# Veranstaltungsübersicht -- Wintersemester 2019/2020

 Freitag, 2019-10-25 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Matthias Paulsen (LMU München): The construction problem for Hodge numbers To a smooth complex projective variety, one often associates its Hodge diamond, which consists of all Hodge numbers and thus collects important numerical invariants. One might ask which Hodge diamonds are possible in a given dimension. A complete classification of the possible Hodge diamond seems to be out of reach, since unexpected inequalities between the Hodge numbers occur in some cases. However, I will explain in this talk that the above construction problem is completely solvable if we consider the Hodge numbers modulo an arbitrary integer. One consequence of this result is that every polynomial relation between the Hodge numbers in a given dimension is induced by the Hodge symmetries. This is joint work with Stefan Schreieder. Freitag, 2019-11-08 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Harry Schmidt (Uni Basel): Uniformization of dynamical systems and diophantine problems This is joint work with Gareth Boxall (Stellebosch University) and Gareth Jones (University of Manchester). We investigate certain number theoretic properties of polynomial dynamical systems, using the notion of a uniformization at infinity. In this talk I will explain how the ideas involved can be used in order to tackle various related problems on diophantine geometry. Freitag, 2019-11-22 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Erwan Rousseau: Automorphisms of foliations We will discuss in various contexts the transverse finiteness of the group of automorphisms/birational transformations preserving a holomorphic foliation. This study provides interesting consequences for the distribution of entire curves on manifolds equipped with foliations and suggest some generalizations of Lang’s exceptional loci to non-special manifolds, in the analytic or arithmetic setting. This is a work in progress with F. Lo Bianco, J.V. Pereira and F. Touzet. Freitag, 2019-11-29 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Dr. Christian Gleißner (Bayreuth): Klein's Quartic, Fermat's Cubic and Rigid Complex Manifolds of Kodaira Dimension One The only rigid curve is $\mathbb P^1$. Rigid surfaces exist in Kodaira dimension $-\infty$ and $2$. Ingrid Bauer and Fabrizio Catanese proved that for each $n \geq 3$ and for each $\kappa = -\infty, 0, 2,\ldots, n$ there is a rigid $n$-dimensional projective manifold with Kodaira dimension $\kappa$. In this talk we show that the result also holds in Kodaira dimension one. Freitag, 2019-12-06 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Amos Turchet (Univ. Washington, Seattle): Special vs Weakly-Special Manifolds A fundamental problem in Diophantine Geometry is to characterize geometrically potential density of rational points on an algebraic variety X defined over a number field k, i.e. when the set X(L) is Zariski dense for a finite extension L of k. Abramovich and Colliot-Thélène conjectured that potential density is equivalent to the condition that X is weakly-special, i.e. it does not admit any étale cover that dominates a positive dimensional variety of general type. More recently Campana proposed a competing conjecture using the stronger notion of specialness that he introduced. We will review both conjectures and present results that support Campana’s Conjecture (and program) in the analytic and function field setting. This is joint work with Erwan Rousseau and Julie Wang. Freitag, 2019-12-13 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Pieter Belmans (Bonn): Deformations of Hilbert schemes of points via derived categories Hilbert schemes of points on surfaces, and Hilbert squares of higher-dimensional varieties, are important and basic constructions of moduli spaces of sheaves. As such they provide a class of interesting yet tractable varieties. In a joint work with Lie Fu and Theo Raedschelders, we explain how one can (re)prove results about their deformation theory by studying their derived categories, via fully faithful functors and Hochschild cohomology, which describes both classical and noncommutative deformations. Freitag, 2019-12-20 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Simon Felten (Mainz): Smoothing Normal Crossing Spaces Given a normal crossing variety $X$, a necessary condition for it to occur as the central fiber $f^{-1}(0)$ of a semistable degeneration $f: \mathcal{X} \to \Delta$ is $\mathcal{T}^1_X \cong \mathcal{O}_D$ for the double locus $D \subset X$. Sufficient conditions have been given famously by Friedman for surfaces and by Kawamata-Namikawa in any dimension. We give sufficient conditions for smoothing more general normal crossing varieties with $\mathcal{T}^1_X$ only globally generated by relaxing the condition that the total space $\mathcal{X}$ should be smooth. Our main technical tool is the degeneration of a spectral sequence in logarithmic geometry that also settles a conjecture of Danilov on the cohomology of toroidal pairs. Freitag, 2020-01-10 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Christian Lehn (TU Chemnitz): A new proof of the Global Torelli Theorem for holomorphic symplectic varieties In a joint work with Benjamin Bakker, we develop a theoretical framework to approach the global moduli theory of certain singular symplectic varieties. Our work is based on new results about the deformation theory of these varieties together with the notion of ergodic complex structures which has been introduced by Verbitsky and used to study for example hyperbolicity questions. I will explain how to use these techniques to prove a Global Torelli theorem for the varieties in question. Our result in particular gives a new proof of Verbitsky's Global Torelli Theorem for irreducible symplectic manifolds as soon as the second Betti number is at least 5. Freitag, 2020-01-17 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Severin Barmeier (Freiburg): Deformations of path algebras of quivers with relations In this talk I will present ongoing joint work with Zhengfang Wang on deformations of path algebras of quivers with relations. Such path algebras naturally appear in many different guises in algebraic geometry and representation theory and I would like to explain how one can obtain concrete descriptions of their deformations. For example, deformations of path algebras of quivers with relations can be used to describe deformations of the Abelian category of coherent sheaves on any quasi-projective variety X, deformation quantizations of Poisson structures on affine n-space, or PBW deformations of graded algebras. Freitag, 2020-01-24 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Mirko Mauri (MPI Bonn): Dual complexes of log Calabi-Yau pairs and Mori fibre spaces Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu conjecture that the dual complex of a log Calabi-Yau pair should be a sphere or a finite quotient of a sphere. It is natural to ask whether the conjecture holds on the end products of minimal model programs. In this talk, we will validate the conjecture for Mori fibre spaces of Picard rank two. Freitag, 2020-01-31 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Prof. Dr. Amador Martin-Pizarro (Freiburg): TBA Freitag, 2020-02-14 10:30 Uhr Raum 404, Ernst-Zermelo-Str. 1 Dr. Fabrizio Barroero (Roma Tre): TBA

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