  In this paper, we consider the nonlinear Schr\"odinger equation $iu_t +\Delta
u= \lambda |u|^{\frac {4} {N-4}} u$ in $\R^N $, $N\ge 5$, with $\lambda \in
\C$. We prove local well-posedness (local existence, unconditional uniqueness,
continuous dependence) in the critical space $\dot H^2 (\R^N) $.
