Scalar curvature and asymptotic Chow stability of projective bundles and blowups

Abstract

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.

15 pages

Cited by in corpus (8)

• Extremal Kähler metrics on projectivised vector bundles
• Relative Algebro-Geometric stabilities of Toric Manifolds
• Quantisation of extremal Kähler metrics
• About projectivisation of Mumford semistable bundles over a curve
• Geometric quantization of coupled Kähler-Einstein metrics
• Stability and canonical metrics on projective spaces blown up along a line
• Quantum moment map and obstructions to the existence of closed Fedosov star products
• Balanced metrics for extremal Kähler metrics and Fano manifolds