Completeness in dimension 4
In dimensions $2$ and $3$ the classification of hyperelliptic varieties is complete: every entry on this site corresponds to an actual variety, and the list is exhaustive (Bagnera–De Franchis; Uchida–Yoshihara, Lange, and Catanese–Demleitner).
In dimension $4$ the situation is different. Demleitner classifies the $79$ finite groups that can occur as the holonomy group, and for each group the tangent representations satisfying the necessary conditions
- $\rho$ is faithful;
- every $g \neq 1$ has eigenvalue $1$;
- $\rho \oplus \overline{\rho}$ is realizable over $\mathbb{Z}$.
These conditions are necessary for a hyperelliptic fourfold, but not known to be sufficient group by group: whether a given representation is realized by an actual fourfold, and the moduli of such, requires a separate analysis of the free actions (the translation parts). This is carried out only in special cases so far, for instance the $16$ rigid fourfolds with holonomy $\mathrm{C}_3^2$ or $\mathrm{Heis}(3)$.
This last example also illustrates what the database counts. Every page here is a tangent representation $(G,\rho)$, not a variety. Exactly two representations are rigid ($q = 0$ and $0$ moduli): one for $\mathrm{C}_3^2$ and one for $\mathrm{Heis}(3)$. Demleitner–Gleissner's $16$ rigid fourfolds are the varieties these two representations realize, counted up to biholomorphism: different tori and translation parts on the same $(G,\rho)$ give different rigid fourfolds. The database stops at the representation and its $(G,\rho)$-invariants, so it shows two entries where the finer classification finds sixteen varieties.
Consequently the dimension-4 list contains every hyperelliptic fourfold, but may contain representations that are not realized. Object pages that are not known to be realized carry a caution mark; those that are known to occur do not.
References
- 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
- F. Catanese, A. Demleitner, The classification of hyperelliptic threefolds, Groups Geom. Dyn. 14 (2020) 1447–1454. MR4186481 doi
- A. Demleitner, The classification of hyperelliptic groups in dimension 4. arXiv:2211.07998
- A. Demleitner, C. Gleissner, The classification of rigid hyperelliptic fourfolds, Ann. Mat. Pura Appl. (4) 202 (2023) 1425–1450. MR4576947 doi
- P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi