\mathrm{C}_{4} \times \mathrm{A}_4 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: $\mathrm{C}_{4} \times \mathrm{A}_4$, order $48$, SmallGroup $[48,31]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, 1, i)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$
$\rho(g_{3}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{4}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho(g_{5}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho \cong$ $\chi_{ 4 }$ $\oplus$ $\chi_{ 13 }$
order of $\omega_X$: 4
number of moduli: 1
irregularity $q = \dim \operatorname{Aut}^0(X)$: 0
Albanese: the Albanese variety is a point; the fiber is $X$ itself (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposability open (why)
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 0, 1, 4, 1, 0, 1, 0, 0)$

Hodge diamond

1
0 0
0 2 0
1 1 1 1
0 0 2 0 0
1 1 1 1
0 2 0
0 0
1

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

1
00
010
1111
00100
0000
010
00
0

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

Since $\operatorname{ord} \omega_X = 4$, the twists by powers of the canonical bundle form a finite package of $4$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
00
020
1111
00200
1111
020
00
1

$\omega_X^{\otimes 1}$

0
01
000
0010
01111
0010
000
01
0

$\omega_X^{\otimes 2}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 3}$

0
10
000
0100
11110
0100
000
10
0

hyperelliptic(.ncag).info