\mathrm{S}_3 \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: $\mathrm{S}_3 \times \mathrm{C}_2$, order $12$, SmallGroup $[12,4]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho(g_{3}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$
$\rho \cong$ $2\,\chi_{ 1 }$ $\oplus$ $\chi_{ 6 }$
order of $\omega_X$: 2
number of moduli: 5
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 $\dim \mathrm{HH}^\bullet$: $(1, 4, 7, 8, 8, 8, 7, 4, 1)$

Hodge diamond

1
2 2
1 5 1
0 4 4 0
0 1 6 1 0
0 4 4 0
1 5 1
2 2
1

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

1
22
151
0440
01610
0440
151
22
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

$\mathrm{S}_3 \times \mathrm{C}_2$ 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