The central result of NA pluripotential theory is the solution to the NA analogue of the Calabi conjecture. A good survey is [5].

Their strategy uses a variational method. There is a concave energy functional $\mathcal{E}$ on the space of continuous semipositive metrics on $L$ (equivalently viewed as continuous $\theta$-psh potentials $\phi$), whose first variation is given by the NA MA measure. One seeks a maximizer of the functional

$$F_{\mu}(\phi)=\mathcal{E}(\phi)-(L^{n})\int_{X_{K}^{an}}\phi d\mu,$$

by first enlarging the space of $\phi$ to a function space $PSH(X_{K}^{an},\theta)$ which is compact modulo the addition of a real constant; this is analogous to the $L^{1}$-compactness of $PSH(X,\omega)/\mathbb{R}$ in the Kähler setting. The notions of the NA MA measure and the energy functional extend naturally to the energy class functions inside $PSH(X_{K}^{an},\theta)$, much like in the complex pluripotential theory setting. One then shows the maximizer is in fact a critical point, namely a weak solution to the NA MA equation. This is subtle since small perturbations of functions in $PSH(X_{K}^{an},\theta)$ may fall outside of the class by losing positivity. One proves the continuity of the weak solution using analogues of Kolodziej’s estimates. The uniqueness of the solution again relies on the concavity of $\mathcal{E}$.

While this strategy shares a very similar logical structure with the complex analytic setting, the technical foundations are built upon intersection theory and vanishing theorems in birational geometry, instead of differential operators.

The main case of interest to us is when $X_{K}$ arises from polarized algebraic maximal degeneration of CY manifolds $L\to X$. Then NA pluripotential theory provides a solution to

$$MA(\left\lVert\cdot\right\rVert_{CY})=(L^{n})d\mu_{0},$$

(7)

where $d\mu_{0}$ is the Lebesgue measure supported on the essential skeleton $Sk(X)\subset X_{K}^{an}$ (cf. section 3.1).

Notice $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$ inherits a natural integral affine structures. Since the restriction of $\phi_{0}$ is convex on these faces by Prop. 3.5, its real MA measure makes sense, and by Prop. 3.7 it satisfies the real MA equation on $\text{Int}(\Delta_{J})$

$$\text{MA}_{\mathbb{R}}(\phi_{0})=\frac{(L^{n})}{n!}d\mu_{0}.$$

(8)

Then the regularity theory of real MA equation (cf. section 2.5) will apply, so we may view the comparison property as a regularity assumption on $\left\lVert\cdot\right\rVert_{CY}$. Some subtleties are discussed in [21, Appendix].