$\mathrm{C}_{6}$ 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 $\omega_X$
- 3
- number of moduli
- 3
- 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 $\dim \mathrm{HH}^\bullet$
- $(1, 2, 5, 10, 10, 6, 2, 0, 0)$
Hodge diamond
1
1 1
1 4 1
1 5 5 1
0 3 8 3 0
1 5 5 1
1 4 1
1 1
1
1 1
1 4 1
1 5 5 1
0 3 8 3 0
1 5 5 1
1 4 1
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
131
1441
02620
0330
020
00
0
11
131
1441
02620
0330
020
00
0
twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$
Since $\operatorname{ord} \omega_X = 3$, the twists by powers of the canonical bundle form a finite package of $3$ diamonds (explained).
$\omega_X^{\otimes 0}$
1
11
141
1551
03830
1551
141
11
1
11
141
1551
03830
1551
141
11
1
$\omega_X^{\otimes 1}$
0
01
021
0341
02631
0341
021
01
0
01
021
0341
02631
0341
021
01
0
$\omega_X^{\otimes 2}$
0
10
120
1430
13620
1430
120
10
0
10
120
1430
13620
1430
120
10
0
References
- A. Demleitner, The classification of hyperelliptic groups in dimension 4. arXiv:2211.07998
- A. Demleitner, C. Gleissner, The classification of rigid hyperelliptic fourfolds, Ann. Mat. Pura Appl. (4) 202 (2023) 1425–1450. MR4576947 doi
- P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi