, linking it to the generalization of the classical notions of du Bois and rational singularities. Additionally, I will present results concerning the local cohomology module of the secant variety in projective space.
15:45-16:45	H4	Guido Kings (University of Regensburg) Deligne’s conjecture for algebraic Hecke characters [Abstract] Deligne conjectured in the middle of the 1970s that the critical values of L- functions of motives coincide with the period of the motive up to a rational number. This conjecture is in general wide open and only in the case of modular forms and algebraic Hecke characters there are significant and far reaching results. For Hecke characters critical values appear only if the number field is either totally real or contains a CM field. The totally real case was settled (up to language) by Klingen and Siegel in the 1960s. In 1983 Blasius was able to prove the case of Hecke characters over CM fields using the theory of Hilbert modular forms. In this talk we will report on joint work with Johannes Sprang and subsequent work of Han-Ung Kufner in his Regensburg thesis, which led to a complete proof in the case of general fields containing a CM field.
Wednesday 17 July, 2024
14:15	H4	Steven Charlton (MPIM Bonn) Goncharov's programme, and depth reductions of multiple polylogarithms (in weight 6) [Abstract] Abstract: Multiple polylogarithms Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics. One of the main challenges in the study of multiple polylogarithms resolves around understanding how on many variables a multiple polylogarithm function (or `interesting' combinations thereof) actually depend (``the depth''), as for example Li_{1,1} can already be expressed via Li_2. Goncharov gave a conjectural criterion (``the Depth Conjecture'') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \zeta_F(m). I will give an overview of multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will try to discuss what is currently known, including recent results in weight 6, and what we are still trying to investigate.
15:45	H4	Xiaolei Zhao (UCSB) Moduli of stable objects on Fano threefolds [Abstract] The derived category of a Fano variety sometimes contains an interesting admissible subcategory, known as the Kuznetsov component. Moduli spaces of Bridgeland semistable objects on the Kuznetsov component provide many examples to study, often generalizing constructions from classical algebraic geometry. In this talk, I will review several cases of Fano threefolds, and explain how moduli spaces on their Kuznetsov components behave similarly to moduli of sheaves on curves. Based on joint work with Chunyi Li, Yinbang Lin and Laura Pertusi.

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The Bielefeld Algebraic and Arithmetic Geometry Seminar 2023

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8 November, 2023

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6 December, 2023

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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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3 July, 2024

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10 July, 2024

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17 July, 2024