\mathrm{C}_{2} \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: $\mathrm{C}_{2} \times \mathrm{C}_{2}$, order $4$, SmallGroup $[4,2]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, -1, -1, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, -1)$
order of $\omega_X$: 2
number of moduli: 6
irregularity $q = \dim \operatorname{Aut}^0(X)$: 1
Albanese: the Albanese variety has dimension $1$; the general fiber has dimension $3$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposability open (why)
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 2, 8, 14, 14, 14, 8, 2, 1)$

Hodge diamond

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

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

1
11
161
1661
021020
1661
161
11
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

$\mathrm{C}_{2} \times \mathrm{C}_{2}$ 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