# Wonderful compactification

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 of each orbit is smooth. C. De Concini and C. Procesi (1983) constructed a wonderful compactification of any symmetric variety given by a quotient *G*/*G*^{σ} of an algebraic group *G* by the subgroup *G*^{σ} fixed by some involution σ of *G* over the complex numbers, sometimes called the **De Concini–Procesi compactification**, and Strickland (1987) generalized this to arbitrary characteristic. In particular, by writing a group *G* itself as a symmetric homogeneous space *G*=(*G*×*G*)/*G* (modulo the diagonal subgroup) this gives a wonderful compactification of the group *G* itself.

## References

- De Concini, C.; Procesi, C. (1983), "Complete symmetric varieties", in Gherardelli, Francesco,
*Invariant theory (Montecatini, 1982)*, Lecture Notes in Mathematics,**996**, Berlin, New York: Springer-Verlag, pp. 1–44, doi:10.1007/BFb0063234, ISBN 978-3-540-12319-4, MR 718125 - Evens, Sam; Jones, Benjamin F. (2008),
*On the wonderful compactification*, Lecture notes, arXiv:0801.0456 - Springer, Tonny A. (2006), "Some results on compactifications of semisimple groups",
*International Congress of Mathematicians. Vol. II*, Eur. Math. Soc., Zürich, pp. 1337–1348, MR 2275648 - Strickland, Elisabetta (1987), "A vanishing theorem for group compactifications",
*Mathematische Annalen*,**277**(1): 165–171, doi:10.1007/BF01457285, ISSN 0025-5831, MR 884653

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