Moduli of Gorenstein Q-homology fake projective planes

Inspired by fake projective planes, this talk concerns normal projective surfaces whose Q-homology equals that of IP^2. 

I shall concentrate on the Gorenstein case of numerically trivial canonical bundle which is closely related to Enriques surfaces. 

In the complex case, I will give a complete classification of their moduli which turn out to be rational, 1-dimensional and defined over Q. 

Time permitting, I will also discuss subtleties in positive characteristic.