Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

Frank Kutzschebauch
Universität Bern
A solution to the Gromov-Vaserstein Problem
DMV Tagung 2011 - Köln, 19. - 22. September
Any matrix in Sln(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 → 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 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.
Christian Liedtke
Stanford Düsseldorf
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 Lohmann
Albert-Ludwigs-Universität Freiburg
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 P 1 , C, C \ {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 \ Supp(D) is isotrivial if the divisor −(KX + 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) leads to 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.
George Marinescu
Universität zu Köln
Equidistribution of zeros of holomorphic sections of high tensor powers of line bundles
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