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Oberseminar WiSe 2010/11 Zusammenfassung des Vortrags von Lars Halle (Oslo, NOR)

Degenerations of Calabi-Yau varieties, and the motivic monodromy conjecture.

Let X be a smooth proper variety with trivial canonical sheaf defined over a complete discretely valued field K. I will explain how one can associate to X a motivic invariant, the motivic zeta function ZX(T), which is a formal power series in T with coefficients in a certain localized Grothendieck ring of varieties. The motivic zeta function encodes many properties connected to degenerations of X, and measures the sets of K'-rational points of X for all tamely ramified extensions K'/K. It can be seen as a global version of Denef and Loeser's motivic zeta functions in the setting of hypersurface singularities. I will formulate a global version of Denef and Loeser's motivic monodromy conjecture, which roughly says that poles of ZX(T) correspond in an explicit way to monodromy eigenvalues, and discuss some evidence for this conjecture, in various directions. This is joint work with Johannes Nicaise (K.U. Leuven).