The LeBrun-Salamon conjecture: positive quaternionic-Kahler manifolds and contact Fano manifolds are homogeneous

Theorem 1.1: every smooth connected complex projective contact Fano manifold $(X,F)$ of dimension $2n+1$, $n\ge1$, is contact-isomorphic to the adjoint variety $Y_{\mathfrak g}$ of a simple complex Lie algebra, with $L\simeq\mathcal O(1)|_{Y_{\mathfrak g}}$. Corollaries: every such manifold is Kahler-Einstein; every smooth projective contact manifold with $b_2=1$ is an adjoint variety and with $b_2\ge2$ is $\mathbb P(T^*Z)$ (via Kebekus-Peternell-Sommese-Wisniewski and Demailly); every closed connected positive quaternionic-Kahler manifold of real dimension $4m\ge8$ satisfies $\nabla Rm=0$ and is homothetic to a Wolf space. It says nothing about negative or zero scalar curvature, about noncompact or incomplete metrics, or about compact Kahler contact manifolds that are not projective beyond the jet criterion it states.

Theorem 1.1: every smooth connected complex projective contact Fano manifold $(X,F)$ of dimension $2n+1$, $n\ge1$, is contact-isomorphic to the adjoint variety $Y_{\mathfrak g}$ of a simple complex Lie algebra, with $L\simeq\mathcal O(1)|_{Y_{\mathfrak g}}$. Corollaries: every such manifold is Kahler-Einstein; every smooth projective contact manifold with $b_2=1$ is an adjoint variety and with $b_2\ge2$ is $\mathbb P(T^*Z)$ (via Kebekus-Peternell-Sommese-Wisniewski and Demailly); every closed connected positive quaternionic-Kahler manifold of real dimension $4m\ge8$ satisfies $\nabla Rm=0$ and is homothetic to a Wolf space. It says nothing about negative or zero scalar curvature, about noncompact or incomplete metrics, or about compact Kahler contact manifolds that are not projective beyond the jet criterion it states.