A famous LeBrun-Salamon conjecture says that every compact positive quaternion-Kähler manifold is symmetric or, equivalently, every contact Fano manifold with Kähler-Einstein metric is a homogeneous space. Quaternion-Kähler symmetric spaces are known as Wolf spaces and, equivalently, homogeneous complex contact manifolds are known to be the closed orbits in the projectivizations of adjoint representations of simple algebraic groups; they are called adjoint varieties. This conjecture was proved for positive quaternion-Kähler manifolds of (real) dimension \(\leq 16\) and for contact Fano manifolds with a Kähler-Einstein metric of dimension \(\leq 9\); moreover, the conjecture is known for contact Fano manifolds \(X\) (even without assuming that they admit a Kähler-Einstein metric) if the first Chern class of the quotient \(L = TX / F\) of the contact distribution \(F \hookrightarrow TX\) does not generate the second cohomology \(H^2(X,\mathbb Z)\), and in some cases with additional assumptions on group actions.
In the paper under review, the authors prove that a contact Fano manifold is the adjoint variety of one of the simple groups if the group of its automorphisms is reductive of rank \(\geq 2\) and the action of its Cartan torus has only isolated extremal fixed points. Moreover, in the main theorem, they improve previous results so that it covers the adjoint varieties of all the classical series of linear groups and almost all the exceptional ones. More precisely, they prove LeBrun-Salamon conjecture in the following setting. If \(X\) is a contact Fano manifold of dimension \(2n + 1\) whose group of automorphisms is reductive of rank \(\geq \max (2, (n-3)/2)\), then \(X\) is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.
We remark that, although motivated by a problem from Riemannian geometry, the present paper depends only on methods from algebraic geometry.