Humboldt Arithmetic Geometry Seminar

Humboldt University Berlin, Winter 2023/24

Time: Tuesdays 13:15 - 14:45

Venue: Room 3.006, Rudower Chaussee 25, 12489 Berlin

— See also the Algebraic Geometry Seminars at HU and FU


31.10.2023 Erwan Rousseau (Brest)
Title: An Albanese construction for Campana's C-pairs
Abstract: We will explain a construction of Albanese maps for orbifolds (or C-pairs), with applications to hyperbolicity such as a generalization of the Bloch-Ochiai theorem. (Joint with Stefan Kebekus).
07.11.2023 Luigi Lombardi (Milano)
Title: On the invariance of Hodge numbers of irregular varieties under derived equivalence
Abstract: A conjecture of Orlov predicts the invariance of the Hodge numbers of a smooth projective complex variety under derived equivalence. For instance this has been verified to the case of varieties of general type. In this talk, I will examine the case of varieties that are fibered by varieties of general type through the Albanese map. For this class of varieties I will prove the derived invariance of Hodge numbers of type \( h^{0,p}\), together with a few other invariants arising from the Albanese map. This talk is based on a joint work with F. Caucci and G. Pareschi.
14.11.2023 Tobias Kreutz (Bonn)
Title: On Simpson's Standard Conjecture for unipotent local systems
Abstract: Simpson's Standard Conjecture predicts that a local system which is defined over \( \overline{\mathbb{Q}}\) on both sides of the Riemann-Hilbert correspondence is motivic. In this talk, I want to discuss this conjecture for unipotent local systems. Conditional on classical transcendence conjectures for mixed Tate motives over number fields, we show that a unipotent local system over \( X = \mathbb{P}^1 \setminus \{s_1,...,s_n\}\) which is defined over \(\overline{\mathbb{Q}}\) on both sides of the Riemann-Hilbert correspondence is the monodromy of a mixed Tate motive over \(X\). This uses the construction of the motivic fundamental group due to Deligne-Goncharov, Borel's computation of the (rational) algebraic \(K\)-theory of number fields, and a homotopy exact sequence for the motivic fundamental group due to Esnault-Levine. For unipotent local systems of small index, we obtain some unconditional results due to transcendence/irrationality results of Baker and Apéry. We also prove a version of the Standard Conjecture over smooth projective curves of genus one, using transcendence results of Chudnovsky and Wüstholz.
05.12.2023 Gabriel Dill (Bonn)
Title: The modular support problem over number fields and over function fields
Abstract: In 1988, Erdős asked: let \(a\) and \(b\) be positive integers such that for all \(n\), the set of primes dividing \(a^n - 1\) is equal to the set of primes dividing \( b^n - 1\). Is \(a = b\)? Corrales and Schoof answered this question in the affirmative and showed more generally that, if every prime dividing \(a^n - 1\) also divides \(b^n - 1\), then \(b\) is a power of \(a\). In joint work with Francesco Campagna, we have studied this so-called support problem with the Hilbert class polynomials \( H_D(T)\) instead of the polynomials \(T^n - 1\), replacing roots of unity by singular moduli. I will present the results we obtained both in the number field case, where \(a\) and \(b\) lie in some ring of \(S\)-integers in a number field \(K\), as well as in the function field case, where \(a\) and \(b\) are regular functions on a smooth irreducible affine curve over an algebraic closure of a finite field.
12.12.2023 Gergely Berczi (Aarhus)
Title: Geometry of the Hilbert scheme of points on manifolds, part I
Abstract: While the Hilbert scheme of points on surfaces is well-explored and extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be exceptionally intricate, exhibiting wild and pathological characteristics. Our understanding of their topology and geometry is very limited. In this series of two talks I will present recent results on various aspects of their geometry. I will discuss

i) the distribution of torus fixed points within the components of the Hilbert scheme (work with Jonas Svendsen),
ii) intersection theory of Hilbert schemes: a new formula for tautological integrals over geometric components, and applications in enumerative geometry,
iii) new link invariants of plane curve (and hypersurface) singularities coming from p-adic integration and Igusa zeta functions (work with Ilaria Rossinelli).

Part II of the talk will be relatively independent from part I and takes place on Wednesday 13 December in the Algebraic Geometry Seminar.
19.12.2023 Bruno Kahn (IMJ Paris)
Title: Universal Weil cohomology
Abstract: In this joint work with Luca Barbieri-Viale, we show that a universal Weil cohomology exists over any field k. The story is actually a bit more complicated: to a suitable class of smooth projective k-varieties (all varieties is the default) we associate 4 universal Weil cohomologies, depending on whether the universal problem concerns targets which are additive or abelian categories, and whether the axioms for the Weil cohomology are plain or if one adds requirements in the style of Weak and Strong Lefschetz. In the latter case, the universal additive category obtained can be used to recover André’s category of motives for ''motivated'' cycles. If time permits, I will explain how the construction extends over a base, and some open problems.
09.01.2024 Michele Pernice (KTH Stockholm)
Title: A stacky Castelnuovo’s contraction theorem
Abstract: In this talk, we are going to discuss a generalization to weighted blow-ups of the classical Castelnuovo's contraction theorem. Moreover, we will show as a corollary that the moduli stack of n-pointed stable curves of genus 1 is a weighted blow-up. This is a joint work with Arena, Di Lorenzo, Inchiostro, Mathur, Obinna.
23.01.2024 Fabien Pazuki (Copenhagen)
Title: Explicit bounds on the coefficients of modular polynomials and the size of \(X_0(N)\)
Abstract: We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials for the elliptic \(j\)-function. These bounds make explicit the best previously known asymptotic bounds. The proof relies on a careful study of the Mahler measure of a family of specializations of the modular polynomial. We also give an asymptotic comparison between the Faltings height of the modular curve \(X_0(N)\) and the height of this modular polynomial, giving a link between these two ways of measuring the "size" of the modular curve. The talk is based on joint work with Florian Breuer and Desirée Gijon Gomez.
30.01.2024 Annette Werner (Frankfurt)
Title: The Hodge-Tate sequence for commutative rigid analytic groups
Abstract: We consider generalizations of Scholze's Hodge-Tate sequence on smooth, proper rigid analytic varieties. These generalizations feature coefficients in commutative rigid groups, which are locally p-divisible. We will also discuss applications to p-adic versions of Simpson's correspondence with coefficients in commutative rigid groups. This is joint work with Ben Heuer and Mingjia Zhang.
13.02.2024 Walter Gubler (Regensburg)
Title: Abstract divisorial spaces and an extension of adelic intersection numbers
Abstract: This is joint work in progress with Yulin Cai. Yuan and Zhang defined an adelic intersection theory over number fields and Yuan used this to give a striking new approach to the uniform Mordell-Lang approach. Recently, Burgos and Kramer extended the arithmetic intersection pairing allowing more singular metrics on the archimedean side. We complete the picture on the non-archimedean side. Using the framework of so called abstract divisorial spaces, we show that Yuan-Zhang's construction is a completion process which works in various situations. In particular, we can extend arithmetic intersection numbers allowing more singular metrics working over any reasonable base field with product formula. In particular, we can do that for proper adelic base curves in the framework of Chen and Moriwaki.


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