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 $\omega_X$
3
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 4, 9, 12, 8, 2, 0)$
Hodge diamond
1
1 1
0 4 0
0 5 5 0
0 2 10 2 0
0 5 5 0
0 4 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
020
0220
02520
1551
242
11
0

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

Since $\operatorname{ord} \omega_X = 3$, the twists by powers of the canonical bundle form a finite package of $3$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
040
0550
021020
0550
040
11
1
$\omega_X^{\otimes 1}$
0
10
220
1521
04521
1521
220
10
0
$\omega_X^{\otimes 2}$
0
01
022
1251
12540
1251
022
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