This paper is an introduction to the theory of wonderful varieties, especially wonderful systems and wonderful diagrams. Let \(G\) be a complex algebraic group. A smooth connected projective \(G\)-variety is called wonderful of rank \(r\) if \(G\) stabilises exactly \(r\) irreducible divisors in \(X\) such that: 1) they are smooth and have a nonempty transversal intersection; 2) two points in \(X\) are on the same \(G\)-orbit if and only if they are contained in the same \(G\)-stable divisors. In such a case one can assume \(G\) reductive (even semi-simple and simply connected).
A wonderful variety is spherical, i.e., it is normal and contains an open \(B\) orbit (we suppose \(G\) reductive and \(B\) a Borel subgroup of \(G\)). Moreover, to any (homogeneous) spherical variety \(Y\) one can associate a wonderful variety \(X\); in this way \(Y\) is determined by the spherical system of \(X\) plus some additional combinatorial data. The wonderful varieties play an important role also in the classification of model homogeneous spaces. Some important examples of wonderful varieties are the flag varieties and the complete symmetric varieties (defined in [C. De Concini, C. Procesi, Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)]).
In the first part of the paper, the general theory is explained. However sometimes the authors give only a sketch of the proofs and assume some hypotheses which are automatically satisfied when \(G\) is of type \(F_4\). In particular, they assume that \(G\) is semi-simply, simply connected and the center acts trivially (this last hypothesis is not satisfied for example in the case of complete symmetric varieties with \(G\) a classic group).
The authors define the spherical system associated to a wonderful variety. These combinatorial data determine uniquely the variety (see [I. V. Losev, Duke Math. J. 147, No. 2, 315–343 (2009; Zbl 1175.14035)]). It is conjectured that each spherical system is associated to a wonderful variety; this conjecture has been proved in many cases. The authors define also the spherical diagrams which are a way to visualize the spherical systems; they are obtained by adding information to the Dynkin diagram of \(G\). The authors assume that the previous conjecture is true for \(F_4\).
The properties of \(X\) are explained emphasizing its spherical diagram. In particular the lines between some automorphisms of \(X\) and some spherical roots (called loose spherical roots) are explained. The authors consider then the combinatorial data associated to an equivariant morphism with connected fibres between two wonderful \(G\)-varieties \(X'\) and \(X\); they also explain the line between the spherical systems of \(X\) and \(X'\). The authors consider especially minimal morphisms, i.e., the morphisms which can be factorized in two morphisms of the previous type.
They also clarify the notion of spherical closure and give a classification of spherical orbits in the projectivization of a simple \(G\) module \(P(V)\) (this part seems to be new). In particular, they show that the generic isotropy group of a spherical variety is spherically closed if and only if it is the stabilizer of a point in \(P(V)\). The authors also give combinatorial conditions on the spherical system so that the generic isotropy group is auto-normalizing, resp. spherically closed.
In the second part of this paper many examples of type \(F_4\) are given to illustrate the general theory and formalism. A particular emphasis is given to the spherical diagrams. Also, the complete list of spherical diagram for \(G\) of type \(F_4\) are given.
The wonderful varieties (of type \(F_4\)) are not very accessible to study. One knows that they can be theoretically realized as closures of orbits in some high dimensional projective spaces, but even the set-theoretical description of these closures seems in general to be out of reach. The authors described the generic isotropy group of many examples; they also describe the generic isotropy groups \(H'\subset H\) of a variety \(X'\) with a minimal morphism \(X'\rightarrow X\). In some examples they explain how to prove that a spherical system is associated to a wonderful variety. They also classify also the spherical orbits in projective fundamental representations of type \(F_4\).
They study some particular class of spherical systems, for examples the ones such that the generic isotropy group of \(X\) is contained in a Borel subgroup or the ones which can be obtained by wonderful varieties of type \(B_3\) (or \(C_3\)) by parabolic inductions. Although \(B_3\) and \(C_3\) are very similar, the structure of their subgroups is very different. Finally they consider some examples not of type \(F_4\).