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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 2, 5, 8, 4, 0, 0)$
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
020
0110
01310
1331
121
00
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
10
110
0311
02321
0311
110
10
0
$\omega_X^{\otimes 2}$
0
00
021
0231
00420
0231
021
00
0
$\omega_X^{\otimes 3}$
0
10
110
0410
03410
0410
110
10
0
$\omega_X^{\otimes 4}$
0
00
111
1221
02220
1221
111
00
0
$\omega_X^{\otimes 5}$
0
11
121
0330
02420
0330
121
11
0
$\omega_X^{\otimes 6}$
0
00
111
1221
02220
1221
111
00
0
$\omega_X^{\otimes 7}$
0
01
011
0140
01430
0140
011
01
0
$\omega_X^{\otimes 8}$
0
00
120
1320
02400
1320
120
00
0
$\omega_X^{\otimes 9}$
0
01
011
1130
12320
1130
011
01
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