  We investigate the recently proposed hybrid inflation models with two stages
of inflation. We show that quantum fluctuations at the time corresponding to
the phase transition between the two inflationary stages can trigger the
formation of a large number of inflating topological defects. In order to study
density perturbations in these models we develop a new method to calculate
density perturbations in a system of two scalar fields. We show that density
perturbations in hybrid inflation models of the new type can be very large on
the scale corresponding to the phase transition. The resulting density
inhomogeneities lead to a copious production of black holes. This could be an
argument against hybrid inflation models with two stages of inflation. However,
we find a class of models where this problem can be easily avoided. The number
of black holes produced in these models can be made extremely small, but in
general it could be sufficiently large to have important cosmological and
astrophysical implications. In particular, for certain values of parameters
these black holes may constitute the dark matter in the universe. It is also
possible to have hybrid models with two stages of inflation where the black
hole production is not suppressed, but where the typical masses of the black
holes are very small. Such models lead to a completely different thermal
history of the universe, where post-inflationary reheating occurs via black
hole evaporation.
