  We show that the metric and Berry's curvature for the ground states of $N=2$
supersymmetric sigma models can be computed exactly as one varies the Kahler
structure. For the case of $CP^n$ these are related to special solutions of
affine toda equations. This allows us to extract exact results (including exact
instanton corrections). We find that the ground state metric is non-singular as
the size of the manifold shrinks to zero thus suggesting that 2d QFT makes
sense even beyond zero radius. In other words it seems that manifolds with zero
size are non-singular as target spaces for string theory (even when they are
not conformal). The cases of $CP^1$ and $CP^2$ are discussed in more detail.
