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Oberseminar WiSe 2010/11 Zusammenfassung des Vortrags von Shabnam Kadir

Motivic singularity conjectures

We consider families of Calabi–Yau manifolds of various dimension and link their singularity structure with the decomposition of their Weil zeta functions. In particular, for deformations of Fermat-type Calabi– Yau manifolds that are hypersurfaces in weighted projected space, we show that the there exists a smooth component of the discriminant locus where the number of isolated A1 singularities (or conifold singularities) is given by precisely by the ‘number of permutations’ of the so-called ‘strong orbits’ which label pieces in the decomposition of the zeta function. This decomposition is related to Fermat motives. Singularities are obtained at specializations of the complex structure corresponding to the discriminant locus. Furthermore, the total Milnor number of all possible singularities which arise is, for all cases checked, always expressible in terms of intriguing combinatorial properties of the ‘strong orbits’. This is joint work with Anne Frühbis-Krüger.