Connexions affines et projectives sur les surfaces complexes compactes

Details Connexions affines et projectives sur les surfaces complexes compactes Connexions affines et projectives sur les surfaces complexes compactes

2008-05-19

arxiv.org/abs/0805.2816v1

Mathematics

Differential Geometry

20 pages, french

Scientific paper

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.

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