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 $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3})$
$\rho(g_{2}) = \operatorname{diag}(1, i, -1, -1)$
order of $\omega_X$
12
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 1, 0, 0, 4, 8, 4, 0)$
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
0000
00000
0220
242
22
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
060
0550
001000
0550
060
11
1
$\omega_X^{\otimes 1}$
0
00
200
2201
04011
2201
200
00
0
$\omega_X^{\otimes 2}$
0
02
022
0060
00440
0060
022
02
0
$\omega_X^{\otimes 3}$
0
10
110
0500
04400
0500
110
10
0
$\omega_X^{\otimes 4}$
0
00
001
0011
00020
0011
001
00
0
$\omega_X^{\otimes 5}$
0
00
020
0320
01500
0320
020
00
0
$\omega_X^{\otimes 6}$
0
00
000
0000
00000
0000
000
00
0
$\omega_X^{\otimes 7}$
0
00
020
0230
00510
0230
020
00
0
$\omega_X^{\otimes 8}$
0
00
100
1100
02000
1100
100
00
0
$\omega_X^{\otimes 9}$
0
01
011
0050
00440
0050
011
01
0
$\omega_X^{\otimes 10}$
0
20
220
0600
04400
0600
220
20
0
$\omega_X^{\otimes 11}$
0
00
002
1022
11040
1022
002
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