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: $\mathrm{G}(3,4,2)$, order $12$, SmallGroup $[12,1]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, i, -1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{3}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$
$\rho \cong$ $\chi_{ 2 }$ $\oplus$ $\chi_{ 3 }$ $\oplus$ $\chi_{ 5 }$
order of $\omega_X$: 4
number of moduli: 2
irregularity $q = \dim \operatorname{Aut}^0(X)$: 0
Albanese: the Albanese variety is a point; the fiber is $X$ itself (details)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposability open (why)
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 0, 2, 8, 5, 2, 4, 2, 0)$

Hodge diamond

1
0 0
0 3 0
1 2 2 1
0 0 4 0 0
1 2 2 1
0 3 0
0 0
1

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

1
00
020
1331
01310
0110
121
11
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
00
030
1221
00400
1221
030
00
1

$\omega_X^{\otimes 1}$

0
01
110
1130
02321
1130
110
01
0

$\omega_X^{\otimes 2}$

0
11
111
0220
02220
0220
111
11
0

$\omega_X^{\otimes 3}$

0
10
011
0311
12320
0311
011
10
0

hyperelliptic(.ncag).info

$\mathrm{G}(3,4,2)$ 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