  The seven and nine dimensional geometries associated with certain classes of
supersymmetric $AdS_3$ and $AdS_2$ solutions of type IIB and D=11 supergravity,
respectively, have many similarities with Sasaki-Einstein geometry. We further
elucidate their properties and also generalise them to higher odd dimensions by
introducing a new class of complex geometries in $2n+2$ dimensions, specified
by a Riemannian metric, a scalar field and a closed three-form, which admit a
particular kind of Killing spinor. In particular, for $n\ge 3$, we show that
when the geometry in $2n+2$ dimensions is a cone we obtain a class of
geometries in $2n+1$ dimensions, specified by a Riemannian metric, a scalar
field and a closed two-form, which includes the seven and nine-dimensional
geometries mentioned above when $n=3,4$, respectively. We also consider various
ansatz for the geometries and construct infinite classes of explicit examples
for all $n$.
