Topological reconstruction theorems over uncountable algebraically closed fields

Working over uncountable algebraically closed fields, we extend the theorems of Kollár-Lieblich-Olsson-Sawin on reconstructing varieties from their Zariski topological spaces.

In particular, we adapt their results to arbitrary quasi-projective varieties in arbitrary characteristic, and thus we give positive answers to each of the relevant `speculations' made by the original authors in our setting.

Our proofs use techniques from model theory: in particular, we employ a general model-theoretic setting for algebro-geometric reconstruction problems, known as the `Zilber trichotomy for ACF-relics'.