  Recently, there have been studies of parametric resonance decay of
oscillating real homogeneous cosmological scalar fields, in both the
narrow-band and broad-band case, primarily within the context of inflaton decay
and (p)reheating. However, many realistic models of particle cosmology, such as
supersymmetric ones, inherently involve complex scalar fields. In the
oscillations of complex scalars, a relative phase between the oscillations in
the real and imaginary components may prevent the violations of adiabaticity
that have been argued to underly broad-band parametric resonance. In this
paper, we give a treatment of parametric resonance for the decay of homogeneous
complex scalar fields, analyzing properties of the resonance in the presence of
out of phase oscillations of the real and imaginary components. For
phase-invariant coupling of the driving parameter field to the decay field, and
Mathieu type resonance, we give an explicit mapping from the complex resonance
case to an equivalent real case with shifted resonance parameters. In addition,
we consider the consequences of the complex field case as they apply to
``instant preheating,'' the explosive decay of non-convex potentials, and
resonance in an expanding FRW universe. Applications of our considerations to
supersymmetric cosmological models will be presented elsewhere.
