Course overview -- Wintersemester 2011/2012 (Archive)

For an ordinary Abelian variety over a perfect field of positive characteristic, Serre and Tate discovered a canonical lift over the Witt ring. Later, this has been generalized to varieties with trivial tangent bundles by Nori and Srinivas. In this talk, we will construct a canonical lift for ordinary varieties with trivial canonical sheaves, which generalizes Serre-Tate as well as Nori-Srinivas. As applications, we obtain a Serre-Tate theory for Calabi-Yau varieties (as anticipated by Stienstra), as well as a Bogomolov-Tian-Todorov unobstructedness theorem for such varieties (building on work of Ekedahl and Shepherd-Barron). We also discuss examples due to Hirokado, Schroeer, Schoen, Cynk and van Straten of non-liftable Calabi-Yau varieties for which unobstructedness of deformations fails.

We formulate and prove a relative version of the K-Z conjecture on periods. In this relative version, numbers (and rational numbers) will be replaced by Laurent series (and rational functions).

It has been a long-standing folklore conjecture going back to Enriques, that on a smooth projective surface over the complex numbers the self-intersection of curves has a lower bound.

In a joint work with Bauer, Harbourne, Knutsen, Müller-Stach, and Szemberg, we disprove the conjecture with the help of certain Hilbert modular surfaces.

We will discuss some Hodge-theoretic aspects of families of cubic fourfolds. Our focus will lie on so-called special cubic fourfolds, which contain an algebraic surface not homologous to a plane. We will show that there is a modular form that counts the special members of a pencil of cubic fourfolds. If time permits, we will go on with some speculations on the derived category of special cubic fourfolds and homological mirror symmetry.

Elliptische Kurven haben sogenannte komplexe Multiplikation, wenn ihr Endomorphismenring nicht nur aus den ueblichen n-Multplikationsabbildungen besteht. Ziel meines Vortrags ist es, einige der grundlegende Eigenschaften sowie den Hauptsatz elliptischer Kurven mit komplexer Multiplikation zu beschreiben, der es ermoeglicht, die Klassenkoerpertheorie eines quadratisch imaginaeren Zahlenkoerpers explizit mit Hilfe der j-Invariante und den Torsionspunkten einer solchen Kurve zu beschreiben. Sofern es die Zeit erlaubt, wuerde ich anschliessend wenigstens auf einige Aspekte des Beweises eingehen und einen Ausblick geben, wie sich die Theorie fuer den Fall abelscher Varietaeten verallgemeinern laesst.