Residues in group completions and the Cech cohomology of BG

Abstract

Let $G$ be a connected affine algebraic group over $\mathbb{C}$, $G\to X$ be an open immersion of $G$-varieties, $Z=X-G$ and $i:Z\to X$ be the inclusion. Let $\alpha \in H^{\ast }(G,\mathbb{C})$ be primitive. We give a method to compute the image of $\alpha$ in $H^{\ast }(Z,i^{!}{\mathbb{C}}_X)$, using a lift of $\alpha$ along the first edge map of the Čech spectral sequence for $H^{\ast }(BG,\mathbb{C})$. We apply it to the wonderful compactification of a centerless semisimple group $G$.

14 pages. Section 2 of the previous version was wrong and has been replaced. Also fixed a ubiquitous sign error