  Using suitable modified energies we study higher order Sobolev norms' growth
in time for the nonlinear Schr\"odinger equation (NLS) on a generic $2d$ or
$3d$ compact manifold. In $2d$ we extend earlier results that dealt only with
cubic nonlinearities, and get polynomial in time bounds for any higher order
nonlinearities. In $3d$, we prove that solutions to the cubic NLS grow at most
exponentially, while for sub-cubic NLS we get polynomial bounds on the growth
of the $H^2$-norm.
