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, ISBN 978-3-540-12319-4, MR 718125, doi:10.1007/BFb0063234 
• 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, ISSN 0025-5831, MR 884653, doi:10.1007/BF01457285