hyperelliptic(.ncag).info

the classification of hyperelliptic varieties in dimensions 2, 3 and 4

$\mathrm{C}_{9}$ 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: $\mathrm{C}_{9}$, order $9$, SmallGroup $[9,1]$ · character table
tangent representation $\rho$: $\rho(g_{1}) = \operatorname{diag}(1, \zeta_{9}, \zeta_{9}^{2}, \zeta_{9}^{4})$
order of $\omega_X$: 9
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)
$\mathbf{D}^{\mathrm{b}}(X)$: indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$: $(1, 2, 1, 2, 8, 10, 4, 0, 0)$

Hodge diamond

1
1 1
0 4 0
0 3 3 0
0 0 6 0 0
0 3 3 0
0 4 0
1 1
1

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

1
11
010
0110
02420
1441
121
00
0

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

Since $\operatorname{ord} \omega_X = 9$, the twists by powers of the canonical bundle form a finite package of $9$ diamonds (explained).

$\omega_X^{\otimes 0}$

1
11
040
0330
00600
0330
040
11
1

$\omega_X^{\otimes 1}$

0
10
120
0411
02411
0411
120
10
0

$\omega_X^{\otimes 2}$

0
10
111
0321
02330
0321
111
10
0

$\omega_X^{\otimes 3}$

0
00
111
1231
02330
1231
111
00
0

$\omega_X^{\otimes 4}$

0
01
021
0240
01520
0240
021
01
0

$\omega_X^{\otimes 5}$

0
10
120
0420
02510
0420
120
10
0

$\omega_X^{\otimes 6}$

0
00
111
1321
03320
1321
111
00
0

$\omega_X^{\otimes 7}$

0
01
111
1230
03320
1230
111
01
0

$\omega_X^{\otimes 8}$

0
01
021
1140
11420
1140
021
01
0

References

A. Demleitner, The classification of hyperelliptic groups in dimension 4. arXiv:2211.07998
A. Demleitner, C. Gleissner, The classification of rigid hyperelliptic fourfolds, Ann. Mat. Pura Appl. (4) 202 (2023) 1425–1450. MR4576947 doi
P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi