In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^2(X,\mathbb{Z})\cong H^4(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(ℰ\right)$, that allows us to obtain the cohomological invariants of $X$ and $ℰ$. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1\left(L\right)$ and its automorphism group $\mathrm{Aut}(V,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case where $\mathrm{Aut}(V,L)$ is not discrete.

Let $X$ be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of $X$ is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.