Scalar curvature and asymptotic Chow stability of projective bundles and blowups

The holomorphic invariants introduced by Futaki as obstruction to the 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. These invariants are calculated in two special cases. The first is a projective bundle over a curve of genus at least two, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the underlying vector bundle is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarization is sufficiently divisible. Moreover it implies that a projective bundle is asymptotically Chow polystable (with every polarization) if and only if it admits a constant scalar curvature Kaehler metric. The second case is a manifold blown-up at points, and new examples of asymptotically Chow unstable constant scalar curvature Kaehler classes are given.Comment: 15 page

On the third coefficient of TYZ expansion for radial scalar flat metrics

We classify radial scalar flat metrics with constant third coeffcient of its TYZ expansion. As a byproduct of our analysis we provide a characterization of Simanca's scalar flat metric.Comment: 14 page