The period map of certain families of singular hypersurfaces
Freitag, 12.6.15, 10:15-11:15, Raum 404, Eckerstr. 1
This is a joint work with Philippe Eyssidieux. We consider a natural Deligne-Mumford stack parametrizing degree \(d\) hypersurfaces of \(\bmathcal P^n\) with ADE singularities, and prove an infinitesimal Torelli property along the stacky strata. This construction gives rise to examples of smooth projective varieties with interesting fundamental groups and universal covers. If time permits, I will discuss the Toledo and Shafarevich conjecture for these examples.
Locally compact abelian groups and their big friends
Freitag, 19.6.15, 10:15-11:15, Raum 404, Eckerstr. 1
I will discuss how to do homological algebra with\nlocally compact abelian groups (which do not form an\nabelian category, so it's not totally obvious), why one\nwould like to do this without local compactness, or without\ntopology, and ideas of Kato and Kapranov how to actually\nachieve this. Recent advances. This is by the way related\nto Drinfeld's ideas on infinite-dimensional vector bundles\nbecause he proposes to model good fibre spaces by a similar\nmechanism.
On the motive of some hyperKaehler varieties
Freitag, 26.6.15, 10:15-11:15, Raum 404, Eckerstr. 1
I will explain why it is expected that the Chow ring of hyperKaehler varieties has a similar structure as the Chow ring of abelian varieties. Examples of hyperKaehler varieties are given by K3 surfaces, and Hilbert schemes of length-n subschemes on K3 surfaces and their deformations. In fact I will introduce the notion of ``multiplicative Chow-Kuenneth decomposition'' and provide examples of varieties that can be endowed with such a decomposition. In the case of curves, or regular surfaces, this notion is intimately linked to the vanishing of a so-called "modified diagonal cycle". For example, a very general curve of genus >2 does not have vanishing modified diagonal cycle, but a result of Ben Gross and Chad Schoen establishes the vanishing of a modified diagonal cycle for hyperelliptic curves.