  We study solutions to the eleven-dimensional supergravity action, including
terms quartic and cubic in the Riemann curvature, that admit an
eight-dimensional compact space. The internal background is found to be a
conformally Kahler manifold with vanishing first Chern class. The metric
solution, however, is non-Ricci-flat even when allowing for a conformal
rescaling including the warp factor. This deviation is due to the possible
non-harmonicity of the third Chern-form in the leading order Ricci-flat metric.
We present a systematic derivation of the background solution by solving the
Killing spinor conditions including higher curvature terms. These are
translated into first-order differential equations for a globally defined real
two-form and complex four-form on the fourfold. We comment on the supersymmetry
properties of the described solutions.
