We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and if it satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviours: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time. A similar alternative was already known to hold for the mean curvature flow of submanifolds on the sphere from previous work by Huisken and Baker. The results are in collaboration with G. Pipoli (Grenoble).