Projective uniformization, extremal Chern classes and quaternionic Shimura curves

Abstract

Projective structures on compact real manifolds are classical objects in real differential geometry. Complex manifolds with a holomorphic projective structure on the other hand form a special class as soon as the dimension is greater than one. In the Kähler Einstein case $${\mathbb P}_m$$ , tori and ball quotients are essentially the only examples. They can be described purely in terms of Chern class conditions. We give a complete classification of all projective manifolds carrying a projective structure. The only additional examples are modular abelian families over quaternionic Shimura curves. They can also be described purely in terms of Chern class conditions.