hyperelliptic(.ncag).info

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

$\mathrm{C}_{6}$ 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
$\mathrm{C}_{6}$, order $6$, SmallGroup $[6,2]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, -1, -1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3}^{2})$
order of $\omega_X$
6
number of moduli
5
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 9, 16, 14, 12, 6, 0, 0)$
Hodge diamond
1
1 1
2 6 2
2 7 7 2
0 4 10 4 0
2 7 7 2
2 6 2
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
252
2662
03830
1551
141
00
0

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

Since $\operatorname{ord} \omega_X = 6$, the twists by powers of the canonical bundle form a finite package of $6$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
262
2772
041040
2772
262
11
1
$\omega_X^{\otimes 1}$
0
12
132
0561
04851
0561
132
12
0
$\omega_X^{\otimes 2}$
0
00
021
0231
00420
0231
021
00
0
$\omega_X^{\otimes 3}$
0
00
000
0110
01210
0110
000
00
0
$\omega_X^{\otimes 4}$
0
00
120
1320
02400
1320
120
00
0
$\omega_X^{\otimes 5}$
0
21
231
1650
15840
1650
231
21
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