Gauged N=8 supergravity in four dimensions is now known to admit a deformation characterized by a real parameter ω lying in the interval 0<ω<π/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 ω=0. For non-vanishing ω, 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

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... nditions. We then construct in detail the consistent truncation of the N=8 theory to give ω-deformed SO(6) gauged N=6 supergravity, again with ω in the range 0<ω<π/8. We show that this theory admits fully N=6 supersymmetry-preserving boundary conditions not only for ω=0, but also for ω=π/8. These two theories are related by a U(1) electric-magnetic duality. We observe that the only three-point functions that depend on ω 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 ω=π/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 ω-deformed N=6 gauged supergravities can be obtained via consistent reductions from the eleven-dimensional or ten-dimensional type IIA supergravities.