\mathrm{C}_{6} \times \mathrm{C}_{2} in dimension 4

This tangent representation satisfies the necessary conditions for a hyperelliptic fourfold, but it is not known whether it is realized by an actual variety. See completeness in dimension 4.

holonomy group: $\mathrm{C}_{6} \times \mathrm{C}_{2}$, order $12$, SmallGroup $[12,5]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, -1, -1, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{3}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3})$
order of $\omega_X$: 6
number of moduli: 2
irregularity $q = \dim \operatorname{Aut}^0(X)$: 1
Albanese: the Albanese variety has dimension $1$; the general fiber has dimension $3$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposability open (why)
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 2, 2, 2, 5, 8, 4, 0, 0)$

Hodge diamond

1
1 1
0 4 0
0 3 3 0
0 0 6 0 0
0 3 3 0
0 4 0
1 1
1

polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$

1
11
020
0110
01310
1331
121
00
0

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

Since $\operatorname{ord} \omega_X = 6$, the twists by powers of the canonical bundle form a finite package of $6$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
11
040
0330
00600
0330
040
11
1

$\omega_X^{\otimes 1}$

0
10
110
0311
02321
0311
110
10
0

$\omega_X^{\otimes 2}$

0
00
011
0121
00220
0121
011
00
0

$\omega_X^{\otimes 3}$

0
00
000
0220
02420
0220
000
00
0

$\omega_X^{\otimes 4}$

0
00
110
1210
02200
1210
110
00
0

$\omega_X^{\otimes 5}$

0
01
011
1130
12320
1130
011
01
0

hyperelliptic(.ncag).info