TITLE: Are compact, complex, affinely flat manifolds geodesically complete?
QUESTION [7 upvotes]: Let $M$ be a real, even dimensional, compact manifold endowed with a symplectic form  $\omega$ and a flat, torsionless connection $\nabla$ compatible with $\omega$, that is $$\nabla \omega=0.$$
Under this assumptions, taking into account a simplified version of the  Markus conjecture, it should follow that $M$ is geodesically complete.
My question is: Under what additional geometric assumptions is the result known to be true?
One concrete example is: If in addition we assume the existence of a parallel almost complex structure $J$ on $M,$ then the result is true for $2$ dimensional and $4$ dimensional manifolds as observed by Misha.
What if instead of the the parallel almost complex structure we assume that there is a nonzero parallel vector field $V$ on $M?$ Does it follow that the manifold is geodesically complete?

REPLY [4 votes]: This is an answer to the last question of the revised version of your question which is 
"What if instead of the the parallel almost complex structure we assume that there is a nonzero parallel vector field V on M? Does it follow that the manifold is geodesically complete?"
The answer is the same as to the   question without the assumption. Indeed, having an incomplete manifold with $\omega$ parallel with respect to a flat torsionsfree connection, we can directly multiply it by $R^2$ with the standard connection and the standard symplectic structure.  The resulting  manifold remains   incomplete but now it has even two parallel vector fields.