From April to July 2025, the Oberseminar "Complex and Algebraic Geometry" takes place on Mondays from 14:15 to 15:45. The room is IA 1/75.

Here is a preview of the speakers and talks scheduled in the near past, present, and future:

• 28.04.2025 : Daniel Plaumann (TU Dortmund), Hyperbolic Curves.

Abstract: Hyperbolic curves are curves in the real projective plane with the maximal number of nested ovals. They arise in several different ways and have been studied extensively, along with their generalizations to any dimension. In this talk, I will give an overview, focussing on examples, determinantal representations, computational questions and open problems. (Partly based on joint work with Mario Kummer, Simone Naldi, Bernd Sturmfels, Cynthia Vinzant).

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• 05.05.2025 : Laurine Weibel (Université de Rennes), Finiteness results for hyperbolic orbifold pairs.

In 1913, De Franchis proved that the number of surjective holomorphic maps from X to Y is finite when X and Y are compact Riemann surfaces and Y has genus at least 2. This result was extended to higher dimensions by Noguchi for certain hyperbolic varieties, and Campana established an analogous statement for hyperbolic orbifold curves. In this talk, we will introduce various notions related to hyperbolicity and orbifolds in order to understand certain finiteness properties of holomorphic maps between hyperbolic varieties or between hyperbolic orbifold pairs, thus generalizing the De Franchis theorem.

• 12.05.2025 : no seminar.
• 19.05.2025 : Rémi Danain-Bertoncini (Université de Rennes), Foliations, multifoliate structures and deformations.

In deformation theory, it is customary to seek to construct for a given structure a family that accounts, at least locally, for all its deformations and in the most "economical" way possible. Kuranishi's theorem, for example, guarantees the existence of such a family for any compact complex manifold. The work of Girbau, Haefliger, Nicolau and Sundararaman reproduces Kuranishi's arguments and constructs such families for holomorphic and transversely holomorphic foliations, and a particular type of deformation for holomorphic foliations. All the above structures are examples of a structure introduced by Kodaira and Spencer: multifoliate structures.

In this talk, I shall present what these structures are, the deformation theory associated with them and, finally, I shall introduce the notion of Calabi-Yau foliations and some remarkable properties of these foliations.

• 26.05.2025 - 16.06.2025 : no seminar.
• 23.06.2025 : Simone Billi (Universita di Genova), Non-existence of Enriques manifolds from OG10 type manifolds.
  
  Enriques manifolds are an higher-dimensional analogue of Enriques surfaces. While Enriques surfaces are all obtained as a quotient of a fixed-point-free involution on a K3 surface, in higher dimension the situation is more intricate: there are examples of Enriques manifolds obtained as quotients of irreducible symplectic manifolds by fixed-point-free automorphisms of order 2, 3 or 4 and, moreover, for some deformation types of irreducible symplectic manifolds it is not known if finite order fixed-point-free automorphisms exist.
  
  We use the Looijenga–Lunts–Verbitsky algebra to describe the action of a finite order automorphism on the total cohomology of a manifold of OG10 type. As an application, we prove that no Enriques manifolds arise as quotients of manifolds of OG10 type. This answers to a question recently raised by Pacienza and Sarti.

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• 30.06.2025 : Louis Dailly (Université de Rennes), Miyaoka--Yau and uniformization of log Fano pairs.

At the beginning of the 20th century, it was known that any compact connected, simply connected Riemann surface is biholomorphic to the projective line. Subsequently, several characterizations of projective spaces were established. For instance, Siu and Yau stated that projective spaces are the only Kähler manifolds with positive holomorphic bisectional curvature, and Mori proved that they are the only projective manifolds that have an ample tangent bundle. In a different direction, projective spaces are the only Kähler-Einstein manifolds with a positive constant satisfying the equality in the Miyaoka-Yau inequality. This result originating from uniformization theory was generalized in the singular setting by Greb, Kebekus, Peternell and Druel, Guenancia, Păun. More precisely, they characterize singular quotients of Pⁿ by finite groups acting freely in codimension 1. The aim of this talk is to discuss a generalization of Greb-Kebekus-Peternell's result in order to characterize quotients of Pⁿ by any group action.

• 07.07.2025 : Flora Poon (National Center for Theoretical Sciences), Brauer twists of K3 surfaces admitting van Geemen-Sarti involution.

For any lattice polarised elliptic K3 surface, van Geemen's Brauer twist construction associates to any order 2 element in its Brauer group another elliptic K3 surface, where the original K3 surface can be recovered by taking the relative Jacobian fibration. We will give explicit geometric constructions of some of the Brauer twists of a very general K3 surface that admits a van Geemen-Sarti involution, as well as their birational models. We also observe the same construction works to geometrically realise the Brauer twists of K3 surfaces in some families of higher Picard ranks. This is an ongoing work with Adrian Clingher and Andreas Malmendier.

• 01.10.2024 -- 31.01.2025 : Gastseminar "Komplexe und Algebraische Geometrie"
• 05.03.2025 -- 07.03.2025 : Workshop "D-Modules in Bochum"