  The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger
equation with periodic boundary data is considered. We show that the problem is
globally well posed in $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and
2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. We use
the ``$I$-method''. This method allows one to introduce a modification of the
energy functional that is well defined for initial data below the $H^{1}({\Bbb
T^{d}})$ threshold. The main ingredient in the proof is a "refinement" of the
Strichartz's estimates that hold true for solutions defined on the rescaled
space, $\Bbb T^{d}_{\lambda} = \Bbb R^{d}/{\lambda \Bbb Z^{d}}$, $d=1,2$.
