  In this paper we prove some multi-linear Strichartz estimates for solutions
to the linear Schr\"odinger equations on torus $\T^n$. Then we apply it to get
some local well-posed results for nonlinear Schr\"odinger equation in critical
$H^{s}(\T^n)$ spaces. As by-products, the energy critical global well-posed
results and energy subcritical global well-posed results with small initial
data are also obtained.
