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: $\mathrm{C}_{3}$, order $3$, SmallGroup $[3,1]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3})$
order of $\omega_X$: 3
number of moduli: 4
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 $\dim \mathrm{HH}^\bullet$: $(1, 4, 6, 8, 17, 24, 16, 4, 0)$

Hodge diamond

1
2 2
1 8 1
0 10 10 0
0 4 18 4 0
0 10 10 0
1 8 1
2 2
1

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

1
22
141
0440
04940
210102
484
22
0

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

Since $\operatorname{ord} \omega_X = 3$, the twists by powers of the canonical bundle form a finite package of $3$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
22
181
010100
041840
010100
181
22
1

$\omega_X^{\otimes 1}$

0
20
441
21042
08941
21042
441
20
0

$\omega_X^{\otimes 2}$

0
02
144
24102
14980
24102
144
02
0

hyperelliptic(.ncag).info

$\mathrm{C}_{3}$ 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