hyperelliptic(.ncag).info

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

$\mathrm{C}_{7}$ 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
$\mathrm{C}_{7}$, order $7$, SmallGroup $[7,1]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, \zeta_{7}, \zeta_{7}^{2}, \zeta_{7}^{3})$
order of $\omega_X$
7
number of moduli
1
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 1, 4, 12, 12, 4, 0, 0)$
Hodge diamond
1
1 1
0 4 0
0 4 4 0
0 1 8 1 0
0 4 4 0
0 4 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
010
0220
03630
1551
121
00
0

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

Since $\operatorname{ord} \omega_X = 7$, the twists by powers of the canonical bundle form a finite package of $7$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
040
0440
01810
0440
040
11
1
$\omega_X^{\otimes 1}$
0
10
130
0521
02611
0521
130
10
0
$\omega_X^{\otimes 2}$
0
10
121
0521
03520
0521
121
10
0
$\omega_X^{\otimes 3}$
0
10
211
1421
04330
1421
211
10
0
$\omega_X^{\otimes 4}$
0
01
112
1241
03340
1241
112
01
0
$\omega_X^{\otimes 5}$
0
01
121
1250
02530
1250
121
01
0
$\omega_X^{\otimes 6}$
0
01
031
1250
11620
1250
031
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