The space of polarized compactifications of an affine variety

Let $X$ be a normal affine variety over an algebraically closed field of characteristic zero. Through the lens of non-Archimedean geometry, we study the space of polarized compactifications of $X$, consisting of pairs $(\bar{X},D)$ where $\bar{X}$ is a normal projective variety containing $X$ as a dense open subset, and $D$ is an effective ample $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor with support equal to the complement of $X$ in $\bar{X}$. We identify this space with a natural class of functions–the Fubini–Study functions–on the Berkovich analytification of $X$. Through a Rees correspondence, we also give a bijection with the space of homogeneous $\mathbb{Q}$-valued, finitely generated negative non-Archimedean norms (or filtrations) on the coordinate ring of $X$. Using the above identifications, we equip the space of polarized compactifications, and numerical classes thereof, with natural metrics and identify the metric completions. The latter include a space of continuous plurisubharmonic functions in the sense of Chambert-Loir and Ducros, and a space of birational curve classes as studied by Dang and Favre.