\mathrm{C}_{2} \times \mathrm{D}_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: $\mathrm{C}_{2} \times \mathrm{D}_4$, order $16$, SmallGroup $[16,11]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho(g_{3}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{4}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho \cong$ $\chi_{ 4 }$ $\oplus$ $\chi_{ 5 }$ $\oplus$ $\chi_{ 9 }$
order of $\omega_X$: 2
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 $\dim \mathrm{HH}^\bullet$: $(1, 0, 3, 6, 4, 6, 3, 0, 1)$

Hodge diamond

1
0 0
0 3 0
1 2 2 1
0 0 4 0 0
1 2 2 1
0 3 0
0 0
1

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

1
00
030
1221
00400
1221
030
00
1

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

Since $\operatorname{ord} \omega_X = 2$, the twists by powers of the canonical bundle form a finite package of $2$ diamonds (explained).

hyperelliptic(.ncag).info

$\mathrm{C}_{2} \times \mathrm{D}_4$ in dimension 4

Hodge diamond

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

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

References