Algebra Seminar 

10/07/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building
Paulo Lima-Filho, Texas A&M University

Deligne cohomology and the Picard-Witt group of real varieties

We introduce a version of Deligne cohomology for smooth proper real varieties which is related to bigraded Bredon cohomology in the same fashion that the usual version of Deligne cohomology is related to singular cohomology. For complex manifolds, the Deligne cohomology group $H_{D/C}^2(X;Z(2))$ can be identified with the group of equivalence classes of pairs $(L,\nabla )$, where $L$ is a holomorphic line bundle and $\nabla$ is a holomorphic connection on $L$. However, when $X$ is a Real manifold, the straightforward generalization of this result does not work due to a certain obstruction related to the set of real points of the variety, and one needs an additional geometric piece which would be a certain real quadratic form on the line bundle. We will provide a gentle introduction to Deligne cohomology and some examples.