  Gauged N=8 supergravity in four dimensions is now known to admit a
deformation characterized by a real parameter $\omega$ lying in the interval
$0\le\omega\le \pi/8$. We analyse the fluctuations about its anti-de Sitter
vacuum, and show that the full N=8 supersymmetry can be maintained by the
boundary conditions only for $\omega=0$. For non-vanishing $\omega$, and
requiring that there be no propagating spin s>1 fields on the boundary, we show
that N=3 is the maximum degree of supersymmetry that can be preserved by the
boundary conditions. We then construct in detail the consistent truncation of
the N=8 theory to give $\omega$-deformed SO(6) gauged N=6 supergravity, again
with $\omega$ in the range $0\le\omega\le \pi/8$. We show that this theory
admits fully N=6 supersymmetry-preserving boundary conditions not only for
$\omega=0$, but also for $\omega=\pi/8$. These two theories are related by a
U(1) electric-magnetic duality. We observe that the only three-point functions
that depend on $\omega$ involve the coupling of an SO(6) gauge field with the
U(1) gauge field and a scalar or pseudo-scalar field. We compute these
correlation functions and compare them with those of the undeformed N=6 theory.
We find that the correlation functions in the $\omega=\pi/8$ theory
holographically correspond to amplitudes in the U(N)_k x U(N)_{-k} ABJM model
in which the U(1) Noether current is replaced by a dynamical U(1) gauge field.
We also show that the $\omega$-deformed N=6 gauged supergravities can be
obtained via consistent reductions from the eleven-dimensional or
ten-dimensional type IIA supergravities.
