  We consider the global regularity problem for defocusing nonlinear
Schr\"odinger systems $$ i \partial_t + \Delta u = (\nabla_{{\bf R}^m} F)(u) +
G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon
{\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$
is a smooth potential which is positive, phase-rotation-invariant, and
homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$,
and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly
supported forcing term. This generalises the scalar defocusing nonlinear
Schr\"odinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1}
|v|^{p+1}$. In this paper we study the supercritical case where $d \geq 3$ and
$p > 1 + \frac{4}{d-2}$. We show that in this case, there exists a smooth
potential $F$ for some sufficiently large $m$, positive and homogeneous of
order $p+1$ outside of the unit ball, and a smooth compactly choice of initial
data $u(0)$ and forcing term $G$ for which the solution develops a finite time
singularity. In fact the solution is locally discretely self-similar with
respect to parabolic rescaling of spacetime. This demonstrates that one cannot
hope to establish a global regularity result for the scalar defocusing NLS
unless one uses some special property of that equation that is not shared by
these defocusing nonlinear Schr\"odinger systems.
  As in a previous paper of the author considering the analogous problem for
the nonlinear wave equation, the basic strategy is to first select the mass,
momentum, and energy densities of $u$, then $u$ itself, and then finally design
the potential $F$ in order to solve the required equation.
