Sharp global well-posedness for a higher order Schr\"odinger equation

Using the theory of almost conserved energies and the ``I-method’’ developed by Colliander, Keel, Staffilani, Takaoka and Tao, we prove that the initial value problem for a higher order Schr"odinger equation is globally well-posed in Sobolev spaces of order $s>1\mathrm{/}4$.