Towards a criterion for slope stability of Fano manifolds along divisors

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Aubin's conjecture on the relation between the anticanonical volume and the existence of a K\"ahler-Einstein metric. Finally, we consider the case that $\dim X=3$; we give a complete answer for slope (semi)stability along divisors of Fano threefolds.