hyperelliptic(.ncag).info

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

$\mathrm{C}_{12}$ 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
$\mathrm{C}_{12}$, order $12$, SmallGroup $[12,2]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, \zeta_{3}, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, i, -1, -i)$
order of $\omega_X$
6
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 5, 8, 5, 2, 1, 0, 0)$
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
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
131
1331
01310
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
141
1441
02620
1441
141
11
1
$\omega_X^{\otimes 1}$
0
01
011
0131
01331
0131
011
01
0
$\omega_X^{\otimes 2}$
0
00
000
0200
02200
0200
000
00
0
$\omega_X^{\otimes 3}$
0
00
020
0220
00400
0220
020
00
0
$\omega_X^{\otimes 4}$
0
00
000
0020
00220
0020
000
00
0
$\omega_X^{\otimes 5}$
0
10
110
1310
13310
1310
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