  Sample path large deviations for the laws of the solutions of stochastic
nonlinear Schrodinger equations when the noise converges to zero are presented.
The noise is a complex additive gaussian noise. It is white in time and colored
space wise. The solutions may be global or blow-up in finite time, the two
cases are distinguished. The results are stated in trajectory spaces endowed
with projective limit topologies. In this setting, the support of the law of
the solution is also characterized. As a consequence, results on the law of the
blow-up time and asymptotics when the noise converges to zero are obtained. An
application to the transmission of solitary waves in fiber optics is also
given.
