Connexions affines et projectives sur les surfaces complexes compactes

TL;DR

This paper proves that holomorphic normal projective connections on compact complex surfaces are flat and classifies affine connections on such surfaces, revealing their local models and special cases.

Contribution

It establishes the flatness of holomorphic normal projective connections and classifies affine connections on compact complex surfaces, including special cases involving elliptic bundles.

Findings

Holomorphic normal projective connections are flat on compact complex surfaces.

Affine connections are locally modeled on translations-invariant connections on c2^2.

Special case: principal elliptic bundles over Riemann surfaces with genus bc a7 2.

Abstract

We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if $\nabla$ is a generic connection on a principal elliptic bundle over a Riemann surface of genus $g \geq 2$, with odd first Betti number. In the last case, the local Killing Lie algebra is of dimension one, generated by the fundamental vector field of the principal fibration.