Not hyperelliptic curves
The word hyperelliptic on this site has nothing to do with hyperelliptic curves. A hyperelliptic variety (also called a generalized hyperelliptic manifold) is a quotient
$$ X = T / G $$of a complex torus $T = V/\Lambda$ by a finite group $G$ acting freely and containing no translations. Equivalently, $G$ acts through a faithful tangent representation $\rho\colon G \to \mathrm{GL}(V)$, and each $g \in G$ acts as the affine map $z \mapsto \rho(g)z + \tau(g)$.
In dimension $2$ these are the classical bielliptic surfaces; calling them "hyperelliptic surfaces" is standard and goes back to Enriques–Severi and Bagnera–De Franchis. The terminology in arbitrary dimension is due to Lange.
See invariants from representation theory for how the numerical invariants are computed.
References
- G. Bagnera, M. de Franchis, Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti, Mem. Soc. Ital. Sci. (3) 15 (1908)
- K. Uchida, H. Yoshihara, Discontinuous groups of affine transformations of $\mathbb{C}^3$, Tôhoku Math. J. (2) 28 (1976) 89–94. MR400271 doi
- H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi