Commensurability Classes of Fake Quadrics

TL;DR

This paper classifies all irreducible fake quadrics by their fundamental group commensurability classes, introducing new techniques to bound arithmetic invariants of these surfaces and related arithmetic manifolds.

Contribution

It provides an explicit classification of fake quadrics based on their fundamental groups and develops novel methods to bound their arithmetic invariants.

Findings

Complete classification of irreducible fake quadrics by commensurability class.

New bounds on arithmetic invariants of fake quadrics.

Techniques applicable to arithmetic manifolds of bounded volume.

Abstract

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of SL_2 over a number field.