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
2
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 2, 9, 16, 8, 0, 0)$
Hodge diamond
1
1 1
0 6 0
0 5 5 0
0 0 10 0 0
0 5 5 0
0 6 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
020
0110
02520
2662
242
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
060
0550
001000
0550
060
11
1
$\omega_X^{\otimes 1}$
0
20
220
0611
04521
0611
220
20
0
$\omega_X^{\otimes 2}$
0
00
122
1342
02440
1342
122
00
0
$\omega_X^{\otimes 3}$
0
11
121
0550
04840
0550
121
11
0
$\omega_X^{\otimes 4}$
0
00
221
2431
04420
2431
221
00
0
$\omega_X^{\otimes 5}$
0
02
022
1160
12540
1160
022
02
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