Houston, TX 77005
4:00 p.m. Tuesday, Nov. 5, 2013
On Campus | Alumni
The Albanese and Picard varieties are classical geometric objects (abelian varieties) that we attach to a smooth proper variety X. For any Weil cohomology theory H^* (e.g., singular cohomology or etale cohomology), these varieties are intimately related to H^1(X) and H^(2d-1)(X), where d is the dimension of X. Starting from this point of view on the Picard and Albanese, and working over a field of characteristic zero to allow resolution of singularities, Barbieri-Viale and Srinivas (2003) define geometric objects (1-motives) for any singular/non-proper variety X that underly its dimension-one and codimension-one cohomology groups. These are the Picard and Albanese 1-motives of X. We will review these results, and then explain how they can be generalized to positive characteristic, as well as how to define Albanese and Picard 1-motives `with compact support'.