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?



Are compact, complex, affinely flat manifolds geodesically complete?
Now that you added the volume requirement, you get close to the Marcus conjecture. In complex dimension 1 you then get a locally Euclidean structure and the answer is clearly positive. In complex dimension 2 it is again positive since linear holonomy then has rank 1 and in this situation Marcus conjecture was proven by Yves Carriere in1989. 


