  We study the structure of linearized field equations in $N = 1$ chiral
supergravity (SUGRA) with a complex tetrad, as a preliminary to introducing
additional auxiliary fields in order that the supersymmetry (SUSY) algebra
close off shell. We follow the first-order formulation we have recently
constructed using the method of the usual $N = 1$ SUGRA. In particular, we see
how the real and imaginary parts of the complex tetrad are coupled to matter
fields in the weak field approximation. Starting from the linearized (free)
theory of $N = 1$ chiral SUGRA, we then construct a Lagrangian which is
invariant under local SUSY transformations to zeroth order of the gravitational
constant, and compare the results with the linearized field equations.
