The classification of surfaces of general type with $q = p_g = 2$, and new progress for the case $K^2 = 7$

I will discuss the classification of minimal surfaces of general type with $p_g = q$. Beauville showed that $q \leq 4$, with equality precisely for products of two curves of genus $2$. For $q = 3$, the classification gives two possibilities up to birational equivalence.
The talk will focus on the case $p_g = q = 2$, where $4 \leq K^2 \leq 9$. The classification is complete for $K^2 = 4, 5$ and $6$, following work of Du, Jiang and Zhang and earlier contributions. The first open case is $K^2 = 7$.
For $K^2 = 7$, let $d$ denote the degree of the Albanese map. Our main result describes the corresponding loci in the moduli space. For $d = 2$, the locus has exactly three connected components, each of dimension two. For $d = 3$, the locus is irreducible of dimension three and consists precisely of the CFPP surfaces.

This is joint work in progress with F. Catanese.