On the canonical real structure on wonderful varieties (Q2878669)

Let \(G\) be a connected reductive linear algebraic group over the complex numbers. Let \(X\) be a complex algebraic variety endowed with an action of \(G\). A real structure on \(X\) is said to be canonical if it is equivariant with respect to the involution defining the split real form of \(G\). The authors study canonical real structures on certain spherical varieties. Recall that in general a spherical variety is a normal algebraic variety endowed with an action of \(G\) admitting an open dense orbit for a Borel subgroup of \(G\).NEWLINENEWLINEMore precisely, they consider two classes of smooth spherical varieties: homogeneous spherical varieties with self-normalizing isotropy groups, and their wonderful embeddings, that is, smooth compactifications satisfying further regularity properties as defined in [\textit{C. De Concini} and \textit{C. Procesi}, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. They show that in both cases a canonical real structure exists and it is unique.NEWLINENEWLINEFurthermore, they give an estimate of the number of orbits for the real form of \(G\) on the set of real points of a wonderful embedding \(X\), provided all isotropy groups of \(X\) are self-normalizing.