$\mathrm{C}_{6} \times \mathrm{C}_{2}$ 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
- $\mathrm{C}_{6} \times \mathrm{C}_{2}$, order $12$, SmallGroup $[12,5]$ · character table
- tangent representation $\rho$
- $\rho(g_{1}) = \operatorname{diag}(1, -1, 1, 1)$$\rho(g_{2}) = \operatorname{diag}(1, 1, -1, -1)$$\rho(g_{3}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$
- order of $\omega_X$
- 2
- number of moduli
- 4
- 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)
- $\mathbf{D}^{\mathrm{b}}(X)$
- indecomposability open (why)
- Hochschild cohomology $\dim \mathrm{HH}^\bullet$
- $(1, 2, 6, 10, 10, 10, 6, 2, 1)$
Hodge diamond
1
1 1
1 4 1
1 4 4 1
0 2 6 2 0
1 4 4 1
1 4 1
1 1
1
1 1
1 4 1
1 4 4 1
0 2 6 2 0
1 4 4 1
1 4 1
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
141
1441
02620
1441
141
11
1
11
141
1441
02620
1441
141
11
1
twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$
Since $\operatorname{ord} \omega_X = 2$, the twists by powers of the canonical bundle form a finite package of $2$ diamonds (explained).
$\omega_X^{\otimes 0}$
1
11
141
1441
02620
1441
141
11
1
11
141
1441
02620
1441
141
11
1
$\omega_X^{\otimes 1}$
0
11
121
1441
14641
1441
121
11
0
11
121
1441
14641
1441
121
11
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