See the paper [http://arxiv.org/abs/1103.4937][1] by M. Villata.    I find this paper extremely interesting and the arguments rather compelling.   It is based upon two ideas.   First, the gravitational mass is not the gravitational charge; rather the gravitational charge is the Energy momentum 4-vector.   Second, there is a kind of correspondence principle, so that CPT which holds exactly in Minkowski space, holds approximately in weak gravitational fields.   The conclusion, that antimatter will fall up, follows.   

I would appreciate comments by others who are more knowledgeable, however.


  [1]: http://arxiv.org/abs/1103.4937