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 we list the tangent representations satisfying the necessary conditions
- $\rho$ is faithful;
- every $g \neq 1$ has eigenvalue $1$;
- $\rho \oplus \overline{\rho}$ is realizable over $\mathbb{Z}$.
Condition (3) is what the lattice imposes: $T = V/\Lambda$ gives $\Lambda \otimes_{\mathbb{Z}} \mathbb{C} \cong \rho \oplus \overline{\rho}$, so $\rho \oplus \overline{\rho}$ comes from a $\mathbb{Z}[G]$-module. Since a $\mathbb{Q}[G]$-module always contains a $G$-stable lattice, (3) is equivalent to realizability over $\mathbb{Q}$, and it implies the weaker condition that the characteristic polynomial of $\rho(g) \oplus \overline{\rho}(g)$ has integral coefficients for every $g$ (equivalently, that the character of $\rho \oplus \overline{\rho}$ is rational-valued). The two differ in general, by the Schur index, but they agree on every candidate representation of every group in this database, which is what makes the character-theoretic test used here legitimate (verification).
The conditions are not sufficient
They are necessary, and they are known not to be sufficient. Already in dimension $3$: the $3$-dimensional irreducible representation of $\mathrm{A}_4$ is faithful, every non-trivial element has eigenvalue $1$ (an element of order $2$ acts with eigenvalues $1, -1, -1$, one of order $3$ with $1, \zeta_3, \zeta_3^2$), and it is realizable over $\mathbb{Z}$: it is generated by the integral matrices
$$ \operatorname{diag}(1,-1,-1), \qquad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, $$hence so is $\rho \oplus \overline{\rho} = 2\rho$. But $\mathrm{A}_4$ does not occur: the holonomy group of a hyperelliptic threefold is $\mathrm{D}_4$ or abelian (Catanese–Demleitner).
The same happens in dimension $4$ at the level of groups. The group with SmallGroup id $[64,20]$ carries $64$ four-dimensional representations satisfying (1)–(3), and it is not among the $79$ hyperelliptic groups, so none of them is realized. (Both examples are due to Demleitner.)
Finally, a group can occur while one of its representations does not: in dimension $3$ the group $\mathrm{D}_4$ has two representations satisfying (1)–(3), and only the one with $q = 0$ is realized. The other is "Case 2" of Catanese–Demleitner, proven not to occur, and this site drops it by hand.
Deciding which representations are realized, and with which moduli, 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 $\mathrm{Heis}(3)$.
What the database counts
That 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 since the conditions are not sufficient it 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
- F. Catanese, A. Demleitner, Hyperelliptic threefolds with group $\mathrm{D}_4$. arXiv:1805.01835
- 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