Residues in group completions and the Cech cohomology of BG

TL;DR

The paper introduces a method to compute residues in group completions using Čech spectral sequences, with applications to the cohomology of the classifying space of semisimple groups.

Contribution

It provides a novel technique to calculate the image of primitive cohomology classes in the cohomology of boundary divisors via spectral sequences.

Findings

Method for computing residues in group completions

Application to the wonderful compactification of semisimple groups

Explicit calculations of cohomology images

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 $\alpha \in H^*(G,\mathbb{C})$ be primitive. We give a method to compute the image of $\alpha$ in $H^*(Z, i^!\mathbb{C}_X)$, using a lift of $\alpha$ along the first edge map of the \v{C}ech spectral sequence for $H^*(BG, \mathbb{C})$. We apply it to the wonderful compactification of a centerless semisimple group $G$.