Kahler-Einstein and Kahler scalar flat supermanifolds

Two results regarding Kähler supermanifolds with potential $K=A+Cθ\barθ$ are shown. First, if the supermanifold is Kähler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with Kähler potential $A$) has constant scalar curvature. As a corollary, every constant scalar curvature Kähler supermanifold has a unique superextension to a Kähler-Einstein supermanifold of one higher fermionic dimension. Second, if the supermanifold is itself scalar flat, then its base satisfies the equation $ϕ^{\bar ji}ϕ_{i\bar j}=2Δ_0 S_0 + R_0^{\bar ji}R_{0i\bar j} - S_0^2,$ where $Δ_0$ is the Laplace operator, $S_0$ is the scalar curvature, and $R_{0i\bar j}$ is the Ricci tensor of the base, and $ϕ$ 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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