A new family of surfaces with $p_g=q=2$ and $K^2=6$ whose Albanese map has degree $4$

TL;DR

This paper introduces a new family of minimal surfaces with specific invariants, constructed via quadruple covers of abelian surfaces, expanding the understanding of the moduli space of such surfaces.

Contribution

It constructs a new family of surfaces with given invariants, showing this family forms an irreducible component of the moduli space and analyzing its geometric properties.

Findings

The family provides an irreducible component of the moduli space.

The component is generically smooth of dimension 4.

It contains a known 2-dimensional family of product-quotient examples.

Abstract

We construct a new family of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$, whose Albanese map is a quadruple cover of an abelian surface with polarization of type $(1,3)$. We also show that this family provides an irreducible component of the moduli space of surfaces with $p_g=q=2$ and $K^2=6$. Finally, we prove that such a component is generically smooth of dimension 4 and that it contains the 2-dimensional family of product-quotient examples previously constructed by the first author. The main tools we use are the Fourier-Mukai transform and the Schr\"odinger representation of the finite Heisenberg group $\mathscr{H}_3$.