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 $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, -1, -1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, 1, \zeta_{3})$
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 7, 12, 7, 2, 1, 0, 0)$
Hodge diamond
1
1 1
1 6 1
1 6 6 1
0 2 10 2 0
1 6 6 1
1 6 1
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
151
1551
01510
0110
010
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
161
1661
021020
1661
161
11
1
$\omega_X^{\otimes 1}$
0
01
011
0151
01551
0151
011
01
0
$\omega_X^{\otimes 2}$
0
00
220
2420
04400
2420
220
00
0
$\omega_X^{\otimes 3}$
0
22
242
0660
04840
0660
242
22
0
$\omega_X^{\otimes 4}$
0
00
022
0242
00440
0242
022
00
0
$\omega_X^{\otimes 5}$
0
10
110
1510
15510
1510
110
10
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