Schottky groups acting on homogeneous rational manifolds
Friday, 4.12.15, 10:05-11:05, Raum 404, Eckerstr. 1
In 1877 Schottky constructed free and proper actions of a free group of rank r on a domain in the Riemann sphere having as quotient a compact Riemann surface of genus r. In 1984 Nori extended this construction to any complex-projective space of odd dimension in order to produce compact complex manifolds having free fundamental group. Larusson as well as Seade and Verjovsky studied further properties of these quotient manifolds such as their algebraic and Kodaira dimensions and their deformation theory. In my talk I will explain a joint work with Karl Oeljeklaus where we have studied the question to which homogeneous rational manifolds Nori's construction can be generalized, and the new examples we have found. If time permits, I will also indicate what we can say about geometric and analytic properties of the quotient manifolds associated with these new examples.
Curves of Genus 2 with Bad Reduction and Complex Multiplication
Friday, 11.12.15, 10:00-11:00, Raum 404, Eckerstr. 1
If a smooth projective curve of positive genus which is defined over a number field has good reduction at some finite place, than so does its jacobian. But the converse already fails in genus 2. To study the extent of this failure we investigate jacobians that have complex multiplication. This forces the jacobians to have potentially good reduction at all finite places by a theorem of Serre and Tate. I will present a result which roughly speaking states that a genus 2 curve whose jacobian has complex multiplication usually has bad stable reduction at at least one finite place. This is joint work with Fabien Pazuki.
On Beauville's conjectural weak splitting property
Friday, 18.12.15, 10:15-11:15, Raum 404, Eckerstr. 1
We present a result on the Chow ring of irreducible symplectic varieties. The main object of interest is Beauville's conjectural weak splitting property, which predicts the injectivity of the cycle class map restricted to a certain subalgebra of the rational Chow ring (the subalgebra generated by divisor classes). For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations. After deducing that this implies the weak splitting property in many new cases, we present parts of the proof.