Hyperelliptic threefolds
In dimension $3$ the classification is complete (Uchida–Yoshihara, Lange, Catanese–Demleitner): the holonomy group is $\mathrm{D}_4$ or abelian of the form $\mathrm{C}_{d_1}\times \mathrm{C}_{d_2}$, giving $17$ groups in all. Each carries one or more tangent representations, listed with the invariants computed from them. The list is not cut out by representation theory alone: $\mathrm{A}_4$ satisfies every necessary condition in dimension $3$ and still does not occur, see completeness in dimension 4.
| group | variety | $\#G$ | tangent representation | $q$ | $\mathrm{h}^{2,0}$ | $\mathrm{h}^{3,0}$ | $\operatorname{ord} \omega_X$ | moduli | $\mathbf{D}^{\mathrm{b}}(X)$ |
|---|---|---|---|---|---|---|---|---|---|
| $\mathrm{C}_{2}$ | no. 1 | 2 | $\operatorname{diag}(1, 1, -1)$ | 2 | 1 | 0 | 2 | 5 | indec. |
| $\mathrm{C}_{2}$ | no. 2 | 2 | $\operatorname{diag}(1, -1, -1)$ | 1 | 1 | 1 | 1 | 5 | indec. |
| $\mathrm{C}_{3}$ | no. 1 | 3 | $\operatorname{diag}(1, 1, \zeta_{3})$ | 2 | 1 | 0 | 3 | 4 | indec. |
| $\mathrm{C}_{3}$ | no. 2 | 3 | $\operatorname{diag}(1, \zeta_{3}, \zeta_{3})$ | 1 | 0 | 0 | 3 | 1 | indec. |
| $\mathrm{C}_{3}$ | no. 3 | 3 | $\operatorname{diag}(1, \zeta_{3}, \zeta_{3}^{2})$ | 1 | 1 | 1 | 1 | 3 | indec. |
| $\mathrm{C}_{4}$ | no. 1 | 4 | $\operatorname{diag}(1, 1, i)$ | 2 | 1 | 0 | 4 | 4 | indec. |
| $\mathrm{C}_{4}$ | no. 2 | 4 | $\operatorname{diag}(1, i, i)$ | 1 | 0 | 0 | 2 | 1 | indec. |
| $\mathrm{C}_{4}$ | no. 3 | 4 | $\operatorname{diag}(1, i, -1)$ | 1 | 0 | 0 | 4 | 2 | indec. |
| $\mathrm{C}_{4}$ | no. 4 | 4 | $\operatorname{diag}(1, i, -i)$ | 1 | 1 | 1 | 1 | 3 | indec. |
| $\mathrm{C}_{5}$ | no. 1 | 5 | $\operatorname{diag}(1, \zeta_{5}, \zeta_{5}^{2})$ | 1 | 0 | 0 | 5 | 1 | indec. |
| $\mathrm{C}_{6}$ | no. 1 | 6 | $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 2 | 1 | 0 | 6 | 4 | indec. |
| $\mathrm{C}_{6}$ | no. 2 | 6 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 6 | 2 | indec. |
| $\mathrm{C}_{6}$ | no. 3 | 6 | $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 3 | 2 | indec. |
| $\mathrm{C}_{6}$ | no. 4 | 6 | $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3})$ | 1 | 0 | 0 | 6 | 1 | indec. |
| $\mathrm{C}_{6}$ | no. 5 | 6 | $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3}^{2})$ | 1 | 0 | 0 | 2 | 1 | indec. |
| $\mathrm{C}_{6}$ | no. 6 | 6 | $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3})$ | 1 | 0 | 0 | 3 | 1 | indec. |
| $\mathrm{C}_{6}$ | no. 7 | 6 | $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3}^{2})$ | 1 | 1 | 1 | 1 | 3 | indec. |
| $\mathrm{C}_{8}$ | no. 1 | 8 | $\operatorname{diag}(1, \zeta_{8}, \zeta_{8}^{3})$ | 1 | 0 | 0 | 2 | 1 | indec. |
| $\mathrm{C}_{8}$ | no. 2 | 8 | $\operatorname{diag}(1, \zeta_{8}, \zeta_{8}^{5})$ | 1 | 0 | 0 | 4 | 1 | indec. |
| $\mathrm{C}_{10}$ | no. 1 | 10 | $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, \zeta_{5}, \zeta_{5}^{2})$ | 1 | 0 | 0 | 5 | 1 | indec. |
