= No-scale identity from degree-one homogeneity
{title2=$K_iK^{-1}_{ij}K_j=3$}
Let $K=-3\log\Gamma$ where $\Gamma>0$ is twice differentiable and homogeneous of degree one. If the Kähler Hessian is invertible, the <Euler theorem for homogeneous functions> identities give $\tau_iK_{ij}=3\Gamma_j/\Gamma$, then $K^{-1}_{ij}\Gamma_j/\Gamma=\tau_i/3$ and $\Gamma_iK^{-1}_{ij}\Gamma_j/\Gamma^2=1/3$. Multiplication by nine gives the no-scale identity. A positive degree-one $\Gamma$ need not induce an invertible or positive <Kähler metric>: $\Gamma=\tau_1+\tau_2$ gives a rank-one metric for $K$. The <Hessian matrix> $(\Gamma_{ij})$ itself always has the radial null vector $\tau$, whereas the <Hessian matrix> of $K$ can be invertible. In a larger theory the relevant full inverse metric, or a decoupled block, must obey the identity.
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