hyperelliptic(.ncag).info

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

$\mathrm{C}_{12}$ 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
$\mathrm{C}_{12}$, order $12$, SmallGroup $[12,2]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, \zeta_{3}, 1, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, i, -1)$
order of $\omega_X$
12
number of moduli
2
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 4, 5, 2, 0, 0, 0)$
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
020
0220
01310
0110
000
00
0

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

Since $\operatorname{ord} \omega_X = 12$, the twists by powers of the canonical bundle form a finite package of $12$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
040
0330
00600
0330
040
11
1
$\omega_X^{\otimes 1}$
0
00
010
0121
00321
0121
010
00
0
$\omega_X^{\otimes 2}$
0
00
110
1210
02200
1210
110
00
0
$\omega_X^{\otimes 3}$
0
01
121
1240
02420
1240
121
01
0
$\omega_X^{\otimes 4}$
0
01
011
0030
00220
0030
011
01
0
$\omega_X^{\otimes 5}$
0
00
100
1210
03210
1210
100
00
0
$\omega_X^{\otimes 6}$
0
11
121
0330
02420
0330
121
11
0
$\omega_X^{\otimes 7}$
0
00
001
0121
01230
0121
001
00
0
$\omega_X^{\otimes 8}$
0
10
110
0300
02200
0300
110
10
0
$\omega_X^{\otimes 9}$
0
10
121
0421
02420
0421
121
10
0
$\omega_X^{\otimes 10}$
0
00
011
0121
00220
0121
011
00
0
$\omega_X^{\otimes 11}$
0
00
010
1210
12300
1210
010
00
0

References

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