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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2022

Time: Wednesday 13:15 - 14:45

Room: 3.007 John von Neumann-Haus

Humboldt Arithmetic Geometry Seminar

Seminar: Algebraic Geometry an der FU

(Cancelled) 12.10.2022 3.007 Fabrizio Catanese (Bayreuth)
Title: Manifolds with vanishing Chern classes and some questions by Severi/Baldassari
Abstract: We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only manifolds with vanishing Chern classes. We exhibit Hyperelliptic Manifolds which are not Abelian varieties (nor complex tori) and whose Chern classes are zero not only in integral homology, but also in the Chow ring.
We prove moreover the surprising result that Bagnera de Franchis manifolds ( quotients \(T/G\) where \(T\) is a torus and \(G\) is cyclic) have topologically trivial tangent bundle.
Motivated by a more general question addressed by Mario Baldassarri in 1956, we discuss the Hyperelliptic Manifolds, the Pseudo- Abelian Varieties introduced by Roth, and we introduce a new notion, of Manifolds Isogenous to a \(k\)-Torus Product: the latter have the last \(k\) Chern classes trivial in rational homology and vanishing Chern numbers. We show that the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: in dimension 2 these are the surfaces with \(K_X\) nef and \(c_2(X) = 0\).
A similar picture does not hold in higher dimension, unless we consider manifolds (isogenous to manifolds) whose tangent (resp. cotangent bundle) has a trivial summand.
We survey old and new results on Kähler manifolds whose tangent (resp. cotangent bundle) has a trivial summand, and pose some open problems.
26.10.2022 3.007 Andrea Di Lorenzo (HU Berlin)
Title: Degenerations of twisted maps to algebraic stacks
Abstract: Line bundles over curves, cyclic covers, elliptic surfaces: what these objects have in common is that they can all be regarded as maps from a family of curves to some moduli stack. Therefore, to have a controlled way to degenerate maps to algebraic stacks means having a controlled way to degenerate all the objects above, and more. How can this be obtained and made precise will be the main focus of this talk, which is based on a joint work with Giovanni Inchiostro.
02.11.2022 3.007 Irene Spelta (University of Barcelona)
Title: Prym maps and generic Torelli theorems: the case of plane quintics.
Abstract: The talk deals with Prym varieties and Prym maps. Prym varieties are polarized abelian varieties associated with finite morphisms between smooth curves. Prym maps are accordingly defined as maps from the moduli space of coverings to the moduli spaces of polarized abelian varieties. Once recalled the classical generic Torelli theorem for the Prym map of étale double coverings, we will move to the more recent results on the ramified Prym map \(P_{g,r}\) associated with ramified double coverings. For most of the values of \((g,r)\) a generic Torelli theorem holds and, furthermore, a global Torelli theorem holds when \(r\) is greater (or equal to) 6. At the same time, it is known that \(P_{g,2}\) and \(P_{g,4}\) have positive dimensional fibres when restricted to the locus of coverings of hyperelliptic curves. But this is not a characterization: the study of the differential \(d P_{g,r}\) shows that there are also other configurations to be considered. We will focus on the case of degree 2 coverings of plane quintics ramified in 2 points. We will show that the restriction of \(P_{g,r}\) here is generically injective. This is joint work with J.C. Naranjo.
09.11.2022 3.007 Omid Amini (Ecole Polytechnique Paris)
Title: The tropical Hodge conjecture
Abstract: The aim of the talk is to present the formulation of the Hodge conjecture for tropical varieties, and to explain a proof in the case the tropical variety is triangulable. This provides a partial answer to a question of Kontsevich.
The proof uses Hodge theoretic properties of tropical varieties established in our companion works, which will be reviewed in the talk. These results generalize to the global setting the work of Adiprasito-Huh-Katz on combinatorial Hodge theory, by going in the local setting beyond the case of matroids, and provide answers to conjectures of Mikhalkin and Zharkov.
Based on joint works with Matthieu Piquerez.
15.11.2022 - 18.11.2022
