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: $\mathrm{C}_{8}$, order $8$, SmallGroup $[8,1]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, \zeta_{8}, i, \zeta_{8}^{3})$
order of $\omega_X$: 4
number of moduli: 1
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)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 2, 1, 2, 9, 12, 5, 0, 0)$

Hodge diamond

1
1 1
0 4 0
0 4 4 0
0 1 8 1 0
0 4 4 0
0 4 0
1 1
1

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

1
11
010
0110
02520
1551
131
00
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
11
040
0440
01810
0440
040
11
1

$\omega_X^{\otimes 1}$

0
10
120
0511
03511
0511
120
10
0

$\omega_X^{\otimes 2}$

0
00
101
1221
03230
1221
101
00
0

$\omega_X^{\otimes 3}$

0
01
021
1150
11530
1150
021
01
0

hyperelliptic(.ncag).info

$\mathrm{C}_{8}$ 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