Two results regarding Kahler supermanifolds with potential $K=A+Cthetabartheta$ are shown. First, if the supermanifold is Kahler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with Kahler potential $A$) has constant scalar curvature. As a corollary, every constant scalar curvature Kahler supermanifold has a unique superextension to a Kahler-Einstein supermanifold of one higher fermionic dimension. Second, if the supermanifold is itself scalar flat, then its base satisfies the equation $$ phi^{bar ji}phi_{ibar j}=2Delta_0 S_0 + R_0^{bar ji}R_{0ibar j} - S_0^2, $$ where $Delta_0$ is the Laplace operator, $S_0$ is the scalar curvature, and $R_{0ibar j}$ 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.
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