  We consider stochastic nonlinear Schrodinger equations driven by an additive
noise. The noise is fractional in time with Hurst parameter H in (0,1). It is
also colored in space and the space correlation operator is assumed to be
nuclear. We study the local well-posedness of the equation. Under adequate
assumptions on the initial data, the space correlations of the noise and for
some saturated nonlinearities, we prove a sample path large deviations
principle and a support result. These results are stated in a space of
exploding paths which are Holder continuous in time until blow-up. We treat the
case of Kerr nonlinearities when H > 1/2.
