\mathrm{C}_{3} \times \mathrm{C}_{3} in dimension 4

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

Hodge diamond

1
1 1
0 6 0
0 5 5 0
0 0 10 0 0
0 5 5 0
0 6 0
1 1
1

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

1
11
010
0220
02420
0220
000
00
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
11
060
0550
001000
0550
060
11
1

$\omega_X^{\otimes 1}$

0
00
020
0221
00411
0221
020
00
0

$\omega_X^{\otimes 2}$

0
00
020
1220
11400
1220
020
00
0

hyperelliptic(.ncag).info