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): On automorphism groups of fields with operators

In 1993 Lacar showed with model-theoretical techniques that the group of field automorphisms of the complex numbers which fix pointwise the algebraic closure of the rationals is simple, assuming the continuum hypothesis. He later on provided a different proof without assuming CH. There are two main ingredients in Lascar's proof: First, isolating those automorphisms such that the image of a point is algebraic over the point, and secondly, amalgamating field extensions with prescribed automorphisms.

In this talk, we will present a sketch of Lascar's proof and explain how the techniques can be used in order to determine the simplicity of the automorphism group of algebraically closed fields (in all possible characteristics) with additional structure (such as a derivation or a transformal map, often arising in algebraic dynamical systems). No prior knowledge of model theory or mathematical logic is required for this talk.

Freitag,
2020-02-07
10:30 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Stefan Schreieder: Torsion orders of Fano hypersurfaces

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have enormously large degree. Our results also hold in characteristic two, where they solve the rationality problem for hypersurfaces under a logarithmic degree bound, thereby extending a previous result of the speaker from characteristic different from two to arbitrary characteristic.

Freitag,
2020-02-14
10:30 Uhr
Raum 404, Ernst-Zermelo-Str. 1
Dr. Fabrizio Barroero (Roma Tre): On the Zilber-Pink Conjecture for complex abelian varieties

The Zilber-Pink conjecture roughly says that the intersection of a subvariety of an abelian variety with its algebraic subgroups of large enough codimension is well behaved. In the case the subvariety has dimension 1, if the abelian variety and the subvariety are defined over the algebraic numbers, Habegger and Pila proved the conjecture, thus showing that the intersection of a curve with algebraic subgroups of codimension at least 2 is finite, unless the curve is contained in a proper algebraic subgroup. Together with Gabriel Dill, using a recent result of Gao, we extended this statement to complex abelian varieties. More generally, we showed that the whole conjecture for complex abelian varieties can be deduced from the algebraic case.


Abonnieren

Die Einträge dieser Veranstaltung können im iCalender Format abonniert werden. die URL dazu lautet:
http://wochenprogramm.mathematik.uni-freiburg.de/ical/WS2019-2020/OS-AlgZTalgGeom.ics