Wonderful compactification

In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group [math]\displaystyle{ G }[/math] is a [math]\displaystyle{ G }[/math]-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 [math]\displaystyle{ G/G^{\sigma} }[/math] of an algebraic group [math]\displaystyle{ G }[/math] by the subgroup [math]\displaystyle{ G^{\sigma} }[/math] fixed by some involution [math]\displaystyle{ \sigma }[/math] of [math]\displaystyle{ G }[/math] 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 [math]\displaystyle{ G }[/math] itself as a symmetric homogeneous space, [math]\displaystyle{ G=(G \times G)/G }[/math] (modulo the diagonal subgroup), this gives a wonderful compactification of the group [math]\displaystyle{ G }[/math] itself.

References

• de Concini, Corrado; Procesi, Claudio (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 
• Evens, Sam; Jones, Benjamin F. (2008), On the wonderful compactification, Lecture notes, Bibcode: 2008arXiv0801.0456E 
• Li, Li (2009). "Wonderful compactification of an arrangement of subvarieties". Michigan Mathematical Journal 58 (2): 535–563. doi:10.1307/mmj/1250169076. 
• Springer, Tonny Albert (2006), "Some results on compactifications of semisimple groups", International Congress of Mathematicians. Vol. II, Zürich: European Mathematical Society, pp. 1337–1348, http://ada00.math.uni-bielefeld.de/ICM/ICM2006.2/ 
• Strickland, Elisabetta (1987), "A vanishing theorem for group compactifications", Mathematische Annalen 277 (1): 165–171, doi:10.1007/BF01457285, ISSN 0025-5831