Fixed angle scattering: recovery of singularities and its limitations

We prove that in dimension $n\ge 2$ the main singularities of a complex potential $q$ having a certain a priori regularity are contained in the Born approximation $q_{\theta }$ constructed from fixed angle scattering data. Moreover, $q-q_{\theta }$ can be up to one derivative more regular than $q$ in the Sobolev scale. In fact, this result is optimal, we construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for $n>3$, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.