Sprecher und Vortragszeiten

Tag
Datum 14:00 - 14:50 15:00 - 15:50 16:30 - 17:20 17:30 - 18:20
Mo 19.9.11 Christian Liedke (Stanford/Düsseldorf) Jörg Winkelmann (Bochum)
Hiro-o Tokunaga (Tokio) Frank Kutzschebauch (Bern)
Di 20.9.11 Thomas Peternell (Bayreuth) Stefan Müller-Stach (Mainz) Christian Böhning (Göttingen) Christian Miebach (Calais)
Mi 21.9.11 Alex Küronya (Freiburg) Martin Möller (Frankfurt) Hannah Markwig (Saarbrücken) Stefan Nemirovski (Moskau)
Do 22.9.11 Sönke Rollenske (Mainz) Daniel Greb (Freiburg) Daniel Huybrechts (Bonn) George Marinescu (Köln)

Bitte beachten Sie, dass wir für alle Vorträge 50 Minuten vorgesehen haben.

Vortragstitel und Zusammenfassungen

Christian Liedke: Rational Curves on K3 surfaces

We show that complex projective K3 surfaces with odd Picard rank contain infinitely many rational curves. Our method of proof is via reduction modulo positive characteristic, where results on Tate conjecture/Weil conjecture provide us with the desired rational curves. We lift these rational curves back to characteristic zero using moduli spaces of stable maps in mixed characteristic. This work is joint with Jun Li and extends the original approach of Bogomolov, Hassett, and Tschinkel.

Daniel Greb: Lagrangian fibrations on hyperkähler fourfolds

In this talk I will report on a joint project with Christian Lehn and Sönke Rollenske. Beauville asked if any Lagrangian torus inside a hyperkähler manifold is a fibre of a (meromorphic) Lagrangian fibration. Building on previous work which settles the question in a non-algebraic situation, we show that the answer to the strongest form of Beauville's question is positive in dimension four. An important ingredient of the proof is a detailed study of almost holomorphic Lagrangian fibrations on projective hyperkähler manifolds using the recent advances in the minimal model program.

Hiro-o Tokunaga: Splitting curves for double covers and the topology of the complements of certain curves on rational ruled surfaces

Let $\Sigma$ be a smooth projective surface. Let $f' : Z' \to \Sigma$ be a double cover, i.e, $Z'$: a normal projective surface, $f'$: a finite surjective morphism of degree $2$.  Let $\mu : Z \to Z'$ be the minimal resolution and we put $f = \mu\circ f'$.

1. An irreducible curve $D$ on $\Sigma$ is called a splitting curve with respect to $f$ if $f^*D$ is of the form  $f^*D = D^+ + D^- + E$  where $D^+ \neq D^-$, $f(D^+) = f(D^-) = D$ and $\mbox{Supp}(E)$ is  contained in the exceptional set of $\mu$.

2. Let $D$ be a splitting curve on $\Sigma$ with respect to $f : Z \to \Sigma$.  If the double cover is determined by the branch locus $\Delta(f)$  (e.g., the case when $\Sigma$ is simply connected), we say that $\Delta(f)$ is a quadratic residue curve mod $D$.

In this talk, we discuss "reciprocity" for quadratic residue curves on rational ruled surfaces under some special setting and consider its application to the topology of the complements of curves.

Frank Kutzschebauch: A solution to the Gromov-Vaserstein Problem

Any matrix in $Sl_n (\mathbb C)$ can (due to the Gauss elimination process) be written as a product of elementary matrices. If instead of the complex numbers (a field) the entries in the matrix are elements of a ring, this becomes a delicate question. In particular the rings of maps from a space $X \to \mathbb C$ are interesting cases. A deep result of Suslin gives an affirmative answer for the polynomial ring in $m$ variables in case the size of the matrix ($n$) is greater $2$. In the topological category the problem was solved by Thurston and Vaserstein. For holomorphic functions on $\mathbb C^m$ the problem was posed by Gromov in the 1980's. We report on a complete solution to Gromov's problem. A main tool is the Oka-Grauert-Gromov-h-principle in Complex Analysis.  This is joint work with Björn Ivarsson.

