Fake quadrics from irreducible lattices acting on the product of upper half planes

TL;DR

This paper constructs and classifies fake quadrics, which are complex surfaces resembling a smooth quadric, by analyzing irreducible lattices acting on the product of upper half planes, and provides explicit examples and classifications.

Contribution

It introduces new examples of fake quadrics from irreducible lattices and advances the classification of such surfaces over real quadratic fields.

Findings

Examples of fake quadrics from irreducible lattices are provided.

A classification of arithmetic lattices over real quadratic fields defining fake quadrics is developed.

General results towards a complete classification of fake quadrics are presented.

Abstract

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of all such fake quadrics.