On second-order characterizability

We investigate the extent of second-order characterizable structures by extending Shelah's Main Gap dichotomy to second-order logic. For this end we consider a countable complete first-order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second-order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second-order logic with Henkin semantics.