Thomas Peternell: Untermannigfaltigkeiten mit positivem Normalenbündel - eine Vermutung von Hartshorne.

Jeder lernt in der Linearen Algebra, dass sich zwei lineare Unterräume $X$ und $Y$ im komplex-projektiven Raum $P$ schneiden, sofern $\dim X + dim Y \geq \dim P$.  Ersetzt man $P$ durch eine beliebige projektive Mannigfaltigkeit, so ist das i.a. natürlich falsch. Hartshorne hat in den 60er Jahren vermutet, dass diese Aussage jedoch richtig bleibt, wenn man eine Positivitätsvoraussetzung an die Normalenbündel von $X$ und $Y$ macht. Letzlich ist dies ein Problem über höherdimensionale Zykel.  In dem Vortrag diskutiere ich den Stand der Vermutung, Beweismethoden in speziellen Fällen und verwandte Probleme über konvexe Räume und semi-positive Geradenbündel.

Stefan Müller-Stach: Abelsche Varietaeten und Thetafunktionen auf kompakten Riemannschen Mannigfaltigkeiten

Wir erklären eine Konstruktion von Moore-Witten (2000), die zu kompakten Riemannschen Mannigfaltigkeiten (insb. Kählermannigfaltigkeiten) mit Zusatzstrukturen Abelsche Varietäten und damit Thetafunktionen konstruieren, und die in der Stringtheorie eine Bedeutung durch Partitionsfunktionen erhalten.  Natürlich erklären wir die Physik nicht, hingegen eine erweiterte mathematische Sichtweise auf diese Konstruktionen, die auch Weilsche Intermediate Jacobians einschließt. Der Vortrag hat eher Überblickscharakter.

(Zusammenarbeit mit V. Srinivas und C. Peters, Preprint 2011 auf arXiv.org)

Christian Böhning: Rationality properties of linear group quotients

In this survey talk we will report on some recent progress on the rationality problem for quotients $V/G$ where $V$ is a linear representation of the linear algebraic group $G$. Most results will represent joint work of the speaker with Fedor Bogomolov, Hans-Christian Graf von Bothmer, and Gianfranco Casnati. Along with general structural results we will give some applications to moduli spaces of plane curves of large degree, Lueroth quartics and tetragonal curves of genus 7. We will also outline elements of the obstruction theory, stable and unramified group cohomology, and some recent results in this direction.

Christian Miebach: Quotients of bounded domains of holomorphy by proper actions of $\mathbb Z$

We consider a bounded domain of holomorphy $D$ in $\mathbb C^n$ with a closed one parameter group of automorphisms. In this situation we have a proper action of the group $\mathbb Z$ on $D$, and we would like to know whether the quotient manifold $D/\mathbb Z$ is Stein. I will speak about a result obtained with Karl Oeljeklaus wich answers this question positively in the case that $D$ is simply-connected and 2-dimensional. As an application we obtain a normal form for such domains in which $\mathbb Z$ acts by translations.

Alex Küronya: Arithmetic properties of volumes of divisors

The volume of a Cartier divisor on an irreducible projective variety describes the asymptotic rate of growth of the number of its global sections. As such, it is a non-negative real number, which happens to be rational whenever the section ring of the divisor in question is finitely generated.

In a joint work with Catriona Maclean and Victor Lozovanu we study the multiplicative semigroup of volumes of divisors. We prove that this set is countable on the one hand, on the other hand it contains transcendental elements.

Martin Möller: Curves in the moduli space of curves: Slopes and Lyapunov exponents.

The numerical invariants of curves in the moduli space of curves have a long history, mainly aiming to describe 'all curves' in the moduli space of curves. One possible precise formulation of such problem is to describe the cone of moving curve in moduli space of curves. This problem and most problems of similar flavor are today still open.

In this talks, we propose to compare two numerical invariants of different origin. The slope is an algebro-geometric invariant and measures the ratio of intersection numbers of the curve with natural divisors. In contrast, Lyapunov exponents are invariants from dynamical systems and measure the growth rate of cohomology classes under parallel transport of some geodesic flow.

We show how these two quantities are related and how to characterize curves where extremal values are attained.

Hannah Markwig: What corresponds to Broccoli in the real world?

