ScalingStacks

While this paper focuses on the algebraic case, one may wonder what happens for ‘transcendental families’. A prototypical examples is the family of degree $n+2$ hypersurfaces in $\mathbb{CP}^{n+1}$:

$$X_{t}=\{\sum_{I}a_{I}t^{\lambda_{I}}x^{I}=0\}\subset\mathbb{CP}^{n+1},$$

where $x^{I}$ denote the degree $n+2$ monomials, $a_{I}$ are coefficients chosen suitably generically, and $\lambda_{I}$ are exponents chosen suitably. When $\lambda_{I}$ are sufficiently irrational, this does not fit into our framework, yet the metric SYZ conjecture makes sense.

There seem to be two natural strategies. One is to make the NA pluripotential theory work over NA fields without a discrete valuation (cf. [7] for the latest progress), and the other is to develop the framework of real MA equation on polyhedral sets such as $Sk(X)$ without explicit reference to NA geometry.