hyperelliptic(.ncag).info

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

$\mathrm{C}_{2}$ 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
$\mathrm{C}_{2}$, order $2$, SmallGroup $[2,1]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, 1, -1)$
order of $\omega_X$
2
number of moduli
10
irregularity $q = \dim \operatorname{Aut}^0(X)$
3
Albanese
the Albanese variety has dimension $3$; the general fiber has dimension $1$ and is an elliptic curve (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 6, 16, 26, 30, 26, 16, 6, 1)$
Hodge diamond
1
3 3
3 10 3
1 12 12 1
0 6 18 6 0
1 12 12 1
3 10 3
3 3
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
33
3103
112121
061860
112121
3103
33
1

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

Since $\operatorname{ord} \omega_X = 2$, the twists by powers of the canonical bundle form a finite package of $2$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
33
3103
112121
061860
112121
3103
33
1
$\omega_X^{\otimes 1}$
0
11
363
312123
11018101
312123
363
11
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