Welschinger invariants count real rational curves on a toric Del Pezzo surface belonging to an ample linear system and passing through a generic conjugation invariant set of points $P$, weighted with $\pm 1$, depending on the nodes of the curve. They can be determined via tropical geometry, i.e. one can define a count of certain tropical curves (which we refer to as Welschinger curves) and prove a Correspondence Theorem stating that this tropical count equals the Welschinger invariant. It follows from the Correspondence Theorem together with the fact that the Welschinger invariants are independent of $P$ that the corresponding tropical count of  Welschinger curves is also independent of the chosen points. However, if $P$ consists of not only real points but also pairs of complex conjugate points, no proof of this tropical invariance within tropical geometry has been known so far.

We introduce broccoli curves, certain tropical curves of genus zero which are similar to Welschinger curves. We prove that the numbers of broccoli curves through given (real or complex conjugate) points are independent of the chosen points. In the toric Del Pezzo situation we show that broccoli invariants equal the numbers of Welschinger curves, thus providing a proof of the invariance of Welschinger numbers within tropical geometry. In addition, counting Broccoli curves yields an invariant in many more cases than counting Welschinger curves. Therefore, it is an interesting question whether there is a meaningful invariant count of real curves that corresponds directly to the tropical Broccoli count.

Joint work with Andreas Gathmann and Franziska Schroeter.

Stefan Nemirovski: Levi Problem and Semistable Quotients

Geometric invariant theory for reductive group actions on Stein manifolds can be applied to function theory using an approach initiated by Tetsuo Ueda in 1980. The talk will discuss these applications and related geometric questions.

Sönke Rollenske: Lagrangian fibrations on hyperkähler manifolds

Hyperkähler (also called irreducible holomorphic symplectic) manifolds form an important class of manifolds with trivial canonical bundle. One fundamental aspect of their structure theory is the question whether a given hyperkähler manifold admits a Lagrangian fibration. I will report on a joint project with Daniel Greb and Christian Lehn investigating the following question of Beauville: if a hyperkähler manifold contains a complex torus $T$ as a Lagrangian submanifold, does it admit a (meromorphic) Lagrangian fibration with fibre $T$? I will describe a complete positive answer to Beauville's Question for non-algebraic hyperkähler manifolds, and give explicit necessary and sufficient conditions for a positive solution in the general case using the deformation theory of the pair $(X, T)$.

Daniel Huybrechsts: Chow groups of K3 surfaces

I review some of the open problems concerning the Chow groups of K3 surfaces over the complex numbers and over number fields. In particular, I shall discuss the case of generic Picard group in the various settings (eg for symplectic automorphisms).

George Marinescu: Equidistribution of zeros of holomorphic sections of high tensor powers of line bundles

We present some equidistribution results for sequences of random sections of high tensor powers of positive line bundles over non-compact manifolds (e.g. Riemann surfaces with cusps, arithmetic quotients or, more generally, quasiprojective manifolds). We also examine the equidistribution of sections of big line bundles endowed with singular Hermitian metrics

Daniel Lohmann: Families of canonically polarized manifolds over log Fano varieties

The Shafarevich hyperbolicity conjecture states that a smooth family of curves of general type is necessarily isotrivial if the base is given by $\mathbb P^1$, $\mathbb C$, $\mathbb C\setminus\{0\}$, or an elliptic curve.  With the aid of the minimal model program we show the following related result.

Let $X$ be a smooth projective variety and $D$ a reduced divisor on $X$. Assume that $D$ is snc, i.e., all components of $D$ are smooth and intersect transversally.  Then any smooth family of canonically polarized varieties over $X\setminus\text{Supp}(D)$ is isotrivial if the divisor $-(K_X+D)$ is ample.

In order to prove this result, we consider the induced moduli map to the coarse moduli space of canonically polarized manifolds. A result by Kebekus and Kovács gives a relation between this moduli map and the minimal model program. In particular, the minimal model program for the pair $(X,D)$ leadsto a fiber space, and the moduli map restricted to a general fiber is constant.  Finally, we apply a generalization of a theorem by Araujo which describes the different minimal model programs for the pair $(X,D)$ in more detail.