# On the cohomology of surfaces with $p_g = q = 2$ and maximal Albanese dimension

## TL;DR

This paper investigates the cohomology of complex surfaces with specific invariants, revealing a relationship with a K3 surface and proving significant conjectures for certain cases.

## Contribution

It introduces a geometric construction of a K3 partner for these surfaces and establishes the Tate and Mumford-Tate conjectures in particular moduli components.

## Key findings

- Cohomology described by Albanese variety and K3 surface
- Construction of a K3 partner and algebraic correspondence
- Proof of Tate and Mumford-Tate conjectures in specific cases

## Abstract

In this paper we study the cohomology of smooth projective complex surfaces $S$ of general type with invariants $p_g = q = 2$ and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface $X$ that we call the K3 partner of $S$. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner $X$ and an algebraic correspondence in $S \times X$ that relates the cohomology of $S$ and $X$. Finally, we prove the Tate and Mumford-Tate conjectures for those surfaces $S$ that lie in connected components of the Gieseker moduli space that contain a product-quotient surface.