  We study the global behavior of finite energy solutions to the
$d$-dimensional focusing nonlinear Schr\"odinger equation (NLS), $i \partial_t
u+\Delta u+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The
nonlinearity power $p$ and the dimension $d$ are such that the scaling index
$s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is
mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$
  For solutions with $\ME[u_0]<1$ ($\ME[u_0]$ stands for an invariant and
conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold
for scattering and blowup is given. Namely, if the renormalized gradient $\g_u$
of a solution $u$ to NLS is initially less than 1, i.e., $\g_u(0)<1,$ then the
solution exists globally in time and scatters in $H^1$ (approaches some linear
Schr\"odinger evolution as $t\to\pm\infty$); if the renormalized gradient
$\g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a
finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite
time.
  This work generalizes the results for the 3d cubic NLS obtained in a series
of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key
ingredients, the concentration compactness and localized variance, developed in
the context of the energy-critical NLS and Nonlinear Wave equations by Kenig
and Merle.
