\mathrm{C}_{4} \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: $\mathrm{C}_{4} \times \mathrm{C}_{2}$, order $8$, SmallGroup $[8,2]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(-1, 1, 1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, i, -1, -1)$
order of $\omega_X$: 4
number of moduli: 3
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 $\dim \mathrm{HH}^\bullet$: $(1, 0, 3, 10, 7, 4, 5, 2, 0)$

Hodge diamond

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

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

1
00
030
1441
01510
0220
131
11
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
040
1331
00600
1331
040
00
1

$\omega_X^{\otimes 1}$

0
01
110
1240
03531
1240
110
01
0

$\omega_X^{\otimes 2}$

0
11
121
0330
02420
0330
121
11
0

$\omega_X^{\otimes 3}$

0
10
011
0421
13530
0421
011
10
0

hyperelliptic(.ncag).info