Balanced Metrics and Chow Stability of Projective Bundles over K\"ahler Manifolds

TL;DR

This paper extends the connection between slope stability and Chow stability from rank 2 bundles over Riemann surfaces to higher rank bundles over higher-dimensional algebraic manifolds with constant scalar curvature, using balanced metrics.

Contribution

It generalizes Morrison's result by establishing Chow stability for higher rank bundles over complex manifolds with constant scalar curvature, employing balanced metrics and recent theoretical advances.

Findings

Chow stability holds for higher rank bundles over certain Kähler manifolds.

Balanced metrics are key to linking slope stability and Chow stability.

Results apply to manifolds with discrete automorphism groups and constant scalar curvature.

Abstract

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group.