# A pair of rigid surfaces with $p_g=q=2$ and $K^2=8$ whose universal cover is not the bidisk

## TL;DR

This paper constructs and classifies a unique pair of complex surfaces with specific invariants, demonstrating their universal cover is not the bidisk, thus completing the classification of such surfaces with given invariants.

## Contribution

It introduces a new pair of rigid surfaces with particular invariants and proves their uniqueness among surfaces with these properties and Albanese map degree 2.

## Key findings

- Constructed two complex-conjugated rigid surfaces with specified invariants.
- Proved these are the only such surfaces with these invariants and Albanese map degree 2.
- Showed the universal cover of these surfaces is not biholomorphic to the bidisk.

## Abstract

We construct two complex-conjugated rigid surfaces with $p_g=q=2$ and $K^2=8$ whose universal cover is not biholomorphic to the bidisk. We show that these are the unique surfaces with these invariants and Albanese map of degree $2$, apart the family of product-quotient surfaces constructed by Penegini. This completes the classification of surfaces with $p_g=q=2, K^2=8$ and Albanese map of degree $2$.