hyperelliptic(.ncag).info

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

$\mathrm{C}_{10}$ 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
$\mathrm{C}_{10}$, order $10$, SmallGroup $[10,2]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, -1, 1, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, \zeta_{5}, \zeta_{5}^{2})$
order of $\omega_X$
10
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 4, 5, 4, 4, 2, 0)$
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
0220
121
11
0

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

Since $\operatorname{ord} \omega_X = 10$, the twists by powers of the canonical bundle form a finite package of $10$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
040
0330
00600
0330
040
11
1
$\omega_X^{\otimes 1}$
0
00
110
1221
02321
1221
110
00
0
$\omega_X^{\otimes 2}$
0
01
021
0140
00420
0140
021
01
0
$\omega_X^{\otimes 3}$
0
00
110
1320
03410
1320
110
00
0
$\omega_X^{\otimes 4}$
0
01
111
1130
02220
1130
111
01
0
$\omega_X^{\otimes 5}$
0
11
121
0330
02420
0330
121
11
0
$\omega_X^{\otimes 6}$
0
10
111
0311
02220
0311
111
10
0
$\omega_X^{\otimes 7}$
0
00
011
0231
01430
0231
011
00
0
$\omega_X^{\otimes 8}$
0
10
120
0410
02400
0410
120
10
0
$\omega_X^{\otimes 9}$
0
00
011
1221
12320
1221
011
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