  We consider the manifold-valued, stochastic extension of the Schr\"odinger
equation introduced by Hughston (Proc.Roy.Soc.Lond. A452 (1996) 953) in a
manifestly covariant, differential-geometric framework, and examine the
resulting quantum evolution on some specific examples of K\"ahler manifolds
with many symmetries. We find conditions on the curvature for the evolution to
be a `collapse process' in the sense of Brody and Hughston (Proc.Roy.Soc.Lond.
A458 (2002) 1117) or, more generally, a `reduction process', and give examples
that satisfy these conditions. For some of these examples, we show that the
L\"uders projection postulate admits a consistent interpretation and remains
valid in the nonlinear regime.
