The Albanese morphism
For a hyperelliptic variety $X = T/G$ the Albanese morphism $\operatorname{alb}_X\colon X \to \operatorname{Alb}(X)$ is described explicitly by Belmans–Demleitner–Núñez. Write $V^G$ for the subspace fixed by $\rho(G)$ and $V_1 = (V^G)^\perp$ for its orthogonal complement with respect to a $G$-invariant Hermitian form, so that $V = V^G \oplus V_1$.
- The Albanese variety $\operatorname{Alb}(X)$ is an abelian variety of dimension $q = \dim V^G$ (the irregularity).
- The general Albanese fiber $F$ has dimension $n - q$, and it is an abelian variety or a (lower-dimensional) hyperelliptic variety. A fiber of dimension $1$ is an elliptic curve.
The fibration is isotrivial: for a fixed $X$ all its fibers are isomorphic (Belmans–Demleitner–Núñez, Theorem A). The fiber is $F \cong (\{a_0\}+A_1)/H$, where $A_1 = V_1/\Lambda_1$ and the fiber's tangent representation is $\rho|_{V_1}$ restricted to $H$; it is abelian when $H$ acts on $A_1$ by translations only and hyperelliptic otherwise.
What $(G,\rho)$ does not determine is the fiber itself. The holonomy $H \trianglelefteq G$ is the stabilizer of a general fiber, $H = \{\, g \in G : t_g^{(0)} \in K_0 \,\}$, the kernel of the translation action of $G$ on the Albanese torus (Belmans–Demleitner–Núñez, Lemma 23). It is a normal subgroup with $G/H$ abelian, so $[G,G] \subseteq H \subseteq \{\, g : \rho(g) \text{ has eigenvalue } 1 \text{ on } V_1 \,\}$, but its precise value, and even whether the fiber is abelian or hyperelliptic, depends on the translation cocycle $\tau$, not on $(G,\rho)$ alone. Moreover not every lower-dimensional hyperelliptic variety occurs as an Albanese fiber, and which do is a complex-structure (CM) question rather than a group-theoretic one (Belmans–Demleitner–Núñez, Proposition 35). For these reasons we record the fiber's dimension and that it is abelian or hyperelliptic, but do not attach a single database entry to it.