\mathrm{C}_{12} \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: $\mathrm{C}_{12} \times \mathrm{C}_{2}$, order $24$, SmallGroup $[24,9]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, -1, 1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, \zeta_{3}, 1, \zeta_{3})$
$\rho(g_{3}) = \operatorname{diag}(1, 1, i, -1)$
order of $\omega_X$: 12
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 $\dim \mathrm{HH}^\bullet$: $(1, 2, 1, 0, 0, 0, 0, 0, 0)$

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
010
0000
00000
0000
000
00
0

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

Since $\operatorname{ord} \omega_X = 12$, the twists by powers of the canonical bundle form a finite package of $12$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
11
040
0330
00600
0330
040
11
1

$\omega_X^{\otimes 1}$

0
00
000
0001
00011
0001
000
00
0

$\omega_X^{\otimes 2}$

0
00
100
1100
02000
1100
100
00
0

$\omega_X^{\otimes 3}$

0
01
011
0230
02420
0230
011
01
0

$\omega_X^{\otimes 4}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 5}$

0
00
000
0100
01100
0100
000
00
0

$\omega_X^{\otimes 6}$

0
00
020
0220
00400
0220
020
00
0

$\omega_X^{\otimes 7}$

0
00
000
0010
00110
0010
000
00
0

$\omega_X^{\otimes 8}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 9}$

0
10
110
0320
02420
0320
110
10
0

$\omega_X^{\otimes 10}$

0
00
001
0011
00020
0011
001
00
0

$\omega_X^{\otimes 11}$

0
00
000
1000
11000
1000
000
00
0

hyperelliptic(.ncag).info