Proposition 21.13.2849A9A66B3A4D

Proof. Let $F_0$ be a smooth convex function on $N_{\mathbb {R}}$ such that $\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^* F_0$ can be extended to a Kähler form on $X$. For example, Guillemin's construction (21.12.9) with respect to a suitable Delzant polytope gives such an example.

Write $\omega $ for a Kähler form on $X$ extending $\mathrm{dd}^{\mathrm{c}}\operatorname{Trop}^*F_0$. Then for any large enough $C>0$, $\theta +C\omega $ is a Kähler form. So we conclude from proposition 21.12.24 that

By the multi-linearity of the non-pluripolar product proposition 13.11.4 6, the left-hand side is a polynomial in $C$. By the expansion formula for the mixed real Monge–Ampère measure theorem 21.11.58 1, the right-hand side is also a polynomial in $C$. We conclude that the same holds for $C=0$. Therefore, (21.13.29) follows.

(21.13.30) is a direct consequence of (21.13.29). $\square$