\mathrm{C}_{6} \times \mathrm{C}_{3} in dimension 3

$\mathrm{C}_{6} \times \mathrm{C}_{3}$ in dimension 3

holonomy group: $\mathrm{C}_{6} \times \mathrm{C}_{3}$, order $18$, SmallGroup $[18,5]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, -1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, \zeta_{3}, 1)$
$\rho(g_{3}) = \operatorname{diag}(1, 1, \zeta_{3})$
order of $\omega_X$: 3
number of moduli: 1
irregularity $q = \dim \operatorname{Aut}^0(X)$: 1
Albanese: the Albanese variety has dimension $1$; the general fiber has dimension $2$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 2, 1, 0, 0, 0, 0)$

1
1 1
0 3 0
0 2 2 0
0 3 0
1 1
1

1
11
010
0000
000
00
0

twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$

Since $\operatorname{ord} \omega_X = 3$, the twists by powers of the canonical bundle form a finite package of $3$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
11
030
0220
030
11
1

$\omega_X^{\otimes 1}$

0
00
001
0011
001
00
0

$\omega_X^{\otimes 2}$

0
00
100
1100
100
00
0

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

P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi

hyperelliptic(.ncag).info