  We study the time of existence of the solutions of the following
Schr\"odinger equation $$i\psi_t = (-\Delta)^s \psi +f(|\psi|^2)\psi, x \in
\mathbb S^d, or x\in\T^d$$ where $(-\Delta)^s$ stands for the spectrally
defined fractional Laplacian with $s>1/2$ and $f$ a smooth function. We prove
an almost global existence result for almost all $s>1/2$.
