  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.
