\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 $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, -1, -1, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{3}) = \operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3}^{2})$
order of $\omega_X$: 6
number of moduli: 1
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 $\dim \mathrm{HH}^\bullet$: $(1, 2, 1, 0, 6, 12, 6, 0, 0)$

Hodge diamond

1
1 1
0 6 0
0 5 5 0
0 0 10 0 0
0 5 5 0
0 6 0
1 1
1

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

1
11
010
0000
01410
1551
141
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
060
0550
001000
0550
060
11
1

$\omega_X^{\otimes 1}$

0
10
110
0501
04411
0501
110
10
0

$\omega_X^{\otimes 2}$

0
00
001
0011
00020
0011
001
00
0

$\omega_X^{\otimes 3}$

0
00
000
0110
01210
0110
000
00
0

$\omega_X^{\otimes 4}$

0
00
100
1100
02000
1100
100
00
0

$\omega_X^{\otimes 5}$

0
01
011
1050
11440
1050
011
01
0

hyperelliptic(.ncag).info