Weightless cohomology of algebraic varieties
Thursday, 6.2.14, 11:00-12:00, Raum 218, Eckerstr. 1
Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups . These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms.\n The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution.
Motives of rigid analyic varieties over perfectoid fields
Friday, 7.2.14, 10:15-11:15, Raum 404, Eckerstr. 1
The aim of the talk is to outline the proof of the equivalence between the category of motives of rigid analytic varieties (defined by Ayoub adapting Voevodsky's construction) over a perfectoid field of mixed characteristic and over the associated (tilted) perfectoid field of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger regarding the isomorphism of the two absolute Galois groups. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces, which will also be briefly discussed.\n
Friday, 14.2.14, 10:15-11:15, Raum 404, Eckerstr. 1
Soulé-elements, p-adic periods & an explicit reciprocity law
Wednesday, 19.2.14, 14:15-15:15, Raum 404, Eckerstr. 1
Intersection theory on singular varieties
Tuesday, 25.2.14, 10:15-11:15, Hörsaal II, Albertstr. 23b
We introduce some ideas from motivic cohomology into the study of singular varieties. Our approach is modeled on the intersection homology of Goresky-MacPherson; our goal is to intersect cycles on a stratified singular variety provided the cycles do not meet the strata too badly. We define "perverse" analogues of Chow groups and motivic cohomology. Properties include homotopy invariance, a localization theorem, and a splitting theorem. As a consequence we obtain pairings between certain "perverse" cycle groups on a singular variety.\n\n