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: $\mathrm{C}_{4}$, order $4$, SmallGroup $[4,1]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, i, -1)$
order of $\omega_X$: 4
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 $\dim \mathrm{HH}^\bullet$: $(1, 4, 7, 10, 15, 16, 9, 2, 0)$

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
151
0550
03930
1771
252
11
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
10
231
1752
05951
1752
231
10
0

$\omega_X^{\otimes 2}$

0
11
242
1661
04840
1661
242
11
0

$\omega_X^{\otimes 3}$

0
01
132
2571
15950
2571
132
01
0

hyperelliptic(.ncag).info

$\mathrm{C}_{4}$ 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