On principal bundles

Abstract EN

We review some recent results on holomorphic principal bundles over a complex projective manifold and more generally over a compact K¨ahler manifold.

One such result is the Atiyah–Krull–Schmidt theorem for principal bundles.

Another result is a necessary and sufficient condition for a holomorphic principal G–bundle over a compact Riemann surface to admit a holomorphic connection.

A principal G–bundle EG over a compact K¨ahler manifold admits a Hermitian–Einstein connection if and only if EG is polystable.

A similar result is proved for principal Higgs bundles, and more generally for principal Higgs G–sheaves.

Let G be a simple linear algebraic group over C.

Fix a parabolic subgroup P G and a nontrivial and antidominant character c of P.

Let EG be a principal G–bundle over a connected K¨ahler manifold M.

Let EG (c) be the line bundle over EG = P is associated to the character c.

The following three are equivalent : (1) The line bundle EG (c) over EG = P is numerically effective.

(2) The G–bundle EG is pseudostable, and c2(ad(EG)) 2 H4 (M ; Q) vanishes, where ad (EG) is the adjoint vector bundle mentioned above.

(3) The vector bundle ad (EG) is numerically flat.