  Two results regarding K\"ahler supermanifolds with potential
$K=A+C\theta\bar\theta$ are shown. First, if the supermanifold is
K\"ahler-Einstein, then its base (the supermanifold of one lower fermionic
dimension and with K\"ahler potential $A$) has constant scalar curvature. As a
corollary, every constant scalar curvature K\"ahler supermanifold has a unique
superextension to a K\"ahler-Einstein supermanifold of one higher fermionic
dimension. Second, if the supermanifold is itself scalar flat, then its base
satisfies the equation $$ \phi^{\bar ji}\phi_{i\bar j}=2\Delta_0 S_0 +
R_0^{\bar ji}R_{0i\bar j} - S_0^2, $$ where $\Delta_0$ is the Laplace operator,
$S_0$ is the scalar curvature, and $R_{0i\bar j}$ is the Ricci tensor of the
base, and $\phi$ is some harmonic section on the base. Remarkably, precisely
this equation arises in the construction of certain supergravity
compactifications. Examples of bosonic manifolds satisfying the equation above
are discussed.
