Moduli of Gorenstein $\mathbb{Q}$-homology projective planes

Abstract

We give a complete classification of complex $\mathbb{Q}$-homology projective planes with numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all moduli curves are rational and defined over $\mathbb{Q}$, and we determine all families explicitly using extremal rational elliptic surfaces and Enriques involutions of base change type.