  Preheating after inflation may over-produce primordial black holes (PBH's) in
many regions of parameter space. As an example we study two-field models with a
massless self-interacting inflaton, taking into account second order field and
metric backreaction effects as spatial averages. We find that a complex quilt
of parameter regions above the Gaussian PBH over-production threshold emerges
due to the enhancement of curvature perturbations on all scales. It should be
possible to constrain realistic models of inflation through PBH over-production
although many issues, such as rescattering and non-Gaussianity, remain unsolved
or unexplored.
