Good Reduction of K3 Surfaces
Friday, 16.1.15, 10:15-11:15, Raum 404, Eckerstr. 1
By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over a p-adic field has good reduction if and only if the Galois action on its first l-adic cohomology is unramified. In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over a p-adic field is unramified, then the surface has admits an ``RDP model'' over the that field, and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction for K3's.) Moreover, we give examples where such an unramified extension is really needed. On our way, we establish existence existence and termination of certain semistable flops, and study group actions of models of varieties. This is joint work with Yuya Matsumoto.\n
Motives, nearby cycles and Milnor fibers
Friday, 30.1.15, 10:15-11:15, Raum 404, Eckerstr. 1
Let k be a field of characteristic zero, and f a regular function on a smooth quasi-projective algebraic k-variety. By analogy to the work of Igusa, Denef and Loeser have associated with the function f a Zeta function which is a power series with coefficients in a Grothendieck ring of varieties. Using motivic integration, they have shown that this power series is rational and defined, in the Grothendieck ring, an element viewed as a motivic version of the Milnor fiber. An analytic avatar in rigid geometry of the Milnor fiber has also been introduced by Nicaise and Sebag. \n\n In this talk I will explain how the theory of motives and stable homotopy theory may be used to recover these Milnor fibers and relate them. These results are joint work with J. Ayoub and J. Sebag. I will also discuss and illustrate the advantages obtained by working with motives instead of Grothendieck rings via some open questions in birational geometry.