Pluricanonical Geometry of Varieties Isogenous to a Product: Chevalley-Weil Theory and Pluricanonical Decompositions of Abelian Covers

Abstract: We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form $ X = (C_1 \times \dots \times C_n)/G $ with $g(C_i)\ge 2$ and $G$ acting freely. We establish the Chevalley-Weil formula for pluricanonical representations of a curve with a finite group action and a decomposition theorem for pluricanonical systems of abelian covers. These tools allow an explicit study of geometric properties of $X$, 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.