Instability of polarised algebraic varieties
By analogy with the definition for sheaves, we define the slope of a polarised algebraic variety and of each of its subschemes. This gives a notion of slope stability, which we show is a necessary condition for K-stability. We also give the modifications needed to get a necessary condition for asymptotic Chow stability. We then give various calculations of slope and concrete examples. These examples have been chosen to be of interest to the conjectured correspondence between K-stability and the existence of K¨ahler metrics of constant scalar curvature. In particular we get new examples of polarised manifolds that do not admit such metrics.