\mathrm{C}_{4} \times \mathrm{C}_{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: $\mathrm{C}_{4} \times \mathrm{C}_{4}$, order $16$, SmallGroup $[16,2]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, i, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, i)$
order of $\omega_X$: 4
number of moduli: 4
irregularity $q = \dim \operatorname{Aut}^0(X)$: 2
Albanese: the Albanese variety has dimension $2$; the general fiber has dimension $2$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 4, 6, 4, 1, 0, 0, 0, 0)$

Hodge diamond

1
2 2
1 6 1
0 6 6 0
0 2 10 2 0
0 6 6 0
1 6 1
2 2
1

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

1
22
141
0220
00100
0000
000
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
22
161
0660
021020
0660
161
22
1

$\omega_X^{\otimes 1}$

0
00
001
0022
00141
0022
001
00
0

$\omega_X^{\otimes 2}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 3}$

0
00
100
2200
14100
2200
100
00
0

hyperelliptic(.ncag).info