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 $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, 1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3})$
order of $\omega_X$
6
number of moduli
4
irregularity $q = \dim \operatorname{Aut}^0(X)$
2
Albanese
the Albanese variety has dimension $2$; the general fiber has dimension $2$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 4, 6, 6, 9, 12, 8, 2, 0)$
Hodge diamond
1
2 2
1 6 1
0 6 6 0
0 2 10 2 0
0 6 6 0
1 6 1
2 2
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
22
141
0330
02520
1551
242
11
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
22
161
0660
021020
0660
161
22
1
$\omega_X^{\otimes 1}$
0
10
221
1532
04541
1532
221
10
0
$\omega_X^{\otimes 2}$
0
01
022
0151
00440
0151
022
01
0
$\omega_X^{\otimes 3}$
0
00
020
0440
02820
0440
020
00
0
$\omega_X^{\otimes 4}$
0
10
220
1510
04400
1510
220
10
0
$\omega_X^{\otimes 5}$
0
01
122
2351
14540
2351
122
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