Remark 18 . [003W]

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Remark 18.

The motivation for toric Kähler metrics on $\mathbb{CP}^{n+1}$ is as follows. The toric property is a natural way to reduce general Kähler potentials to convex functions. The idea of extension to an ambient space, is based on

Proposition 6.8.

([19, Thm. B]) Let $(X,\omega)$ be a projective manifold with a Kähler form representing an integral class, and $Y$ be a smooth subvariety of $X$ . Then any $\phi\in PSH(Y,\omega|_{Y})$ extends to $\phi\in PSH(X,\omega)$ .