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 $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, i, i, -i)$
order of $\omega_X$: 4
number of moduli: 5
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 $\dim \mathrm{HH}^\bullet$: $(1, 2, 9, 16, 15, 14, 7, 0, 0)$

Hodge diamond

1
1 1
2 6 2
2 7 7 2
0 4 10 4 0
2 7 7 2
2 6 2
1 1
1

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

1
11
252
2662
03930
1661
151
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
262
2772
041040
2772
262
11
1

$\omega_X^{\otimes 1}$

0
12
132
0661
05951
0661
132
12
0

$\omega_X^{\otimes 2}$

0
00
141
1551
02820
1551
141
00
0

$\omega_X^{\otimes 3}$

0
21
231
1660
15950
1660
231
21
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