The canonical base property

The canonical base property (CBP) is a certain stability-theoretic property whose formulation was
influenced by an "algebraicity" theorem of Campana for compact complex manifolds, and by an analogous
theorem for differential (and difference) algebraic varieties in char. 0 by myself and Ziegler (yielding a fast
account of function field Mordell-Lang in char. 0).
I will discuss this background, describe the CBP, and then prove some positive and negative results:
(i) The CBP holds for aleph-1 categorical theories under a "rigidity" assumption on definable automorphism
groups, (ii) there is an aleph_1-categorical theory without the CBP, witnessed in a "group-like" manner.  
((i) and (ii) are joint work with E. Hrushovski and D. Palacin.)