hyperelliptic(.ncag).info

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

$\mathrm{C}_{4}$ 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
$\mathrm{C}_{4}$, order $4$, SmallGroup $[4,1]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, i, i, i)$
order of $\omega_X$
4
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 1, 0, 15, 30, 15, 0, 0)$
Hodge diamond
1
1 1
0 10 0
0 9 9 0
0 0 18 0 0
0 9 9 0
0 10 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
010
0000
03930
312123
393
00
0

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

Since $\operatorname{ord} \omega_X = 4$, the twists by powers of the canonical bundle form a finite package of $4$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
0100
0990
001800
0990
0100
11
1
$\omega_X^{\otimes 1}$
0
30
330
01201
09911
01201
330
30
0
$\omega_X^{\otimes 2}$
0
00
303
3333
06060
3333
303
00
0
$\omega_X^{\otimes 3}$
0
03
033
10120
11990
10120
033
03
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