According to Mori Theory, the geometry of Fano varieties is governed
by their effective curves. Wisniewski proved that, for small
deformations of a smooth Fano manifold, curves "essentially" move
along the deformation. One can interpret this result as a rigidity
property of Fano manifolds. Wisniewski's result is of a topological
nature, relying on the Hard Lefschetz Theorem. Using completely
different methods, we investigate to which extend this property holds
for deformations of singular Fano varieties. Simple examples show that
the conditions on singularities we consider are optimal. There is a
surprising connection between this problem and a generalization of
Siu's theorem on the invariance of plurigenera. This is joint work
with Christopher Hacon.