Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers

We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form $X=(C_1\times \cdots \times C_n)\mathrm{/}G$ with $g(C_i)\ge 2$ and $G$ acting freely. For this purpose, we provide a technical result which is of general interest: a decomposition theorem for pluricanonical systems of abelian covers. This theorem provides an effective tool for the explicit study of geometric properties, such as base loci and the birationality of pluricanonical maps. For threefolds isogenous to a product, we prove that the 4-canonical map is birational for $p_g\ge 5$ and construct an example attaining the maximal canonical degree for this class of threefolds. In this example, the canonical map is the normalization of its image, which admits isolated non-normal singularities. Computational classifications also reveal threefolds where the bicanonical map fails to be birational, even in the absence of genus-2 fibrations. This illustrates an interesting phenomenon similar to the non-standard case for surfaces.