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 $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, -1, 1)$
$\rho(g_{2}) = \operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3}^{2})$
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 5, 10, 10, 8, 6, 2, 0)$
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
1441
02620
0440
141
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
11
141
1441
02620
1441
141
11
1
$\omega_X^{\otimes 1}$
0
01
121
1441
04631
1441
121
01
0
$\omega_X^{\otimes 2}$
0
11
131
0440
02620
0440
131
11
0
$\omega_X^{\otimes 3}$
0
00
121
1441
03630
1441
121
00
0
$\omega_X^{\otimes 4}$
0
11
131
0440
02620
0440
131
11
0
$\omega_X^{\otimes 5}$
0
10
121
1441
13640
1441
121
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