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: $\mathrm{C}_{6}$, order $6$, SmallGroup $[6,2]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, \zeta_{3})$
order of $\omega_X$: 6
number of moduli: 9
irregularity $q = \dim \operatorname{Aut}^0(X)$: 3
Albanese: the Albanese variety has dimension $3$; the general fiber has dimension $1$ and is an elliptic curve (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 6, 15, 20, 15, 6, 1, 0, 0)$

Hodge diamond

1
3 3
3 10 3
1 12 12 1
0 6 18 6 0
1 12 12 1
3 10 3
3 3
1

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

1
33
393
1991
03930
0330
010
00
0

twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$

Since $\operatorname{ord} \omega_X = 6$, the twists by powers of the canonical bundle form a finite package of $6$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
33
3103
112121
061860
112121
3103
33
1

$\omega_X^{\otimes 1}$

0
01
033
0393
01991
0393
033
01
0

$\omega_X^{\otimes 2}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 3}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 4}$

0
00
000
0000
00000
0000
000
00
0

$\omega_X^{\otimes 5}$

0
10
330
3930
19910
3930
330
10
0

hyperelliptic(.ncag).info

$\mathrm{C}_{6}$ 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