Homology stability for special linear group and scissors congruence groups.
Friday, 7.12.12, 10:00-11:00, Raum 404, Eckerstr. 1
The Scissors Congruence group - or pre-Bloch group - P(F),\nof a field F is a group presented by generators and relations which derive from the five-term functional equation of the dilogarithm.\n\nThe Bloch group is a subgroup of P(F) which, by a result of Suslin, is naturally a quotient of the indecomposable K3 of F, and this in turn is a quotient of the group H3(SL(2,F),Z). Up to some possible 2-torsion, the kernel of the map H3(SL(2,F),Z) to K3^ind(F) coincides with the kernel of the stabilization map from H3(SL(2,F),Z) to H3(SL(3,F),Z). We will describe how, for fields with valuations, lower bounds - and even exact computations - of this latter kernel can be expressed as direct sums of pre-Bloch groups of residue fields.
L2-dbar-cohomology of singular complex spaces
Friday, 14.12.12, 10:00-11:00, Raum 404, Eckerstr. 1
We will explain how one can use a resolution of singularities to understand the L2-Dolbeault-cohomology of a singular Hermitian complex space \(X\)\n(in the sense of finding a smooth model for the cohomology). The central tool is an L2-resolution for the Grauert-Riemenschneider canonical sheaf of \(X\).\nWe obtain particularly nice results if X is a Gorenstein space with canonical singularities, including an L2-representation of the cohomology of the structure sheaf.\nTo attack the more general case, we introduce a new kind of canonical sheaf, namely the canonical sheaf of square-integrable holomorphic\ntop-degree-forms with some (Dirichlet) boundary condition at the singular set of X. An L2-resolution for this sheaf allows to describe the L2-cohomology of arbitrary isolated singularities.
Pulling back differential forms
Friday, 21.12.12, 10:00-11:00, Ort noch nicht bekannt