On semistable principal bundles over a complex projective manifold

TL;DR

This paper characterizes semistable principal G-bundles over complex projective manifolds via the numerical effectiveness of associated line bundles, extending Miyaoka's results from vector bundles on curves to higher dimensions.

Contribution

It generalizes Miyaoka's characterization of semistability from vector bundles on curves to principal bundles over higher-dimensional manifolds.

Findings

Semistability characterized by numerical effectiveness of line bundles over E/P

Second Chern class of adjoint bundle vanishes in rational cohomology

Criteria extended to reductive algebraic groups and parabolic bundles

Abstract

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.