| $\mathrm{C}_{12}$ | no. 1 | 12 | $\operatorname{diag}(1, i, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 12 | 1 | indec. |
| $\mathrm{C}_{12}$ | no. 2 | 12 | $\operatorname{diag}(1, i, -1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 12 | 1 | indec. |
| $\mathrm{C}_{12}$ | no. 3 | 12 | $\operatorname{diag}(1, i, -i)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3})$ | 1 | 0 | 0 | 3 | 1 | indec. |
| $\mathrm{C}_{12}$ | no. 4 | 12 | $\operatorname{diag}(1, i, i)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3}^{2})$ | 1 | 0 | 0 | 2 | 1 | indec. |
| $\mathrm{C}_{2} \times \mathrm{C}_{2}$ | no. 1 | 4 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, -1)$ | 1 | 0 | 0 | 2 | 3 | indec. |
| $\mathrm{C}_{2} \times \mathrm{C}_{2}$ | no. 2 | 4 | $\operatorname{diag}(-1, 1, -1)$, $\operatorname{diag}(1, -1, -1)$ | 0 | 0 | 1 | 1 | 3 | indec. |
| $\mathrm{C}_{4} \times \mathrm{C}_{2}$ | no. 1 | 8 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, i)$ | 1 | 0 | 0 | 4 | 2 | indec. |
| $\mathrm{C}_{4} \times \mathrm{C}_{2}$ | no. 2 | 8 | $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, i, i)$ | 1 | 0 | 0 | 2 | 1 | indec. |
| $\mathrm{C}_{6} \times \mathrm{C}_{2}$ | no. 1 | 12 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 6 | 2 | indec. |
| $\mathrm{C}_{6} \times \mathrm{C}_{2}$ | no. 2 | 12 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3})$ | 1 | 0 | 0 | 6 | 1 | indec. |
| $\mathrm{C}_{6} \times \mathrm{C}_{2}$ | no. 3 | 12 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, \zeta_{3}, \zeta_{3}^{2})$ | 1 | 0 | 0 | 2 | 1 | indec. |
| $\mathrm{C}_{12} \times \mathrm{C}_{2}$ | no. 1 | 24 | $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, i, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 12 | 1 | indec. |
| $\mathrm{C}_{3} \times \mathrm{C}_{3}$ | no. 1 | 9 | $\operatorname{diag}(1, \zeta_{3}, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 3 | 1 | indec. |
| $\mathrm{C}_{6} \times \mathrm{C}_{3}$ | no. 1 | 18 | $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, \zeta_{3}, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 6 | 1 | indec. |
| $\mathrm{C}_{6} \times \mathrm{C}_{3}$ | no. 2 | 18 | $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, \zeta_{3}, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 3 | 1 | indec. |
| $\mathrm{C}_{4} \times \mathrm{C}_{4}$ | no. 1 | 16 | $\operatorname{diag}(1, i, 1)$, $\operatorname{diag}(1, 1, i)$ | 1 | 0 | 0 | 4 | 1 | indec. |
| $\mathrm{C}_{6} \times \mathrm{C}_{6}$ | no. 1 | 36 | $\operatorname{diag}(1, -1, 1)$, $\operatorname{diag}(1, 1, -1)$, $\operatorname{diag}(1, \zeta_{3}, 1)$, $\operatorname{diag}(1, 1, \zeta_{3})$ | 1 | 0 | 0 | 6 | 1 | indec. |
| $\mathrm{D}_4$ | no. 1 | 8 | $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, -1, -1)$, $\operatorname{diag}(1, -1, -1)$ | 0 | 0 | 1 | 1 | 2 | indec. |
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