Scalar curvature and asymptotic Chow stability of projective bundles and blowups

TL;DR

This paper links Futaki invariants to Mumford weights, showing that projective bundles over curves are asymptotically Chow stable if and only if their vector bundles are slope polystable, and provides new examples of instability in blowups.

Contribution

It establishes a connection between Futaki invariants and Mumford weights, proving a conjecture relating Chow stability to slope polystability and scalar curvature.

Findings

Projective bundles over curves are asymptotically Chow stable iff the vector bundle is slope polystable.

A projective bundle admits a constant scalar curvature Kähler metric iff it is asymptotically Chow stable.

New examples of asymptotically Chow unstable manifolds are provided in blowup cases.

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.