Conference: Resonance, topological invariants of groups, moduli
23.11.2022 3.007 Alex Suciu (Northeastern University)
Title: Topological invariants of groups and tropical geometry
Abstract: There are several topological invariants that one may associate to a finitely generated group \(G\) -- the characteristic varieties, the resonance varieties, and the Bieri–Neumann–Strebel invariants -- that keep track of various finiteness properties of certain subgroups of \(G\). These invariants are interconnected in ways that makes them both more computable and more informative. I will describe in this talk one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.
30.11.2022 3.007 Laurentiu Maxim (University of Wisconsin)
Title: On variants of the Singer-Hopf conjecture in complex geometry
Abstract: The conjectures of Singer and Hopf predict the sign of the topological Euler characteristic of a closed aspherical manifold. In this talk I will discuss various generalizations (e.g., for singular spaces or Hodge enhancements) and partial results concerning the conjectures of Singer and Hopf in the context of Kähler geometry.
07.12.2022 3.007 Jieao Song (HU Berlin)
Title: Geometry and moduli of Debarre-Voisin hyperkähler fourfolds
Abstract: Debarre-Voisin varieties are one of the few known locally complete families of projective hyperkähler fourfolds, constructed inside the Grassmannian \(\mathrm{Gr}(6,10)\). Motivated by the case of varieties of lines on cubic fourfolds, we study their geometry as well as that of two associated Fano varieties, focusing on some special divisors in the moduli. This is a joint work with Vladimiro Benedetti.
04.01.2023 3.007 Leonid Monin (MPI Leipzig)
Title: Inversion of matrices, ML-degrees and the space of complete quadrics
Abstract: What is the degree of the variety \(L^{-1}\) obtained as the closure of the set of inverses of matrices from a generic linear subspace \(L\) of symmetric matrices of size \(n\times n\)? Although this is an interesting geometric question in its own right, it is also motivated by algebraic statistics: the degree of \(L^{-1}\) is equal to the maximum likelihood degree (ML-degree) of a generic linear concentration model. In 2010, Sturmfels and Uhler computed the ML-degrees for \(\mathrm{dim}(L)\) less than 5 and conjectured that for the fixed dimension of \(L\) the ML-degree is a polynomial in \(n\). In my talk I will describe geometric methods to approach the computation of ML-degrees which in particular allow to prove the polynomiality conjecture. The talk is based on a joint works with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, and Jaroslaw A. Wisniewski.
11.01.2023 3.007 Robert Lazarsfeld (Stony Brook)
Title: Measures of association for algebraic varieties
Abstract: I will discuss joint work with Oliver Martin addressing the following (vague) question: given two projective varieties \(X\) and \(Y\) of the same dimension, how far are \(X\) and \(Y\) from being birationally isomorphic?
18.01.2023 3.007 Joshua Lam (HU Berlin)
Title: Infinitely many rank two motivic local systems
Abstract: A natural question in the study of motivic local systems is whether there are infinitely many such with fixed rank and determinant on a fixed proper curve, or a punctured curve if we further fix the conjugacy classes at the punctures. I'll discuss joint work with Daniel Litt, in which we answer this in the positive in the punctured case. More precisely, we construct all the rank two motivic local systems on P^1-4 points, with unipotent monodromy at three points, and "1/2-unipotent"-monodromy at the remaining point. Time permitting, I'll show how this implies several conjectures of Sun, Yang and Zuo coming from the theory of Higgs-de Rham flow.
25.01.2023 3.007 Angel Rios Ortiz (MPI Leipzig)
Title: Asymptotic base loci on Hyperkähler varieties
Abstract: Hyperkähler varieties can be thought as higher dimensional analogues of K3 surfaces. As such, it is generally expected that properties that hold for K3 surfaces can be generalised (appropriately) for this class of varieties as well. In this talk I will discuss ongoing joint work with Francesco Denisi (Bologna University) that characterizes the so-called asymptotic base loci of a big divisor in a Hyperkähler variety in terms of rational curves on it.

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