Singular twisted sums generated by complex interpolation

Although our methods are quite general, we are mainly concerned with the choice of X as either a Hilbert space or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces (which includes the only known example so far: the Kalton-Peck space Z_2). In the second case we obtain the first example of an H.I. twisted sum of an H.I. space.