  We consider the 2-dimensional focusing mass critical NLS with an
inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From
standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions
with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow up
singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we
consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and
sufficient condition on $k$ to ensure the existence of critical mass finite
time blow up elements. Moreover, we give a complete classification in the
energy class of the minimal finite time blow up elements at a non degenerate
point, hence extending the pioneering work by Merle who treated the pseudo
conformal invariant case $k\equiv 1$.
