---
tag: 84MN3SXH6Q
kind: concept
name: Wonderful compactification
aliases: [de-concini-procesi-compactification]
sources:
  - wikipedia: en:37580247@r1298736868
    locator: Wonderful compactification
    note: adopted
    mention: wikipedia://en/37580247
    sections: [Summary]
same_as:
  wikidata: Q8031858
  wikipedia: Wonderful compactification
inventory: [G1XS39GNWC]
---

## Summary

In algebraic group theory, a **wonderful compactification** of a variety acted on by an algebraic group $G$ is a $G$-equivariant compactification such that the [[closure|0QEAJB91GZ]] of each orbit is smooth. (txt, 1–44) constructed a wonderful compactification of any symmetric variety given by a [[quotient|0KYX2N69J0]] $G/G^{\sigma}$ of an algebraic group $G$ by the subgroup $G^{\sigma}$ fixed by some involution $\sigma$ of $G$ over the [[complex numbers|09FTAET5Z4]], sometimes called the **De Concini–Procesi compactification**. (txt) generalized this construction to arbitrary characteristic. In particular, by writing a group $G$ itself as a symmetric homogeneous space, $G=(G \times G)/G$ (modulo the diagonal subgroup), this gives a wonderful compactification of the group $G$ itself.