Scalar curvature and asymptotic chow stability of projective bundles and blowups

The holomorphic invariants introduced by Futaki as obstruction to asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme of ℙ n. These invariants are calculated in two special cases. The first is a projective bundle ℙ(E) over a curve of genus g ≥ 2, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only if the bundle E is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarization is sufficiently divisible. Moreover it implies that ℙ(E) is asymptotically Chow polystable (with every polarization) if and only if it admits a constant scalar curvature Kähler metric. The second case is a manifold blown up at points, and new examples of asymptotically Chow unstable constant scalar curvature Kähler classes are given. © 2012 American Mathematical Society.