Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Hannah Markwig
Universität des Saarlandes, Saarbrücken
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 ±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.
Christian Miebach
Université du Littoral Côte d’Opale, Calais
Quotients of bounded domains of holomorphy by proper actions of Z
We consider a bounded domain of holomorphy D in C n with a closed one parameter group of automorphisms.
In this situation we have a proper action of the group Z on D, and we would like to know whether
the quotient manifold D/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 Z acts by translations.
Martin Möller
Goethe-Universität Frankfurt
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.
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