Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2021/22

Das Forschungsseminar findet dienstags in der Zeit von 13:15 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.

02.11.2021 Salvatore Floccari (U Hannover)
Isogenous hyper-Kähler varieties
Abstract: The Torelli theorem for hyper-Kähler varieties explains to which extent such a variety can be recovered from its integral second cohomology, together with its pairing and Hodge structure. I will address a variant of this problem: How much of a hyper-Kähler variety is determined by its rational second cohomology? Work of Huybrechts and Fu-Vial provides the answer for K3 surfaces. In higher dimension we expect this rational cohomology group to control the full motive of the variety; the main result of the talk confirms this in the realm of André motives.

09.11.2021 Ruijie Yang (HU Berlin)
Vanishing cycles for divisors
Abstract: Given a holomorphic function $f$ on $\mathbb{C}^n$ with an isolated critical point at the origin, the vanishing cycles associated to $f$ are homology cycles of the level set $\{f=t\}$ for $|t|\ll 1$. If the singularity is non-isolated, the vanishing cycles glue to complexes of vector spaces. In algebraic geometry, divisors arise more frequently than functions. A natural question is, how do the vanishing cycles associated to local defining functions glue together? In this talk I will explain a global construction using $\mathscr{D}$-modules and discuss its relation to singularities of divisors. This is based on joint work in progress with Christian Schnell.

16.11.2021 Antareep Mandal (HU Berlin)
Uniform sup-norm bounds for Siegel cusp forms
Abstract: Let $\Gamma\subsetneq \mathrm{Sp}(n,\mathbb{R})$ be a torsion-free arithmetic subgroup of the symplectic group $\mathrm{Sp}(n,\mathbb{R})$ acting on the Siegel upper half-space $\mathbb{H}_n$ of degree $n$. Consider the $d$-dimensional space of Siegel cusp forms $\mathcal{S}_k^n(\Gamma)$ of weight $k$ for $\Gamma$ and let $\{f_j\}_{1\leq j\leq d}$ be a basis of $\mathcal{S}_k^n(\Gamma)$ orthonormal with respect to the Petersson inner product. In this talk we present our work regarding the sup-norm bound of the quantity $S_k^{\Gamma}(Z):=\sum_{j=1}^{d}\det (Y)^{k}\vert{f_j(Z)}\vert^2\,(Z\in\mathbb{H}_n)$. We show using the heat kernel method, for $n=2$ unconditionally and for $n>2$ subject to a conjectural determinant-inequality, that $S_k^{\Gamma}(Z)$ is bounded above by $c_{n,\Gamma} k^{n(n+1)/2}$ when $M:=\Gamma\backslash\mathbb{H}_n$ is compact and by $c_{n,\Gamma} k^{3n(n+1)/4}$ when $M$ is non-compact of finite volume, where $c_{n,\Gamma}$ denotes a positive real constant depending only on the degree $n$ and the group $\Gamma$. Furthermore, we show that this bound is uniform in the sense that if we fix a group $\Gamma_0$ and take $\Gamma$ to be a subgroup of $\Gamma_0$ of finite index, then the constant $c_{n,\Gamma}$ in these bounds depends only on the degree $n$ and the fixed group $\Gamma_0$.

30.11.2021 Igor Burban (U Paderborn)
Tame non-commutative nodal curves, gentle algebras and homological mirror symmetry
Abstract: In my talk, I am going to introduce a class of finite dimensional associative algebras, which are derived equivalent to certain tame non-commutative nodal curves. Following the approach of Lekili and Polishchuk, the constructed derived equivalence allows to interpret such non-commutative curves as homological mirrors of appropriate graded compact oriented surfaces with non-empty marked boundary. This is a joint work with Yuriy Drozd.

07.12.2021 Feng Hao (KU Leuven)

14.12.2021 Constantin Podelski (HU Berlin)
The degree of the Gauss map on Theta divisors
Abstract: The Gauss map attaches to any smooth point of a theta divisor in an abelian variety its tangent space translated to the origin. For indecomposable principally polarized varieties this is a generically finite map. In this talk, I will compute its degree for a generic ppav on some irreducible components of Andreotti-Mayer loci that have been introduced by Debarre in terms of specific Prym constructions. The computation will rely on the technique of Lagrangian specialisation.

18.01.2022 !! Online talk on zoom !!
Christian Lehn (TU Chemnitz)
Singular varieties with trivial canonical class
Abstract: We will present recent advances in the field of singular varieties with trivial canonical class obtained in joint work with Bakker and Guenancia building on work by many others. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau (ICY) and irreducible symplectic varieties (ISV). The proof uses a reduction argument to the projective case which in turn is possible due to advances in deformation theory and a certain result about limits of Kähler Einstein metrics in locally trivial families.

25.01.2022 Tobias Kreutz (HU Berlin)
Arithmetic and $\ell$-adic aspects of special subvarieties
Abstract: We define an $\ell$-adic analog of the Hodge theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that the two notions are equivalent. In this talk, I want to discuss some properties of these subvarieties and prove this equivalence for subvarieties satisfying a certain monodromy condition. This builds on work of Klingler, Otwinowska and Urbanik on the fields of definition of special subvarieties.

15.02.2022 Alex Torzewski (King's College London)
Abstract: tba


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