The author studies the connected automorphism group of the wonderful varieties. Let \(G\) be a reductive group. A smooth \(G\)-variety \(X\) is wonderful if: i) it has an open \(G\)-orbit \(\mathcal{O}\); ii) the complementar set \(X\backslash \mathcal{O}\) is a smooth normal crossing divisor; iii) the irreducible components of \(X\backslash \mathcal{O}\) have non-empty intersection; iv) the closure of a non-open orbit is the intersection of the \(G\)-stable prime divisors containing such orbit. One can suppose that \(G\) acts effectively on \(X\).
The connected automorphism group of a wonderful variety is been previously studied by M. Brion [J. Algebra 313, No. 1, 61–99 (2007; Zbl 1123.14024)]. The author uses some of the results of this work, together with the classification of wonderful varieties of rank 1 (due to D. N. Akiezer [Ann. Global Anal. Geom. 1, No.1, 49–78 (1983; Zbl 0537.14033)]) and rank 2 [due to B. Wasserman, Transform. Groups 1, No. 4, 375–403 (1996; Zbl 0921.14031)]. The author uses also the fact that a wonderful varieties is determined by some combinatorial data, called its spherical system and introduced by Luna. The proof that two wonderful varieties corresponding to the same spherical system are isomorphic is due to I. V. Losev [Duke Math. J. 147, No. 2, 315–343 (2009; Zbl 1175.14035)].
Brion has proved the following fact: given any wonderful variety \(X\) and any connected group \(G\subsetneq G'\subseteqq \mathrm{Aut}^{0}(X)\), \(G'\) is semisimple and \(X\) is again wonderful with respect to the action of \(G'\). The author determines all the wonderful \(G\)-varieties such that the identity component of the automorphism group contains properly \(G\). The author determines also the spherical system of such varieties under the action of their connected automorphism group. Moreover, he gives also a geometrical description of most of such varieties.
First, the author classifies the wonderful varieties \(X\) such that exists a group \(G\subsetneq G'\subseteqq \mathrm{Aut}^{0}(X)\) which acts on \(X\) with the same orbits of \(G\). In particular, he shows that in the case where \(G'=\mathrm{Aut}^{0}(X)\), \(X\) is a product of wonderful \(G\)-varieties such that at least one factor \(X'\) is a homogeneous \(G\)-varieties with \(\mathrm{Aut}^{0}(X')\) strictly greater than \(G\) (the projective homogeneous \(G\)-varieties are exactly the wonderful \(G\)-varieties with rank 0).
In a second time, he studies explicitly the wonderful varieties with rank 1. Finally, he uses the following fact proved by Brion: if \(\mathrm{Aut}^{0}(X)\) does not fix all the \(G\)-orbits, then \(X\) contains a wonderful \(G\)-variety \(X'\) with rank 1 which is homogeneous under \(\mathrm{Aut}^{0}(X')\). Observe that a rank 1 wonderful variety contains exactly two orbits.
Furthermore, the author reformulates his main theorem using certain smooth morphism with connected fibers between wonderful varieties. This second formulation is more concise and evidences the crucial role of wonderful varieties with rank 1 in the theory of wonderful varieties. In particular, if \(G\subsetneq \mathrm{Aut}^{0}(X)\) then there is a \(G\)-equivariant morphism from \(X\) to a product of wonderful \(G\)-varieties where at least one factor \(X'\) has rank 1 and is homogeneous under the action of \(\mathrm{Aut}(X')\).