# Motives Seminar-SS 2012

The motives seminar this semester will be a workshop on the representation theory of reductive groups with emphasis on the algebraic geometry of orbit spaces and orbit closures.

We will meet on Tuesdays, 14-16 h, in T03 R04 D10.

Here is a preliminary program; this contains a more detailed description of the planned lectures.

Schedule:

Tues. April 10-Lecture 1(Weyman): Seminar overview and introduction to the representation theory of Lie algebras

Tues. April 17-Lecture 2(Weyman): Representation theory continued

Tues. April 24-Lecture 3 Weyman): Representation theory continued

Friday May 4-Lecture 4 (Kumar): Groups generated by pseudo-reflections (meeting in the Tea-room)

Tues. May 8-Lecture 5(Weyman): Representations of type I

Tues. May 15-Lecture 6(Weyman): More on type I representations; representations of type II

Tues. May 22-Lecture 7 (Weyman) Vinberg theory

Tues. May 29-Pfingsten

Tues. June 5-no meeting

Tues. June 12-no meeting

Tues. June 19-Pablo Pelaez (Rutgers): The motivic spectral sequence via birational invariants

Tues. June 26-Yu-Jong Tzeng (Harvard): Universal polynomials for singular sections of vector bundles

Abstract: Let `S` be a smooth projective complex smooth and `L` be a line bundle on `S`. For any collection of analytic and topological singularities, we prove the number of curves with prescribed singularities in the linear system of `L` is given by universal polynomials in the Chern numbers of `L` and `S`. Moreover, if `E` is a vector bundle on a smooth variety `X` of arbitrary rank and dimension, the universal polynomials which counts the zero locus of sections with fixed analytic singularities also exist. We will also discuss the properties of the universal polynomials and its generating series. This is a joint work with Jun Li.

Tues. July 3-Two lectures:

Joe Ross (USC): Cohomology theories with supports. We show that, for `E` a presheaf of spectra on `Sm/k`, satsifying Nisnevich excision, and for `X` smooth and either affine or projective over a field, the Friedlander-Suslin ``equidimensional supprts" condition on `X x Delta ^{*} x A^{q} is a model for the qth term in Voevodsky's slice tower for E.`

Jinhyun Park (KAIST): A^{1} flasque sheaves and Nisnevich excision for semi-topological theories. We describe the "semi-topologicalization" of a presheaf of spectra on smooth varieties over the complex numbers, and show that this process preserves the Nisnevich excision property.

Tues. July 10-Florian Ivorra (Rennes): Reciprocity functors and their `K`-groups [joint work with K. Rülling]

Abstract: Let `k` be a perfect field. As shown by Y. Nesterenko-A. Suslin (1989) and B. Totaro (1992), the usual Milnor `K`-groups provide a purely algebraic description of the groups of higher zero cycles of `k`. Independently, following an idea of K. Kato, M. Somekawa (1990) has generalized Milnor `K`-theory by attaching a `K`-group to a family of semi-abelian varieties and shown that by taking all the varieties to be the multiplicative group one recovers the usual Milnor `K`-groups. M. Spiess-W. Raskind (2000) and R. Akhtar (2004) have built upon this idea and shown that, by using a generalization of Somekawa's `K`-groups, we may also recover the higher Chow groups of zero cycle of a smooth projective `k`-variety. In a recent work B. Kahn-T. Yamazaki (2011) have interpreted (and recovered) these results in terms of sheaves with transfers and the triangulated category of effective motivic complexes.

In this talk we introduce reciprocity functors and construct the associated `K`-group functor of a family of reciprocity functors, which is itself a reciprocity functor. This work may be seen as a first attempt to get close to the notion of reciprocity sheaves imagined by B. Kahn. Homotopy invariant Nisnevich sheaves with transfers, cycles modules, algebraic groups or Kähler differentials provide different type of examples of reciprocity functors. As algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves. They may be used to recover the above mentioned results but are also related to the non homotopy invariant world e.g. Kähler differentials and (probably) to the additive Chow groups of Bloch-Esnault.

Tues. July 17-Mikhail Bondarko (St. Petersburg): Fat hyperplane sections, weak Lefschetz and Barth-type theorems for etale cohomology.

For a smooth complex projective variety `X`, a quasi-finite morphism `s:X'→ X` and a hyperplane section `Z` of `X`, Goresky and MacPherson proved the following weak Lefchetz-type result: if `X'` is locally a complete intersection, then the lower homotopy groups of `X'` are isomorphic to those of the preimage (with respect to `s`) of a small neigbourhood of `Z` in `X` (this could be called a 'fat hyperplane section' of `X'`). Fulton and Lazarsfeld used this statement in order to deduce the following Barth-type result: the lower homotopy groups of a smooth closed local complete interesection `X⊆ CP ^{N}` are isomorphic to that of

`CP`. The speaker will describe cohomological ('algebraic') analogues of these statements, that are valid for varieties over arbitrary fields (of any characteristic).

^{N}