We discuss the question of lifting an abelian variety $A$ over a finite field $k$ to an abelian scheme $B$ over a local domain $R$ with generic characteric $0$ which has sufficiently many complex multiplications. The main result is: if one requires $R$ to be normal, then there is an obstruction coming from the residue field of the reflex field; this turns out to be the only obstruction. (joint work with B. Conrad and C.-L. Chai)