Wonderful compactification

In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group ${\displaystyle G}$ is a ${\displaystyle G}$-equivariant compactification such that the closure of each orbit is smooth. Corrado de Concini and Claudio Procesi (1983) constructed a wonderful compactification of any symmetric variety given by a quotient ${\displaystyle G/G^{\sigma }}$ of an algebraic group ${\displaystyle G}$ by the subgroup ${\displaystyle G^{\sigma }}$ fixed by some involution ${\displaystyle \sigma }$ of ${\displaystyle G}$ over the complex numbers, sometimes called the De Concini–Procesi compactification, and Elisabetta Strickland (1987) generalized this construction to arbitrary characteristic. In particular, by writing a group ${\displaystyle G}$ itself as a symmetric homogeneous space, ${\displaystyle G=(G\times G)/G}$ (modulo the diagonal subgroup), this gives a wonderful compactification of the group ${\displaystyle G}$ itself.

References

• de Concini, Corrado; Procesi, Claudio (1983), "Complete symmetric varieties", in Gherardelli, Francesco (ed.), 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 0718125
• Evens, Sam; Jones, Benjamin F. (2008), On the wonderful compactification, Lecture notes, arXiv:0801.0456, Bibcode:2008arXiv0801.0456E
• Li, Li (2009). "Wonderful compactification of an arrangement of subvarieties". Michigan Mathematical Journal. 58 (2): 535–563. arXiv:math/0611412. doi:10.1307/mmj/1250169076. MR 2595553.
• Springer, Tonny Albert (2006), "Some results on compactifications of semisimple groups", International Congress of Mathematicians. Vol. II, Zürich: European Mathematical Society, 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 0884653