hyperelliptic.ncag.info

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: $\mathrm{Q}_8$, order $8$, SmallGroup $[8,4]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, i, -1, -i)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, i, -i)$
$\rho(g_{3}) = \operatorname{diag}(1, 1, -1, -1)$
$\rho \cong$ $\chi_{ 1 }$ $\oplus$ $\chi_{ 2 }$ $\oplus$ $\chi_{ 5 }$
order of $\omega_X$: 2
number of moduli: 3
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 $\dim \mathrm{HH}^\bullet$: $(1, 2, 5, 10, 12, 10, 5, 2, 1)$

Hodge diamond

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

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

1
11
131
1441
03630
1441
131
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

$\mathrm{Q}_8$ 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